Thermodynamics: Principles Characterizing Physical and Chemical Processes [2 ed.] 0123550459, 9780123550453

Citation preview

PREFACE T O T H E FIRST EDITION

The publication of yet another text on the well-explored topic of thermodynamics requires some commentary: such a venture may be justified on the grounds that as scientists our perceptions of any subject matter continually change; even as traditional and established an area as chemical thermodynamics is not exempt from such a subtle transmutation. Thus, there appears to be merit in a continuing series of expositions of the discipline of thermodynanics that differ perceptibly from linear combinations of discussions found in prior texts and monographs. In the present volume there occur several departures from conventional treatments, among them: (i) the presentation of the Second Law based on a simplified approach to Carath(~odory's method; (ii) a reasonably comprehensive treatment of thermodynamics of systems subjected to externally applied fields~ special emphasis has been placed on the systematics of electromagnetic fields and on gas adsorption processes, concerning which there has been much confusion; (iii) detailed investigations on the uniqueness of predictions of properties of solutions, in the face of a bewildering array of standard states, of methods for specifying composition, and of equilibrium constants; (iv) a rationalization scheme for the interpretation of phase diagrams; (v) a discussion of the thermodynamics of irreversible processes, centered on the macroscopic equations. Most of the above topics are not covered in detail in existing texts. Throughout, emphasis has been placed on the logical structure of the theory and on the need to correlate every analysis with experimental operating conditions and constraints. This is coupled with an attempt to remove the mystery that seems so often to surround the basic concepts in thermodynamics.

xi

oo

Xll

PREFACE TO THE FIRST EDITION

Repeatedly, the attention of the reader is directed to the tremendous power inherent in the systematic development of the subject matter. Only the classical aspects of the problem are taken up; no attempt has been made to introduce the statistical approach, since the subject matter of classical thermodynamics is selfconsistent and complete, and rests on an independent basis. The course of study is aimed at graduate students who have had prior exposure to the subject matter at a more elementary level. The author has had reasonable success in the presentation of these topics in a two-semester graduate class at Purdue University; in fact, the present book is an outgrowth of lecture notes for this course. No worked numerical examples have been provided, for there exist many excellent books in which different sets of problems have been worked out in detail. However, many problems are included as exercises at various levels of difficulty, which the student can use to become facile in numerical work. The author's indebtedness to other sources should be readily apparent. He profited greatly from fundamental insights offered in two slim volumes: Classical Thermodynamics by H. A. Buchdahl and Methods of Thermodynamics by H. Reiss. Also, he found instructive the perusal of sources, texts, and monographs on classical thermodynamics authored by C. J. Adkins, I. V. Bazarov, H. B. Callen, S. Glasstone, E. A. Guggenheim, G. N. Hatsopoulos and J. H. Keenan, W. Kauzmann, J. Kestin, R. Kubo, P. T. Landsberg, F. H. MacDougall, A. MOnster, A. B. Pippard, I. Prigogine, P. A. Rock, and M. W. Zemansky. Specific sources that have been consulted are acknowledged in appropriate sections in the text. He is greatly indebted to Professor L. L. Van Zandt for assistance in formulating the thermodynamic characterization of electromagnetic fields. Most of all, he has enormously profited from the penetrating insight, unrelenting criticism, and incisive comments of his personal friend and colleague Professor J. W. Richardson. Obviously, the remaining errors are the author's responsibility, concerning which any correspondence from readers would be appreciated. It is a pleasant duty to acknowledge the efforts of several secretaries, Jane Biddle, Cheryl Zachman, Nancy Holder, Susan Baker, and especially Konie Young and Barbara Rosenberg--all of whom cheerfully cooperated in transforming illegible sets of paper scraps into a rough draft. Special thanks go to Hali Myers, who undertook the Herculean task of typing the final version; without her persistence the manuscript could not have been readied for publication. Dr. Madhuri Pai contributed greatly by assisting with the proofreading of the final manuscript. In a matter of personal experience, it is appropriate to acknowledge several meaningful discussions with my father, the late Richard M. Honig, who was an expert in jurisprudence and who readily saw the parallels between scientific methodology and the codification of law. He persisted with questions concerning the nature of thermodynamics that I could not readily answer and was thereby indirectly responsible for the tenor of the present volume. Last, it is important to thank my immediate family, particularly my beloved wife, Trudy, for much patient understanding and for many sacrifices, without which the work could have been neither undertaken nor completed.

j. M. Honig

July 1981

PREFACE T O T H E S E C O N D EDITION

The present volume is an upgraded version of a reference text published by Elsevier under the same title in 1982. The goals of the presentation have remained unaltered: to provide a self-contained exposition of the main areas of thermodynamics and to demonstrate how from a few fundamental concepts one obtains a whole cornucopia of results through the consistent application of logic and mathematical operations. The book retains the same format. However, Section 1.16 has been completely rewritten, and several new sections have been added to clarify concepts or to add further insights. Principal among these are the full use of thermodynamic information for characterizing the Joule-Thomson effect, a reformulation of the basic principles underlying the operation of electrochemical cells, and a brief derivation of the Onsager reciprocity conditions. Several short sections containing sample calculations have also been inserted at locations deemed to be particularly instructive in illustrating the application of basic principles to actual problems. A special effort has also been made to eliminate the typographical errors of the earlier edition. The author would appreciate comments from readers that pertain to remaining errors or to obscure presentations. It remains to thank those whose diligence and hard work have made it possible to bring this work to fruition: Ms. Virginia Burbrink, who undertook much of the enormous task of converting the typography of the earlier edition to the present word processor format; Ms. Gail Shively, who completed this onerous task and patiently dealt with all of the unexpected formatting problems; and Ms. Sophia Onayo, who compiled the index and the table of contents.

ix

X

PREFACE TO THE SECOND EDITION

Purdue University has provided a very comfortable milieu in which both the writing and the later revision of the book were undertaken. It is a pleasure to express my appreciation to various individuals at Academic Press who encouraged me to prepare the revised text and who were most cooperative in getting the book to press. Last, this task could not have been completed without the support of my beloved wife, Josephine Vamos Honig, who gave me much moral support after the death of my first wife, as well as during the book revision process, and to whom I shall remain ever grateful.

j. M. Honig Purdue University

Chapter i

FUNDAMENTALS

I. 0

INTRODUCTORY COMMENTS

In this chapter the fundamental concepts of thermodynamics are introduced in the form of the four basic laws.

The procedure

is

with

reasonably

concept heat

axiomatic,

of temperature

flow;

(ii)

the

so

that

without

definition

one

can deal

initially having of

energy

as

a

(i)

the

to refer

to

function

of

state,

after a general introduction to the concept of forces;

(ill)

the

derivation

Carath~odory's performance

of

Theorem,

the

entropy

rather

than

characteristics

functions

of

state

emphasis

is

placed

mathematical homogeneous

are on

properties. functions

of heat

then

by

use

of

generalization

of

engines.

introduced

systematically Lastly,

is

function

the

introduced,

by

A variety

and

considerable

exploiting

important and

used

of

their

concept

of

to provide

framework for analyzing the properties of open systems.

a

We end

with a discussion of stability problems. The reader should examine not only the individual steps in the derivation, presentation, relation

i. I

also

the

internal

structure

the exploitation of Maxwell's equations,

between

experimental

but

commonly

information,

used

thermodynamic

of every and the

functions,

and

such as equations of state.

INTRODUCTORY DEFINITIONS

To understand the concepts of thermodynamics we must agree on the meaning of some basic terms discussed below:

2

I. FUNDAMENTALS

(a) System" A region of the universe set apart from the remainder of the cosmos for special study. (b)

Surroundings"

Regions

immediately

outside

of

and

contiguous to the system under study. (c) Boundaries"

Partitions that separate the system from

its surroundings. Commentary"

It is of the utmost importance to delineate

a system adequately surroundings. paradoxical actual

and to distinguish

Failure results.

physical

to do The

this

boundaries

constraints

such

it properly

leads as

from

its

to nonsensical

and

may

represent

walls,

or

either

conceptual

designs such as geometric surfaces. (d)

Body"

The

total

physical

contents

of

a

specified

region of the universe. Commentary"

In order

to be able

to describe

physical

properties of the body or to carry out measurements that do not disturb

the characteristics

of the body,

its volume

must be

large compared to the atomic scale, typically at least 10 -15 cm 3 in s ize. (e) Homogeneous and Heterogeneous system is uniform throughout

System" A homogeneous

in all its physical properties.

A heterogeneous system comprises several different homogeneous parts in close proximity. (f) Subsystem"

A portion

of a system,

singled out

for

special study. (g) Continuous System" One whose physical properties vary continuously as a function of position within the system. (h) Open (Closed) Systems"

Systems in which an exchange

of matter with its surroundings is (is not) allowed to occur. (i) Permeable Boundaries" Boundaries that enclose an open system. (j)

Semipermeable

Boundaries"

Boundaries

that

permit

passage of certain chemical species, but that are impenetrable to others. (k)

Diathermic

Boundaries"

Boundaries

impermeable

to

matter flow but which permit other changes to occur inside the enclosed system through heating or cooling of the surroundings.

INTRODUCTORY DEFINITIONS

3

(I) Adiabatic Boundaries and Systems" Boundaries that are impervious

to matter flow and that render the enclosed system

totally unresponsive to processes in which the surroundings are heated or cooled. (m) Isolated System" One that is surrounded by boundaries which

render

the enclosed system totally unresponsive

to any

changes in the surroundings. Commentary"

For

a

distinction

between

adiabatically

isolated and completely isolated systems, see Exercise One

should

also

note

external

manipulations

magnetic,

radiation,

perturbations

systems

of

applied

or

be

perturbed

fields

gravitational

through

(e.g. ,

electric,

fields).

These

excepting the

Boundaries of a system may be physical,

or,

imaginary ones.

(n) Thermodynamic are selected

may

can be achieved in all situations

isolated system. in some cases,

that

i.i.7.

Properties"

for a description

Physical attributes

that

of a system on a macroscopic

scale. (o)

Thermodynamic

Equilibrium"

A

state

of

the

system

where, as a necessary condition, none of its properties changes measurably over a period of time exceedingly long compared to any possible observations on the system. Commentary" checked

out,

through

external

properties.

as If

In each a

instance

sufficiency

manipulations the

response

the

system needs

condition, to is

all out

by

sorts of

to be

subjecting of

changes

proportion

to

it in such

applied deformations and the system cannot be returned to its earlier

state on lifting the deformations,

been at equilibrium"

it could not have

If the system is restored to its initial

state on completion of the checking process, without incurring any other changes in the universe, the system is at equilibrium with respect to the tests that have been conducted. In practice

it may be

difficult

to decide

equilibrium state has been achieved or not,

system is subject to very sluggish processes. sufficient response

whether

especially

In particular,

time must be allowed for a system to complete to

external

perturbations.

the

if the

Generally,

it

its is

4

I. FUNDAMENTALS

acceptable to ignore processes which are so slow as to produce no significant changes over time periods that are long compared to

the

observation

ignore

the

laboratory

effect

For

example,

cosmological

to

the

interval

rule

where

of

one

changes

A reasonable

time of any process:

compared

system,

of

experiments.

the relaxation large

interval.

the

may

usually

on

everyday

of thumb

involves

the latter

is very

measurements

on

the

one may regard a state of equilibrium or steady state

to have prevailed during the experimentation. As an example of problems arising in connection with the sufficiency condition,

consider the case of a mixture of H 2 and

02 in a balloon at room temperature.

The fact that there is no

important change in the content of the balloon over many hours does

not

establish

introduction disturbance

that

equilibrium

of platinum black as a catalyst of

the

system

in

this

manner

irreversible change in its properties. H2,

02,

and

prevails,

H20

are

formed

in

shows:

leads

to

small

excursions

of

appropriate

the

system

the

A small a

large,

On the other hand, once amounts,

introduced by heating and cooling are reversible only

as

from

its

changes

and lead to newly

formed

state. (p) matter

Reservoir:

with

the

A

source

system

or

or

for

sink

used

altering

suitable manipulations

to be discussed later.

always

of

assumed

to be

characteristics

remain

such

large

size

essentially

for

the

exchanging

properties

by

A reservoir

is

that

unaffected

its

physical

during

any

exchange process. (q)

Coordinates,

Variables

(of

State),

Degrees

of

Freedom: All three refer to quantities needed to characterize the state of a system at any instant. Commentary:

Ordinarily

independent

set of variables

For

at equilibrium

with

systems a

minimum

number

of

one

selects

for the above

this

linearly

characterization.

characterization

coordinates

a

whose

is achieved values

are

independent of the history by which the equilibrium state was reached. As examples one may cite variables such as temperature or pressure whose alteration will change the state of a system.

INTRODUCTORY DEFINITIONS

(r)

Intensive

(Extensive)

Variables"

Variables

whose

value is independent of (depends on) the size and the quantity of matter within the region which is being scrutinized. (s)

Thermodynamic

interrelation

between

Function several

of

State"

variables

describes a property of the system.

A

of

mathematical state,

which

The suitable manipulation

of such functions provides useful information to the observer of a system. Commentary"

Generally,

such functions

are useful

only

where a system is at or very close to equilibrium, because only in these circumstances can a simple functional interrelation be anticipated. (t) State Space, Configuration Space, or Phase Space" An abstract

space constructed as an aid in the visualization

processes.

Each variable

is assigned

an axis

on which

point represents a numerical value of the variable. space

is then formed by a mutually orthogonal

of any

A (hyper-)

disposition

of

these axes about a common origin. (u) Representative reflects

Point" Any point in phase space that

the physical properties

of the system for which

the

phase space was constructed. (v) Macroscopic

Process"

Any change

in the system that

leads to an alteration of its large-scale properties. (w) system

in

Path"

The

succession

its

passage

from

a

of

states

specific

traversed

initial

by

state

the to

a

specific final state. (x) succession

Quasistatic

Process"

of equilibrium

states,

One

path,

altered

consists

of

each of which differs

infinitesimally from its predecessor. irretrievably

that

a

only

The surroundings may be

in such a transition,

and

in a return

the universe may end up in a final state which differs

from the initial state. (y) Reversible

Process"

One whose

path may be

exactly

reversed by a succession of infinitesimal changes in operating conditions;

the original

state of a system undergoing

such a

process may be restored without incurring any other alterations in the universe.

I. F U N D A M E N T A L S

(z) Irreversible Process: One that occurs spontaneously, generally

while

equilibrium. undergone

the The

system

exhibits

original

an irreversible

state

change

severe of

departures

the

system

from

that

has

can be restored only at the

expense of other permanent changes in the universe. Commentary: system

could

Not

in

all

conceivable

principle

be

processes

subjected

to which

can

realized; we shall encounter such cases later.

actually

a be

Operationally,

the distinction between reversible and irreversible processes introduces

an element of patience.

time between successive alterations changes.

Only

a long

steps that involve just infinitesimal

in the system can one hope Clearly,

if one waits

all

dissipative

to produce

effects

such

reversible as

friction

must be avoided also. The

distinction

between

quasistatic

processes may be illustrated by an example.

and

reversible

In a paramagnetic

material a slow increase and subsequent decrease of the applied magnetic field leads to a gradual magnetization and subsequent gradual

restoration

ferromagnetic field

of the initial unmagnetlzed

medium,

the

to a demagnetized

magnetization

of

the

gradual

application

specimen will

system,

but

on

lead

state. of

In a

a magnetic

to a progressive

gradual

removal

of

the

applied field, hysteresis effects cause the system to go, in a succession of quaslstatic in which

a degree

absence

of

the

states,

to a different

of magnetization field.

necessarily quasistatic,

Thus,

is preserved a

reversible

final state, even

in the

process

is

but the converse may not hold.

(aa) Steady State Process: Processes which induce no net alterations

in the state

of a system but which

do alter

the

state of the surroundings. Commentary:

Steady

state

conditions

are distinguished

from equilibrium by the occurrence of processes in which inputs and outputs for the systems remain in balance, alterations occur within the system. do cause alterations

However,

in the surroundings.

so that no net those processes

For example,

in the

passage of a constant electric current through a wire, which is the system,

the number of electrons in any section of the wire

INTRODUCTORY DEFINITIONS

"I

remains constant because of the requirement of conservation of charge.

The

ultimately

rise

be

surroundings. process.

in temperature

balanced The

wire

However,

off is

due

to Joule

by

increased

then

involved

the battery

heating

heat

flow

in

steady

that produces

a

will

to

the

state

the current and

the surroundings to the wire undergo irreversible changes. (bb) Fictitious Processes: These are changes that can be forced

on

a

system

in

a

thought

experiment

but

cannot

be

performed in actuality. Commentary: visualization

Such

and

processes

didactic

are

purposes;

useful

an

early

aids

for

example

is

provided in Section 1.5. The

preceding

understanding.

discussion

points

It is presumed

intuitive

grasp

such as volume V or pressure P. to

explained

the

transfer

through

they are utilized. supply

better

processes,

the

concept of

the various

of

the

variables, relating

of

that

up

energy laws

a matter

tacit

reader possesses

mass must of

of

and

of

mechanical

By contrast, be

very

an

items

carefully

thermodynamics

before

We shall also be able at a later point to

definitions

for

adiabatic

and

diathermic

and for steady-state conditions.

EXERCISES i.i.I Is it appropriate to classify a definition as being correct or incorrect? Conventional or unconventional? Complete or incomplete? Consistent or inconsistent? 1.1.2 If a definition is labeled as differing from the accepted norm, does that make the definition incorrect? 1.1.3 Are there any conceivable circumstances under which a proffered definition should be rejected? Explain. 1.1.4 A solid is connected to a battery by wires and a current is allowed to flow. Must the battery be considered as part of the system? Explain. 1.1.5 An electrical heater is embedded in a solid, and connecting wires are attached to a II0 volt line. Is the heater necessarily part of the system which includes the solid? Explain. i.I.6 Provide several examples which illustrate the distinction between quasistatic and reversible processes.

8

I. FUNDAMENTALS

i. i. 7 Distinguish clearly between boundaries which enclose an adiabatic and those which enclose an isolated system. For this purpose, enumerate several changes in surroundings which can alter the properties of an adiabatically enclosed system.

1.2

TEMPERATURE AND THE ZEROTH LAW OF THERMODYNAMICS

The Zeroth Law of Thermodynamics emphasizes

the

transitive

is based on a statement which

property

that

two

bodies

in

equilibrium with a third are in equilibrium with each other. Here we

restrict

ourselves

to the case where

the mechanical

variables, namely pressure P and volume V, suffice to describe a system at equilibrium:

however,

this approach can easily be

enclosed

by

generalized. Consider explained matter

a

system

in Sec.

but

do

manipulations

i.I,

permit of

these

enclosures

changes

the

diathermic

in

As

exchange

of

the

system

by

example

of

prevent

state

surroundings.

walls.

of

An

this

situation is provided by a Bunsen burner that is placed below a flask containing ice, water and vapor;

the diathermic

glass

walls of the beaker permit the ice to melt in response to the application

of a

flame

steam) and boundary

exterior

to

the

system

(ice,

water,

(flask).

Let two different systems be brought in contact with each other

through

system

I be

diathermic

walls.

characterized

In a field-free

by pressure

region

P1 and volume

Vl,

let and

system 2 by pressure

P2 and volume V 2. Surround the composite

system

an

i

+

equilibrium. four

2

by

isolating

It is then the experience

quantities

P1,

VI,

P2,

arbitrarily and independently. variables

at

adjustable others.

boundary

but

specific is

V2

no

establish

of mankind longer

that

be

the

varied

Rather, having set three of the

values,

determined

can

and

by

the

fourth

the

chosen

is

no

values

longer for

the

This constraint may be manifested mathematically as a

functional relation among the variables:

~3(P1, VI, P2, V 2 ) - O,

(1.2.1)

TEMPERATURE OFTHE ZEROTH LAW OFTHERMODYNAMICS

where ~3 expresses appearing

the interrelation among the four variables

in the argument.

In principle,

Eq.

(1.2.1) may be

solved for any one of the quantities in terms of the remaining three. We now repeat the discussion for the distinct systems i and 3 which are to be combined into a composite and allowed to equilibrate:

at equilibrium

another

functional

interrelation

holds, namely

~2(P1, Vl, P3, V 3 ) Likewise,

O.

combining

equilibrium,

(1.2.2) systems

2

and

3

into

a

composite

at

we obtain

~l(P2, V2, P3, V3) - O.

(1.2.3)

The Ze~oth Law of ThermodynamiGs

asserts

that

if

two bodies are in equilibrium with a third they are in equilibrium with each other. As a result of the Zeroth Law, Eqs. imply Eq.

(1.2.3).

More generally,

(1.2.1) and (1.2.2)

the applicability

of any

two of these relations implies the validity of the third. As with

any other

law in science,

the above

statement

cannot be accepted or rejected by a priori arguments.

It is,

instead, necessary to examine the physical consequences of the statement

and to ascertain whether

are compatible with the law. Inasmuch remained Therefore,

as

equilibrium

unaltered we rewrite

in

the

all physical

This we proceed to do below. has

prevailed,

union

with

dependent

system

systems

i

3

has

or

2.

(1.2.2) and (1.2.3) as

P3-@I(P2, V2, V3)- @2(PI, Vl, V3), in which we have

observations

(1.2.4)

'solved e for P3, w h i c h thereby becomes

variable.

From

function termed ~, namely

(1.2.4)

we

obtain by

the

difference

a

J0

I. FUNDAMENTALS

~I(P2, V2, V3) " ~2(PI, VI, V a ) "

Comparing ( 1 . 2 . 5 )

A(PI, VI, P2, V2, V a ) -

O.

(1.2.5) and invoking the Zeroth

with ( 1 . 2 . 1 )

Law one notes a glaring inconsistency,

in that there appears to

be a functional dependence of A on V 3 which is absent from the function ~s. system

Clearly,

3 when

isolation.

it makes no sense

considering

the property

to have of

to refer

systems

i +

to

2 in

To preserve consistency we must demand that V 3 occur

in ~i and ~2 in such

a manner

(1.2.5) is constructed.

that

it cancels

out when

Eq.

This goal may be achieved in the most

general form by writlng

#i-

(1.2.6)

~I(P2, V2, V3) - f 2 ( P 2 , V2) h(V3) + q(V3)

# 2 " ~2(PI, VI, Va) - f 1 ( P 1 , where

f,

h,

variables;

and

note

both equations.

q that

are

general

the

(1.2.7)

Vl) h(V3) + q(Va), functions

functions

Substitution

of

the

h and q are

in (1.2.4)

indicated

identical

thus yields

f1(P1, Vl) - f2(P2, V2). Similarly,

f1(P1, V I ) -

Eqs.

(1.2.6)

specific

(1.2.8)

consistent with the Zeroth Law,

(1.2.9)

f3(P3, V3).

Observe

that as a result of the restriction and

(1.2.7),

variables

observation. and

V

for

Equations any

us

to regard

functions

is now made for

each

only

system

to

in the

under

they also force the recognition that mutually

at

equilibrium

cannot

be

Moreover, Eq. (1.2.8) and (1.2.9) permit

f1(P1, VI) as a reference

f2, f3 or, by induction,

are to be compared.

inherent

(i. 2.8) or (I. 2.9) may thus be used to

systems

independently varied.

reference

appropriate

characterize equilibrium; P

in

f•

standard

to which

for systems

the

2, 3, or i

TEMPERATURE OFTHE ZEROTH LAW OF THERMODYNAMICS

The

experimental

~

realization

procedure

consists

in

establishing diathermic contact between system i and systems 2, 3,...,

i,...

in succession.

If the value

of the function

does not change in this comparison process,

fl

then by the Zeroth

Law, systems 2, 3, ..., i, ... are all in equilibrium with each other and with I. if any

of

the

standard.

The same conclusion would have been reached

other

systems

had been

chosen

as

a reference

Clearly, the functional interrelation between Pi and

V i in fi(Pi, Vl) for a given system is of great significance. It fi(Pi,

is

Vi)

therefore by

temperature

the

convenient

compact

(function);

to

designate

I"i,

symbol

termed

the

functions

the

empirical

the reason for this choice of name is

clarified in the next section.

Equilibrium between two or more

systems is thus characterized by equality of all fi(Pi, Vl), all

such

i.e. , by equality of the empirical

systems.

equilibrium

then

More their

explicitly" empirical

the pertinent

temperature

If two

systems

temperature

for

are

functions

at are

equal. The function state.

Note

determining

~i - fi(Pi, Vl) is known as an equation

that

the

latter

provides

T through measurements

function fl is arbitrary,

us

with

of P and V.

a

of

means

However,

of the

in the sense that any single-valued

function which increases or decreases monotonically with I" can be used"

it does not matter what

system

is utilized

specification of the empirical temperature. looseness ...

are

in general

variables

(PI, VI),

I, 2,

(P2, Vz),

functions ....

of I".

new function ~[fi(Pi, Vl) ] which

of

their

However,

systems at equilibrium the corresponding the same numerical value

For the

... the various quantities

different

the

One can turn this

into an advantage in the following manner"

different systems i -

for

fl, fz,

independent

for two distinct

functions must yield

We may therefore is to have

introduce

a

such a functional

dependence for different fl that ~i - ~(Pi, Vl) thereby becomes a universal function. terms

of

variables been used;

mechanical

While we have couched our discussion in variables,

any

other

set

of

conjugate

for characterizing a system could have equally well this is discussed more fully below.

I '2

I. FUNDAMENTALS

In the next section we briefly discuss practical ways of establishing temperature scales. EXERCISE

1.2.1 Discuss ways in which the temperature of an adiabatically insulated system may be changed by manipulation of equipment set up in the surroundings. (See also Exercise 1.1.7).

i. 3

TEMPERATURES,

TEMPERATURE SCALES, AND TEMPERATURE

MEASUREMENTS The use of the word ~temperature' state T(P, V)

is clearly meant

in describing the equation of to establish a link between

a

physical property of any system and human sensory perceptions of what Serrln I has termed 'hotness levels'.

These correspond

to our primitive notions of bodies being ~hot' or 'cold'.

The

very fact that two bodies not in equilibrium are generally at different

temperatures

reflects

ordering of hotness levels; properties

systems,

this

quantification

through

the

scales

need

to

establish

some

to be useful in characterizing the

of

temperature

the

I- which

ordering

must

be

establishment should

functions of increasing hotness

be

levels.

capable

of

of

empirical

monotonic

increasing

Thermodynamic

theory

is related to human experience that whenever both pressure and volume increase in a system subjected to changes in its thermal state,

this

corresponds

level of the system.

to

a perceived

rise

in the

hotness

For this reason the equation of state is

well suited to function as an indicator of the hotness under

investigation.

However,

observations

of

level

physical

properties other than functions of state are equally useful: In a

generalization

physical

variables

of

earlier

which

concepts,

change

these

perceptibly

encompass as

a

system

any is

Ij. B. Serrin, Lectures at NSF-CBMS Regional Conference on Mathematical Foundations of Thermodynamics, July 1978, Ball State University.

TEMPERATURES: SCALESAND MEASUREMENTS

advanced such

to

different

as pressure

however, use

|

hotness

Mechanical convenient

quantities; such as the

resistivity,

thermoelectric

are very

variables

there are myriads of other possibilities,

of

and volume

levels.

viscosity,

voltage,

susceptibility,

etc.,

spectral

sound

each

emissivity,

velocity,

of which

responds

in a different

way to alterations in thermal states of the body. experimental

measurement

involves

a

correspondence between hotness levels, property

selected

empirical

for monitoring

temperature

state, T(P, V).

magnetic Each type of

distinct

one- to- one

the particular physical

the hotness

level,

and

the

function I" specified by the equation of

We can thereby correlate the measure values of

the chosen physical variables with the values obtained from -

t-(P, V).

The

arbitrariness

function allows us enormous and

procedures

for

in

the

latitude

temperature

empirical

temperature

in the choice of systems

measurements.

Here

we

must

ultimately be guided by experience and theory to pick from all of the possible

temperature

functions

those few) of greatest convenience be appropriate that

serves

that particular

and utility.

as a reference

standard

of hotness

which other physical variables can be calibrated.

(or

It will also

to single out from available materials

well

one

one set against

This we now

proceed to do. For simplicity one searches for a material

in which the

system provides a natural linear scale for hotness levels. has been known for three centuries

It

that certain gases such as

He, Ar, H2, and the like closely approximate to the relation PV (Boyle's Law) when these gases are kept at a given

-constant

empirical

temperature well

liquified.

In

view

of

above

the

the range where

flexibility

empirical temperature functions

in

they can be

construction

of

(as explained in Section 1.2),

let us adopt the product PV as a direct measure of ~" ~(P, V) - PV.

This

is the

simplest possible

procedure;

as shown

in

Section 1.17 this approach is actually firmly grounded in the basic laws of thermodynamics. temperatures henceforth

serves

shall

as

call

an a gas

Thus, He gas at all but lowest

excellent

test

thermometer,

system,

which

for monitoring

we the

|4

I. FUNDAMENTALS

values Once

of

I" of

the

any

system

mechanical

temperature

is e a s i l y

the v o l u m e

conjugate note

that

only

so

of

the

subsequent

as

gas

the

relation

this

of

PV-

is

is a s s u m e d

which

the

r - 0 when

w i t h water,

ice

interval V100, and

in I" is termed V

represent

V - V0

-

or

centigrade

pressure

the

system

at

I" -

O,

low value),

x i00

to set up

scale,

for

is e q u i l i b r a t e d is in e q u i l i b r i u m

Celsius of

(when the p r e s s u r e

the

is p a r t i c u l a r l y

volume

respectively

at some arbitrary,

a

in

one w h i c h

the

with

low;

of one atmosphere.

degree

equilibrium

satisfied

to be the case.

9 - I00 w h e n w a t e r

the

should

can be a s s i g n e d

Celsius

w i t h steam u n d e r a total p r e s s u r e

is

the

values

at a t m o s p h e r i c

and for w h i c h

let

one

it is n e c e s s a r y

are in use:

so-called

to

I". Also,

sufficiently

measurements,

such scales is

empirical

and

constant

some k i n d of a scale on w h i c h n u m e r i c a l

convenient

equilibrium.

the

constant

as a m e a s u r e

pressure

discussion

Many

in

measured

thermometer

serve

For q u a n t i t a t i v e

to I".

is

specified.

empirical

long

are

it

it is s i m p l e s t to h o l d e i t h e r the p r e s s u r e

the

variable

which

variables

Experimentally or

with

The unit

(~ gas ~

If n o w Vo, thermometer

-

I00,

of the gas t h e r m o m e t e r

and

in I",

is f i x e d

then ~ is m e a s u r e d by

(P fixed),

(1.3.1)

V1o o - V o

or analogously,

if the v o l u m e

is h e l d fixed

(and all p r e s s u r e s

are low),

P - P0

-

P1oo-

where

P0,

x I00

(V fixed),

P100, and

respectively.

P

These

correspond

to

I" - O ,

are to be d e r i v e d

The above m a y be r e w r i t t e n

P

~ -

in E x e r c i s e

I00,

and

r,

1.3.1.

as a d i f f e r e n c e

P0

-

x i00 P1oo"

(1.3.2)

Po

Po

-

x I00 P1oo"

Po

(V fixed),

(1.3.3a)

TEMPERATURES: SCALESAND MEASUREMENTS

I

- T - To,

wherein

(1.3.3b)

we have

T - IOOP/(PI00

" P0);

a new set of t e m p e r a t u r e s

To - lOOP0/(P100

T O is a constant;

in w h i c h shift

defined

T o in zero.

(1.3.3c)

" P0),

the r and T scales

Experiment

has

shown

differ

that

only by a

the q u a n t i t y

To -

iOOP0/(P100 " P0) for the gas t h e r m o m e t e r has the v a l u e of 273.15 degrees

Kelvin

define

the

(now

generally

absolute

called

temperature

kelvlns,

scale

by

T

K). -

Thus, ~

+

we

273.15

(ke ivins ) . It

will

Kelvin

scale

merely

the

turn has

out

a

later

much

that

more

convenience

the

introduction

deep-seated

of

of

significance

establishing

the

the than

direct

proportionality

T-

IOOP/(PIoo

in place clear

of

the

linear

relation

Using establish

Eqs.

one

(1.3.4)

or

calibration

it is e a s i e s t

T1

that

coexistence later, and

It

should

also

be

only

ice,

one

water,

gaseous

fields

the

Celsius

principles

of t h e r m o d y n a m i c s

K and P1 - 4.579 mm Hg.

are

T 1 and

water

with

K,

is n e c e s s a r y

T 1 for

the

which

and vapor.

Conformity

(P/P1)273.16

it

gas

fixed

to select

characterizes as will

then

there

pressure coexist

scale

and w i t h

P1 at in later

to

or T/T 1 ffi

For,

can

only

thermometer.

It turns out v e r y c o n v e n i e n t

temperature

and

point

temperature

if all external

liquid,

(1.3.5)

to set T/T 1 - P/P1 at fixed v o l u m e

single

of

(1.3.5)

(P cons tant).

- V0)

V/V 1 at fixed pressure. for

(1.3.2).

(I. 3.4)

that alternatively,

T - lOOV/(V~00

Here

(V constant)

- Po)

be

the shown

exists

which

one

solid,

equilibrium. fundamental

is a t t a i n e d by s e t t i n g T I ffi 273.16

Accordingly,

for a gas thermometer,

T - (V/V1)273.16 where

K or T -

V 1 and P1 are

the

16

i. FUNDAMENTALS

volume

and

pressure

of

the

gas

established both a temperature

at

273.16

K.

We

have

scale and a means of measuring

the temperature. Gas thermometers tend to be awkward in use. practice,

therefore,

is

to

employ,

where

The general

possible,

other

thermometers that exhibit a near-linear variation of some other property x, with the empirical for this purpose

is the mercury

expands in volume as ~ rises. i

-

temperature

~. A common device

or alcohol

thermometer which

For many liquids the quantity

(I/V)(aV/aT)e remains approximately constant.

(aV/aT)p

Thus

aV, which shows a direct proportionality between AV and AT -

A~,

to the extent to which ~ is truly constant and V is very

large (and nearly constant)

compared to AV. In thermometers of

this type one measures AV as a change in length of the liquid column maintained in a capillary attached to a large reservoir of liquid. Many other temperature sensors are in use. convenient

and

reproducible

resistance

thermometer.

is

Here

a bifilar, the

One which is

annealed

resistance

platinum

R and

absolute

temperature T are related by the expression R - R0(l + aT + bT 2 +

cT 3) ,

where

established

by

R0

a,

b,

c,

are

calibration.

constants

Quite

commonly

thermocouples are used as thermometers; -

which

must

be

two-junction

these produce an emf E

aiAr + a2A~ z + a3A~ 3, where Ar is the difference

in empirical

temperature between the two junctions,

one of which is held at

a

semiconductors,

standard

temperature.

Crystalline

carbon

resistors, paramagnetic salts, and optical pyrometers are also in common use in their appropriate temperature ranges indicated in Fig.

1.3.1.

A detailed discussion of various experimental

techniques that have been used to measure temperatures would go too

far

afield;

the

reader

is

advised

to

consult

the

very

extensive literature on the subject for further details. For convenience

in calibration of thermometers

in their

appropriate ranges, several secondary fixed temperature points have been agreed upon. hydrogen,

Among these are:

the triple point of

the boiling point of hydrogen,

the boiling point of

neon, the triple point of

oxygen, the boiling point of oxygen,

TEMPERATURES: SCALES A N D M E A S U R E M E N T S

| 7

Expansion of l i q u i d s Magnetic su sc ept i bi l i t y

--~~Thermocoup[es

Semiconductors

~ C a r b o n

resistance

~ Ptat~num -~-

~---

~

Gas T(on logarithmic i 0.1

scale)

l lo

i 1

FIGURE 1.3.1.

the

triple

-----1

k 100

N.B.R He 4

Pyrometers~

resistance

N.B.P. N 2

~

1

i

1000

10,000

N.B.P. w a t e r

Useful range of different kinds of thermometers

point

of water,

freezing point of zinc, freezing point of gold. agreed-upon temperatures

the boiling

point

of water,

the freezing point of silver, For details

and for values

the

and the of these

the reader is referred to appropriate

reference books and monographs.

EXERCISES 1.3.1 From the relation 9 - PV derive Eqs. (1.3.1) and (1.3.2). i. 3.2 Look up and then describe the following temperature scales in relation to the Kelvin scale: Celsius, Fahrenheit, Ranklne, and R~aumur. 1.3.3 Look up and then describe the physical principles on which the operation of the following thermometers is based: (a) semiconductors, (b) paramagnetic salts, (c) optical pyrometers, (d) thermocouples. 1.3.4 Describe several feedback systems commonly used to regulate the temperature of a system within a range of + 50 K near room temperature. 1.3.5 Consider the question as to what changes must be made in the approach of the present section if a nonideal gas is to be used as a thermometer.

J8

I. FUNDAMENTALS

1.3.6 What set of physical principles and experimental apparatus is used to measure the temperature of materials between 10 .3 and i K? Of galaxies in outer space? Of the temperature inside the head of a house fly? 1.3.7 The coefficient of linear expansion of Pyrex glass in the range 0-I000 ~ is 3.6 x 10 .8 deg -I, and for mercury V~ V0(r + 0.18182 x 10-31- + 0.0078 x lO-~z). If O~ and IO0~ are taken as fixed points on a mercury-ln-glass thermometer, what does this thermometer read when an ideal gas thermometer reads 40 ~C? 1.3.8 Let a benzene temperature scale (~ be defined by T0~ at the freezing point (5.56~ of benzene, and T IOO~ at its normal boiling point (80.15~ Set up a formula by which you can convert from ~ to ~ and from ~ to K. I. 3.9 'Derlve' the Perfect Gas Law from the combination of Boyle's Law (1660) PV~(T), where I- is a parameter independent of pressure P and volume V; also 'derlve' Charles' Law (1787) P ~(V)T, where ~ is a parameter independent of pressure P and absolute temperature T. 1.3.10 Provide a concise discussion of the types of substances which are suitable as standards in thermometers that are to operate within + I00 K of room temperature. Why must a common material such as water or mercury be excluded from this listing? -

i. 4 We

MATHEMATICAL APPARATUS need

to

investigate

repeatedly

in

equations.

These

several

subsequent

mathematical

manipulations

of

procedures

used

thermodynamic

results must be thoroughly mastered before

one can proceed with the unified description of thermodynamic principles.

Readers are advised to consult mathematical texts

on calculus and geometry for a more thorough exposition of the subject matter.

(a) The first deals with the transformation of variables. The theorems are illustrated below in three dimensions but may readily be generalized.

Consider a function F(x,y,z) of three

independent variables x, y, z, each of which can be expressed in terms of three alternative independent variables u,v,w that are of greater convenience

in characterizing the system:

MATHEMATICAL APPARATUS

x-

I

x(u,v,w),

y-

y(u,v,w),

The variables x,y,z

z

-

(1.4.1)

z(u,v,w).

(as well as the original variables u,v,w)

are not generally to be identified with position coordinates, but

represent

appropriate

independent

interest.

On substitution we find

F(x,y,z)-

F[x(u,v,w),

variables

y(u,v, w), z ( u , v , w ) ] -

of physical

G(u,v,w)

- F(u,v,w). The change

(1.4.2)

in notation from F(x,y,z)

to G(u,v,w)

emphasize that in passing from x,y,z,

to u,v,w,

to write

functional

symbol

procedure because

may

out the relation as

the

result

the physical

original

in

However,

for u,v,w with function

confusion

but

to

one generally

converts F to a different functional form, G. customary

is meant

it is

the

of x,y,z.

remains

same This

convenient

interpretation of the dependent variable

remains unaltered by coordinate transformations. Now

differentiate

(1.4.2)

with

respect

to

u;

by

the

'chain rule' one obtains

~

--

V,W

~

y,z

~

+

V,W

X,Z

with similar relations for aF/av

dF-

~

dx + y,Z

aN

X,Z

dy +

~

+

V,W

X,y

V,W

(1.4.3)

and aF/aw.

aH

,

dz

Next, write out

(1.4.4)

X,y

as

dF-

Xdx + Ydy + Zdz,

(1.4.5)

where X I (aF/ax)y,z ' etc. Then substitution in (1.4.3) leads to the important result

I. F U N D A M E N T A L S

~O

aH

OH

-X

V,W

aH

+Y

V~,W

with comparable

aN

+ Z

V,W

equations

(1.4.6)

,

V,W

8F/av and 8F/aw. From a purely

for

intuitive viewpoint

it appears as if in the differentiation

F with

u

respect

constant

and

(1.4.6).

to

we

the

'differentiated'

The

reader

to

(1.4.3),

equivalent

held should

'coefficients'

only note

which

dx,

dy,

that

explains

in

of

(1.4.5)

dz

to

obtain

(1.4.6)

is

really

the

origin

of

the

apparent anomaly.

(b) A special case of (a) arises when one sets u -- x and eliminates variables

z,w,

thereby

to two.

reducing

Equation

the

(1.4.3)

number

of

independent

then leads directly

to the

important relation

~

~

v

aN ~

y

(1.4.7)

x

v

which shows how one may pass from a quantity constant

v,

constant

to

the

y.

same

This

partial

result

thermodynamics,

particularly

derivatives

the

in

last

8F/ax evaluated at

derivative

finds

many

evaluated

applications

at in

if the product of the two partial

term

of

Eq.

(1.4.7)

may

be

readily

evaluated.

(c) A special alerts

third

equation us

relation

to the may be

result F(x,y) fact

solved

of

-C,

importance where

is obtained

C is constant.

that x and y are not in principle

from This

independent;

for y = y(x)

the form the

so that we

find

dy -

~

dx; F

(1.4.8)

21

MATHEMATICAL APPARATUS

y is now specified as a function of the second variable and its differential

is known as well.

We may then determine

dF =

dx + y

whence

Na

dy = x

~

dx + y

[ [ dx '~81X-Lax/ F

aM

one finds another result of importance,

=

O,

(1.4.9)

namely

8{~xx) = _ (8F/ax)y , F (8F/Oy)x which

reexpresses

quantity

F

derivatives

any

is h e l d in

(1.4.10)

constant

which

differentiated.

partial

F

This

is

differential in

terms

treated

formulation

is

of as

in which

a ratio the

a given

of

partial

quantity

often

needed

in

to

be

later

sections.

(d) Next,

eliminate x by solving F(x,y) - C as x - x(y).

It should be obvious

that this leads to the result

airy] _ _ (aF/ay)x F (aF/ax)y From a c o m p a r i s o n

(1.4.11)

of (1.4.10)

and

(1.4.11)

we

then obtain

the

Recip. rocal Theorem,

(1.4.12) F

(@y/@X)F

The procedure involving the equation F(x,y,z) = C is more complex;

this

relation

longer independent.

shows

that

the

three variables

are no

That is, x,y,z may be interrelated either

through the symbolic equation x = x(y,z) or through y = y(x,z). Then

the differentials

are likewise

of the two dependent

interrelated:

variables

x or y

~,~-

I. FUNDAMENTALS

dx -

dy +

aN

~

z,F

dy

Na

-

dx

Na

+

z,F

Substitute

dz

(1.4.13a)

dz.

(1.4.13b)

y0F

x,F

from (1.4.13b)

I -

into

(1.4.13a)

and collect

dx z,F

terms"

+

z,F

z,F

dz.

x,F

y,F

(1.4.14)

Because Thus,

of (1.4.12)

the right-hand

the left side of (1.4.14)

side may be readily rearranged as

-z,F

x,F

i,

(1.4.15)

y0F

which is known as the Reciprocity

(e) circumstances where

8y/au

vanishes.

The

procedure

where and

Theorem.

described

the evaluation

8x/au

are

easily

F(x,y) - C and set x ffi x(u,v),

below

of 8y/ax obtained.

y = y(u,v).

is

useful

in

is difficult,

but

We

begin

with

Then, by the chain

rule,

~

-~ v

+ y

which may be rearranged

(ay/au)v,F

(aF/ax)y

( ax/au)v,F

( aF/ay)~

v,F

, x

0

(1.4.16)

v,F

to read

(1.4.17)

MATHEMATICAL APPARATUS

2

On introducing

m

F

(ay/au)v.F (ax/au)v.F

(f)

We

relationships

(1.4.10) one obtains the useful result

9

next in

(1.4.18)

consider

mathematics

one as

of

the

applied

most to

fundamental

thermodynamics,

namely Euler's Theorem of Homogeneous Functions.

This theorem

is motivated by a thought experiment in which one container at constant T and P containing nl moles of species

i in volume V

is adjoined to another container of identical size and content under

the

same

conditions.

On

partition

the combined

system has

the

numbers

all

mole

parameters

of

removal twice

species

have been maintained

i,

of

the

internal

the volume

and twice

while

intensive

the

at constant values.

This

is

but one example of a more general situation in which one looks for a function F(x,y,z,...) which has the property that if all independent variables x,y,z,.., are replaced by ~x, ay, ~z,..., then one requires

that the original function F be replaced by

aF: F(ax,ay,az,...)

(1.4.19)

- aF(x,y,z,...).

A function satisfying the requirement (1.4.19) is said to be

homogeneous

reexpress set ~ -

in ~,

of

degree

this requirement I + ~ and examine

one.

It

is

of

interest

to

in another form.

For this purpose

the special

for which

case

E ~ 0.

According to (1.4.19) we demand F[(I + ~)x,

(I + ~)y,

(I + ~)z,...] - (i + E)F(x,y,z,...). (1.4.20)

We can

apply

Taylor's

theorem to the left-hand side; we also

rewrite the right-hand side as

~-4

I. FUNDAMENTALS

F(x,y,z,...)

+ ~x

'N

+ ~y

- F(x,y,z,...)

+ ~z

+

...

+ ~F(x,y,z,...),

(1.4.21)

where we have terminated the series after expansion in the variable ~ ~ O. Then,

F(x,y,z,...)

Thus,

- x

+ y

a homogeneous

'M ~ + z

function

very stringent requirement

of degree

+

the

first-order

....

one

specified by Eq.

(1.4.22)

must

satisfy

(1.4.22),

the

w h i c h is

known as Eu!er's Theorem. A n alternative

d e r i v a t i o n proceeds as follows:

a function F(xl,x2,...Xr) +

for which we write

... + FrdXr, with F i - (@F/axl).

variables

proportlonally:

dF-

Now change

dx I - xldA, FdA;

ratios

unaltered,

of

the

homogeneous F2x2dA +

various

xi's

remain

of

degree

one

in all

... +

FrxrdA,

from which

all independent

dx 2 - x2dA,

xrdA and impose the constraint d F -

Consider

F1dx I + F2dx 2

...,

this implies

xi's.

Then

Euler's

dx r -

that the

and

that

FdA

- F1xldA

theorem

F

follows

is + at

once.

The results obtained so far are s u m m a r i z e d in Table 1.4.1 below.

(g) In conjunction with the foregoing we b r i e f l y discuss thermodynamic

functions

in the m a t h e m a t i c a l systems time.

one

requires

Moreover,

experimentalist (f)

versus

measurement, from

the

of state.

description

This concept arises b e c a u s e of e q u i l i b r i u m

single-valued

functions

properties

of

independent

of

when the e q u i l i b r i u m state is altered what an cares

about

initial

is the c o n f i g u r a t i o n (i)

state

that

is

of the final subject

to

not the specific path by w h i c h the change of state

i to f was

accomplished.

selection of appropriate

This must be r e f l e c t e d

mathematical

w h e n the transition from i to f is

functions,

in the

F, such that

d e s c r i b e d by a s u m m a t i o n of

25

MATHEMATICAL APPARATUS

,i

Table

i

1.4.1

Summary

of R e s u l t s

Listed

in this

Section

Given dF-

X dx + Y d y + Zdz

x-

x(u,v,w)

y-

y(u,v,w)

z - z(u,v,w)

then (I)

~H

I

~H

X

V,W

(2)

( 3)

~

a[~X]

I

~H

Y

+z Ha V,W

V,W

V,W

~

v

+

~ ~

+ y

x

_ (aF/ax)y

(aF/ay)x

F

-i

(4)

Ma Na ~N ~ ~H

when

F(x,y)

- C

F

(5)

z

0 (< 0) w h e n

(compression).

force.

For c o n s i s t e n c y w i t h later d e v e l o p m e n t s

I. 8. I] we now set # - - M, gz

so that the increment

(rather

than u s i n g

of w o r k a s s o c i a t e d with

the

of mass dM, from infinity to height z is given by

gzdM,--

gzMdn,

(1.6.12)

42

I. FUNDAMENTALS

in which M is the gram-atomlc or gram-molecular mass and dn is the increment another

in mole numbers.

possible

entire

One should contrast this with

formulation,

~rw, -

- M, gdz,

for

which

the

system of constant mass M, is simply displaced upward

through the distance dz. physical

processes;

The two versions

one cannot argue

more fundamental than the other.

describe different

that one

formulation

is

Because we ultimately wish to

consider changes in the internal constitution of the systems it turns

out

to

be

easier

to

thread

Eq.

(1.6.12)

into

the

thermodynamic theory than to start with the expression for ~W'. Thls

is

related

to

the

fact

that

in

(1.6.12)

the

internal

thermodynamic coordinates are changed for the system as a whole as additional mass

is inserted at position z, whereas

expression

only

whole

for ~W'

is changed.

For

the elevation further

z for

discussions

the

in the

system as a

relative

to

this

point the reader is referred to Section 6.2. (f)

The work

in subjecting

a charge

electrostatic potential de is ~W' - - qdr

q to a change

in

Here, the situation

is analogous

to the motion of a body through an infinitesimal

distance

a

in

gravitational

field.

An

increase

de

while

preserving q does not change the deformation coordinates of the system.

An

alternative

formulation

is

possible:

Here

an

element of charge is taken from infinity and placed on a body at electrostatic potential r

We will find this approach much

more useful and will generalize it immediately, for

later

equivalent

applications. to

the

The

formulation

result, ~W-

Eq.

in preparation

(1.6.13)

- Cdq.

below,

Consider

is

a body

maintained at a specified electrostatic potential which has a value r

at the position r where its charge density is p(r).

Now let additional charge dq - d3rdp(r)

contained in a volume

element d3r be brought from infinity to position r. The element of work involved in this process then is - d3rr total work for the entire body now reads ~ W -

and the

- ~vd3rr

where the integration is carried out over the volume V of the body. V-D-

According

to Maxwell's

equation for a material medium,

4~p, where D is the electric displacement vector.

Thus,

THE CONCEPT OF WORK

43

dp = (4ff)-iV.dD, 1.4.2,

line

and by use of the vector

(d), we

may

- - (4if)-I fd3r{V.(OdD)

now write ~ W -

- dD.VO(r)},

theorem

in Table

-fd3r(4~)-1(V-dD)r

where the integration must

be carried out over the total region in which the displacement vector has nonvanishing values.

By Gauss' Theorem,

Table

of the

1.4.2,

the first portion

mathematically the surface

equivalent

small;

above integral

is zero.

the

field.

integral

identity

accordingly,

for ~IW is

- fd2r(n.OdD)_ _ over

the field exists.

The

that the surface

flux

may be taken to be so remote

is vanlshlngly use

an

expression

of the region within which

boundaries

we

to

llne (j) of

the

first portion

In the second portion of the integral

E-

- VO,

where

E

is

the

electrostatic

Then (1.6.13a)

~W = - (I/4x)fd3rE. dD. For

a

more

elementary

specialized

medium

field

and

interacts

system,

is referred

derivation

It is important

electric

derivation

the reader

sophisticated 1.7.

of the

which

the

more

1.6.2.

A more

in the Appendix,

Section

and

fields

so that an integration

much

that E and D are the local

displacements,

with

also

to Exercise

is provided

to recognize

is

that

that

over

are

the

dielectric

external

all space

to

is called

the for.

These two factors render Eq. (I. 6.13a) awkward in manipulation. A formula inherently better suited for later use was derived by Stratton (1941) and Heine increasing

~WHere

(1956)"

the applied electric

The work done on a system by field E 0 in the amount dE 0 is

(1.6.13b)

fvd3rP.dE0 . P

is

the

Operationally, interrelations

polarization

this

quantity

following

must

be

described specified

below. through

of the type P = ~E or P • ~0E0, where ~ is the

electric susceptibility. Appendix;

vector

Equation

(1.6.13b)

a special case is considered should

be

noted"

(i)

is derived in the

in Exercise

Equation

1.6.4.

(1.6.13b)

solely to the work performed on the system of interest;

The

refers P = 0

44

I. F U N D A M E N T A L S

outside

the system.

(1.6.13b)

extends

(ii)

only

that of (1.6.13a)

Correspondingly,

over

the volume

the

integration

of the

is taken over all space.

system,

in

while

(iii) For the total

work involved one must combine Eq. (1.6.13b) with the integral - fd3r(4~)-iE0.d{_0. represents

The

latter

(1.6.13a).

specified

over

all

space,

and

the work involved in energizing space prior to the

insertion of the dielectric. Eq.

extends

by

(iv)

the

P

is

This combination is equivalent to the polarization

interrelation

D-

vector

E + 4~P

which

, (v) E 0 is

is the

applied electrostatic field in vacuo and is to" be held constant on insertion of the system into the field.

(g) which

A similar set of statements applies

a magnetic

placing

the

energized.

field

material

is

applied

in

an

to

a

system,

electromagnet

In one formulation,

to the case in such

which

be ~W' - - •dH; here H is the magnetization. shall,

instead,

involved

provide

in changing

is

then

field to

In this case the

of the system remain unaltered.

a derivation

the

by

one may regard the element of

work associated with the increase dH of the magnetic thermodynamic coordinates

as

field

such

We

for the element of work that

the

thermodynamic

coordinates are altered. The work involved in subjecting matter to magnetic fields H may be determined by considering a local current density J(r) subjected

to a vector potential A(r).

Changes

in the latter

are associated with work according to ~W--

(i/C)fvd3r(J-dA),

(1.6.14)

where c is the velocity of light. VxH-

(4~/c)J which holds

no time variation ~W-

Now use the Maxwell relation

for the special case where there is

in the electric displacement.

- (4~) -I ~d3r{dA.(VxH)}.

By the vector

One obtains

identity line

(g)

of Table 1.4.2 the work increment becomes HI~ = - (4~)-Ifd3r{H. (VxdA + V.(HxdA)}. all regions

The

integration must be carried out over

to which the magnetic

field extends.

portion of the integral can be rewritten by Gauss'

The

second

Theorem so

THE CONCEPT OF WORK

45

that it involves the quantity HxdA which is to be evaluated at

the

fringe

vanishes.

boundaries;

this

part:' of

the

integral

therefore

In the first part we set VxA i B; accordingly,

-- - (I/4~r)fd3rH *dB. In the foregoing, more

elementary

rather

(1.6.15a)

B is the magnetic

derivation

specialized

case,

of

induction vector.

the above

see

Exercise

results, 1.6.3,

Eq.

(1.6.15a)

based

while

elegant derivation is furnished in the Appendix, Again,

For a on

a

a

more

Section 1.7.

is awkward in actual use because H and B

are local fields which reflect

the reaction of the medium

to

the applied field, and because the integration extends over all space.

For practical applications it is more convenient to use

Heine's

(1956) formulation

--

-

(1.6.15b)

~ d3rHo*dM

which is derived in the Appendix, is

considered

in Ex.

1.6.4.

Section 1.7.

Note"

(i)

A special case

Equation

(1.6.15b)

refers solely to the work involved in placing a magnetic medium into

a constant

applied

outside the system. only

over

the

magnetic

induction

(il) Correspondingly,

volume

V

of

the

field

H0;

M-

0

the integral extends

magnetized

material.

(iii)

Equation (1.6.15a) is equivalent to (1.6.15b), augmented by the expression

- ~d3r(4~)-iH0.d~_0, integrated over all space.

(iv)

In the preceding, H 0 is the applied magnetic field vector prior to insertion of the sample;

this quantity is to be held fixed

when the samples are moved into position. moment vector specified by M are magnetic

xH or by M -

susceptibilities;

(v) M is the magnetic x0H0, where X and X0

alternatively,

(vi) Note the difference in sign when comparing

H = B

- 4~M.

(1.6.15b) with

(1.6.13b). (h) When surface effects are important one deals with a geometry Section

where 5.2

is

these devoted

particular to

a

effects

detailed

are

emphasized.

discussion

of

this

46

I.

particular

topic

in the context

of gas adsorption

FUNDAMENTALS

phenomena.

Fig. 1.6.4 shows a setup where in a very thin, elastic film is stretched

across

a wire

frame,

one part of which

is movable.

There is a natural tendency on the part of the film to minimize the

surface

area

Correspondingly,

by

pulling

the

movable

wire

to

the

left.

a force f is required to overcome the surface

force;

the work in moving the wire to the right by dx is ~IW =

- fdx.

The film area A increases by dA = 22dx, where account

must be of the formation of a film on both frame.

Then

~W = -

(f/22)dA;

tension or surface force by ~ -

~W

=-

on

sides of the wire

defining

f/2~

the

!n.terfacial

we obtain

(1.6.16)

-ydA.

(i)

In what

follows

one can always determine

it will henceforth the work performed

be

assumed

that

in any process by

use of the general relation (1.6.1) which involves an integrand formed

as

differential

a

product of

an

of

an

intensive

extensive

variable

variable.

with

Basically,

the this

relation acts as a bootstrap equation by means of which one can specify

the

elements

of

work

which

enter

the

various

thermodynamic quantities of interest that are introduced later. It

is

elements

therefore

of

the

utmost

of work correctly;

importance

failure

to

to do so has

specify

the

in the past

led to problems when dealing with the thermodynamic properties of materials

subjected to externally applied fields.

() G) I I I I i I I I KS k_/

(lh k_J dx

FIGURE 1.6.4 frame.

Setup for expanding a film stretched over a wire

THE CONCEPT OFWORK

47

EXERCISES

1.6.1 (a) In subsection 1.6.1(i) it was stated that the form ~ -fd2 always obtains. Yet Eqs. (1.6.13) or (1.6.15) appear to have two factors such as P and E 0 both of which are intensive variables. Resolve the apparent anomaly. (b) Which is the intensive and which is the extensive variable in each case? 1.6.2 (a) By elementary methods derive the relation -VEdD/4~ for a parallel plate condenser of such size that fringing fields may be neglected; here V is the volume inside the condenser plates, E is the local electrostatic field, D E + 4~P is the electric displacement, and P the electric polarization per unit volume. (b) What feature in this system allows one to confine the derivation to the volume inside the parallel condenser plates? (c) Show that in the absence of any material between the plates ~ J -VEdE/4~-(V/8~)dE z and interpret this result. (d) Prove that the reversible work done in inserting a neutral dielectric material into the condenser is ~ W -VEdP. (e) Explain the significance of the negative sign in these results. 1.6.3 (a) By elementary methods derive the relation ~W - -VHdB/4~ for an infinite solenoid of such size that fringing fields may be neglected; here V is the volume inside the solenoid, H B - 4~M, where H is the magnetic field, B the magnetic induction, and M the magnetic moment per unit volume. (b) What feature in this system makes it possible to restrict the results to the volume inside the solenoid? (c) Show that in the absence of any material inside the solenoid ~ - V H d H / 4 ~ - -(V/8~)dH 2 and interpret this result. (d) Prove that the insertion of an unmagnetized, non-ferromagnetlc material into the previously empty solenoid requires an element of work ~W. (e) Explain the significance of the negative sign in the above relation. 1.6.4 (a) For the elementary case of a very large parallel plate condenser prove that Eq. (1.6.13a) may be rewritten as ~ W VEodDo/8~ + VPdE O. Here E 0 and D o are the electric field and displacement inside the condenser resulting from fixed charges prior to the insertion of a material into the condenser, and E and D are the corresponding quantities after insertion while keeping the fixed charges unaltered. The derivation hinges on the important question as to how D changes in the insertion process while the charges on the condenser remain fixed. (b) Repeat the procedure for the long solenoid, showing that Eq. (1.6.15a) may be rewritten as -VHodBo/81r -VBodM. -

48

I. FUNDAMENTALS

1.6.5 (a) Let the equilibrium state of a simple elastic system be described by three variables: stress a, strain e, and temperature T and choose e as the dependent variable. Introduce Young' s Isothermal Modulus of Elasticity by Y(8o/8e)~ and the coefficient of thermal strain by A (8e/aT)=. Write out the equation of state in differential form for do. (b) State under what conditions e may be considered as a function of state for the system. If de is an exact differential find a differential interrelation between y-Z and A. (c) Using appropriate reciprocity conditions, prove that ( 8 a / a T ) , - Bz, termed the coefficient o~ thermal stress, obeys the relation B z - -AY. (d) Prove that do - Yde + BzdT. (e) Show that o - Y e is a formulation of ~.ooke's Law. 1.6.6 Can one conceive of processes where friction plays a role and for which the forces are conservative? Why or why not? 1.6.7 Calculate the reversible work of compression on I00 g of a liquid when the pressure is increased from I0 to I00 atm at a constant temperature of 20~ The compressibility is - 82 x I0 -B arm -I. Assume that ~ is constant within the pressure range 1 to I00 arm, and that the density of the liquid, p, at 20~ and I atm pressure is 0.792 g/cm 3. What assumptions have you introduced carrying out these particular calculations? 1.6.8 Determine the magnetic field in gauss (oersted) inside a long solenoid containing I0 turns of wire per mm through which a current of 3 amps is passed. I. 6.9 Write down the equation which correlates the applied magnetic field, magnetic induction, and magnetic intensity. Using this relation define the magnetic susceptibility and write an equation expressing magnetic induction solely in terms of the applied magnetic field. On this basis define the permeability and relate it to the susceptibility. 1.6.10 A vertical cylindrical tank of length 77 cm has its top end closed by a tightly fitting frictionless piston of negligible mass. The air inside the cylinder is at an absolute pressure of i atm. The piston is depressed by pouring mercury on it slowly, so that the temperature of the air is maintained constant. What is the length of the air column when mercury starts to spill over the top of the cylinder? 1.6.11 Write down an expression for the potential energy of a magnetic dipole which is at an angle 8 with respect to the applied magnetic field. Recognizing that the magnetic intensity is also the magnetic moment per unit volume, write down an expression for the work required to increase the magnetic intensity infinitesimally.

APPENDIX: WORK IN THE PRESENCEOF ELECTROMAGNETIC FIELDS

4~

1.6.12 Write down an expression for Curie's law pertaining to paramagnetic materials and for the Curie-Weiss law pertaining to ferromagnetic or antiferromagnetic materials. 1.6.13 It was stated in the text that the element of work always involves the product of an intensive variable with a differential of an extensive variable. Yet Eqs. (1.6.13) and (I. 6.16) involve two intensive variables. Explain the apparent discrepancies, and show explicitly how the problem is resolved when the relevant fields are uniform throughout the sample. 1.6.14 Look up in an appropriate text how the differential of the mechanical work term must be modified when a medium with anlsotropic properties is considered; write down the result in appropriate tensorial notation.

1.7

APPENDIX. DERIVATION FOR THE ELEMENT OF WORK IN THE PRESENCE OF ELECTROMAGNETIC FIELDS

We provide an alternative derivation leading to Eqs.

(1.6.13a)

and (1.6.15a) by starting with the Maxwell field equations +

(tic)

aB_lat

Vx~/ - ( 1 / c ) a D / a t In the

above,

-

(l.7.1a)

o

- 4~Jf/c. E,

D,

H,

electric displacement,

(l.7.1b) B,

Jf represent

the

Form

the

field,

magnetic field, magnetic induction,

free current density vectors respectively; light.

electric

scalar

product

of

and

c is the velocity of

(l.7.1a)

with

H

and

of

(l.7.1b) with E and subtract to find

-

-

Next,

-

+

+ (1/c)s

(llc)x.(as/at)

(1.7.2)

(4x/c)E'Jf.

introduce

the

vector

identity

H. (Vxs

-

E- (VxH)

-

( 1.7.3

)

V. (FxH), llne (g), Table 1.4.2. Then

V.(Ex/~) + ( 4 x / c ) E - J t : + ( l / c ) E - ( a D / S t ) +

(1/o) x. (as/a t)

-

O.

~0

I. FUNDAMENTALS

Now integrate over all space. By the vector theorem (j), Table 1.4.2, and with v - ExH, f d a r V . v - fdZr v.n for any vector v, n being the outer surface normal, to

a

surface

boundaries

integral

the first term is transformed

extending

and thus vanishes.

over

infinitely

remote

This leaves

- fd3r E-J~dt - (i/4~)fdar(E.dD + H.dB_).

(1.7.4)

The left-hand side may be rewritten as

J_~.~- pzv~._~- [p~E + c-Zp~!~xs].!~- _f'!~,

(1.7.5)

in which p~ and vf are the density and drift velocity free carriers which constitute

the current J~.

of the

Notice that in

the third relation we introduced the vector identity v~.(vfxB) - 0.

The quantity in square brackets

as formulated by Lorentz.

is the force density f,

It follows that

- fd3rJ~.E dt - - fdZrf.vfdt - ~W, which

is

Section system.

(1.7.6)

clearly

identifiable

as

1.6) when

the electric

field transfers

Equations

Helne's

(1.6.13a),

(1956)

an

increment

+ H.~

+ (V0"~0

reformulation

+ (v-

- E0"~0 + H0"~0 + (E'~0 - S0"~0)

D0)'~0

+ (S0"~0

+ S0"(~

energy

(see

to the

(1.6.15a) now follow directly. of

the

established by taking note of the mathematical

~.~

of work

- H0"~0)

- ~_0) + ( H -

- D'~0) + ~'(~H0)'~.

above

may

be

identity

+ (H0"~

- ~0"~)

~_0) (1.7.7)

In the context used below E0, Do, H0, B 0 are to be regarded as electromagnetic

field vectors that arise from fixed charge and

current distributions existing prior to insertion of the system of interest;

E, D, H, B are the corresponding quantities when

APPENDIX: WORK IN THE PRESENCEOF ELECTROMAGNETIC FIELDS

the body

is present

in the same

5 |

external

charge

and current

distribution. We now the

fields

integrate

penetrate.

Eq.

(1.7.7)

The

third

over all space term

into which

is equivalent

to

(E -

D).d~_0 - - 4zP-E0, because prior to insertion of the body dD 0 dE

in free

volume

of

space; the

4~B0.dM.

integration

system because

system P B 0. -

the

in

may be

the

free

restricted space

to

the

outside

the

The fourth term with B 0 B H 0 reads B0.(dB- - dH)

Again,

the integration may now be restricted to the

volume of the system since in free space M - 0.

The fifth and

sixth terms vanish automatically since in free space B 0 - H 0 and D O - E 0.

When the integration is carried out each of the last

four terms vanishes on account of a vector theorem derived by Stratton 3 (1941).

Thus it follows that

+ fvd3rB0 9d~. The reader

(i. 7.8)

should specifically note

that the integral

on the

left and the first one on the right of (1.7.8) are taken over all space, whereas the remaining integrals on the right extend solely over the volume of the system.

Equation (1.7.8) is thus

very useful in that the first integral on the right represents the work increment associated with changes dD 0 = dD and dB - d ~ in

free

space,

additional

whereas

work

the

increment

remaining arising

integrals

involve

specifically

from

the the

presence of the system.

In general only this latter quantity

is

thermodynamic

of

interest

in

the

analysis

of

systems.

Finally, attention is directed to the difference in signs which occurs in Eq.

(1.7.8) between the third and fourth integrals,

and to the unsymmetric disposition of the quantities

H0, E 0 and

P, M in these same integrals.

.

.

.

.

.

.

.

3J.A.

,

,

,

,

Stratton,

York, 1941), p. 111.

Electromagnetic

Theory

(McGraw-Hill,

New

5 ~,

1.8

I. FUNDAMENTALS

THE FIRST LAW OF T H E R M O D Y N A M I C S

(a) The First Law of T h e r m o d y n a m i c s experience

of

mankind"

adiabatically initial

Let

i,

thermodynamic characterized

xI

{xl}1,

depends

solely on the initial

path

which

is

here

above

from

designated

as E, with

is now p e r f o r m e d

situation

fact

the

that

no

longer

other

adiabatic

The answer

set

of

state

2,

is invariably

found

In

other

words,

involved

Wa

in the

and final values

EI

a.

holds.

This

conditions,

fact

would

seem

of the function E, were it not

changes

process.

initial

also

The

occur

question

which

were

arises

absent

whether

the

in these altered circumstances.

is in the affirmative.

The

First

L.aw of

existence

of

internal

energy

with

any

final depends

a

Thermodynamics

function

of

any

between

state

(2),

solely

characterizing

of

(function)

state

whatsoever,

on

such a concept

later how the quantity

state

system;

E,

for

initial

difference

the

these

guarantees termed

process

AE

the

any

process

state

(i) and

AE

coordinates

states

and

m

E2

xI

-

EI

and

x2

is independent

of

them.

is incomplete E is actually

until

it is e s t a b l i s h e d

to be determined.

The First Law contains an important corollary: an adiabatic

the

which may be c o r r e l a t e d

a given the

the path connecting

processes

it from an

final

under n o n a d i a b a t i c

function E still remains useful

Clearly,

a

and final states

greatly to impair the usefulness the

to

an

In v i e w of Section 1.4 one may specify a f u n c t i o n of

If work

for

on

independent

taken.

and E2, such that AE m E 2 - E I - W

the

following

performed

transforming

Such a change

adiabatic

state,

on the

be

the

OF HEAT

of the type of adiabatic work done and of the

precise

process.

by

m

by x 2 - {xl} 2.

to be independent

Wa

thereby

characterized

variables

is b a s e d

work

isolated system,

state

AND THE CONCEPT

- W a - 0, one can compute

from a m e a s u r e m e n t

or c a l c u l a t i o n of W a.

Since for AE for such

If this same

THE FIRSTLAW AND THE CONCEPT OF HEAT

change

is

now

brought

remains unaltered, Since

AE

- W

53

~

about

via

a nonadiabatlc

but the w o r k performed, 0,

it

is

expedient

to

process,

W, differs introduce

AE

from W a.

a

deficit

function Q such that Q - (AE - W) - 0.

Note

for any of the adiabatic or nonadlabatlc

changes w h i c h connect

the

same

initial

and final

state,

AE

carefully

is a b s o l u t e l y

that

the

same,

but it is the nature of the interaction of the system with its surroundings this

that is altered

sense

different

Q

and

W

may

causative

internal

energy

for the different

be

agents

of

the

considered that

system

as

produce

via

its

processes.

In

measures

of

the

change

in

the

with

the

a

interaction

surroundings. The quantity Q just Basically,

it is seen to be

that attends

Henceforth,

sufficient The

AE

a bookkeeping

we

shall

to

Q

the

type of a d j u s t m e n t

experimental

is incomplete is

to be

until

determined

initial

the process of interest? to locate another and Wa'' to

I

corresponding difference

2,

changes

between

state

i with

in energy, final

and

and

from a knowledge

-

- W.

it has Thus,

are

always

the

be

1.6.

is no adiabatic final

state

path 2 for

initial

of w o r k W a' from state

AEI0

let

W a' + AE21 - W a'' - 0,

W

W a can

Let

equivalently,

quantity

as

experimentally;

adiabatically

E is a function of state,

Historically,

0

In that event it is usually possible

Then, because

calculated

AE;

if there

respectively.

the

Q-

directions

state 0 such that by p e r f o r m a n c e

and

special

For the present

in Section

the system can be brought

states

of

outlined

How may AE be calculated the

for

W a as the m e t h o d par excellence

determination

a s c e r t a i n e d by the methods

that connects

Clearly,

condition

these matters are dealt with in Section 1.19.

the

flow.

adiabatic processes.

of heat

show how

regard

we cling to the relation A E for

the heat

- W a - 0, and Q - 0 for this

to characterize concept

furnished

is called

to the difference b e t w e e n AE and W.

an adiabatic process, case.

introduced

and

AE21 be state

AE20

the

of

0 be

energy

interest.

AEI0 + AE21 + A E 0 2 -

0;

so that AE - AE21 can be

of W a' and Wa''. been

Q ffi AE

the

custom

+ W;

then

to the

deal

with

First

Law

the of

54

I. FUNDAMENTALS

Thermodynamics

finds its analytical expression in the relation

AE - Q - W,

(1.8.1)

which now implies the following"

(a) Q and W are two different

manifestations

transit

separating

of

the

energy

system

in

from

its

surroundings;

changes

in E which

correspond

dynamic

coordinates

of the system,

energies

per

se

in

the

across Q

to alterations but

system.

boundaries and

of

the

they do not

(b)

AE

is

W

incur

thermo-

represent

increased

rendering Q > O, in which case one states conventionally heat flows int0 the system;

that

it follows that Q < 0 represents

flow of heat out of the system and a concomitant the internal energy of the system.

reduction

energy;

it

o__nnthe system,

follows

characterized

that

of

taken

state

state

(their values

in carrying

heat

E-

or work

E I, because

different

processes

defined

by

experimentally

the

energies

can

such

process,

of

the

heat

do

first

depend

their

a

on

the

algebraic

it makes no sense to inquire

state

can

different

Furthermore, law

be

in the

reached

amounts

by

many

of heat

flow

only a difference (or,

there

for

is no

that

such

fixed

in energy

matter,

thing

as

is

'the'

This is reflected in the well-known fact

only be

An interesting W r be

is

Q and W are not usually

system possesses

determined

constant which has no fundamental

and

system

to a reduction

generally

Also,

a given

accessible);

energy of the system. that

that

its internal

b_,y the

out a given process),

involving

and work performance. is

increases

leads

that although

sum is a function of state. how much

in

energy of the system.

It is emphasized

path

which

performed

by W > 0 and that this

the internal

functions

work

a

(c) E is also increased by

rendering W < O, in which case one states conventionally work is performed

by

to within

sidelight

should be brought

flux

work

and

carried out reversibly,

+ AW be the corresponding

an

arbitrary

significance.

associated

out: with

Let Qr a

given

and let Q - Qr + AQ and W - W r

quantities

when the same process

is

THE FIRSTLAW AND THE CONCEPT OF HEAT

55

carried out irreversibly. -

O, so that always A E -

A W -

(b) A comparable differential dE

-

Then, according to the first law, AQ

dQ

Qr - W r -

situation

Q - W.

obtains when we consider

form of (1.8. i), namely

(1.8.2)

dW.

-

the

Here dE is quite properly

the differential

increment

of

the function of state E; given E(xl,... , xl) one finds n

(1.8.3)

dE - ~ (aE/ax~)dx i. i-i By contrast,

the

written

down

in the same manner

changes

in Q or w from

symbol

dQ

and

quantities

dW.

This

can

adiabatic

such

that

as dE we use

both

be

small

these

the

latter

in value on the path one takes however,

according

to

constitutes

an

sum dQ - dW does not.

This

the

concept

is defined

of what to be

system such that at every stage ~ Q this particular

usually

To distinguish

of the system;

now broaden

process.

as dE.

implies

depend

displacement

the algebraic

We

of Q and W cannot

infinitesimals

individually

in a slight (1.8.2),

differentials

any change

O, so that d E -

in the - dW a in

case.

(c) Next, we hark back to the fact that Q and W represent energies

in transit across boundaries.

occurring in

its

that

Thus,

is a paradigm

the

principle

for

It

is

of

surroundings

when

isolated

Conservation

no matter what

solely

an

system

for such a situation.

corollary of the First Law" constant,

any process

totally within a system does not produce

energy.

universe

Therefore,

-

0;

the

It is then seen

Energy

follows

as

a

in isolated systems the energy is

processes

there

of

any change

AE

are

occur

entirely

interactions

within

which

involve

that the energy of a system can be altered.

the energy E for an interactive

system is not fixed;

it.

Thus,

however,

if the system and all of its interactive component surroundings

~6

I. FUNDAMENTALS

are grouped

into an isolated

unit,

continues

to hold for the enlarged

-Q

must

- W,

always

be

then the condition system.

interpreted

The first

to mean

that

AE - 0 law,

AE

changes

in

energy E of the original system are precisely m a t c h e d by energy flows Q and W out of or into heat the

interfacing

boundaries.

and work

Once

reservoirs

more:

energy

across

is

never

generated locally w i t h i n the boundaries of the original system. This concept

is more fully illustrated

in Chapter

At this point we can again provide perspective

on Eq.

A

(1.8.3):

A

Since

E-

a more

6. sophisticated

E(xl,x2,...,Xn)

A

A

we may

A

set _E" - _eiE~ + e2E~ + ... + _enE~ and dx_ - eldx I + egdx 2 +

...

A

+ endxn, with E~ " 8 E / a x i. A

In requiring

A

A

satisfy

el.e j - 0

(1.8.3)

in the shorthand notation:

discussed

(i ~ j)

in Section

1.4(b)

that the unit vectors

A

and el-e • -

i, we

dE-

can

rewrite

Eq.

E" .dx.

The m a c h i n e r y

may now be invoked.

Since E is a

function of state ~E"-dx_ - 0 for any cyclic process;

by Stokes'

theorem,

sufficient

V

x

condition. form

we

E"

-

When

obtain

definitions

0

follows

as

a necessary

this requirement 8E[/axj

-

is w r i t t e n

8E'j/ax i

-

O,

or,

and

out in component on

adopting

for E~, E~, we obtain @ 2 E / a x l a x j - 8 2 E / a x j a x l ,

is a familiar

the

which

result.

(d) The foregoing calls for a more p r o t r a c t e d d i s c u s s i o n of the interchange surroundings. of

state

It is extraordinarily

brought

distinguish

of work and energy b e t w e e n a system and its

about

clearly

surroundings,

and

through

between

the

the

what

execution

performed setting system

up under

performance calculated.

or a

by

the

"work study, of

work

system

to

(lii) can be

be

in the system w h e n w o r k to recognize determined

completely in such

either

them;

the

of work it is not n e c e s s a r y

operating

can

system,

to

to

reservoir," and

(i)

(ii)

on it or by it; and

on

the

link b e t w e e n

be informed about the internal changes is p e r f o r m e d

in all changes of work

constitutes

connecting

note that for the determination

important

that w o r k only

external

to

the

that

the

measured

or

a manner

readily

after

THE FIRSTLAW AND THE CONCEPT OF HEAT

57

The problems which arise are perhaps best illustrated by reference

to

Fig.

insulated enclosure S Mp.

1.8.1,

which

depicts

an

adiabatically

equipped with a movable piston P of mass

The latter rests on release pins r I and r2; the container

is also furnished with stops s I and s 2 which ultimately the

downward

motion

earth's gravitational

of

the

piston

field.

under

the

of

the

Let the space in the enclosure be

totally evacuated and let the release pins the piston now

action

arrest

is accelerated

through

then be retracted;

a vertical

distance

h,

until arrested by the lower stops, s I and s 2.

The volume of the

enclosure is

The work involved

thereby reduced

from V i to Vs

in moving P through the distance h is M~gh (for the problem of sign,

see

Exercise 1.8.5).

~r I

If the system is considered to be

r2

h

-7Sl

S2 [ ~

FIGURE 1.8.1 Schematic diagram depicting a system S, which is an adiabatically insulated enclosure, provided with a movable piston P; r I and r 2 are retractable release pins, and s I and s 2 are stops which arrest the downward motion of the piston under the influence of the earth's gravitational field.

S~

I. FUNDAMENTALS

constituted solely of the space inside the adiabatic enclosure, no work has

been performed

on

the

system

in the process

of

diminishing its volume from V i to V~; however, heat in the form of radiation

crosses

the boundaries

of the system

since

the

surrounding walls are heated through the friction of the moving cylinder and by its sudden arrest. enclosure wall

If, on the other hand,

is included as part of the system,

temperature

indicates

the

then the rise in

that the energy of the system has

been raised by the descent of the piston;

for this adiabatic

system the increase in energy is given by AE - M~gh. Next,

let the container be filled with a gas of pressure

Pi sufficiently low that the released piston compresses the gas to pressure Ps when ultimately arrested by the stops; i.e., the pressure of the gas satisfies the relations Pi -< P -< P~ < M~g/A, A being the cross-section of the piston.

Let the system again

include the adiabatically insulated walls. misconceptions, walls, gas

is still Mpgh, even though there now exists an opposing

pressure

present

P

in the

situation

the

prevailing in vacuum; to

Contrary to popular

the work performed on the system, including the

convection,

stage;

also,

than before, ultimate

range

Pi -< P -< P~.

acceleration

of P

is

However, lower

in

than

the that

at the same time the gas is now subject

turbulence,

and

heating

the walls are heated,

in

the

compression

though to a lesser extent

by the friction of the moving piston and by its

sudden

deceleration.

In

the

absence

of

detailed

information it is impossible to ascertain how much of the work, during

the

descent

of P,

is

converted

into

an

increase

in

energy of the gas, and how much appears as increased energy of the walls.

For nonequilibrium situations this type of analysis

would require the specification of a host of nonthermodynamic variables.

However,

as far as the ultimate equilibrium state

is concerned, none of this matters; after the system (walls and gas), has come to equilibrium,

the increase of energy of the

system is again AE - M~gh, regardless of how AE was originally partitioned between that

the

gas

specification

through without

and walls. of

reference

the

It is clearly

final

state

to the complex

can

fortunate

be

carried

internal changes

of

THE FIRSTLAW AND THE CONCEPT OF HEAT

~9

the system during the process. As the next experiment, let enough gas be admitted to the system

to

support

relation P s -

th e

M~g/A.

piston

Also,

by

virtue

of

for simplicity

the

balancing

consider

where frictional effects may be neglected.

the case

Now, add one grain

of sand at a time to the top of the piston,

thereby forcing P

to move downward slightly at each step against the increasing opposing gas pressure.

Continue this process until the piston

is gently nudged against the lower arrests. the amount - g~~

This time, work in

has been performed, where m(z) represents

the mass of the plston-plus-sand which is present at the height z above the arrests.

A new feature results from this process,

which consists of a series of quasistatic

steps"

The applied

external and resistive internal pressures very nearly balance, and at each stage the gas is only infinitesimally removed from equilibrium.

In

this

circumstance

one

equation of the state of the gas P(T,V) both the work

may

introduce

the

and thereby evaluate

- ~ P(T,V)dV and the energy increase A E -

- W;

appropriate equations of state for this adiabatic process are provided in Section 1.17. Finally,

one

should

note

the

need

for

correspondence

between the independent variables involved in the specification of dE and dW; both must be formulated internal

coordinates.

If

dW

also

in terms of changes

includes

any

changes

of in

potential energy that leave the internal coordinates unaltered, then this quantity must be correlated with dU, where U is the total energy of the system.

These matters are illustrated

in

Chapter 6. Another

instructive

exercise

consists

free expansion of a gas, depicted initially totally opened

confined

to a space

evacuated.

A hole

in the diaphragm

in Fig.

of volume the

involved

in

this

straightforward"

process? If

encompassing only the

the

The

two volumes;

system

volume VA,

answer is

V B being

dimensions

ultimately occupies the total volume V^ + V B. is

taken

the

The gas is

VA, volume

of macroscopic

separating

in examining

1.8.2.

is now the

gas

What is the work not

absolutely

with

boundaries

complications arise because

60

I. FUNDAMENTALS

vA

VB

FIGURE 1.8.2 Schematic diagram pertaining to the free expansion of a gas. The gas is initially confined in compartment A of the system and is allowed to expand into compartment B through a hole opened in the diaphragm separating the two regions.

in the expansion process the internal coordinates of the system so defined the hole.

are altered,

in

of the motion

of gas

through

With respect to the system of volume V A one now deals

with an open system, up

because

Section

encompasses

with new ground rules

1.23.

If,

on

the

that will be taken

other

hand,

system

the total volume V A + V B plus the inside portion of

the walls,

no work is executed by the process.

this point

it is important

system

its

in

the

equilibrium

original

state.

to restrict

equilibrium

To understand

considerations

state

and

There has been no change

in

to the

its

final

in surroundings,

nor has any work reservoir been needed to run the process. differently, within

the process

the completely

of expansion has taken place

isolated compound

or work crosses the boundaries.

totally

system A + B; no heat

It is worth noting again that

there is no need to worry about the complex interactions jets

of expanding

gases

exert

Put

forces

first in one direction as they emerge

on

the

container

as the walls

from chamber A and then

in other directions as they strike the various walls of chamber

THE FIRST LAW AND THE CONCEPT OF HEAT

B.

The

final

state

of

6 I

the

system

is

reached a totally quiescent state,

that

in which

it has

and AE for the process

of

reaching this final state from the initial state, vanishes. As yet

another

example,

suppose

head to drop under the earth's adiabatic enclosure.

one

allows

gravitational

a cylinder

field inside an

A gas initially at pressure Pi is being

compressed in this process.

The cylinder ultimately comes to

rest

pins.

on

a

set

of

retaining

The

change

in potential

energy of the cylinder head has been transmitted in part to the walls and to the retaining pins and in part to the gas which is being compressed, by

so that only a portion of the work performed

the externally

gas.

applied

This has serious

field has been

transmitted

to

the

implications with respect to the work

concept and would indicate that performance of work is affected by

the

opposing

circumstances

internal

the work

pressure

of

the

should be written

gas.

In

these

in the form mgdz

-

in conjunction with

a

PidV. The

fallacy

of

this view

concept of adiabatic processes. partitions from

which

the

separate

remainder

of

arises

Adiabatic enclosures are ideal

regions

of

thermodynamic

the universe;

interest

in particular,

no

transfer of any type can occur across those boundaries. present

example,

however,

the walls

of the container

are

intimate contact with the gas which is being compressed. these

walls

cannot

be

considered

as

part

of

the

heat

In the in

Thus,

adiabatic

partition which separates the container plus contents from the remainder of the universe. Note slammed

further that the final state of the piston having

into

the

arrest,

is

not

an

equilibrium

state :

temporarily the piston, walls, and arrests are at temperatures different various

from

that

subsystems

of

are

the

gas;

in different

the final system of the assembly. between until

the the

calculating initial

inside

adiabatic

equilibrium work

state

performed,

equilibration

at

state

the

time

of

impact

states which differ

the from

Heat exchange will now occur

walls

and

arrests

develops. one

must and

For thus

end

and

the

gas

purposes

of

start

with

with the

the final

62

I. FUNDAMENTALS

equilibration state.

The fact that the system passes

intermediate

wherein

potential does

stages,

energy

not

of the piston

matter:

irrelevant

in

not

Details

are

of

thermodynamic

all

of

the

communicated

the

intervening

analyses

through

alterations

based

to

the

in gas,

processes on

are

equilibrium

states. The above

examples

illustrate

the necessity

of dealing

appropriately with the concept of work and with the placement of boundaries which delineate a system. one

must

distinguish

perturbations surroundings, a process

that

carefully

affect

the

Concerning the former,

between

system

or

work the

state

it is completed,

boundaries

the

must

be

such

that

no

The placement

frictional

or

other

effects occur in the immediate vicinity or across

the boundary; be

of

and the work

involved must be calculable by standard methods. of

other

one must deal with a system at equilibrium before

is started and after

dissipative

and

clearly

regions

included

surroundings.

One

in which such phenomena take place must either should

in

the

note

system

again

or

that

else

in

the

changes

in

the

specification of the boundaries may totally alter the work that accompanies a given process. (e)

It may not be

out of place

to provide

illustration concerning the concept of work.

one

further

Consider a point

of mass m attached by a weightless spring (See Exercise 1.8.10, which deals with changes

required

mass)

This

to

a rigid wall.

1.8.3.

Note

system,

which

that will

the

spring

later

if the spring

situation

is depicted

and mass

point

placed

in

be

itself has

in Fig.

constitute contact

mechanical work reservoir through which mechanical

a

with

the a

forces may

be exerted on the mass point and spring. Initially the spring is in an equilibrium configuration; in the absence of any net force length x -

x o.

it is extended

to its normal

In this initial state the force exerted on the

mass point by the spring and by the surroundings both vanish" msprlng m F - 0 and Fsur m Fexc - 0.

Furthermore,

the mass point

M remains stationary at x = Xo; its velocity is v = O.

Now

THE FIRST LAW AND THE CONCEPT OF HEAT

6~

~F I~!

X~

FIGURE 1.8.3 Schematic attached mass point.

apply an external so that F spring.

diagram

compression

xo) , and that the force F points x.

quasistatically, infinitesimal for

the

massless

- xo), where k is the force

that under

(increasing)

of

X

spring

with

force on M, assuming that Hooke's Law holds,

- k(x

Note

~

Let such

constant

(extension)

force

that v m 0;

compression

The w o r k done by the mechanical

x < x o (x >

in the direction of decreasing

this

first

then

be

Fext -

or

exerted

- F.

excess or deficit of IF,xtl relative

quasistatic

of the

expansion

Only

an

IFI is n e e d e d

of

the

spring.

reservoir on the spring is then

given by

W-"

~~o F e x t . d+x -_

~1

_F'dx--_

~1

0

where

x I is the extension

final

value

adiabatic the

-

is

located

given

This

stored

is compressed

leads by

spring

x I.

Note

to the

final

AE-

energy

in the

at

of the

hk(x I

difference

spring,

that

appears

its

in

result

- xo) 2

no matter

to

if

an

that k

is

as

the

whether

the

or extended.

let external

to counter

(compression)

(1.8.4)

is

x.

energy

Next, needed

of

M

Eq.

change

independent

spring

which

process

energy

potential

at

(I.8.4)

k(x_ - x0)-dx,_

0

force be applied

the force of the spring.

- F + Fa, where F a is the excess force.

in excess of what is We now set up F,x t Note carefully

that

64

I. FUNDAMENTALS

the

definition

context,

Eq.

(1.6.1)

still

applies

in

the

present

but now

O

The first

term is treated just as before,

-

The

Xo) z.

resulting -

second

first

in W reads

reaches

law,

(d2x/dt2).dx-

W--

reflects

from the u n c o m p e n s a t e d

0 at x - x o and

Newton's

term

F-

~k(x~

the

force F a.

- md2x/dt 2.

( d v / d t ) . v d t - v-dv.

Let

- v;

- x0) 2 - h m ~ ,

first

term

is (1.8.7)

is

clearly

term represents

the mass point because

then

(1.8.6)

energy involved in the elongation and the second

to

the second term

AE - hk(x I - Xo) 2 + h m ~ , the

is X

According

dx/dt

M

(1.8.5) becomes

and the energy change of the system

where

of

The v e l o c i t y

Accordingly,

Then Eq.

- h(x I

acceleration

X - vl at x - x I.

ma

- hm(v a - 0).

yielding W -

the

increase

in

or compression

the kinetic

the outside

applied

potential

of the spring

energy

acquired by

force exceeded

the

force exerted by the system in the amount F a. The fact

that

replace

foregoing only

is a very

in quasistatic

elementary processes

illustration

is it permissible

the externally applied force by an oppositely

internal force.

If this can be done the internal

Proper

is invited

to work Exercises

on his or her u n d e r s t a n d i n g thought

delineate

should

be

appropriately

surroundings consideration.

to

directed

amenable

to

1.8.5-1.8.10

to

analysis.

The reader enlarge

the

force becomes

a function of state of the system itself and thus, thermodynamic

of

in

any

given the

to

of the foregoing concepts. the

system

particular

problem as

physical

as

to h o w

opposed problem

to

to the

under

THE FIRST LAW AND THE CONCEPT OF HEAT

65

EXERCISES

1.8.1 On the basis of the First Law of Thermodynamics, show that for the sake of consistency with the conventions adopted in the present section, it is necessary to define the element of work for moving a system through a height of dz in the earth's gravitational field as -M, gdz rather than +M, gdz. Repeat the argument for the electrostatic case where one may write ~W r - Q d ~ . What is the corresponding expression where a system is subjected to a magnetic field? 1.8.2 Discuss methods of determining changes in the internal energy of a system which do not rely on measurements of work carried out adiabatically. I. 8.3 A 2 :1 mixture of H 2 and 02 gases was ignited and exploded in a bomb of constant volume and the temperature rose 20 ~ before any energy had time to escape. What are the values of Q, W, AE? Discuss the reasons for your findings. 1.8.4 A I00 g lead ball at 27~ is dropped from a height of i0 m. (a) Find the kinetic energy and speed of the ball just as it reaches the ground. (b) Find the temperature rise of the ball is all of its kinetic energy is transformed into internal energy as the ball is suddenly stopped after I0 m, given that the specific heat Cv of lead is 0.03 cal/deg-g and assuming that its volume change is negligible. [In anticipation of later results, note that AE = CvAT ] . The constant of gravitational acceleration, g, is 980 cm/s 2. 1.8.5 Carefully consider the various signs attached to expressions for AE for the various processes described in subsection (d), and justify the presence or absence of minus signs for each case. 1.8.6 Consider a system consisting of a vertical hollow, adiabatically insulated tube to whose bottom is attached an inflatable, insulated balloon. Let a piston of mass m slowly move through a height h within the tube by the process of gradually adding sand to the top of the system, thus inflating the balloon. What is the final pressure relative to the initial pressure? The final volume relative to the initial volume? The work performed on the system if the latter is considered to be the gas within the enclosure? What has happened to the work performed by the plston-plus-sand? 1.8.7 A piston of cross section A moving horizontally in a hollow tube (adiabatically insulated) is propelled by a force F A against a gas held at constant pressure Ps < FA/A" What is the initial acceleration of the piston? How far is the energy of the gas raised after the piston has been moved a distance d? How much is the energy of the walls increased in this process? What can you say about the work transfer after the piston has been stopped and the entire system is allowed to equilibrate?

~6

I. FUNDAMENTALS

1.8.8 (a) Consider a fast irreversible expansion of a gas in an adiabatic enclosure which results in the upward motion of a piston that supports a platform onto which is bolted a weight of mass M v. The piston and support system is massless and slides upward without friction, until arrested by a set of stops. How much work is performed by the system if the total travel of the piston is over a vertical distance h? (b) The experiment is now repeated with the m a s s not bolted onto the platform. When the piston hits the arrests, the mass continues to travel vertically upward through an additional distance h' before reversing its motion. It is caught at that point and stored on a shelf. How much work has the system performed in this operation? 1.8.9 Consider again the experiment in which grains of sand are gradually added to and subsequently removed from a piston supported against gravity by a gas at pressure P contained in an adiabatic enclosure. Analyze the processes that occur, taking into account the frictional effects of the piston as it slides up and down the walls of the container. How does this situation differ from the case where frictional effects are assumed to be negligible? What effects does Irreverslbillty have? Consult Footnote 4 if necessary. 1.8.10 Two what extent must the discussion of Section 1.8(d) be modified if the spring has a mass of its own? Work out the detailed expressions. I. 8. II Explain why an equation of state cannot be derived on the basis of thermodynamic considerations but must be obtained by experimentation. 1.8.12 Use the well known Einstein relation to determine the energy in calories liberated when I tool of helium nuclei of mass 4.0028 are formed from two protons, of mass 1.0076 each, and from two neutrons, of mass 1.0090 each. 1.8.13 Using the well known Einstein relation check whether the change of mass inherent in a typical chemical reaction is experimentally detectable. As an exampe, select the combustion of on mole of sucrose for which the heat liberated is approximately 1.35 x I06 cal. 1.8.14 For each of the processes considered hereafter determine whether an ideal gas used as a working substance does negative work, positive work, or no work: (a) the gas contained in a balloon is cooled in an ice bath and the balloon shrinks, (b) the gas is placed in an rigid container and warmed in sunlight, (c) the gas is transferred reversibly from one rigid container to a second which is half its size; a piston is employed in the transfer.

4J.A. Beattie and I. Oppenheim, Principles dy~.amlcs (Elsevier, Amsterdam, 1979) Chapter 3.

of Thermo-

THE FIRSTLAW AS A PARABLE

67

1.8.15 Two what extent must the discussion of Section 1.8(d) be modified if the spring has a mass of its own? Work out the detailed expressions.

i. 9

THE FIRST LAW OF THERMODYNAMICS AS A PARABLE

Callen 5 has

provided

Thermodynamics

an

interpretation

of

the

Law

of

in terms of a parable that bears repeating.

A

certain man owns a rectangular swimming pool; to enter or to leave and and daily measures

First

he allows water

the height of the

water level as a means of monitoring the volume of water. water inlets and outlets to the pool are

The

equipped with water

meters. The noting

owner

can

the height

now

check

in the water

volume change of the water,

on

conservation

level,

of

computing

water

thereby

by the

and comparing result this with the

water meter readings at the inlet and outlet.

In general the

owner finds no correlation that is immediately evident, but he notices

that on rainy days

anticipated

the pool contains

from a balance

more water

of inflow and outflow

than

through

water meters, and on very dry days the pool contains less.

the So,

the owner eventually is led to cover the pool with a tarpaulin; at

this

changes

stage

he

notices

that

in the amount of water

he

can balance

precisely

in the pool with the rates

the of

water passage through the inlet and outlet. The analogy with the energy of a system should be fairly clear.

The pool covered with a tarpaulin

adiabatic energy.

system;

changes

Water

flowing

is analogous

in water level represent changes through

the

inlets

and

represents negative and positive work, respectively. tarpaulin removed

is

removed,

through

the

evaporation

water is

added

through

analogous

to

negative heat flow in a diathermic system. a water balance

to an

cannot be achieved unless

expanded so as to keep track of changes

outlets When the

rainfall

a positive

Thermodynamics

or and

In the latter case the bookkeeping

is

in pool level arising

from the rainfall and evaporation processes.

5H. B. Callen,

in

(Wiley, New York,

1960).

~8

I. F U N D A M E N T A L S

Notice water;

one

finally cannot

that

the

water

differentiate

in

the

between

collected from the rainfall and that which pool

via

the

contention

inlet.

in

manifestations the system,

the

This earlier

of energy

should

in transit

more Heat

across

but are not identifiable

is

simply

liquid

which

was admitted to the

make

section:

pool

the

plausible

the

and

are

work

the boundaries

of

as two types of internal

energy contained within the system. All parables, and

to

some

parable

being analogies,

extent,

misleading.

are to be explored

are by necessity The

in Exercise

inexact,

shortcomings

of

this

1.9.1.

EXERCISE

1.9.1

I.i0

Discuss

the shortcomings

of the above parable.

THE CARATHEODORY APPROACH TO THE SECOND LAW OF THERMODYNAMICS :

PRELIMINARIES

We have seen in Section 1.8 that under suitable conditions

the

performance

the

energy.

of work can be related to a function of state,

The

question

arises

whether

a similar

option

exists

for the transfer of heat, again under suitable conditions. answer is in the affirmative; unfortunately, is not so easily demonstrated.

The

the correspondence

A fair amount of mathematical

groundwork must be laid to establish the link between heat flow and a new

function

mathematical

of state.

converts

differential termed

the

proceed

not

interested

in the

niceties can assume the implication of the Second

Law of Thermodynamics, which

Readers

the

through empirical

namely that there does exist a function inexact the

differential

relationship

entropy

dQ/A-

function.

The

to Section 1.13, beginning with Eq.

dQ

into ds,

We now begin with the more rigorous

where

reader (1.13.1),

loss of continuity. analysis.

an

can

exact s is then

without

THE

FIRSTLAW AS A

(a) The here

69

PARABLE

for

discussion

use

in

Thermodynamics.

of

Section

conjunction

1.4 will

with

the

be

generalized

Second

F.u.nctions o.f state of interest

Law

of

are analytic

relationships e of the form

R-

(z.io.I)

R(xl,...,xl,...,x~),

in which the dependent variable R is uniquely prescribed by the values

of the independent variables

x i which characterize

thermodynamic properties of a system. differential

the

The corresponding linear

form is given as

n

(1.1o.2)

dR - ~. (8R/ax i) xj~idxl. i=l In thermodynamics linear differentials

we have

frequent

occasion

to examine

of the type

n ~L " ~ X i(xl,...,ym)dxl, i=l which

resemble

Pfaf.f.i.an state,

Eq.

forms.

(i.zo.3)

(1.10.2);

If

it

these

so happens

quantities that

L

is

are a

then (I.i0.2) and (I.I0.3) are equivalent:

interchangeable,

known

as

function

of

L and R are

subject to the definition X i ~ 8R/Ox i. On the

other hand, ~L may not be the differential of something;

~L is

then simply a shorthand notation for ~(1)Xidxl, and the X i are no longer

partial

quantities

derivatives

are

not

of

directly

a

function

useful

in

of the

state.

Such

thermodynamic

characterization of systems of interest. (b)

If a given

differential possible,

of

the

quantity form

under conditions

L does

(1.10.2)

not

it

constitute an exact differential. are of special

6Nonanalytlc

points.

may

of

properties

an exact

nevertheless

to be established later,

a function q(xl,... ,xn) with the property such a property

admit

be

to specify

that ~L/q ~ dR does

Pfaffian forms that display

interest;

they are said to be

are allowed at isolated singular

70

I. FUNDAMENTALS

holonomlc or inte_~rable; otherwise they are nonholonomic. obvious

reasons

For

I/q is termed an inte_gratln~ factor and q, an

integratln~ denominator. _

w

(c)

The particular

equation

n

(1.10.4)

~L - ~ Xldx i - X.dx - 0 i-1 is of special

interest.

Eq.(l.10.2)

applies,

equlvaleDt

of the form

R(x~,...,x~)

If it is an exact

~L-

the

x,.

and

there

exists

(1.10.5) Recall

a hypersurface If

~(1)Xldxl

(Section 1.4) that Eq. (1.10.5)

in the dimensional

is

nevertheless holonomic, such that q d R -

not

an

exact

space spanned by

differential

but

is

then an integrating factor may be found

Z(1)Xldxi, whence

X i(x i,...,x a) - q(x 1,...,xn)(aR/axl)xj~i. Further,

then

an algebraic

- C,

where C is a constant. represents

dR,

differential

relation

(i. I0.4),

(1.10.6)

~(1)Xldxl -

0,

then

implies

that

(l/q) ~(1) Xldxl - 0, and hence, dR - 0, thereby establishing the existence of an algebraic equivalent

(I. i0.5) to Eq. (I. i0.4).

It is necessary here and later to distinguish carefully between the

Pfaffian

form

~L

diffe, rential equation

-

Z(1)Xidxl

and

the

corresponding

~(1)Xldxl - 0.

(d) By analogy to Section 1.4(g), any particular solution of the differential the hyperspace additionally are

equation Zci)Xidxl-

R n spanned by the

0 forms a curve c n in

'coordinate

the Pfaffian form is holonomic

functions

of

8R/ax i by

(1.10.6),

may

of

the

function

R.

Similarly,

the

{xi}.

If

then the X•

which

be

of

components of a vector X which is proportional VR,

axes'

thought

as

to the gradient,

various

dx i may

be

considered as components of the displacement vector dx m {dx i}.

THE FIRST LAW AS A PARABLE

Finally,

7 I

again by extension of the discussion of Section 1.4,

the algebraic equivalent R(xl,... ,x=) - C may be represented by a hypersurface ~= in R=. Now since VR is everywhere orthogonal to the surface R C, X is always in the direction of the normal to the surface of ~n"

The requirement

(i. i0.4),

met by having dx perpendicular to the surface R - C .

abbreviated

as X-dx- 0, may be

to X, i.e.,

everywhere

Therefore,

solution curves C n to X . d x -

0, formed by adjoining the various must

lle

Finally, of

entirely

on

the

tangent

infinitesimal

surface

~n

segments dx,

defined

by

(I.I0.5).

it should be evident that there exist a vast multitude

points

(1.10.4),

in

Rn

not

accessible

by

solution

curves

Cn

of

namely all points not lying on the surface ~n.

All of the above geometric imagery is of great importance to the formulation of the Second Law which is to follow. (e)

We

holonomlc

have

and

sufficient

if

so

far

(i. I0.4)

to guarantee

demonstrated

that

if

holds,

these

conditions

that

then

in any neighborhood

point x= in R n there exist other points surface

~n

accessible

corresponding

(1.10.3)

to

Eq.

are

of a given

[namely those not on the

(I.I0.5) ]

which

are

not

from x via solution curves of X-dx - 0.

Is the converse also true;

i.e.,

from the assumption of

nonaccessibility can one deduce that ~(1)Xidxl is holonomic? affirmative

is

answer is provided by Carath4odory's

If every neighborhood a hyperspace

contains

An

Theorem"

of an arbitrary point x= in points

x not accessible

from

x o via solution curves of the equation ~(1)Xldxl - 0, then

the

Pfaffian

form ~L-

~(i)Xidxi i s h o l o n o m i c .

The proof of this important theorem which establishes the necessary Appendices, interested assume

conditions given in

the

in

for

holonomlcity

Sections

mathematical

its correctness,

i.ii

niceties

and hence,

inaccessibility and vice versa.

and

is

provided

1.12. of

in

two

Readers

not

this

proof

that holonomicity

may

implies

The stage is then set for the

7~

I. FUNDAMENTALS

statement Section

i. II

of

the

Law

Second

of

Thermodynamics

provided

in

1.13.

APPENDIX:

MATHEMATICAL

OF THERMODYNAMICS.

Before p r o v i d i n g

a proof

PREPARATIONS

FOR THE SECOND

LAW

HOLONOMICITY

of Carath~odory's

Theorem we digress

to specify necessary and sufficient conditions for e s t a b l i s h i n g whether

the Pfafflan

form ~ L -

We first prove variables

~(1)Xidxl is holonomic

that any linear differential

is integrable:

or not.

form in two

given

~L -- Xldx I + X2dx2,

(1.11.1)

we seek a function q(xl,x2) such that ~L/q = dR, where dR is an exact differential;

BL-

i.e,

qdR,

(1.11.2)

dR-

(aR/axl)dxl + (aR/@x2)dx 2.

(1.11.3)

From

(I.II.I)-(i.ii.3)

where

we require

(1.11.4)

Xldx I + X2dx 2 - q (aR/axl) dXl + q (OR/ax2) dx 2 , which holds

X I

-

for arbitrary

q(SR/Sxl),

X2 -

dx I and dx 2 if and only if

q(aR/Ox2).

(1.11.5)

These two relations can in principle always be solved for q and for R; X I and X 2 are known functions

x~

(aR/ax~),~

lax2]

X2

(OR/Ox2)x 2

[OxJ x

of x I and x 2. Thus,

(1.11.6)

7~

APPENDIX: MATHEMATICAL PREPARATIONS FOR HOLONOMICITY

as

is

consistent

displacement

with

vector

(1.11.3)

and

(I.ii.I).

components

are

related

Note to

how

one

the

another

t h r o u g h X I and X 2. We

conclude

integrable

that

since

if one were

the

(1.11.5)

to examine

Pfaffian

specifies

form

(i. II.i)

q as well

a sum of three

terms

to a d j o i n

r e l a t i o n X 3 - q(aR/Sx3).

It is then no longer

q(xl,x2,x3)

and

a

second

(1.11.5)

function

as R.

However,

the a d d i t i o n a l clear w h e t h e r

R(xl,xg,x3)

found that s a t i s f y all three X i - q ( S R / a x i) equations. we shall n o w address

(b)

In

q(xl, . . . ,Xn)

ourselves

precisely

the

more

general

case

such

that

(I/q)WL

-

R(xl,... ,Xn) is a f u n c t i o n w h i c h a total d i f f e r e n t i a l n

dR-

We

can

a be

In fact,

to this question.

one

dR

always

as Xldx I + X2dx 2 +

Xsdx 3 - ~ L one w o u l d have

function

to

is

or

does have

seeks

a

~L

qdR,

where

(in c o n t r a s t

to ~L)

-

function

of the form n

7. (8R/axl)dxl " 7. Yidxl. i-i i-i now

examine

(1.11.7)

the n e c e s s a r y

conditions

that m u s t

be met

for

this scheme to work.

If indeed the f u n c t i o n q w i t h the d e s i r e d

properties

found

(i. II.7)

has

shows

been

then

comparison

of

~L

qdR

with

that

(1.11.8)

Xl - qYi - q(aR/axl)

m u s t hold.

-

For the m o m e n t c o n s i d e r

the special

case w h e r e n -

3, so that

(8R/axl)

- YI - X j q ,

(8R/axa)

(8R/ax2)

- Y2 - X2/q,

(1.11.9)

" Y3 = Xa/q.

Since the order of d i f f e r e n t i a t i o n find

a

(I.II.9)

relation

between

according

any

two

of

to the f o l l o w i n g

the

is immaterial, functions

procedure:

we can

listed

in

~4

I. FUNDAMENTALS 8ZR

8Y1 I

8x28xl

8xz

aYz

- a~

The

1

8X1

X1

8q

q

8xz

q~-

8x2

8ZR

(1.11.1o)

I

I 8X~ - 8 a~

central

and

X " ~

last

8xlSx2

aq a~"

term

in

the

above

sequence

can

be

recombined to obtain

pX

q [.~x2

-

8Xz} 8xl

-

X1

8q 8xz

-

Xz

8q 8xl

(1.11.11a)

Using similar procedures for the remaining second partials of R, one finds

pXa q L~xl

.

8XI "I 8xa

SXz 8Xa} q[axs'-'7-- 8xz

.

X 3

8q cqxl

-

X I

8q 8xa

(l.ll.llb)

-

Xz

8q 8xs

-

Xa

8q 8xz

(l.ll.llc)

Now multiply both sides of (l.lla,b,c) by X3, X2, and X I respectively and add the resultants. found to vanish identically,

The rlght-hand side is

as a result of which the common

factor q on the left may be dropped.

x~t~

-

~j

+

x, 6E

-

ax.

+X

This leaves

3

6~

-

7xj

-

0

"

(1.11.12) We see,

then,

that if an integrating factor I/q exists

that converts the Pfaffian form ~L, into an exact differential, Eq.

(i. II.7),

then the coefficients

X i in the relation ~L -

X1dx 2 + X2dx 2 + Xadx 3 must obey (1.11.12). The latter relation may readily be generalized to the more general case n > 3, by replacing the subscripts i, 2, 3 in Eq. (1.11.12)with i, j, k, respectively.

APPENDIX: MATHEMATICALPREPARATIONSFOR HOLONOMICITY

(c) We n o w

seek

Given a Pfaffian

form

to prove

75

the

inverse

of the

foregoing:

n ~L

X~dx~,

- Y~

(1.11.13)

i-i in w h i c h we

the X i obey

prove

factor linear

that

(1.11.12),

there

I/q w h i c h

the

+

Xndxn ;

demonstrated

well

form

Eq.

is

(a).

a total

as an i n t e g r a t i n g

indices

exist

an

~L

let

then

one

a

first the

form

integrable

as

was

must

be

able

form

to

find

(1.11.7),

a as

such that

(1.11.14)

(xl,... ,xn_2 constant).

form Xn_ I and X n are b o t h

likewise

hold

to

of the

~,

us

reduces

certainly

denominator

of all x i, ~ and H will

into

(1.11.7).

Thus,

in their most general

i, J, k,

integrating

differential

(1.11.13)

differential

Xn_Id~_ I + Xndx n - ~ d H

Since

inexact

(i. Ii. 13)

this

in part

function H with

the

always

of the form

Pfaffian

xl, . . . ,xn_2 constant; Xn_idxn_ I

then

converts

differential In

will

with cyclical

depend

functions

on x 1,...,xn_ 2,xn_ 1,x n.

If we n o w relax the c o n d i t i o n on xl,...,xn_ 2, then we must w r i t e n dH - ~ (8H/axl)dx i. i-I To

go

back

(1.11.14)

from

(i.ii.15)

(1.11.15)

we replace

to

the

the r i g h t - h a n d

more

restrictive

side

in (1.11.14)

condition as s h o w n

below"

Xn_ldXn_l + Xndx n - ~ dH -

(1.11.16)

(@H/Bx i)dx i , i-I

w h e r e we have n o w d r o p p e d the r e s t r i c t i v e (1.11.14), been

because

effectively

right.

The

differential

dH form

the effect 'subtracted

appearing (I. ii. 15).

in

c o n d i t i o n r e q u i r e d in

of the v a r i a b l e s out'

by

the

(I.ii.16)

xl,...,xn_ 2 has

second is

the

term

on

the

full-fledged

76

I. FUNDAMENTALS

C o n s i d e r next the P f a f f i a n

form (1.11.13),

rewritten

as

n-2 Xtdx i + Xa_Idxn_1 + Xadx n - ~ L

(i.ii.17)

i-I and substitute

from (1.11.16).

O[x

This yields

for d L -

0

}

t/~ - (8H/Bxt) dxt + dH - ~L/~ - 0.

(1.11.18)

i-I

At this stage it is convenient Yt -

xt

for

(i-

1,2,...,n-2,n)

and

This makes Yn-1 a function of all permissible

operation.

Then Eq.

set

Yn-1 "

xt, but this

Furthermore,

for i - 1,2,...,n-2.

to switch variables.

H(xl,.-.,Xn)is a p e r f e c t l y

set Yt " Xi/~

(I.ii.18)

Let

- (SH/axt)

takes the form

n-2 Ytdyt + dyn_ I - ~L/~ - O,

(i.ii.19)

i-i

containing however, entire

one

fewer

differential

than

does

(ii. Ii. 17) ;

the Yt at first glance w o u l d all seem to depend on the

set of variables Since

we

have

Yl, 9 9 9 ,Yn-

passed

algebraic

transformations

integrable

if and only

from the

(1.11.17) former

if the latter

to

(1.11.19)

equation

is.

by

will

be

By hypothesis,

Eq.

(1.11.12) applies to (1.11.17); hence, a c o r r e s p o n d i n g e q u a t i o n will have

to h o l d relative

8Yj

Yt [By k " 8yj Now

+Yj

8Yt/ay= - 0

+Yk [8~y~

specifically

(1.11.19),

to (1.11.19),

set j - n

8y k

namely

PaYt

8Yj ] -

ayj

- i, k - n.

-

0.(i.

11.20)

ayl

Since,

according

Yn-1 " I and Y= - O, the above e q u a t i o n reduces (all i),

w h i c h shows that,

to to

(1.11.21)

in fact, none of the Yi in (1.11.19)

depends

APPENDIX: MATHEMATICAL PREPARATIONS FOR HOLONOMICITY

on Yn.

Thus,

Eq.

Yl,''', Yn-l" We now

(1.11.19)

repeat

the

77

involves only the n -

entire

process

I variables

beginning

anew

with

(1.11.14); each time we do so we arrive at a form equivalent to (1.11.19) variable

which than

can be

its

demonstrated

equivalent,

Eq.

to

depend

(i. Ii.17).

on

one

fewer

In

the

final

iteration the analogue of (1.11.19) contains only two variables and

thereby

definitely

becomes

integrable.

Thus,

we

have

demonstrated that the original equation (1.11.17) is integrable if Eq.

(I.ii.12) holds. In view of the above development,

to be

both

a necessary

and

a

Eq.

sufficient

(1.11.20)

is found

condition

for

the

integrability of the Pfaffian form (I.ii.13).

EXERCISES

i. Ii.i Which of the following forms is holonomic? Find integrating factors for those which are holonomic. (a) (3xZy y)dx + (3x + x)dy, (b) xdy - ydx, (c) ydx - xdy + yZxdx, (d) y z d y - zydz. 1.11.2 Consider the relation d L - R(r).dr If dL is an exact differential prove that dL is the diff4rential of a scalar function F(r) satisfying the relation R(r) - V.F(r) and show that the necessary and sufficient condition for existence of a gradient of a scalar function is that it be irrotational" V x R - 0. Show that this latter requirement coincides with the necessary and sufficient conditions rendering ~L an exact differential. 1.11.3 Show that if there exist a scalar function G(r) v such that V x (GR) - 0 then the relation d L R(r).dr Is holonomic. (This may be most readily accomplished by examining R-V x (GR) and noting what happens when this quantity is forced to vanish with R, G ~ 0.) 1.11.4 (a) Is the linear differential expression 3yzdx + xzdy + 2xydz an exact differential? (b) Show that x2z is an integrating factor for the foregoing expression. (c) Show that x2z(xSyz 2) is an integrating factor. i. Ii.5 What is the geometric interpretation of the quantity y d x - xdy? [Hint" Make the transformation to cylindrical polar coordinates x - r cos 8, y - r sin 0. ] Find an integrating factor for ydx - xdy.

"~

I. FUNDAMENTALS

1.12

APPENDIX:

M A T H E M A T I C A L PROOF FOR THE N E C E S S A R Y CONDITION

OF CARATHEODORY' S THEOREM

In

Section

sufficient

i. I0

we

conditions

We establish also

had

provided

arguments

establishing

in regard to Carathdodory's

Theorem.

in this Section that Carathdodory's

formulates

a

necessary

statement,

by

Theorem

proving

the

following:

If in every n e i g h b o r h o o d defined

by

the

set

of any point

(x)i,

i -

in the space

l,...,n,

there

are

points

that are inaccessible along solution curves n of the equation.~ Xidx i - O, then the c o r r e s p o n d i n g

Pfaffian

forml--~L

-

~(1)Xldxl

is

necessarily

integrable.

As

is

well

known,

a

curve

C

in

a

hyperspace

may

be

defined as the collection of all points generated by equations of the form

x i - fi(u)

(i-

where the fl are specified, valued

functions

correspond parameter

to

(1.12.1)

1,2,...,n), continuous,

of a parameter the

choice

of

such that u a _< u _< u b.

intersects

C

at

infinitesimally

u

-

from C.

u,

and

u. two

differentiable, Let

the points

specific

single-

Pa and Pb

values

of

the

Let C a be another curve w h i c h which

otherwise

differs

only

Then C a may be specified by equations

of the form

xl - fi(u) + ~41(u)

(i = l,...,n),

in which

~ is a small

function

analogous

41(u,) ~ 0

positive

quantity

(1.12.2) and ~i(u)

is another

to fi(u), with the constraint

(all i).

(1.12.3)

APPENDIX: PROOF FOR CARATHI~ODORY'STHEOREM

Moreover, curves,

since

C

"79

and

C a are

defined

they must satisfy Eq. (i.i0.4),

to be

solution

i.e.,

n im

(1.12.4)

Xi(xl)fi(u) - O, i-1

where

we

shall

emphasize

the

write

Yi(u)

- Xi(fi(u))

whenever

we

want

to

dependence of the specified X i functions on u.

The same condition must apply to the displaced curve" n

(]..].2.5)

7. Xi(fi(u) + e41(u))[f[(u) + '4'i(u)] - 0. i-I

We write out (1.12.5) by expanding the arguments of the X i in a Taylor's series in all the 41, keeping only terms of order ~0 and ~I. This yields n

(1.12.6)

Z Yi(u)[f[(u) + ~4[(u)] + i-I n n [0Xll

+Z

Z

i"'-i J-i

~0(u)

I,.~jJ

On account of (1.12.4), n

n Z

Xi41 +

t-t

n ~

fi

'(u) - 0

9

(1.12.6) simplifies to

(1.12.7)

(0xl/axjl4jf [ - 0.

t-].j-1

Equation

(1.12.7)

may be satisfied by choosing n - I of the

functions ~j arbitrarily and requiring the last, ~n, to conform to the requirements of (1.12.7). of this statement,

Xm ~n +

~ pXi]

t-1

rewrite

n-I

j -i

To examine the implications

(1.12.7) as n

x~

t-1

n-i p X • I j-1

f'i~0.

La~xjj

(I 12 8) "

"

The first-order differential equation may be solved by seeking a function A(u), with which both sides of (1.12.8)

are to be

~

I. FUNDAMENTALS

multiplied, such that this will convert the left-hand side into the differential

d(lYn~n) du

i AXn~ n + I E i-I

Comparison

of

finn + A'Xn~n"

(1.12.9)

with

the

(1.12.9)

left-hand

side

of

(1.12.8)

multiplied by A shows that if an identity is to be obtained, then A must satisfy the relation

;~, _x~il ~ [ax~ ax~] ,

(1.12.1o)

i-1 [8xn " 8-xxi fi-

Now multiply both sides of (1.12.8) by A and introduce (1.12.9) to eliminate X,4 n.

Also, eliminate A' in the intermediate step

by use of (1.12.10).

Integration of the resultant expression

leads to

~x,~,,

-

-

i

~

xa~ a +

9

I

n pX,]flea]du

~

j-I~-i ta-~TaJ

1

(i 12.11) "

'

where (1.12.3) has been used on the left. Next,

we examine

the first

integral

on the right

and

carry out an integration by parts. n-I - [a ~

n-i A ~Xj~ a

Y Xa~,~ d u - j-1

+

n-I

+ ~ua A' ~ Xj~j

j-1

du

j-z

fan-i

n

[8Xj1

g

I

ta-~-j f;~jdu,

.I-i

~-i

(1.12.12)

where (1.12.3) was again applied in obtaining the first step on the right, and (1.12.1) was introduced to obtain the last term. Now substitute Eq. yields

(1.12.10)

for A' in (1.12.12);

this

APPENDIX: PROOFFOR CARATH[:ODORY'STHEOREM

n-i

8 [

n-i

. [ax. a~ ]

nn-i

I ~ j -r.1 x~,;~u- - ~ j -r~1 x~,~. ~ ~ ~.

-

I E ~-D-1

--..

+

fi~j [ a ~

o

- -

Xj

OXi

(1.12.13)

du, axl

This

is

now

ready

to

be

substituted

into

(ii.12.11) ;

on

simplifying one obtains

. nn r]

~.--I

~j-

I

[x,{;x ,x.I Lastly,

observe

)

I

x

~

fox.

that

on account

(1.12.14)

of

(1.12.4)

one may

add

the

te rms

without altering equation

(1.12.14)

in any way.

This yields,

finally,

n-I Xj~j(u) ~.(u)

-

-

I

j-I

-

Xn

i

n

AX n

~

n-I ~ ~

f'i~j(u)

Ju,

Fi3n du, (1.12.15)

in which

Fijn

defines

the left-hand side of (1.11.20) with Y

X and y ~ x. Now examine

(1.12.15)

limit in the integral.

after setting u - Ub as the upper

Observe that ~n(Ub) on the left depends

not only on all ~j(Ub) occurring in the first term on the right but also on the values

these ~j take over the entire

u a _< u _< u b within the integral on the right.

interval

It follows

that

8~-

I. FUNDAMENTALS

one can choose the arbitrary functions ~j(u) that

from

the point Pa one can reach

neighborhood,

in such a manner

any other point

by choosing u b sufficiently

close

in its

to u a and by

permuting

indices so that all n functions ~j are successively

assigned

the

hypothesis;

we

neighborhood from Pa.

terminal started

n.

But

out under

of Pa there were

this

is

contrary

the assumption

points which were

that

to

in the

inaccessible

The only way to avoid the impasse is to cause all the

integrals to vanish; attempting else

index

clearly, we should not accomplish this by

to set l, ~j, or 1/X, equal

would

be

lost.

However,

we

to zero,

can

achieve

for then all this

goal

by

setting n l

(1.12.16)

Fij,fi - 0; i-i for,

in that event 4n(Ub) depends

only on the specific values

which the n - i arbitrary functions 4j take on at the point u - u b.

This

clearly limits

the number of curves which

constructed to pass through Pa, and hence restricts

can be

the number

of points Pb that can be reached from Pa. l

From (1.12.16) it is clear that either we must set all fl equal

to

zero

or

else

all

Fijn must

vanish.

The

former

alternative is unattractive because it places a restriction on the set of functions fl which should have been left arbitrary; hence

we

must

discussion

require

of Sections

instead

that

all

I.i0 and I.ii

Fijn E

it follows

0.

From

at once

the that

n

iZiX~dxl - ~ L

is integrable.

This proves the second part of the

Carath~odory statement :

If in the neighborhood of a given point others are n inaccessible along a solution curve of i_ZiX• 0, then

this

represents

both

a

necessary

and

a

sufficient condition for the integrability of the corresponding differential Pfaffian form. This completes the mathematical background exposition required for the Second Law.

THE SECOND LAW OF THERMODYNAMICS

1.13

The

~3

THE SECOND LAW OF THERMODYNAMICS

discussion

certain

of the

sense,

Second

analogous

to

Law

of Thermodynamics

that

of

the

First

is,

Law.

in a

We

saw

earlier that, under certain conditions, we could correlate with an

element

of

differential

work,

~W,

a

function

of

of the energy function.

state,

namely,

the

The question now arises

whether a similar step can be taken with respect to the element of heat transfer, ~Q, and if so, what kind of function of state corresponds

to it.

To preserve

a reasonably

this problem will have to be addressed

in a rather

way, based on the discussion of previous On

the

experiments

basis

of

carried

out

the

outcome

on

logical

roundabout

sections.

of

systems

approach

a

of

vast

number

greatly

of

different

complexity the following law is established as an experience of mankind:

The

SecoDd

eve ry

L.a.w of Thermodynamics

ne i ghbo rhood

adiabatically

of

isolated

any system

states that are inaccessible

The

above

Carath~odory's adiabatic in

the

immediately

an

exist

other

from S.

for

are inaccessible

differential

in

S

theorem to the equation ~Q - 0, which holds

form ~ Q -

to

there

in

of

systems.

According

s tate

that

application

leads

to

the

Since the heat flow is related

thermodynamic

Pfaffian

asserts

coordinates

~(1)Xidxl,

of

the

this means

system

to changes through

there are states

the that

from S along the solution curves ~(1)Xidxl - 0. the

discussion

form ~(•

i must

of

Sections

therefore

I. i0- I. 12

be holonomic,

the i.e.,

one must be able to find an integrating denominator l(xl,... ,Xn) such that

~Q/A - ds,

where ds is an exact differential

(1.13.1)

of a quantity

s, termed the

empirical entroov (function), which must be a function of state

84

I. FUNDAMENTALS

of the system. and for s;

It remains

two d i s t i n c t

systems

n o w relax the a d i a b a t i c f u n c t i o n of state of

as

function long

maintained.

the

function

adiabatic

state

s to the r e v e r s i b l e

and

by t

the

that w h e r e a s

of the

here

we

transfer

in

relate

under

all are

in the First

Law

the

of w o r k

function

of

of heat.

let two c l o s e d

A and

xl, .... Xn_1,t

contact,

empirical

let the t o t a l i t y of these c o o r d i n a t e s

systems

coordinates

diathermic

at a c o m m o n

set

conditions

E to the p e r f o r m a n c e

deformation

be

they are

to exist

Note

of the foregoing,

Yl, 9 9 9 ,Ym-1,

equilibrium

continues

equilibrium

conditions,

characterized

of

Note that we

once the e x i s t e n c e

as

of state

under

In light

the p r o c e s s

into a composite.

constraint:

said

conditions

we r e l a t e d

s o l e l y by

for A

(1.2.1) has b e e n p o s i t e d u n d e r a s p e c i f i c

circumstances,

appropriately

B,

an i n t e r p r e t a t i o n

this will be a c c o m p l i s h e d

combining

other

to p r o v i d e

so

that

temperature

characterize

at

t, and

the c o m p o u n d

sys tern C. Because surroundings

exchange

is additive,

involving

the

system

and

its

we find that

(1.13.2)

~QB.

~0~ - ~ 0 ~ +

Further,

any heat

from the h o l o n o m i c

character

of ~Q

it follows

that

(1.13.3)

ds c -- (AA/Ac)ds A + (AB/Ac)dSB.

Since s A and s B are functions of state they must d e p e n d on xl,...,xn_1,t and on yl...,ym_1,t respectively. can

invert

these

relations

to

solve

for

In p r i n c i p l e Xn_ I

in

terms

Xl,...,Xn_2,SA,t and for Ym-1 in terms of Yl,...,Ym-2,SB,t, shifting

SA and s B into the set of i n d e p e n d e n t

The

following

in its simplicity, elegant steps :

in

argument sweep,

science. [I]

sc

It

must

coordinates

xl, 9 9 9 ,Xn-2;

conjunction

with

Eq.

which

now

we of

thereby

variables.

identifies

~ and

s is,

and c o m p e l l i n g nature,

among the m o s t

proceeds

the

be

according

independent

Yl, 9 9 9 ,Ym-2; (1.4.4),

if

of

to the

t:

As

this

were

following

deformation discussed not

so,

in the

THE SECOND LAW OF THERMODYNAMICS

differentials

of

85

these

particular

coordinates

appear on the right-hand side of Eq. AA/AC and AB/Ac are likewise

(1.13.3).

independent

for,

if

it

to

did,

[I].

quantities

to

[2] The ratios

depend on this set,

[3] ~c cannot depend on Yl,...,Ym-z;

then

hA

deformation coordinates

have

of these coordinates:

if this were not so, s c would necessarily in c o n t r a d i c t i o n

would

would

have

to

in the same manner

depend

on

these

in order for these

to cancel out from the ratio AA/Ac, as is necessary

for consistency with [2] and [I] . But h A cannot possibly depend on the deformation coordinates initial assertion. the deformation

of system B, thus v e r i f y i n g the

By similar reasoning,

coordinates

xl,...,xn_ 2.

Ac cannot depend on [4] h A cannot

depend

on xl,...,xn_ 2 and AB cannot depend on Yl,...,Ym-2; for, according to [3], these x i and Yl are not functionally must

therefore

[2].

However,

quantity

be missing is

common

functions hA, ~B, and and

(SA,SB,t),

of assertions to

in

[5]

[I ] - [4] .

AA/AC and AB/Ac. would

have

hA-

4A(SA)T(t);

B,

and

C.

[5]

The

three

this is an elementary consequence

the

form

4A(sA)T(t),

in which T(t)

temperature

t which

4B(SB)T(t),

and

is a common function

cancels

from the ratios

If this were not so, the ratios AA/A c and AB/AC

necessarily

finally follows

A,

to the

[6 ] The functional dependences referred

respectively,

of the empirical

to

to satisfy

does not apply

~c are at most functions of (sA,t) , (sB,t)

respectively;

must

4C(SA,SB)T(t)

from h A and AB in order

this chain of arguments

t which

involved in Ac and

depend

on t,

in c o n t r a v e n t i o n

to

[2].

It

that

AB -- 4B(sB)T(t);

Ac - 4c(SA,SB)T(t)

(1.13.4)

and

~QA-

T(t)~A(SA)dSA;

~QB-

T(t)~B(SB)dSB;

~Qc- T(t)~c(sA,SB)dscIt turns that

out

function

convenient for which

to define dS(s)

(1.13.5) the metrical

= ~(s)ds;

then,

entropy

S as

provided

only

8~

I. FUNDAMENTALS

that

~(s)

is

integrable

so

that

S = f~(s)ds

is defined,

one

obtains

~QA = T(t)dSA

~Qn = T(t)dSn

(1.13.6)

~O~ = T(t)dSc,

where dSc(SA,SB) m ~c(Sa,Sn)dsc(Sa,Sn) " Inasmuch temperature

scale

regardless study,

as

of

it

is

function

T

specific

called and

the

may

properties

absolute

be

used

of

of

an

denominator the

temperature

as

empirical of ~Q,

system

under

(functlon)

a universal

in

function

for

temperature.

Note (1.13.1)

same

serves as an integrating

the

thermodynamics measuring

the

finally

that

the

substitution

leads to the additivity

of

(1.13.6)

relation

(1.13.7)

dS c = d S a + dS B.

It follows

that S c = SA + Sn, where we have

constant of integration Notice restricted

into

that

to zero.

in

ourselves

set the a r b i t r a r y

the to

foregoing reversible

we

have

necessarily

processes.

For,

an

irreversible process cannot be c h a r a c t e r i z e d by a small number of

deformation

coordinates

(see

Section

I.I).

To

emphasize

this feature we therefore write

~Q[rev

(1.13.8)

= TdS.

It should also be noted that a reversible necessarily as well. process

i sentropic" The

If ~ Q l r e v -

question whether

is to be considered

Q-

adiabatic process

0 then n e c e s s a r i l y 0 implies

in Exercise

is

dS = 0

AS = 0 for

any

1.13.1.

One additional feature of the Second Law must be taken up at this point, which relates a

finite,

possibly

isolated system.

to the entropy change incurred in

irreversible

process

in

an

adiabatically

Consider such a system in an initial state ~I

c h a r a c t e r i z e d by the set of d e f o r m a t i o n variables x I and entropy S I and

its

transition t o

a final

state

~2 a s s o c i a t e d

with

x2

87

THE SECOND LAW OF THERMODYNAMICS

and S2; the possible

achievable values

for S 2 cover some range

R, such that if ~,(x2,Sa) and ~b(X2,Sb) are two states accessible from

~I,

then

so

are

between S, and Sb. entropy for

all

states

Further,

~2(x2,$2)

for

which

S 2 lies

S I must lle in R, since this is the

every final state that is reached from ~l(xl,sl) in

any adiabatic

quasl-statlc

process.

that S I must be on an end point

It is important

to note

in R; if this were not so then

the final state ~2(x2,$2) would involve an entropy S 2 which can have

any

value

changing

the

sufficiently

deformation

close

ways

without

processes

it

neighborhood Principle.

altering

is

then

of

~i;

S I.

coordinates

adlabat~c quasi-static processes, of

to

this

subsequently

through

a

variety

of

one can adjust x in a variety

S 2.

Through

possible

By

to

flatly

this

reach

combination

every

contradicts

state

in

of the

Carath~odory's

The only way to avoid such an impasse

it to demand

that S 1 form an end point of R; then either S 2 _> S I or S 2 >_ SI, independent

of what the initial state might be.

Having once settled on one of the two alternatives, cannot

switch

to

situation held,

the

other"

For,

suppose

so that for any adiabatically

the

one

contrary

isolated system

all states ~ accessible from an initial configuration ~i had an entropy S >_ S I.

Suppose further,

that for any such system all

states ~ accessible from an initial state ~2 had an entropy such that

S _< S 2.

Then

for

a particular

state

for

which

corresponding

entropy happens

< $I, neither

the state ~I nor the state ~2 is accessible

~.

to satisfy

~

This then means, however,

the condition

that ~ is accessible

the

S2 < S from

from both ~I

and ~2; otherwise we would contradict the requirement that S be intermediate system. states

between

of

both

accessible, Principle. the

again

S 2 in the

~Q[=,vthat

larger in

and

adiabatically

clear

smaller

of

the

the choice. TdS; dS

inequality The

by universal _> 0,

so

that

convention

of

isolated

and

would

the

one must

once

selection

an

entropy

violation

To avoid contradictions

direction

abide by

and

S 1 and

That is to say that from the initial configuration

Carath~odory

therefore

for

is based

all,

in Q

choose

and

then

on the relation

one requires

increase

become

in a

that T >_ 0 reversible

~8

I. FUNDAMENTALS

process S.

involving heat exchange

This

now

leads

is matched with an increase

to a new corollary

to the

second

in

law for

processes which do not involve heat exchange between the system and the surroundings: The

entropy

of

the

final

carried out adiabatically

state

for

is never

any

process

less than that

of the initial state.

Herein, property

then,

one

of the entropy

can

detect

function"

a very

important

It provides

useful

an indication

whether or not a contemplated process can proceed spontaneously in an adiabatically process

or

totally

isolated

to occur it is necessary

system.

For

such

a

that the entropy of the final

state of the system be greater than that of the initial state. If

the

reverse

operates

situation

in the opposite

Thus,

for

any

isolated system,

holds,

the

process

under

sense.

process

dS >_ 0

whatsoever

in

an

adiabatically

and AS _> 0.

Implicit in the foregoing are several features" events occur in a totally isolated system, of constraints such processes surroundings

study

(a) When

through the removal

(for example, by unlocking a sliding partition), remain unaffected

by any manipulations

and are therefore uncontrollable

in the

and spontaneous.

The entropy of a spontaneous process in such an isolated system must

always

course

and

~eached an

the

has

system

the

process

set

its maximum value.

in.

a

decreases

for

conditions

where in

a

such

situation system the

has At

run

that

its

natural

point,

S has

(b) For any process whatsoever

equality

is carried out reversibly

encounter

However,

until

equilibrium

isolated

process can

increase

in in

system

dS

-

holds

only

if

and adiabatically. which

the

processes

interacts

circumstances

0

one

carried

with

must

entropy

its

in the

(c) One actually out

under

surroundings.

encounter

somewhere

else in the universe a compensating process such that the total entropy change of the system and that of all its surroundings where compensating processes

occur is always nonnegative.

THE SECOND LAW OF THERMODYNAMICS

We

further

pressure-volume carried

note work

that is

dE - ~Q - ~W - TdS

represents

developments,

special

and

case

where

one may combine

where

all

only

changes

the First

are

and Second

into the form

(1.13.9)

- PdV.

the

starting

one must note

the energy and the entropy, stressed

the

point

for

all

subsequent

beginning with Section I. 17.

Finally,

As

for

involved

out reversibly,

Law of Thermodynamics

This

89

earlier,

the

an important

difference

between

considered as a function of state. energy

of

a system

can be

altered

solely by transfer of heat or work or (as shown later) matter across

the boundaries.

By contrast,

the entropy

can also be

changed by processes occurring totally within the system. as

the

performance

adiabatically

can

system,

so the

can

used

be

instance

one

be

of used

transfer

to

track

must

be

work to

(reversibly track

of heat the

to

change. avoid

or

energy

reversibly

entropy

careful

the

Just

otherwise) change

of

a

across boundaries In

changes

this that

latter occur

totally within the system in order to keep an accurate account 9of entropy changes.

EXERCISES

1.13.1 Is it correct to state that for any process for which Q 0, one must have AS - 0 as well? If there exist exceptions, can you classify such cases? 1.13.2 Prove that the postulate rendering AS > 0 for any spontaneous process in isolated systems is consistent with making the universal temperature function T positive. 1.13.3 Various recourses to experiment have been proposed to settle the question whether AS increases or decreases in the course of an irreversible process. (a) Consider an adiabatic change in internal energy produced by frictional effects. In which direction does the energy change? What can you then conclude about AS? (b) Consider the adiabatic change in pressure and temperature of an isolated gas when its volume is changed irreversibly. How does AS change in

90

I. FUNDAMENTALS

this process? (c) Give reasons why this approach is considered to be relatively 'inelegant' as compared to the discussion in the text. 1.13.4 An adiabatic enclosure is filled with supercooled water and allowed to stand. After a while some ice is observed to form. Obviously, the process is spontaneous, yet it appears that there is an increase in order, hence, a reduction in entropy. Can you think of a way that gets you around this apparent violation of the Second Law? 1.13.5 A supersaturated salt solution is placed in an adiabatic enclosure. After a while a very small amount of salt precipitates out. One would argue that the separation of a single phase into two would result in a system that is less 'mixed up' at the end than at the beginning. This seems to violate the Second Law. Provide a concise argument which circumvents the conceptual difficulties by writing out an expression for AH and for AS accompanying the precipitation of An moles of salt and the concomitant change AT in temperature. 1.13.6 Is an adiabatic process necessarily reversible? Is a reversible process for which dS - 0 at every stage necessarily adiabatic ?

I. 14

AN ENTROPY ANALOGY

As was

the case

that

for the First Law,

illuminates

the

follow

the

change

is to be likened

two

example

Second

points

mountain

A

and

climber

provided

B,

Law by

one may provide

of Thermodynamics. Spalding

and

to a difference

ASAB, on

undertakes

a parable

rough

Cole 7.

ASAB

in

we

Entropy

in elevation

mountainous

to measure

Here

between

terrain. terms

of

A the

number of steps nf, each one-foot in height, which he must take to get from level A to level B. a circuitous elects

to

route

take

along

a direct

a firm path, route

which is full of dirt and gravel, step. than

Now the increase the number

taken:

If the climber elects to take

along

then ASAB -- nf. a

steep,

slippery

path

then slippage occurs at every

in altitude of the mountaineer

of one-foot

If he

incremental

The climber can now only establish

steps,

nl,

is less

that were

the inequality ASAB

7D.B. Spalding and E.H. Cole, EnEineering Thermodynamics (Arnold, London, and McGraw-Hill, New-York, 1959).

91

CYCLIC PROCESSES

< n e. Considerable

work has been wasted

in the

form of heat

supplied irreversibly to the countryside when the climber, dirt and gravel

resettles

yet another

method

at every step.

Obviously,

of establishing

AS~

there exists

namely by consulting

geodetic survey maps on which altitude contours are entered. It should be clear change, that

that ASAB corresponds

to an entropy

that n~ and n, simulate J~Q=/T and J~QI/T respectively,

the

firm

irreversible

and

loose

processes,

paths

correspond

to

and that contour

reversible

and

on a map

are

lines

analogous to tabulations of entropy values.

The shortcomings

of the parable should be explored by the reader.

1.15

CYCLIC PROCESSES IN RELATION TO REVERSIBILITY AND IRREVERSIBILITY.

CARNOT EFFICIENCY

(a) The considerations powerful tool in Consider

two

of cyclic processes

final deductions

points

in

based on the

configuration

infinitesimally close to one another, 2 in Fig.

1.15.1.

constitute

Choose

Second Law.

space

which

as represented

a particular

a very

quasistatic

are

by i and process

which takes a given system from state I to state 2; during this step let the heat exchange between system and surroundings be given as ~Q~-2 (in Exercise 1.15.1 we deal with the question as to

whether

it

positive).

matters

if

this

quantity

is

negative

Select a second, irreversible path that effects the

same i~2 change and that involves a heat exchange ~Q~-2. that

this

latter

irreversible appropriate

path

process,

_

that

is

dotted

the path

on

lies

the

diagram:

outside

to quasistatic processes.

dEi-2 + ~I-2_..rand ~Q~-2

Note

or

functions of

energy

Being

the phase

an

space

By the First Law, ~Q~-Z

dE I"2 + _..~ , so that

(1.15.1)

_

the

Note

differentials

have

canceled

out;

state, they are the same for both processes.

being The

92

I. F U N D A M E N T A L S

2 \\ r

1

x2

FIGURE 1.15.1 lllustration of a reversible path (solid curve) connecting states I and 2, and of an irreversible path (dashed curve) connecting states 2 and i. The dashed curve is meant to convey the fact that the irreversible process involves many thermodynamic coordinates which fall outside the domain of the indicated configuration space.

differences

in Eq.

(1.15.1)

cannot vanish.

If they did

reversible and irreversible paths would have to coincide, ~Q and ~W depend on the chosen paths. contradiction of terms.

Thus,

This would

the algebraic

the

since

lead to a

sums in (1.15.1)

must be either positive or negative. Suppose

first ~Q~-2 _ ~Q~-2 m ~Q~-2 + ~Q~-Z > 0; then the

work involved in completing the cycle by going from I to 2 via path i and returning to I via path r is given by ~ - 2 ~-2

+ ~W~-Z > 0.

expended

That is to say"

(i.e., put into the system)

~Q~-I > 0 and have obtained

_ ~z-2_..=_

In executing a cycle we have an amount of heat ~Q~-2 +

from the system

(i.e.,

the system

has performed) an exactly equal amount of work ~W~ ~2 + ~2~z > 09 w.. r No other changes have occurred in the universe.

Suppose next that ~Q~-2 _ ~Q~-2 m ~Q~-2 + ~Q~-Z < 0, so that by Eq. (1.15.1) ~ ~ 2

_ ~-2

_ ~W~2 + ~ - i

< 0.

A similar line

of reasoning now shows that in going around the cycle once we

CYCLICPROCESSES

have

93

expended

(i.e. ,

put into the system)

an amount of work

~WiI"2 + ~W z'1 < 0 and we have obtained from the system (i.e., the system has

transferred

to its surroundings)

amount of heat ~Q~+2 + ~Q~*I < 0.

an exactly

equal

No other changes have occurred

in the universe. Which of these alternatives do we choose? Let us simplify ~fQr1".2 - TdS I~2 - TdS.

the notation by setting ~ Q ~ 2 One must now consider

, ~Qi and

the possibilities

(a) ~Qi - TdS > 0, i.e., TdS < ~IQi, or (b) ~IQi - TdS < 0, i.e., TdS

> ~Qi.

The

present

discussion

must

apply

to

all

cases

including the special situation where the irreversible process is carried out adiabatically.

Under alternative

obtains TdS > 0 for the adiabatically contradictions are uncovered. the

corollary

to the Second

isolated system,

Under alternative

require TdS < 0 for an irreversible, Law,

this possibility must be ruled out.

(b) one then (a) one would

adiabatic process,

discussed

and no and by

in Section

1.13,

We thus claim that (I.15.2a)

~Qi < TdS - ~Qr.

This result is eminently sensible:

If we take the system

from state I to state 2 by a reversible

sequence of steps the

entropy change must be matched by a heat transfer of magnitude dQrev/T.

If

the

same

change

is achieved

by

an

irreversible

operation,

part of the overall entropy change

is generated by

processes

totally

amount

required

to

within

match

the

correspondingly smaller.

the

system.

remaining Also,

The

entropy

change

of will

heat be

from (i.15.7a) we find

~W i < ~Wr,

(I.15.2b)

and for any finite changes

S2

- S1 >

Equation

(1.15.3)

~QI/T.

(1.15.3)

open the question

must as

be

used with

care,

for it leaves

to what temperature is to be employed in

~4

I. FUNDAMENTALS

conjunction with the irreversible process that takes the system from state (1.15.3)

I to state 2.

In later arguments we shall use Eq.

solely in situations

where

the system under study

thermally properly anchored to a reservoir; temperature

is

T then is the fixed

of the surroundings.

Consider

next

a different

taken from some state

process

wherein

through a macroscopic

a system

is

cycle back to the

same initial state by means of a reversible process.

Denoting

such a path by the integral sign ~, we then find that

(1.15.4)

~dS - ~IQr/T - 0. On

the

other

process,

~QI/T

hand,

(1.15.3)

shows

that

in

this

cyclic

for which S I - S 2,

< 0.

(1.15.5)

Both statements

~Q/T

Eq.

may be combined to read

~ 0,

(1.15.6)

which is known as the Clausius In

this

irreversible

connection

note

the

an

no

isolated

system.

direction

We emphasize a

means

adiabatically for which system;

occurs

in an

If an irreversible process proceeds at all,

in the spontaneous

as

that

to

of

there is

a process

return

nature

point without incurring other changes in the universe; influencing

can never

very

starting

of

one

again

the

way

process:

inequality.

of

once more that the state

monitoring

isolated

S decreases conversely,

that increases

whether

a

the entropy. function S serves

given

process

system is indeed possible. can occur

any process

it must go

in

an

No process

in an adiabatically for which S increases

isolated in such

a system will be spontaneous. For a system A which exchanges heat with surroundings

B

we consider an enlarged system in which the original system and all of

its surroundings

form a composite isolated system;

then

95

CYCL,C PROCESSES

dSto t -

dS A +

d S B >_ O.

(b) Consider useful

now the operation

as a continuous

source

being o p e r a t e d in cycles,

of a heat engine.

of power

it must be

in the universe only. as

to

whether

engines

operate

temperatures that heat

All

In Exercise 1.15.2 the question is p o s e d made

next

apply

the

engine

T h and T c.

Let

between

The

two

to

noncyclic

reservoirs

do not appreciably

Qh be

the

heat

reservoirs are

alter

transfer

(Qh > 0 or Qh < 0 according

as heat

engine at the hot junction),

per

across

reservoir

the

so

at

large

the temperatures cycle

across

the

T h and the engine

into or out of the

and let Qc be the heat transfer per

boundary

at temperature

flows

kept

assumed

b o u n d a r y b e t w e e n the reservoir at temperature

flows

it started.

the operation of the engine thus occur

statements

transfers

each.

cycle

of

such as rockets.

We

of

the

capable

so that at the end of every cycle the

engine gets back to the same state from which net changes accompanying

To be

separating

the

engine

and

the

T= (Qc > 0 or Qc < 0 according as heat

into or out of the engine

at the cold junction).

If now Qh > Qc, or Qh - Qc > 0, then in the course

of one

cycle we must have W > 0 in order that the First Law applied to the

cyclic

process

arranging matters

(Qh - Qc)

- W-

so that heat

from the hot reservoir the cold reservoir,

0 may hold.

is transferred

and heat

That

is,

into the engine

is rejected from the engine

the difference

by

Qh " Qc is transformed

to

into

work W > 0 performed b_.y the engine. We now ask how efficiently this energy c o n v e r s i o n process can be carried

out.

The efficiency

is m e a s u r e d by a quantity

defined as i W o r k out/Heat

To

get

at

(1.15.6).

in .

this In

(1.15.7)

quantity

the

present

recall case

reduces

to a sum of processes

Qh from

the hot

reservoir

the

the

integration

occurring

into

the

Clausius

inequality in

during heat

engine

and

(1.15.6) transfers

- Q= from

the

~

I. FUNDAMENTALS

engine

into the cold reservoir.

Thus,

7.(i)Qi/Ti - Qh/Th - Qc/Tc _< 0,

(z.15.8)

Qh/Th --< Qc/Tc,

(1.15.9)

where

Qh,

Qc

are

both

positive

quantities,

inasmuch

as

a

negative sign precedes Qc whenever we consider the heat removed from the engine.

Hence

(1.15.9) may be rewritten as

(1.15.1o)

qo/qh >- T o/mh, or

-

Qc/Qh -
(YT) ~jaj. I. 18.9 (a) Show that the adiabatic and isothermal compresslbilitles ~s i - v-l(av/aP)s and ~T " - V-I(aV/aP)T are related by ~s i 7-I~T, where 7 i Cp/Cv. (b) Prove that ~s - ~T -- (T/CpV)(aV/aT)~. (c) Prove that for an elastic system C= - Ce ,

,

~,&

,

-

-

'IV

,,

.

j~,~

.

.

,

.

.

.

.

.

~,Bial.

1_18.10 Show that for any material whose equation of state is specified according to P - P(T,V) the following equation is obeyed: Pap - ~av, wherein ap - P-l(aP/aT)v, av i v-1(aV/aT)p and ~ - - v(aP/aV) T. 1.18.11 Show that for any material that is representable by an equation of state of the form P f(V).T the energy is independent of the volume, so long as the function f(V) depends on V alone. 1.18.12 (a) Prove that (aS/aP)B > 0 and (as/av) E < o, by noting that V, P, T are all positive. (b) Prove that (as/av)p CpTVa, where ~ - V-l(aV/aT)p. Is (as/av)p always positive or always negative ? 1.18.13 Relate the quantity (aV/aT) s to quantities that may be experimentally determined. (b) From the Gibbs equation TdS - dE + PdV prove that for all adiabatic processes (aT/aP)s - (T/Cp) (8V/aT)p. 1.18.14 For a van der Waals gas determine the Joule Thomson coefficient and inversion temperature T i in terms of a, b, P, T, and Cp. Make a sketch of T i versus T; note and comment on the double-valued nature of the function. 1.18.15 Let the force constant of a spring depend on temperature as k cT with c > 0, where c depends only on length L. (a) Show that E is a function of T only and that S decreases with increasing length. (b) Show that T rises if the string is stretched adiabatically. (c) For a spring which follows Hooke's Law the element of work a'Wr is kdx, where dx is the incremental elongation and where the spring "constant" k is p a r a m e t r i c a l l y dependent on temperature. Find the energy, heat capacity Cx, entropy, and Helmholtz free energy for this system. -

131

FUNCTIONS OF STATE

1.18.16 The equation of state of a spring is given by T (a - bT) ( L - Lo(T)), and the increment of work is given by a~w= - ~dL. Here I" is the tension, L the length, L 0 the extension of the spring under zero tension, a, b are constants, T is the temperature. (a) Determine AG, AH, AS, and AE for a process in which the spring is extended reversibly to a length 2L 0 at constant T and P. Assuming d V - 0, express your results in terms of = ~ IZ1(dL0/dT), a, b, T, and L 0. (b) Determine Q=,v for the process of stretching. Assuming A V - 0, determine W=,v for the process of stretching. (c) Let the spring snap back to its original position. Now calculate AS and AE, and the entropy change in the surroundings during this latter process. What is the overall entropy change in the system and surroundings during the cycle just described? Hint" Use Maxwell's relations to obtain a partial derivative involving S and L. 1.18.17 (a) Determine expressions for Cv, E, S, F, and G for a van der Waals gas. (b) Show that for an adiabatic change T(V - b) 7-I and (P + a/V 2) (V - b) 7 are constants. 1.18.18 (a) Determine (8E/SV)~ and (8H/aP)~ for a gas obeying the van der Waals equation of state. (b) Show how the entropy changes with pressure and with volume at constant temperature for the above type of gas. (c) Prove that C v is independent of V for this type of gas, provided the van der Waals "constants" are taken as independent of T. 1.18.19 A glass ampoule containing i mol of liquid water at 100~ is inside an evacuated container of volume V. The ampoule is broken, while the system is maintained at 1000~ Calculate AS for the change in state (a) for the case that V I0 liters, and (b) for the case that V - i00 liters. Make any reasonable approximations required. 1.18.20 The specific heat of CC14(2) for t < 25~ is 0.20 cal deg-Zg -I at a constant pressure of I atm. Find its specific heat at constant volume at 25~ at this temperature, the isothermal compressibility is given by (107.70 + 99.4 x 10-aP + 70 x 10-6p2)x 10 -6 bar -1, its density is 1.58455 g cm -3, and its coefficient of expansion is 0.00124 deg -1. 1.18.21 By how much does the enthalpy of 3 tool of Cl2(g) change when its temperature is increased at a constant pressure of I atm from 300 to 400 K? Its heat capacity at a constant pressure of i atm in this temperature range is 7. 5755 + 2.4244(I0-3)T 2 - 9.650 (10-7)T 4 cal deg -1 tool-I. 1.18.22 The Berthe!ot equation of state as applied to many gases reads -

PV - RT

i +

I - 6 128 P=r

-~-] ~ '

where T= and P= are the critical temperature and pressure.

132

I.

FUNDAMENTALS

Derive an equation showing how the enthalpy change for such a gas is related to a pressure change from P - PI to P - P21.18.23 (a) Determine the change in Cp for N 2 gas when the pressure is changed from 0.I to 400 atm at 270~ The van der Waals constants for this gas are a - 1.39 llter 2 atm/mol 2, b - 3.92 x 10 -2 llter/mol. (b) Repeat the calculation for the change in C v. 1.18.24 Calculate the entropy change in irreversibly converting i mol of supercooled water at -10~ to ice. Hint" Calculate this quantity by determining the change in entropy involved in heating the water reversibly to 0~ freezing the water at 0~ and freezing the ice reversibly to -10~ Set o C~(2) - 18 cal/deg-mol, Cp(s) - 9 cal/deg-mol, Allo 2 7 s - -1436 cal/mol. The solid or liquid is maintained at a pressure of I atm throughout. For liquid water (@V/aT)p = 0.015 cm3/deg. Calculate the change in entropy when the pressure over the liquid is reduced by 1/2 atm. Check your sign carefully. 1.18.25 The accompanying Table (Table 1.18.1) provides a listing of the Joule-Thomson coefficients for N2, ~j (aT/aP)H , over a large range of T and P. "

Table 1.18.1

Joule-Thompson

Temperature

Coefficient

for Nitrogen in deg arm -I Pressures (atm)

I

20

60

I00

200

200~

0.0540

0.0460

0.0365

0.0260

0.0075

I00

0. 125

0. 114

0.0955

0.0760

0. 0415

50

0. 179

0. 166

0. 141

0. 115

0. 066

25

0. 214

0. 200

0. 169

0. 138

0. 078

0

0. 257

0. 242

0. 204

0. 166

0.090

-50

0. 384

0. 362

0. 299

0. 231

0.094

-i00

0. 628

0. 578

0.443

0. 281

0.062

(a) Through self-consistent procedures determine the final temperature achieved by the throttled expansion of N 2 gas initially at 25~ from 200 to i atm. (b) Express the Joule-Thomson coefficient for as gas subject to the approximate

ILLUSTRATIVE EXAMPLE: J O U L E - T H O M S O N

133

COEFFICIENT

equation of state PV - RT + B(T)P, where B is the so-called second virial coefficient. (c) Using (b) and the accompanying graph (Fig. 1.18.2) for CO 2 determine the approximate temperature at which Nj changes sign. (d) Calculate the Joule-Thomson coefficient for CO 2 at 125~ (e) Derive an expression for (8Cp/aP)~ in terms of the Joule-Thomson coefficient of the gas. (f) Is it not it a self-contradiction to compute (8Cp/aP)T when C e implies a heat capacity measured at constant pressure? Why then do we not set ( a c e / a P ) T - 07 Explain these matters carefully. (g) for N 2 gas C p - 6.45 + 1.41 x 10-aT - 0.81 x 10-7T 2 cal/deg-mol. Using Table 1.18.1, estimate (8Ce/aP) T for N 2 at 300 K, and estimate the change in heat capacity with a pressure change from I to i01 atm.

Temperature 200~

20

1 CO2

l

~

Pressure 60 0. 0365

i 0. 0540

20 0. 0460

i00 0. 0260

200 arm 0. 0075

i00

0.125

0.114

0.0955

0.0760

0.0415

50

0. 179

0. 166

0. 141

0. 115

0.066

,

I0 0

25

0. 214

0. 200

0. 169

0. 138

0. 078

-10

0

O. 257

0. 242

0. 204

0. 166

0. 090

~ -20 U 0'~- :30

-50

0. 384

0. 362

0. 299

0. 231

0. 094

-i00

0. 628

0. 578

O. 443

0. 281

0. 062

-150

1.225

1.097

0.062

0.0215

-0.0255

-50 -60 / ! -

70

-8(300 400

5~)0 6(~0 700 T (K)

8~) 900

FIGURE 1.18.2

1.18.26 The standard entropy of Ar(g) as a function of temperature is given by S ~ (eu tool-I) - 8 . 6 8 + 11.44 log T. Find the change in Gibbs free energy for I mole of Ar(g) when its temperature is changed from 298 K to 348 K at a constant pressure of i atm.

I.A

ILLUSTRATIVE

EXAMPLE"

DEDUCTIONS

BASED

ON

THE

JOULE-

THOMSON COEFFICIENT (a) We illustrate the power of the general thermodynamic approach by examining a set of deductions based on the use of the Thomson coefficient.

We begin with Eq. (1.18.35b), which

134

, FUNDAMENTALS

we rewrite

as

(SH/aP)~--

~jCp.

(I.A.I)

From dH - TdS + VdP, (l.18.10b),

(SH/SP)~-

Next,

with

the M a x w e l l

relation

we obtain

-

T(SV/ST)p

substitute

(I.A.2).

in c o n j u n c t i o n

the

+ V-

-

equation

r/jCp. of

(I.A.2)

state

for

V

-

As an example we adopt the van der Waals

V(P,T)

in

equation

of

state" P - nRT/(V-nb)

Brute

force

dlfflcult;

inversion

RT/P

In

first

fourth V-

of

accordingly,

both sides by

V-

(I.A.3)

- an2/V 2.

the

to

solve

for

we resort to approximations.

V(P,T)

is

Multiply

(V - nb)/P: (I.A.4)

- a/PV + b + ab/PV 2.

approximation

replace

terms on the right,

RT/P

above

V

by

RT/P

in

the

second

and

to obtain

(I.A.5)

- a/RT + b + abP/R2T 2.

Then (aV/ST)p-

Solve Eq.

R/P + a/RT 2 - 2abP/R2T 3.

(I.A.6)

(I.A.I) for R/P and insert the result into (I.A.6)

to

find (8V/aT)p-

When

(V-b)/T + 2a/RT 2 - 3abP/R2T 3.

(I.A.7)

and (I.A.5)

are p l a c e d

in (I.A.2)

(I.A.7)

one finds

ILLUSTRATIVEEXAMPLE:JOULE-THOMSON COEFFICIENT

J3 5

112a 3P1 1(2a ) ~-~ - b - RZTZ ] = -~p ~-~ - b ,

(1.A.8)

~j - -~p

where we have neglected

the higher

order

term

in P/T 2 on the

right. Note

that

~j is positive

for

'large'

a and

'small'

b,

corresponding to large interatomic attractive forces and small interatomlc decreasing

repulsive P across

the temperature, of the gas. prevails, (I.A.8)

For

the porous

which

~j

>

0,

plug of Fig.

(aT/aP) H >

1.18.1

0:

decreases

is the effect we need for c o n d e n s a t i o n

For 'small' a and 'large' b the opposite situation

which

Generally,

forces.

is of no

interest

as the gas pressure

causes

negative.

~j

to

Further,

is increased

decrease, at

for possible

pass

'low'

the third term in

through

pressure

applications.

zero,

and

~j increases

as

turn T

is

diminished (Cp also decreases with T); higher pressures are then needed to effect a sign reversal in ~j. a change b(T)

Thus, one anticipates

in sign in ~j as P and T are altered and as a(T)

are changed by selection

of different

types

and

of nonideal

gases. (b) These matters determining ~j-

0.

The

satisfies

are addressed

the so-called central

inversion

term

in

more

quantitatively

temperature,

Eq.

(I.A.8)

vanishes

when

curve

of T i versus pertains

experiment. critical

the

left

to

One

value

temperatures equation.

Ti

the quadratic equation

(I.A.9)

T~ - (2a/Rb)T i + 3aP/R z - 0. A plot

by

Ti, at which

P is shown N2

sees Pc

(=

in Fig.

gas

and

that

at

is

300

atm)

a

in

given there

I.A.I; fair

the

agreement

pressure exist

that are derived from solutions

indicated with

Po b e l o w

two

a

inversion

of the quadratic

To achieve liquefaction of the gas one must stay to of the curve

larger this quantity,

in Fig.

I.A.I,

for which

the more effective

the gas through the porous plug.

Wj > 0.

the pressure

According to Eq.

The

drop of

(I.A.8) one

136

I. FUNDAMENTALS

500~

"~.~

._. 400~ ~r,...)

o

"~, "x

~ 300"

"x x\

~

",,

.o= lOO~ >

,

~=

/

oOJ-

/////

o

--I00

20

FIGURE I.A.I

should large

100

200 Pressure

Joule-Thomson

therefore and b

60

is

select small,

300 Atan.

inversion curve for nitrogen.

a van

der Waals

and Cp is as

small

gas

for which

as possible.

should also operate at the lowest temperatures, the operating Fig.

pressure,

while

a

is One

consistent with

remaining within

the limits

of

I.A.I. (c) Another quantity of interest is the set of isenthalps

that

may

be

constructed.

Begin

with

dH

-

(8H/aT)pdT

+

(8H/aP)TdP and write dH - Cp(T, P) dT

-

CprIJdP 2a

-

Cp(T,P)dT

+

- ~-~ +

3abP RZTZ

+ b] dP,

(I.A. i0)

ILLUSTRATIVE EXAMPLE: JOULE-THOMSON COEFFICIENT

where

the

gases,

second dH

derivatives

l a__~C1 ~ 'apl T

is

an

refers

exact

specifically

dlfferentlal;

to van hence,

der

Waals

the

cross

must match:

2a ~

llne

[ ,3"I

6abP

(I.A.II)

- ~ .

Next, carry out an indefinite integration over pressure on both sides at constant T:

cp

2aP (3ab ~ p2 - ~ - k-z~ l + C(T),

(I.A.12)

where C(T) is an arbitrary function of temperature. (I.A.12) i r/j - - ~

Then, from

and (I.A.10) (8H) 2 a / R T - 3abP/R2TZ - b ~ T - C(T) + (2a/Rr z)P - (3ab/RZT ~)Pz

2a/RT - b = C(T) + (2a/RTZ)P '

(I.A.13)

where we have neglected higher order terms in abpn/R2T m. Consider next a process taken

from an initial

state

in which a van der Waals T1,P 1 to a final

state

convenient sequence of steps involves the following:

gas is

T2,P 2.

A

(i) Cool

the gas reversibly from T I to T2, keeping the pressure fixed at P1.

(ii) Expand the gas reversibly from P1 to P2 while holding

the

temperature

isenthalplc

at

T z.

conditions

In we

executing

find,

on

use

these

steps

of

(I.A.12)

under and

(Z.A.Z0), 0 - All - [ T2 CpdT T1

-r ~~

[

"

I

P1 + /P1 dH T2 - 3abP

+

2a 3abP ] RT 2 + R ~ 2 + b dP.

]dT

(I.A.14)

138

FUNDAMENTALS

I.

On adopting as

the approximations

sensibly

constants,

mm

H

mm

.

independent

of

already mentioned, T,

and

taking

a

regarding

and

b

to

C be

we find 2aPt [ i R

. I

+ C(T2"

TI) + -

~2

+ b

(P2 " PI),

(I.A. 15) or

GT I + bP 1 - (2a/R)(P1/TI)

= CT 2 + bP 2 - (2a/R)(P2/T2) (I.A.16)

- H(T1,PI) - H(T2,P 2) m H, a constant.

The

quantity

- H(T,P)

m

(2a/R)(P/T)

- bP

- CT,

which

is

a

constant under the c o n t e m p l a t e d conditions and approximations, fs the desired

P

-

isenthalp.

We see that the e x p r e s s i o n

[ "

RT

2a

-

"

provides us with a family of curves P -

P(T;a,b,C,H)

"

such that

for a given TI,PI,C and for a given gas with specific a and b we arrive

at

a

specific

value

of

H,

as

determined

from

Eq.

(I.A.17). Figure set of values

I.A.2 shows a schematic of HI,...,H 8.

point represents values. curves Fig.

The

T for a

The shape of these curves

for any

the value ~j for that specific

dashed

HI,...,H8; I.A.I.

plot of P versus

llne

this In

passes

through

schematically

numerical

work

regions

applications,

accurate

applications. proceed

llke

maxima

of

the

curve

one

should

(I.A.16),

Fig.

I.A.2

the of

recall

the

(I.A.17).

As

indicate

of T and P for which a given gas provides

possible Wj. more

plots

the

duplicates

approximations used in arriving at Eqs. concerns

set of T and P

the

the largest

With more realistic equations of state one can get plots The

that

serve

present

in such cases.

as

guidelines

discussion

shows

in e n g i n e e r i n g how

one

would

I 39

HEAT, CALORIMETRY, AND THERMOCHEMISTRY

HB ~

~H7

\\

H5

T

\ \\1~--------

/-/4

////

H3

////

H2 H1

~

0

FIGURE I.A.2

I. 19

HEAT, (a)

Schematic

diagram for isenthalps.

CALORIMETRY, Up

etherlal

P

to

now

quantity,

AND THERMOCHEMISTRY

heat

having

has

been

been

treated

introduced

as

deficit function which covered the difference, between AE and - W. with

a physically

both

a

set

of

We now complement

more

units

meaningful

and

a method

somewhat as

a

in any process,

these earlier concepts

discussion, for

a

originally

by

metering

introducing out

and

for

monltoring heats. The heat developed in a circuit by an electric current of I - I ampere flowing through a resistance of R. ~ I ohm (across a potential is defined

difference

of V , -

to be i joule.

i volt)

One calorie

for a time t -

i second

is defined as equal

to

4.1840 joules and corresponds very closely to the heat required to raise 15.5~ a circuit

the

temperature

More generally, is specified by

of

I g of pure

water

from

14.5

to

the heat developed by current flow in

140

, FUNDAMENTALS

(1.19.1)

Q I ~12R.dt I ;V.ldt.

(b) A convenient method for measuring heat flux utilizes calorimetry. derives internal

The

thermodynamic

from the First Law: energy

E I

for

has

the

likewise,

differential

(aS/aV)TdV.

procedure form

dE

I

the entropy S considered as

a function of T and V has the differential dT +

this

We saw in Section 1.18 that the

E(S,V)

(aE/aS)vdS + (aE/aV)sdV;

basis

(aS/aT)v

form dS =

If this latter relation is substituted

into

dE, one finds

(aE/aS)v (aS/aT)v dT + [(aE/aS)v (as/av)T + (aE/aV)s]dV.

dE =

(i.19.2a) Using the

the chain rule of differentiation

right,

and

Eqs.

(1.4.7),

in the first

(1.18.25),

(l.18.10a)

term on in

the

second, yields

dE I

(aE/aT)vdT + (aE/aV)TdV ,

(i.19.2b)

which shows that E may also be considered an implicit function of T and V.

However, as is to be shown in Exercise 1.19.1, Eq.

(i. 19.2b) is incomplete, though it remains extremely useful for our further development. At fixed volume and in the absence of other work dE I ~Q

i (aE/aT)vdT; thus, on introducing the definition for Cv, Eq. (1.18.12), we find

dE - ~Q = CvdT

(1.19.3a)

(V constant).

For finite processes

in which the temperature

is changed from

TI to T 2:

CvdT - AT wherein

the

average

defined by < C v > i

heat

(V constant), capacity

at

(i.19.3b) constant

volume

~ T2 TI CvdT /~~T~ dT, and where AT m T 2 - TI.

is

14 I

HEAT, CALORIMETRY, AND THERMOCHEMISTRY

Equation

(1.19.3)

adiabatically monitored

isolated

through

its

Adiabatlclty

is

surroundings

while

process

may

shows

be

system

the heat

at

to

volume

rise

prevent

is being

conveniently

transfer

constant

temperature

essential AT

that

if

heat

monitored.

executed

by

into an may

is

leakage The

be

known. to

the

calibration

supplying

a known,

small amount of electrical energy Q . - Jl2R, dt to the system and measuring

the

small

resultant

ATe;

this

establishes

and

permits Eq. (I.19.3b) to be used to determine Q for any process of interest.

Naturally,

calibration indicated

by

numerous flows, on

and

operation

the

other

the correct experimental procedures of are

superficial

techniques

much

discussion.

for measuring

as indicated by reference

calorimetry.

The

more

main

complicated Also,

than

there

and quantifying

are heat

to any of the numerous books

point

of

discussion

here

is

to

demonstrate at least one method by which Q can be measured. Analogous is held

remarks apply when the pressure of the system

fixed.

In

Exercise

1.19.2

the

reader

is

asked

to

establish that dH-

KQ -

CpdT

(P constant)

(I.19.4a)

and AH-Q-~

2 CpdT - AT

(I.19.4b)

(P constant).

1

The remarks of the previous paragraph also apply to the present case, except that P rather than V is held fixed. (c) In chemical thermodynamics the preceding concepts are largely

applied

One widely

to measure

used procedure

the heats

of

is to carry

very rapidly and adiabatically

chemical

out

reactions.

a combustion

in a bomb calorimeter,

step

thereby

changing the system from the state T - T o , P - P0, and V = V 0 to the

state

T - TI,

P - P2,

and V = V0;

in this

step

AE -

0

because no work has been performed and no heat flux has taken place

in the time scale of the first step.

The products

are

142

,

then slowly

cooled

so that

the

system ends up in the c o n f i g u r a t i o n T - To, P - Pi, V - V 0.

In

this step A E z z -

T

to their

initial

temperature,

FUNDAMENTALS

CvdT, where C v includes

c a l o r i m e t e r that chlanges in temperature. of state, AE - AEII. + A(PV).

the p o r t i o n of the

Since E is a f u n c t i o n

To convert to enthalples we write AH - AE

W h e n e v e r condensed phases are involved A(PV) tends to

be negligible,

but for reactions

of the a p p r o x i m a t i o n

A(PV)

involving gaseous species use

= RT An is generally

satisfactory,

where An is the increase in mole numbers of the gaseous species at the average is

asked

to

temperature,

carry

out

T.

In Exercise

corresponding

1.19.3

analyses

the reader

for

reactions

carried out at constant pressure. The foregoing m e t h o d o l o g y applies, of

process,

particular, standard

including

to cases where

conditions,

one atmosphere quantities

equilibration

i.e.,

the

reaction

with

at the temperature

all

zero,

e.g.,

to any type

procedures, is carried

species

and,

in

out under

at a pressure

of interest.

obtained for these conditions

by a s u p e r s c r i p t

of course,

of

Thermodynamic

are usually indicated

E ~ 9 C o 9 S o 9 and the like

EXERCISES

1.19.1 Provide a thoroughgoing d i s c u s s i o n to show in what sense the differential equation dE (8E/aS)vdS + (8E/aV)sdV is more fundamental than the relation d E - (8E/aT)vdT + (8E/aV)TdV. 1.19.2 Provide a d e r i v a t i o n of Eq. (1.19.4). 1.19.3 Discuss the basic calorimetric method for d e t e r m i n i n g AH for a reaction carried out at constant pressure. 1.19.4 For a fluid any two of the three variables P, V, t (empirical temperature) are independent. An increment of heat may now be described in several alternative ways ~ Q - Cvdt + LvdV - Cpdt + LpdP - MvdV + MpdP. (a) Express in words the m e a n i n g of the various C, L, M coefficients. (b) Derive expressions for Mv(Lv,Cp, Cv), Mp(Lp,Cv,Cp) and show that Mv/L v + M p / L p - I. (c) Derive expressions for (@P/at)v and (@V/at)p in terms of Cp, Cv, Lp, or L v. 1.19.5 The value of AH for the formation of FeS from its elements at 25~ is -22.72 kcal/mol. Determine AE per mol for the process at i atm. The densities of Fe, S, and FeS are 7.86 x 103 , 2.07 x 103 , and 4.74 x 103 kg/m 3 respectively. 1.19.6 One hundred grams of lead at IO0~ are put in 200

HEAT, CALORIMETRY,AND THERMOCHEMISTRY

143

g of water at 20~ the resulting temperature at equilibrium is 21.22~ Calculate the specific heat of lead. 1.19.7 Consider a system surrounded by a heat reservoir, both of which are in an adiabatic enclosure. The system can exchange work with surroundings outside the enclosure. (a) Suppose the system is taken from state A to state B. Express the entropy change of the system in terms of Q, AEAB , W and do the same for the bath. What is the overall change in entropy? (b) Allow the system to return spontaneously to state A without any work being done. Express ASBA and AEBA in terms of the entropy and energy change for the inverse process. Explain how any energy is dissipated in the process. On this basis obtain AS for the reservoir, and the overall entropy change for the process B ~ A. (c) Show how the above may be applied to determine the entropy change of a freely dropping object of mass m. (d) Repeat, for a spring whose equation of state is - ~ ( L - L0) where T is the tension, L the length, ~ the force constant, and L 0 the length at zero tension. Assume that when completely extended, Lf - 2L0, and that the stretching is accomplished isothermally. 1.19.8 Calculate the changes in the entropy of the univers~ as a result of the following processes" (a) A copper block of mass 400 g and thermal capacity 150 J deg -i at IO00~ is placed in a lake at 10~ (b) The same block at 10~ is dropped from a height of i00 m into the lake. (c) Two similar blocks at 100~ and 10~ are joined together. (d) A capacitor of capacitance I ~F is connected to a 100-volt battery at 0~ (e) The same capacitor, after being charged to i00 V is discharged through a resistor at 0~ (f) One mole of a gas at 0~ is expanded reversibly and isothermally to twice its initial volume. (g) One mole of gas at 0~ is expanded irreversibly and adiabatically to twice its initial volume. 1.19.9 Consider two identical springs, one at length L 0 (and, therefore, not under tension), and the other at length 3L 0. If these two springs are dissolved in aqueous acid solution at temperature t, will there be any difference in the heats of solution for the two springs, and if so, by how much will these heats differ? 1.19.10 One gram of zinc is placed in excess dilute H2SO 4 at 15~ contained in a cylinder fitted with a weightless, frictionless piston of cross-sectional area 490 cm 2. As the reaction proceeds the piston moves outward against the external pressure of I atm. The liberated heat of reaction is allowed to dissipate and eventually the temperature of 25~ is restored. The total amount of heat lost by the reaction mixture during this time is found to be 36.43 kcal, and the piston has moved outward a distance of 50 cm. Find AE for the contents of the cylinder.

144

, FUNDAMENTALS

1.19.11 Two identical bricks, one at 700 K and the other at 300 K, each weighing 2000 g, are placed in contact. Find ASbri=k,, if the heat capacity of brick is 0.20 cal deg-lg -I. 1.19.12 Two moles of an ideal monatomic gas at 37~ and I atm are compressed reversibly and adiabatically until the pressure is doubled. What final temperature is attained? Find W, Q, AE, and AH for this process. 1.19.13 (a) Three moles of an ideal gas (Cv - 4.97 cal deg-lmol -I) at I0.0 atm and 0~ are converted to 2.0 atm at 50~ Find AE and AH for this process. (b) Why cannot the value for W be calculated from these data? 1.19.14 If the change described in Exercise 1.19.13 is carried out in two stages, (a) a reversible adiabatic compression to 50~ and (b) a reversible isothermal expansion, find AE, AH, W, and Q for each stage and also for the overall process. Compare AE and AH with the answers to Exercise 1.19.13. 1.19.15 If the change described in Exercise 1.19.13 is carried out in two stages, (a) an adiabatic compression to 50~ using a constant external pressure of 18.1 atm, and (b) an isothermal expansion against a constant external pressure of 2.0 atm, find AE, AH, W, and Q for each stage and also for the overall process. Compare the overall values with those of Exercise 1.19.14. Why is Q ~ AH in (a), even though P.x was constant? 1.19.16 A certain gas obeys the equation of state P(V nb) - n R T and has a constant volume heat capacity, Cv, which is independent of temperature. The parameter b is a constant. For i mol, find W, AE, Q, and AH for the following processes" (a) Isothermal reversible expansion. (b) Isobaric reversible expansion. (c) Isochoric reversible process. (d) Adiabatic reversible expansion in terms of TI, Vl, V2, Cp, and Cv; subscripts of i and 2 denote initial and final states, respectively. (c) Adiabatic irreversible expansion against a constant external pressure P2, in terms of PI, P2, TI, and 7 ~

(cp/Cv). 1.19.17 Ten liters of nitrogen at 27~ are compressed reversibly and adiabatically to a volume of 2 liters. Calculate the final temperature" (a) with Cv - 4.95 cal/mol-deg; (b) with C v - 4.51 + 0.0010T cal/mol-deg. Assume that nitrogen is an ideal gas. 1.19.18 The heat combustion (AH) of tungsten carbide at 300 K is -285.65 kcal/mole WC, and the reaction is WC(s) + (5/2)Oz(g) - W O 3 ( s ) + CO2(g). Compute the heat for the same reaction if it takes place in a constant-volume bomb calorimeter at 300 K. Assume that the gases behave ideally and that corrections to standard states are negligible for solid species.

145

NUMERICAL CALCULATION OF ENTROPIES

1.19.19 In an adiabatic enclosure at 298 K 2.00 moles of CH4(g) are mixed with 5.00 moles of O2(g). A spark is produced in the mixture and the CH 4 is completely burned in the oxygen to CO 2 and H20. Assume ideal gas behavior and compute the final temperature of the gas mixture. What approximations are used in arriving at the final result? The following data are relevant: O2(g )" C ~ - 7.16 + 1.00 x 10-aT cal deg -I tool-I CO2(g )" C ~ - 10.57 + 2.10 x 10-aT H20(g)" C ~ - 7.30 + 2.46 x 10-aT CH 4"(g) C~ - 5.65 + Ii.44 x 10-aT

1.20

NUMERICAL CALCULATION OF ENTROPIES AND ENTHALPIES

(a) One the

of the

important problems

determination

experimental

of

heat

transition.

the

in thermodynamics

entropy

capacity

of

a

given

measurements

involves

substance

and

from

from

heats

of

Before considering this matter it is necessary to

examine the thermodynamic characterization of transitions. Suppose

a

given

fixed temperature T t. vaporization

at

material

undergoes

Examples

the

a

transition

at

a

that come to mind are fusion,

boiling

point,

first-order

phase

transitions, and allotropic modifications,

all of which will be

discussed

change

later

in

by

a heat

accompanied

detail. flow

Each into

study at a fixed temperature t. holds

will

be

discussed

in

such

or

out

of

the

is

normally

system

under

The reason why this condition

Chapter

2.

The

energy

change

accompanying the transition is then given by

(1.20.1)

AEt - Qt - Wt - Qt - ~tPdv. Two special cases are of interest: experimentally conditions,

be

constrained

If the transition may

to occur under

constant volume

then (1.20.2)

AEtlv- Qtlv, which

shows

that

in this

instance

a heat

flow

is equal

to a

change in a function of state, namely the energy of the system.

146

, FUNDAMENTALS Usually

constant

it is difficult

during

antifreeze however,

a

transition

to counteract

of constant pressure.

where

P.xt -

reversibly.

The

P

so

Since

- AH[p-

entropy

determlned

the

widespread

of engine

simple matter

In this event,

blocks).

to m a i n t a i n Eq.

change

long

use

of

It is,

conditions

(1.20.1)

becomes

(1.20.3)

as

the

P is fixed,

transition

PAV-

A(PV),

is

carried

out

and

(1.20.3)

may

to read

Q~[p .

(1.20.4)

accompanying

the

transition

may

now

be

as

AS t - AEt/T t

or

the volume p r e c i s e l y

(P fixed),

then be rearranged

AE + A(PV)

(recall

cracking

a comparatively

AE - Qt - PAV

to m a i n t a i n

(V fixed),

(I. 20.5a)

(P fixed),

(1.20.5b)

as

AS t - AHt/T t

depending on whether volume or pressure

(b)

We

turn

to

the

is being h e l d constant.

determination

experimental

data involving measurements

of

transition

heats

of

at

constant

of

the

of heat

pressure.

entropy

from

capacity This

and

method

depends on the use of Eq. (1.20.5b) and of the relation (@S/ST)p - Cp/T,

Eq.

(1.18.26).

S - rTf Ti

(Cp/T)dT - rTfCp d2n T. Ti

For purposes

of

The latter may be rewritten

as

(1.20.6)

illustration

consider

a material

which

undergoes a transformation from ~ to ,8 at temperature T=~, melts at the temperature

T m > T~,

T m.

to determine

It is desired

and boils

at the temperature

the entropy

of this

Tb >

material

NUMERICAL CALCULATION OF ENTROPIES

147

at a temperature T > T b. temperature

range

[O,T]

For convenience we divide the total into segments 0 ~ T a ~ T ~

~ Te ~ Tb

T, where T a is some low temperature in the range 0.I-I0 K, below which it is impossible

to measure Cp accurately,

the numerical values become

small,

both because

and because of problems

in

maintaining low temperatures during heat capacity measurements. In the range theory and

0 ~ T a one generally

first proposed by Debye

extended

temperature

by

Sommerfeld

(1907) (1926)

resorts

to use

for nonmetallic to

metals.

of a

solids At

low

and in zero order approximation,

Cp - aT s + 7T,

(1.20.7)

where a and ~ are parameters which in principle are determined by the theory.

However,

these quantities long as Eq. straight

it is general practice

empirically

(1.20.7)

by plots

of Cp/T versus

llne of slope _a and intercept

plots

are shown in Fig.

which

Eq.

extrapolation

T z.

So

remains valid such a plot should yield a 7 (for nonmetals

usually, though not always, extremely small). (1.20.7)

to determine

1.20.1,

applies

which

in these

indicate

cases

7 is

Examples of such and

the degree

to

the extent

of

that is involved.

On the basis of Eqs. (1.20.6) and (1.20.7) we thus obtain (in the range 0 < T _< Ta)

ST.

- S o - ~Ta(aTZ + 7)dT -

(aTe/3) + 7T a.

(1.20.8a)

0

(Note

that

So - 0

only

under

the

conditions

prescribed

in

Section 1.21 dealing with the Third Law.) In the range T a to T=~ we obtain S (T=~)

-

S (Ta) - rjTc,~ (C~/T) dT,

(1.20.8b)

Ta

which requires an empirical determination of C = for phase ~ as P

a function of T before the integration

is carried out. At the

transition point T=~ under constant pressure

148

I. F U N D A M E N T A L S

1.6 O~

1.2

E o

0.8 Copper 0.4

O0

I 2

,

I 2

,

1

~

4

1 6

L

I 8

~

l

L

1 10

~

I 10

~

1 12

~

1 14

1

1 16

,

l 16

x

1 1

8

4.0 "0

o3.0 E

E. 2 . 0 I--

1.0 0

. 0

1 4

~

1, 6

8

t , 12 4 1

/

l

l 18

10

'0

7

8

0

E

6

-)

E

4

L.)

1.

4

2

1

l

6

L

I

8

,

b ,

1

[ 12

t

l 14

I

[

16

1

[

18

T2deg 2

FIGURE 1.20.1 Heat capacity m e a s u r e m e n t s of elemental metals. After W.C. Corak, M.P. Garfunkel, C.B. Satterthwaite, and A. Wexler, Phys. Rev. 98, 1699 (1955).

(Sfl - S=)T= ~ - Q=fl/T=p - AH=fl/T=fl. The c o n t r i b u t i o n S(Tm)

- S(T=p) = I ~"

in the range T=~ to T m is given by

(Cpfl/T)dT.

It is easy to see that the remaining (S I - Ss)~m - Q~/T m

(1.20.8c)

(at the m e l t i n g point)

(1.20.8d)

contributions

are

(1.20.8e)

NUMERICAL CALCULATIONS OF ENTROPIES

S(Tm)

S(Tb)

jb

149

(C$/T)dT

(for the liquid phase)

(1.20.8f)

Tm

(S v - S,)Tb -- Qb/Tb S(T)

-

=

S(Tb)

(at the b o i l i n g point)

(1.20.8g)

(C~T)dT

(for the gas phase).

ST

by

(1.20.8h)

Tb Thus,

we

obtain

(I. 20.8h).

-

So

addition

Eqs.

(1.20.8a)-

Wherever phase changes occur the increment

given by a term of the type Q/T; contributions (1.20.8a)

of

are

of the

in S is

in the intervening ranges the

form J(Cp/T)dT,

and

for T < T a, Eq.

is used.

The problem with this approach is that the difference

ST

- S o depends on the pressure under which the experiment is done, since Cp, Qt, and T t change as this parameter is altered. therefore results

conventional

to measure,

at a standard pressure

users of such information such as (l.18.10b), It

is

It is

or at least to report,

of one atmosphere,

introduce

all

and to let

corrections via equations

in order to evaluate S T for other pressures.

conventional

to

standard conditions

denote

values

of

ST

reported

under

(generally I atm) as S~.

It should be evident that a similar approach may be used in

the

determination

(1.20.4)

of

the

enthalpy.

and with the relation C p -

Beginning

with

Eq.

(8H/aT)p we see that at a

pressure of i atm,

.o

_ Ho o

-

aT~ 4

vT~

+ -T

+

f T=#C=OdT + Ta

P

Q%#

[P

+

j. o "a#

Note now that H~ is an arbitrary constant any convenient value. in principle,

By contrast,

+

. .. (1.20.9)

that may be assigned

S~ is determined,

at least

by the methods of statistical thermodynamics,

that S~ is uniquely determined. quantities

P c~

We may now combine

so

the above

to find the Gibbs free energy:

(1.20.10)

ISg

i.

FUNDAMENTALS

Tabulations involving the quantities shown in (1.20.10) are of great

importance

applications.

and

play

They are

large

generally

for (GT~ - Ho~/T,

listings

a

role

supplied

in

industrial

in the

form

of

Ho~/T, and S~~ over a specified

(~-

temperature range. To illustrate exhibit

the type of analysis

a representative

1.20.2

for

oxygen,

representation

set of heat

as

a

plot

is useful

for

of

the

that is involved we

capacity

Cp

data

versus

direct

log

exists

in three allotropic

T;

calculation

entropy of oxygen from the area under the curves. the element

in Fig. of

this the

Note that

modifications

in the

solid state, with transition temperatures near 23.6, 43.8, and 54.4 K, the last being the melting point of solid phase I. The boiling

point

of

liquid

oxygen

is

near

90. I

K.

An

extrapolation procedure was used below 14 K. As reader

an

illustration

is again

cryogenic

heat

of

referred

temperature

to Fig.

capacity

1.20.1,

measurements

analyzed according to Eq.

14

low

on

techniques

which Cu,

pertains Ag,

and

the to Au,

(1.20.7) as plots of Cp/T versus T 2.

I

1. . . . . Liquid

_

02

S.I

10

-6 E

s 8

Gas

i c~

\

1:3

6

5i j u

1:3.

t 4 So

I

0 -- --""

J

"~I1

1 2

1 Log

T

FIGURE 1.20.2 The molar heat capacity of oxygen. After W.F. Giauque and H.L. Johnston, J~ Am. Chem. Soc. 51, 2300 (1929).

NUMERICAL CALCULATIONS OF ENTROPIES

In conformity straight

lines

coefficient

with with

earlier

|~ |

statements,

nonzero

the

three

intercepts

which

~ of the linear c o n t r i b u t i o n

to Cp.

plots

represent

yield the

EXERCISES

1.20.1 For SO2(g) in the range b e t w e e n 300 and 1800 K the following relation is found to hold" Cp11.895 + 1.089 x 10-3T - 2.642 x 10-ST 2 cal/mol deg. Determine AH and AG for a process in which I mol of SO 2 is h e a t e d from 300 to 1500 K. 1.20.2 In the range 300-1500 K one finds that for O2(g) , Cp - 6. 148 + 3. 102 x 10-3T - 9.23 x 10-TT 2 cal/mol deg. Determine AH and AG for i tool of oxygen when the gas is h e a t e d from 300 to 400 K. 1.20.3 Find the change in entropy for H20(s,-10~ arm) H20 (s,0~ atm); take for ice as 9 cal deg -I tool-I. 1.20.4 Find AS for H20(s,-10~ arm) ~ H20(2,10~ atm), given that is 9 and 18 cal deg -I mole -I for H20(s) and H20(2), respectively, and that the heat of fusion is 1440 cal deg -I mole -I . 1.20.5 The molar heat capacity at constant pressure of gaseous NH 3 varies with the temperature according to the empirical equation Cp - a + bT - cT 2, where a - 8.04 cal/mol-deg, b 7.00 x 10 .4 cal/mol-deg 2, c 5.10 x I0 -s cal/mol-deg 3. What is the entropy change of 2 tool of NH 3 that is h e a t e d at constant pressure from 20 to 200~ 1.20.6 By how much does the enthalpy of 3 tool of Cl2(g) change when its temperature is increased at a constant pressure of I atm from 300 to 400 K? Its heat capacity at a constant pressure of i atm in this temperature range is 7.5755 + 2.4244 (10-3)T - 9. 650 (10-7)T 2 cal/deg-mol. 1.20.7 The normal boiling point of benzene is 80~ and the heat of v a p o r i z a t i o n at this temperature is 7353 cal tool-I. For CsHs(~) and CsHs(g), Cp is 36.2 and 20.3 cal deg -I tool-I, respectively. (a) Find AS for CsHs(2,70~ i arm) C6Hs(g,70~ atm), (b) Find AS for CsHs(~,70~ atm) CsHs(g,90~ atm). i. 20.8 Taking -~ Cp - 7.11 + 6.00 x 10-3T - 0.37 x 10+ST -2 cal/mol-deg for NH3(g), calculate H~ - H2s8.15. values at T = 200, 500, 700, and i000 K. Obtain an expression for C$ from the preceding C~ expression and use it to find E~ - E2s8.15- at the above temperatures. 1.20.9 For the reaction CO + (1/2)O ~ CO 2 at 25~ S~ - 20.74 cal/deg-mol. The heat capacities @in cal/deg-mol) are given as

152

I. FUNDAMENTALS

Oz(g)" C p - 7.16 + 1.00 x 10-3T - 0.40 x 10+ST -2 CO(g)" C p - 6.79 + 0.98 x 10-3T - 0.II x 10+ST -2 CO~(g)" C p - 10.57 + 2.10 x 10-3T - 2.06 x 10+ST -2. Calculate AS ~ for the reaction at 1000~ 1.20.10 Between 0 and 15 K the molar heat capacity in cal/deg-mol of a certain insulating substance is computed according to the extrapolation formula Cp(s) - 5.78 x 10-4T 3. For the solid above 15 K, C p - 0.0085T + 0.00030T2; for the liquid, Ce(~) - 14.50 + 0.0043T; and for the gas, Ce(g) - 12.00. The heat of fusion at 200 K is 1.837 kcal/mol; the heat of vaporization at the boiling point of 350 K is 7.14 cal/mol. Find the molar entropy of the substance in gaseous form at 400 K and at a pressure of 0.I arm, assuming the gas to be ideal. 1.20.11 Use the data given below to calculate $i00 - S O for NazSO 4 by graphical integration. Use the T 3 law to extrapolate Cp below I0 K. Temperature, K _Cp, cal/de~ Temperature,K Cp, cal/deg 13.74 0.171 52.72 7.032 16.25 0.286 68.15 10.48 20.43 0.626 82.96 13.28 27.73 1.615 95.71 15.33 41.11 4.346 1.20.12 For a certain substance A, Cp for the solid form is 0.196 cal deg -I mol -I at 15 K. From this temperature to the normal meltln~ point, 200 K, it varies with temperature (T,K) according to Ce(s) - 0.0085T + 0.00085T 2 cal deg -I tool-I. The heat of fusion is 1.800 kcal tool-I. The heat of vaporization is 7.05 kcal mol -I at the normal boiling point, 300 K. Find the entropy of A(g) at 350 K and 0.5 atm, assuming the gas to be ideal. 1.20.13 (a) In the range 0 to 15 K, the Debye relation C ~ - (12~4/50a)RT 3 is valid for the heat capacity at 1 atm. For CO, 8 - 103.3 K. (b) CO undergoes an allotropic modification of solid CO(a) - solid CO(~) at 61.5 K, with an associated heat of transltlon of 0.633 kJ/mol. (c) The melting point occurs at 68.0 K; the heat of fusion is 199.7 cal/mol. (d) The boiling point occurs at 81.0 K; the heat of vaporization is 1443.6 cal/mol. (e) The molar heat capacity at constant pressure of 1 atm has been tabulated as shown below for the low temperature range" cal C~

4.0

6.0

8.0

9.0

ii.0

13.0

14.0

deg-mol T(K) 22.5 30.5 39.5 45.2 52.3 57.6 60.0 An empirical equation for the heat capacity between 61.5 and 68.0 K reads C~ - 7. 1621621 + 0.0775629T cal/deg-mol. The

153

NUMERICAL CALCULATIONS OF ENTROPIES

liquid is c h a r a c t e r i z e d by a constant heat capacity of C ~ 14.445 cal/deg-mol. For gaseous CO an empirical equation is given by C~ - 6.79 + 0.98 x 10-ST -_ 0.ii_ x 10-ST 2 (T in K) - ~ - H~ -S ~ , G ~ - G o O at i00, 200, 300 cal/deg-mol. Determine H 400, and 500 K. 1.20.14 The accompanying graph (Fig. 1.20.3) specifies Cp/T versus T z for pure Ti metal and for Ti-Mo alloys at low temperatures. The Jump in Cp for the alloys is due to a s u p e r c o n d u c t i n g transition. (a) For the Ti(85)Mo(15) alloy determine AS c and AH c for the s u p e r c o n d u c t l n g - n o r m a l transition at temperature T c, assuming that S o for the s u p e r c o n d u c t i n g phase is the same as for the normal phase if the latter existed at 0 K. (b) Determine the difference in entropy for the Ti(85)Mo(15) alloy and the Ti(95)Mo(5) alloys at 3 K and at 5 K. [Recent microscopic theories predict that above the superconducting transition (T > T=), C p - ~T 3 + 6T, that at the critical temperature T c, ACp(T c) - 2.436Tc, and that for T < T c, C p - ~T 3 + a6T c exp (-bTc/T)]. 1.20.15 (a) Refer to Exercise 1.18.22 and show that w h e n the pressure is altered from an initial value Pi to a final i

!

i

3K

I I 1 I 4K 5K Atomic percent m o l y b d e n u m

20--

~

~ 5

I I i

J

~

_'

_

/~/

12--

"

1

9 ~o

,

_@ o

E

)E. 8 - ~

4 ---~

-

/

1 FIGURE I. 20.3

1

10

1 T2,K 2

!

20

i

1

30

154

, FUNDAMENTALS

value Pz, the enthalpy change for a Berthelot gas reads H~ - H i (9 RT=/128 P=)(I - 18 T c / T z ) ( P z - Pi). (b) Determine the q u a n t i t y Cp - C v for a Berthelot gas. (c) Show that Ce(P ) - Ce(O ) - (81R/32)(T=/T)3P. 1.20.16 For H 2 gas one finds the following values for C~/R at i a t m " T(K) 40 60 80 I00 120 150 C~/R(K) 2.56 2.56 2.61 2.72 2.86 3.06 Determine the change in molar entropy for H z when the temperature is altered from 60 to 140 K. 1.20.17 (a) At very low temperature, T _< 0/10, the molar heat capacity of a monatomlc solid insulator is given by the relation C v - 233.782 R(T/e) 3, where e, the Debye temperature, is a parameter that must be specified. Determine E, S, and F as a function of temperature for the solid in the range of a p p l i c a b i l i t y of this expression. (b) In the temperature range T _> 100 the relation C v - 3 R [ I - (I/20)(0/T) 2 + (I/560)(0/T) 4] applies. Repeat the derivation for this case. 1.20.18 The following data apply to zinc; all energies are s p e c i f i e d in cal/mol" C(s)- 5.35 + 2.4 x 10-3T AH~ - 1.765 at 692.7 K C(2) - 7.50; Ce(g) - 4 . 9 7 A H ~ - 27.430 at 1180 K Determine the entropy and G~ - Ho~ for zinc at 1250 K. 1.20.19 For CI 2 gas C~ may be represented by the empirical relation 8.85 + 1.5 x IO-4T - 6.8 x I04/T cal/mol above 25~ Determine H~ - H~9s, S~ - S~9e, G~ - G~ge for CI z. -

I. 21

THE THIRD LAW OF THERMODYNAMICS

The Third

Law of Thermodynamics

deals

with

events

at T -

0;

clearly, problems arise here because the integrating factor I/T of ~Q diverges

at T -

0.

We first investigate T -0. mass

For this purpose, which

coordinate

may z.

generalized.) the

case

where

be

the question of the a t t a i n a b i l i t y of consider a special system of constant

characterized

(The

following

by

a

single

argument

may

Let Z be the conjugate variable. only

mechanical

work

correspond to P and V, respectively.

is

deformation be

readily

Note

that in

performed,

Then,

TdS - dE + Zdz - (SE/ST)dT + (Z + dE/8z)dz.

Z

and

z

since E - E(S,z),

(1.21.1)

THETH'RD LAW OFTHERMODYNAMICS

In an adiabatic

15 5

(isentropic)

process with dS = 0, Eq.

(1.21.1)

becomes

(aElaz)]/(aElaT)}dz.

dT I - {[Z +

This

situation

conditions hope

to

is

of

particular

which prevent obtain

alteration

a

(1.21.2)

interest,

a heat

maximal

flow

change

into in

since the

only

system

under

can one

temperature

through

in z, and thereby diminish T to its lower limit.

Now observe integrable,

T-I(aE/aT)dT + T-I[Z + (aE/az)]dz is

that dS -

as follows from the fact that the second derivative

of S with respect to z and T in either order must be the same. It

then

follows

(aE/az)]}.

(a/az)[T-1(aE/aT)]

that

(a/aT){T-I[Z

This condition may be met by requiring

independent variables)

z +

=

(aE/az)

- T

Inserting

(T and z are

that

(1.21.3)

(az/aT).

(1.21.3)

+

into

(1.21.2)

we

have

for

an

isentropic

process

dT I - T [.(aZ/aT)z-.]dz C ' ~ I T ( a Z / a T )Iz d-z , ( a E / a T ) z

where

Cz

(aE/aT),

m

maintaining contains

the

is

system

an extremely

adiabatic

process

inexorably

at

heat

capacity

constant

important

z.

measured

Equation

lesson.

It shows

any

chiange

in

a

a

change

in

temperature,

produces

Such a change

the

(1.21.4)

deformation

in d e f o r m a t i o n coordinate

while

(1.21.4)

that

in a______nn

coordinate

of

the

z

system.

z can be achieved,

for

example, by changing the volume of the system or by imposing or altering isolated low

the

electromagnetic

systems.

temperatures,

(aZ/aT)z

remained

possible

to attain

Note as

that

it

nonzero the

field

on

if C z were

does

at

in

this

absolute

high

adiabatically

to remain constant

at

temperatures,

and

if

would

be

range,

zero

the

then

temperature

it by

suitable

156

I. FUNDAMENTALS

changes

in

z.

investigated

for

all

it is found that C~ varies

In

fact,

with

w h e n T ~ O, w i t h ~ a I. on w h e t h e r

however,

The a t t a i n a b i l i t y

(aZ/aT) z approaches

- T =-I, so as

to p r e v e n t

a runaway

F - E - TS for w h i c h

Integrability

(az/aT),

far

as T =

of T - 0 thus h i n g e s

situation Consider

dF - dE - TdS

of dF then demands

as T ~ O.

This

the f u n c t i o n

- SdT - - Zdz

that a 2 F / a T a z

of

- SdT.

- a2F/azaT,

(as/az)T,

I

so

temperature

zero more r a p i d l y than does Cz/T

q u e s t i o n m a y be r e s t a t e d as follows" state

materials

or

(1.21.5a)

so that

(1.21.5b)

dT - - (T/C,) ( a S / a z ) T d z .

In other words, approaches

T-

unless,

for all z,

(as/az)r

zero more r a p i d l y than does Cz/T in the limit of low

temperatures. linked

0 is a t t a i n a b l e

to

The

the

attainability

behavior

of

problem

the

is

thus

metrical

intricately

entropy

at

low

namely

that

temperatures. We n o w

invoke

another

for e v e r y s i t u a t i o n

experience

s t u d i e d so far,

of mankind,

( a Z / a T ) z or ( a S / a z ) T -~ 0 as

T ~ 0, and that for any T in this range 0;

clearly,

satisfied chemical

is

a necessary that

reaction

condition

(aS/az)T involving

-~ 0

as

changes

a A S / a n i - 0 (this will be further this

experience

we

see

that T -

rise to the s o - c a l l e d p r i n c i p l e zero. be

The s t a t e m e n t s

included

of

particular, its

lowest

approaches any

condition

0.

Further,

dealt w i t h

later).

of u n a t t a i n a b i l i t y

finite,

to be

for

B a s e d on this

gives

of a b s o l u t e

-~ 0 as T ~ 0 m a y

law:

asserts

that

degrees

system

as the

Kelvin,

tends

toward

the a

least value w h e n the s y s t e m is in

possible

energy

state.

any

n i at 0 K,

0 is u n a t t a i n a b l e ;

zero

given

T/C z ~ 0 as T

in mole n u m b e r s

Law of T h e r m o d y n a m i c s

temperature entropy

T ~

this

( a S / a z ) ~ -~ 0 and ( a s / a n ) ~

as part of a n o t h e r

The T h i r d

for

(aS/az)~

The

entropy

LAW OFT H E R M O D Y N A M I C S

THE THIRD

reaches

15 7

such a value with zero slope for the

limit T ~ 0. One

should

carefully

qualifications: zero. in. the

to energy attends

excited configurations

When considering most

the

statement

and

its

Nothing is said about S approaching a value of

The reference

temperatures

note

stable

to the fact that at low

may accidentally be frozen

the state of lowest energy we deal with

configuration.

For

it,

the

entropy

at

the

absolute zero of temperature has the lowest possible value and is independent By

of all z or n.

convention

the

lowest

value

of

the

entropy,

So,

is

generally taken to be zero, except for circumstances enumerated below.

This

statistical

convention

accords

thermodynamics.

is the possibility given material

with

Exceptions

of conflgurational

may be

predictions

in more

based

arise whenever

disorder,

than one

state

i.e.,

examples

that come

to mind often

involve

there when

associated

the same or nearly the same lowest energy at 0 K.

on a

with

Elementary

cases where

for one

reason or another a certain degree of disorder is maintained at 0 K:

(a) Helium,

high pressure.

which remains a liquid at T (b) Solid CO, H20 , or N20:

In these materials

two adjacent units may be in configurations NNO-NNO

that

temperature

differ is

ultimately

energy difference units

only

is so small

very

slightly

reached

at

energy.

which

even

When

this

a

small

is important the rate of rotation of the N20 that random

orientations

Glasses or mixtures such as AgBr + AgCI: supercooled

such as ONN-NNO or

in

the solid cannot reach the lowest possible

as

0 except under

liquids

or

solutions,

residual entropy of mixing.

are

frozen

in and

energy state.

(c)

These can be regarded respectively,

(d) Finally,

with

in principle,

a

one has

nonzero values of S O in materials where isotropic distributions occur latter

or

in which

factors

experimental

nuclear

are

usually

conditions

such

the temperature is changed. done in setting S O -

spins

deviate

ignored

because

randomness In

from

I -

0.

under

remains

These

ordinary

unaltered

as

these circumstances no harm is

0; by contrast,

in for cases

(a)-(c) the

158

, FUNDAMENTALS

disorder

is changed by rising temperature.

setting

So

-

experiments

0

in

case

(d)

needs

dealing with nuclear

A very

important

to

The c o n v e n t i o n be

revised

of

in

any

transformations.

consequence

of the T h i r d Law is that

it

denies the existence of a perfect gas at low temperatures.

For

a perfect gas we had seen earlier that

(1.21.6)

S - C ~n T + R ~n V + S c.

It is clear that (8S/8V) T does not v a n i s h as T ~ 0, in v i o l a t i o n of the Third Law. use

of

the

This conclusion does not, of course,

perfect

thermodynamic

gas

law

calculations,

experimental

results.

well

as

away

a

However,

from

means

it

of

does

T

prevent =

0

in

approximating

make

for

logical

defects in conventional discussions of the Second Law, b a s e d on Carnot

cycles

substance,

which

employ

a

whose very existence

perfect

gas

as

a

working

is denied by the Third Law.

One may adopt another viewpoint c o n c e r n i n g the Third Law" Consider

a general

process

state A to state B. entropies for S B.

by which

a system

A c c o r d i n g to Eq.~.~l.18.12),

is altered

from

the respective

are given by SA - SA(0) + J

CpAd~n T, and similarly 0 Let the system initially be in state A at temperature

T I, and allow the process to occur a d i a b a t i c a l l y and r e v e r s i b l y so that the system ends up in c o n f i g u r a t i o n B at temperature T 2. Since SA - S B for the reversible

SA(0) +

where

CpAd2n T -

SA(O)

SB(0) +

adiabatic process,

and SB(0 ) are the entropies

two states at T -

0.

adiabatic

can be found w h e r e b y

system

changes

achieved,

then

> S^(O), that

process

then

the

tntegrand through

it

above is

this

T2 >

0

to

T2 -

SB(O ) _ SA(O ) _

is possible equation

positive. process

for the system

in the

The question now arises as to w h e t h e r an

from

AS I

(1.21.7)

CpBd2n T,

AS

Therefore, the temperature

0.

If

1CeAd2n T.

to choose for

the temperature

the

is

initial

satisfied

with Ta - 0

this

this

of the

could

Thus,

if

value

SB(O)

T1 s u c h

because value

be

of

can be reached.

the

T1 and But:

THE THIRD LAW OF THERMODYNAMICS

this

violates

the

temperature.

I 59

principle

of

the

unattalnabillty

The only way to avoid this

impasse

of

zero

is to demand

that SB(0 ) cannot exceed SA(0);

then there exists no value T -

TI

process

for

which

the

adiabatic

A

~

B

reaches

zero

temperature. The leads

same

to

reasoning,

the

requirement

conclusion

based that

on the S^(0)

inverse

cannot

exceed

SA(0 ) -- SB(0 ) is the only one

two contradictory

claims.

process

conditions

must hold in all other circumstances.

AS

-

The

SB(O)

requirement

of

their

-

is

0

in

the

special

cases

the

original

application either

AS

version

of equilibrium

already

discussed

here.

One

cannot

of

assume

Eq.

be

conditions

state

Heat

One must be quite because

computed,

or

should

in

else,

a

These matters be

supplemented wherein

to an e q u i l i b r i u m

are nonaccessible

some

the

state

to e q u i l i b r i u m

changes do not fall w i t h i n

the

(1.21.8).

have stated that

T~(~

an important point.

(aS/azl) ~ 0 for all d e f o r m a t i o n

including mole numbers;

Cz

famous

AS - 0 for any process

so that corresponding

-

the

interferes.

but

In closing one should reemphasize

C,

of

(1.21.8)

states

earlier,

from a metastable

because metastable

fact

It therefore

that are reversible

in 1906.

cannot

nonattainment

scope of Eq.

adiabatic

(t.2t.8)

were

system passes

under

.

Theorem proposed by Walther Nernst careful

the

that

S^(0)

foregoing

theories,

equality

that for all general processes

-

The

Note that since SA(0 ) and SB(0 ) are

the

we may ascertain

SB(0 ) .

that reconciles

constants, now follows

B ~ A,

We

coordinates,

this in turn hinged on the experimental

>_ i).

In

Section

1.18

it

was

shown

that

- T(aS/@T)z, 9 therefore , (aS/aT), - T =-I must likewise approach

zero.

In other words, near the absolute zero of temperatur.e the

entropy,

for any process

whatsoever,

as far as its dependence

on temperature T is concerne.d, approaches i t s lowest value with zero slope.

,. FUNDAMENTALS

160

EXERC I S ES

1.21.1 It was stated earlier that no perfect gas can persist to T - 0 K because this would incur a violation of the Third Law. Yet helium gas, which closely approximates to a perfect gas, does not solidify on being cooled toward 0 K under pressure of less than 26 atm. The liquid helium phase also appears to violate the Third Law. What contradictions might be involved and how may they be resolved? I. 21.2 Is a van der Waals gas admissible as an appropriate thermodynamic entity? Back up your answer with proofs. 1.21.3 Specialize Eq. (1.21.4) to the case of an ideal gas and show how T varies as the pressure and as the volume are altered adiabatically. 1.21.4 From the results of Sections 1.18 and 1.19, establish that it is necessary to have Cp/T and Cv/T approach zero as T ~ O.

1.22

THE GIBBS-DUHEM

(a) An elementary the

problem

number

of

of

EQUATION AND ITS ANALOGS

application

determining

moles

of

the

of Euler's

the

volume

constituent

specified

for constant

a special

case a solution containing

chloride

dissolved

temperature

in water.

of moles each of H2SO4, NaCI,

theorem

of

a

and pressure.

the

constant)

the been as

acid and sodium the number

at fixed T and P, then

generalization

that

at

mole numbers n i of all species of the system. of the state of aggregation

This leads quite

fixed

volume V of the system should be a homogeneous is independent

in

and H20 (note how this forces us

we must also double the volume of the system. to

when have

Consider

If we wish to double

to maintain all concentrations

naturally

system

components

sulfuric

is found

T

and

P,

function

the

in the

This requirement

of the system.

In view of the foregoing we write out Euler's Theorem (see Section 1.4)

in the form

V - ~i n i(Sv/8n

i)T ~

To

Euler' s

simplify

notation

(1.22.1)

~i equation

we

introduce

the

shorthand

161

GIBBS-DUHEM EQUATION AND ITS ANALOGS

(aV/Oni)T,p,nj#i,

V i -

(1.22.2)

with which

v - E i

(1.22.3)

where V i is known as the Partial mola! vo!ume. defined

by

Eq.

(1.22.2)

may

physically

be

The quantity

regarded

as

the

effective volume that is to be assigned to one mole of species i

in

an

infinite

conditions.

For,

copy

of

the

mixture

under

prevailing

one must take account of the fact that the

volumes of the various components in solution generally differ from their volumes when pure; components

is not

usually

i.e., the sum of volumes of pure

equal

to

the

final

volume

of

the

mixture.

We will learn later how to determine V i and thereby,

to render

(1.22.3) physically significant.

We

now

turn

to

a

conceptual difficulties:

matter

double

distinguish numbers,

the

a

of

a

system.

functional

We

rendering V a homogeneous

of

functionally

on all

V

on

mole

To satisfy (1.22.3) dependence

of the

it is necessary to demand that

all V i also depend parametrically on T and on P. to maintain full generality,

therefore

function of the nl, and a

while retaining a functional and parametric V(T,P; nl,...,nl,...),

must

dependence

parametric dependence of V on T and on P. type V -

presents

yet doubling of T or of P will not

volume

between

frequently

Obviously, the total volume V depends

on T, P, and mole numbers; usually

that

In addition,

every V i must be allowed to depend

the n i.

Thus,

we

rewrite

(1.22.3)

more

explicitly as

V(T,P;

How

nl,...

is

the

,ni,...

)

"-

dependence

i

niV i(T

of

reconciled with requirement

Vi

P;

nl,

on

. . . ,ni,

. . . ) .

T,P,nl,...,nl,...

(1.22.4) to

be

(1.22.2), according to which T, P,

and n3#i are to be held constant during partial differentiation? The answer lles in the fact that the partial derivative depends on the fixed values assigned to T,P,n3 at which they are to be

162

,

maintained sets

of

during differentiation" different

derivatives values.

values

V i will

in

When T and P are a s s i g n e d

(TI,PI)

general

FUNDAMENTALS

or

assume

(Tz,Pz) ,

etc.,

different

numerical

The fact that T and P are to be h e l d fixed in (1.22.2)

does not preclude V i from altering

in value as one passes

one

a

set

remarks

of

parametric

values

to

different

set.

from

Similar

apply to the nj~ i.

(b) We now have one formulation for V in Eq. wish to introduce nl,...,nl,...)"

dV -

the

a second formulation From Eq.

(SV/ST)p,.j,~dT

(1.4.4)

(1.22.3).

We

derived from V - V(T,P;

it n e c e s s a r i l y

follows

+ (8V/aP)~,nj$ idP + ~ Vldnl, i

that

(1.22.5)

which is to be compared with the result derived from (1.22.3), name ly, dV - ~ Vi d n i i Consistency

i

+ ~ n Idv i. i is achieved

(1.22.6) only through

the requirement

nldV i - (@V/aT)p nj#idT + (8V/aP) T,nj#idP ,

which

(1.22.7)

reduces

to the form L nldV i - 0 w h e n T and P are fixed. i (1.22.7) is a variant of the G i b b s - D u h e m relation.

Equation

(c) The preceding unless

operations

are of limited

one knows how to determine

stage we begin with a definition

V " V/

where

that

the various

significance

V i.

To set the

of the molar volume"

nj,

(1.22.8)

the sum runs over all components

introduce

the

definition

component,

x i - nl/ .~ nj, and Eq. 3

for

the

in the system.

mole

(1.22.3)

fraction

of

to obtain

We now the

_ith

16:3

GIBBS-DUHEM EQUATION AND ITS ANALOGS

- ~i xIVi"

(1.22.9)

Note that if the system contains V

-

nIV/n I -

special

niV I and

Vl

-

only one species we may write

(SV/anl)T,P

-

Vl,

whence

in

this

case

VI - VI

(one-component

system).

(1.22.10) m

For a b i n a r y s o l u t i o n we p r o c e e d to determine follows"

m

V I and V 2 as

Starting w i t h V - xiV I + x2V 2,

(~V/~x2)T,p-

XI(~VI/~X2)T,P + X2(~g2/~X2)T,P +

Now at fixed T and P Eq.

(1.22.7)

reduces

V2 -

Vl"

(1.22.11)

to (1.22.12)

xldV I + xzdV z - O , whence

8Vi ) [ 8+ Vx2 2~aXl

~[aV1

Xl "a'vl dxl +----ax2 dx2 T P

aV2 dxl + ax2

] dx 2

But, w i t h 8/8x I r e w r i t t e n as

- a/Ox 2 and

D

-O. T,P

~

dx I -

- dx 2,

the

m

(1.22.14)

2x1(avl/ax2)T, P + 2x2(av2/ax2)T, P - o, whence

(1.22.11)

(aV/ax2)~.p

Inserting -

(I

(1.22.13) above may be

- v2

reads -

vl.

(1.22.15)

- x2)Vl

+

into

(1.22.15) (1.22.9)

x2[(aV/0x2)T.p

yields (1.22.16)

+ gl],

or

VI -

V

- x2(SV/Sx2)~,

In Exercise s i m i l a r l y that

(1.22.17)

P.

1.21.1

the

reader

is

asked

to

demonstrate

164

Vz-

,. FUNDAMENTALS

V + x~(aV/ax2)T,p.

(1.22.18) m

Note the p r e s c r i p t i o n for d e t e r m i n i n g V I and V 2"

Analysis

of the various solutions p r e p a r e d in the laboratory yields sets of values of x I and x2; for each such set the molar volume V of the

solution

obtains

to

be

determined

at

fixed

a plot of the type shown in Fig.

For V(x2)

is

any

curve

According

particular yields

to

value

the

(1.22.17),

- 0 yields VI, while

the

of

P.

One

then

1.22.1.

of x 2 - b,

value

T,

the

tangent

(8V/ax2)r, P

intercept

for

to

x2

-

of the tangent

the b.

at x 2

its intercept at x 2 - i (x I - 0) yields V 2

according to (1.22.18).

This is a very simple graphical m e t h o d

for finding V I and V 2.

(d)

An

alternative

approach

to

the

above

draws

on

the

d e f i n i t i o n of an apparent molal volume 4, via the r e l a t i o n

V - nlV 01 + n 2 ~ ,

(1.22.19)

/

/

v~

J J

~O ~~O

~O~q

~

/

O

b X2

FIGURE 1.22.1 be determined.

Plot illustrating how partial molal volumes may

165

GIBBS-DUHEM EQUATION AND ITS ANALOGS

where ~0 is the molar volume of component I in pure form.

Now

switch to molalitles by referring all quantities

at I000 g of

pure

mass

solvent

(i.e. , pure

i).

Then

the

total

of

the

solution of density p is given as Vp - i000 + raM2, where m is the molality and M2, the gram molecular mass of solute. pure

solvent

solve

we

(1.22.19)

have

correspondingly,

for 4 and

relations just derived.

substitute

V~nip 0 for V

For the

I000.

Next,

and ~0 from

the

This yields

lP(m)"po " P~

i000

(1.22.20)

m

Measurements

of all the quantities

on the right-hand side now

permit ~ to be calculated for any solution of interest. Once ~ is known one may obtain V 2 via (1.22.19)

as

I

V 2 - (OV/@n2)nl,T,p- 4 + n2(@~/@n2)nl,T,p- ~ + m(O~/Om)T, P. (1.22.21) VI may then be found from the expression niV I + n2V 2 - niV ~ + n2~ as

VI-

~o + (n2/nl)( ~ _ V2) - ~ o

-

where

(mzH

-

MI

(1.22.21)

is

the

/1000) (a

gram

. m(n2/nl)(a@/am)T,e

atomic

and (1.22.22)

(1.22.22)

/am)T.p,

mass

of

solvent.

are the relationships

For many types of solutions

Equations

of interest.

it is found empirically

(and

will later be shown to follow from theoretical considerations) that ~ is proportional to ~ , a law

is

encountered

(i/2~)(a4/a~). (1.22.22) -

§

it

Now

i.e., ~ - b + a ~ .

is expedient

one obtains

to

Whenever such

rewrite

in place

of

(a4/am)

(1.22.21)

as and

the relations - b +

(1.22.23a)

166

FUNDAMENTALS

,

and _

~

mSl~M

V I - V~ -

The

I

2000

advantage

(84/8~)T'P"

of

(1.22.23)

(1.22.23b)

is

that

a plot

of

4 versus

obtained from (1.22.20) after making measurements of solutions should

of different molalities

yield

a

straight

line

~,

in a variety

for two fixed compounds,

of

intercept

b

and

slope

(8~/8~)Top, which remain fixed, along with MI, V~, and numerical constants.

It is then a straightforward matter to calculate V 2

and V I for any molality m of interest. (e) What has been stated with respect to V may be carried over to other thermodynamic quantities. of these Section

is the 1.16;

so-called Gibbs

free

energy

G,

described

in

also be reintroduced

later.

We assume G to be a function of T, P, and mole numbers.

If it

is to be

this quantity will

Among the most useful

a homogeneous

function

of the

conformity with the Euler criterion,

latter

set,

in

we must demand that

G - ~ ni~i, i where

then,

(1.22.24)

~i " Gi "

(8G/anl)T,P,nj#i is also known

as

the chemical

potential, for reasons explained in Section 2.4.

This quantity

occurs so frequently in chemical thermodynamics

as to warrant

use

of the

special

symbol ~i.

We now recognize

that G is a

function of T, P, and n i; then

n dGWe

(SG/ST)p,nldT + (8G/SP)~ nldP + ~ ' i-i can

thus

relation,

immediately

write

out

(@G/@nl)T,P,nj#i dn i. (1.22.25) the

famous

Gibbs-Duhem

in analogy to (1.22.7)"

~ nld~i - - SdT + VdP. Since contrast

G

or to

G

is

V or

V,

(1.22.26)

not

easily

it

becomes

accessible

to

correspondingly

measurement, more

difficult

in

THERMODYNAMICS OF OPEN SYSTEMS

167

to measure or determine ~i-

We will deal extensively with this

problem at a later stage.

EXERCISES

1.22.1 Prove Eq. (1.22.18) explicitly. 1.22.2 Cite an example of a system for which the following claim is incorrect" Consider two systems at the same temperature and pressure containing the same chemical materials in identical amounts. When these systems with identical amounts and chemical constitution are combined, the total energy is twice that of each subsystem. What does this counterexample teach you? Explain in some detail and show what conditions must be met so that the above statement will always be correct. 1.22.3 Find an expression for the partial molal volume of a gas in an ideal gas mixture at total pressure P and temperature T. Use it to find the partial molal volume of argon in an ideal mixture of 0.50 tool Ar and 1.50 tool Ne at 0~ and a total pressure of 0.20 atm. 1.22.4 When 1.158 mol of water is dissolved in 0.842 mol of ethanol the volume of the solution is 68.16 cm 3 at 25~ If Vs2o - 16.98 cm 3 mol -I in this solution find Vc~ssoH. Compare partial molal volumes of the components with their molar volumes. The densities of H20(~) and C2HsOH are 0.9970 and 0.7852 g c m -s, respectively, at this temperature. I. 22.5 At O~ the density p of aqueous calcium ferrocyanlde can be represented by the relation p - 1.000 + 0.245c - 0.0215c 2, where c is the molar concentration. Calculate the partial molar volumes of solvent and solute, and the total volume of solution for c - 1/2, i, and 2. 1.22.6 The apparent molar volume, 4, of KCI in an aqueous solution at 25~ is given by the equation ~ - (V - I002.94/m) - 27.17 + 1.744 m, where V is the volume in cm 3 of a solution containing I000 g of water and m mol of KCI. Find expressions for the partial molal volumes of KCI and HzO as functions of the molality. Calculate the numerical values of VI and V 2 for m O, 0.I, 0.2, and 0.5.

1.23

THERMODYNAMICS

OF OPEN SYSTEMS

(a) In Section 1.16 we considered number

of moles

now generalize

of various

closed systems

components,

in which

nl, remains

fixed.

the We

to the case where the various n i are allowed to ~

168

, FUNDAMENTALS

vary. F,

This

means

and G must

components

n o w be

(see

the system.

that

functions

functions

Section

2.2

For e x a m p l e ,

of s t a t e

of m o l e

such

numbers

for a d e f i n i t i o n

we set G -

as V,

G(T,P;

of

E,

S, H,

the v a r i o u s

of this

term)

in

nl,...,nc) , so t h a t

c

dG-

For f i x e d nl, it

(SC/8P)T,nldp + ~. (8C/Snl) T pnj#• i-I ' '

(@G/ST)p,nldT +

is

also

chemical

(8G/aT)p,nl -

very

useful

potential

- S, a n d

to

(8C/8P)T,n i -- V,

reintroduce

the

as b e f o r e ;

symbol

termed

/'~.t " (8C/ani)T,p,n3#.. L " Ci. One

should

energy, be

note

that

(1.23.2)

~i H

in the n o t a t i o n

rewritten

the

#i b y

G i is

really

of S e c t i o n

1.22.

a partial Equation

molal

free

(1.23.1)

may

as c

dG-

- S d T + V d P + i~l~idnl;_~

this

is

an

relations designed closed,

extension F

to one

-

G

of

- PV,

apply

(1.23.3)

Eq. H

-

regardless

(1.18.4). G

+ of

TS,

Bearing E

whether

-

F a

+

in

TS,

system

mind

the

which

are

is

open

or

obtains c

dF -

- SdT

- PdV +

~. #idnl i-I

(1.23.4)

c

dH - TdS

+ VdP +

~. ~Idnl i-i

dE - TdS

- PdV +

~. #Idn• i-i

(1.23.5)

c

(1.23.6)

in w h i c h

#i -

(SF/Snl)T,V,no# i " Fi

(1.23.7a)

#i -

(SH/Snl)s,P,nj# i " Hi

(1.23.7b)

#i -

(8E/anl)s,v,nj# i " El,

(1.23.7c)

THERMODYNAMICS OF OPEN SYSTEMS

169

and where

(aE/aV)s.,,j

P--

I

(aF/aV)t.nj

-

(1.23.8a)

v I (aH/aP)s.nj I (ac/aP)T.nj T-

S--

(1.23.8b)

(OE/OS)v..j- (OH/OS)p,,j (aF/OT)v,nj-(OC/OT)p,nj.

(1.23.8c) (1.23.8d)

We examine in Exercise 1.21.1 the question whether or not a different for

each

(1.23.2)

symbol

of

the

(say #e, ~F, .us, ~-) should have been used

quantities

G i, F i, H i , and

it is simplest

adding dn i moles of component the

in

the process

of

and (1.23.7).

Experimentally and

E i appearing

nj~i

conventional relations

are

kept

to prefer

to realize

i under conditions where T and P

constant. Eq.

For

(1.23.2)

that

rather

in (1.23.7) as a representation

reason

than

any

it of

is the

for Pl.

(b) One should note again the topology of the structure of thermodynamic

theory.

The four basic

thermodynamic

functions

of Section 1.18 have been enlarged to allow for variations

in

composition of the system.

P,

Correspondingly,

the variables

V, T, S may be found from Eq. (1.23.8), and ~i may be determined via (1.23.2) or (1.23.7). matter the

of convenience

system.

Again,

The choice in each case is simply a

relative

to the constraints

E and H are

adiabatic conditions prevail.

functions

most

P, V,

S,

T,. n)

Table

1.23.1.

(82X/axlax3) - (82X/axjaxl) leads As

to

the Maxwell

in Section

useful

For constant T processes

and G that are the useful functions of state. of holonomicity

imposed

1.18,

(X ffi E, H, F, G; x specified

in

a limited number

of

these is useful,

among them the partial differentials

entropy in parts

(A) and (B) of the table.

Items extensions systems.

[i]

and

of earlier

[4]

in

Maxwell

Table

1.23

relations

when

it is F

The requirement

equations

only

on

represent to the

Item [2] may be rewritten in the form

case

for the trivial of open

,. FUNDAMENTALS

170

TABLE 1.23.1 Maxwell

Relations

for O p e n Systems

(A) See Eq. (1.23.3)

Na

i T,x i

-

Na [I]

[aT J P,x i

~:,p.=j

P,x i

T.P,nj

[o--P-JT,x i

-

[31

[2]

(B) See Eq. (1.23.4)

T,x i

V,x i

[aT ; V,x i

T,V,n~

[4]

T,V,nj

[~---J

T,x i

[6]

[5]

(C) See Eq. (1.23.5)

[~v] S,x i

S,P,n0

P,x i

[7]

[~--)

~ni S,P,n j

P,x i

_ ra~il -- [~'-p j

S,x i

[9]

[8]

(D) See Eq. (1.23.6)

~~ Na S,x i

S,V,nj

V,x i

A subscript

The s u b s c r i p t fixed.

x i means

nj means

that all mole n u m b e r s

that all mole n u m b e r s

[aV ;S,x i [12]

[11]

[lo]

Note"

S,V,nj

[~-S--) V,x i

are

except

to be h e l d

n• are

fixed.

to be h e l d

I 71

THERMODYNAMICS OF OPEN SYSTEMS

(1.23.9)

S i = (8~i/aT)p,x• and item

[3],

in the form

(a~/aP)T,x~,

V~-

(1.23.10)

w h i c h shows h o w Eqs. partial

molal

appropriate

(1.18.8)

quantities

to e m p h a s i z e

should be a l t e r e d S i,

V i.

.fixed. which

of state)

in w h i c h

By contrast, some

are k n o w n

(c) Several definition

G-

all

stage

such as aS/anl,

fixed

quantities

as d i f f e r e n t i a l

additional

quantities

represent

entropies,

abbreviations

immediately

W

-

extension

of

Y.(1)Didnl) v a n i s h

earlier

the

that

where

only

introduce

E be

mechanical

Hi -

(aH/Onl)T,P,no#i and

definitions,

identically

this is e q u i v a l e n t

Law

From the

that

function needed

consideration; First

etc.

Si =

introduced.

v i e w e d as the excess or deficit +

extensive

volumes,

(1.23.11)

the c u s t o m a r y

an

are he!.d

points will n o w be made.

H - TS it follows

(aS/ani)T,p,nj#i have b e e n As

is

a V / a n i, . . . , in

~i = Hi " TSI,

where

it

of the type a Y / a n i (Y is any

intensive

quantities

or all of the

variables,

this

the fact that the term "partial molal"

is r e s e r v e d for p a r t i a l d e r i v a t i v e s function

At

to apply to the

a

in the usual m a n n e r

is

any

may

of

state.

involved

we

the total e n t r o p y

now

be

to have Q - (AE process

to the r e q u i r e m e n t

function

work

for

Q

For may

under

i m p o s e d by the

case

therefore

f u n c t i o n S such

that

f f Q r - TdS.

(1.23.12a)

Then

aT;r - PdV

- ~ ~idnl. i

(1.23.12b)

172

I.

FUNDAMENTALS

There exists, however, an alternative identification.

One

may switch from the set of independent variables S, V, n• to the set T, V, n i by reformulating

the First Law of Thermodynamics

as

dE - T[ (8S/aT)v,.~ dT + -

-

(SS/SV)T,xldV

(SS/8nl)r,v,nj#idnl]

+

PdV + ~ #idnl i

(SE/8T)v,xldT +

(1.23.13a)

(8E/aV)~,x i dV +

~(SE/Snl)~,v,nj~idni.

(1.23.13b)

i As

usual,

match.

the

coefficients

of

all

differential

terms

must

Thus,

- T(8S/8T)v,xi

(8E/8T)v,xi

.. Cv.xi

(1.23.14a)

(SE/HV)T,x i - T(BP/HT)v,x i - P

(1.23.14b)

( S E / S n i ) T , v , n j , ~ i " ~i + T(SS/Sni)T,v,nj,~ i.

(1.23.14c)

Note that in (1.23.14a) an enlarged definition of heat capacity has been introduced.

Use of the Maxwell relation [4] of Table

1.23.1 has yielded (1.23.14b); this is a slight modification of the

caloric

Writing

equation

of

state

introduced

(8Z/Snl)T,v,nj~i- Z i, Eq.

(1.23.14c)

in

Section

1.18.

shows that ~ i -

El

- TS i. An alternative derivation of this relation is called for in Exercise 1.23.19.

Thus,

finally, (1.23.15)

dE - Cv,xldT + [T(SP/ST)v,xl - P]dV + ~ Eidnl. i One

should

carefully

note

that

it

is

not

permissible

to

identify ~Qr with Cv0xldT for reasons to be explored in Exercise 1.23.22. The

reader

should

be

able

to construct

the

analogs

of

(1.23.14) involving the enthalpy, and to derive the expression

dH - Cp,xidT + [- T ( a V / a T ) p . x i

+ V]dP

+

~. Hidn i. i

(1.23.16)

173

THERMODYNAMICS OF OPEN SYSTEMS

(d)

Lastly,

extensively (1.23.3)

in

we

cite

later

with respect

two

relations

sections.

If

to P keeping

we subsequently differentiate

we

will

be

used

differentiate

Eq.

T and all n i fixed,

and if

the resultant with respect to nl,

we obtain the first result of interest part

which

(see also Table 1.23.1,

[A] ) :

(a~/aP)z,.~

(1.23.17)

= v~. m

Starting with Eq.

(1.23.9) we write ~i/T - HI/T - Si, whence

[a(~,i/T)/aT]p,x i - (- HI/T 2) + (i/T)(aHi/aT)p,x•

- (a-si/aT)p,x i. (1.23.18) m

On

introducing

Eq.

(1.23.5)

in

the form (i/T)(aHi/aT)p,x•

I

+ (aSi/0T)p,xl, it is seen that the last two terms cancel; [a(~i/T)/aT]p,xl - - HI/T z, which

is a basic

relation

thus,

(1.23.19) that

finds

frequent

application

in

later sections.

EXERCISES

1.23.1 (a) The chemical potential has been v a r i o u s l y identified as (aE/ani)s,v,na , (@H/.@ni)s,p,n~ , (aF/ani)T,V,na , (aG/anl)T,Pn' 3 " Are all these quantltles identical or do they differ? Document your answers fully by means of appropriate derivations. (b) The chemical potential has also occasionally been defined as (aE/ani)T,v, n and (aH/ani)T,P,nl. Are these quantities the same as for (a~? Document your answer fully. 1.23.2 working with the Helmholtz free energy function show that : (a) ( a E / a n ) T , V - ~ - T ( a ~ / a T ) v , n (b) (an~aT)v, n = (an/a~)T, v {~- (a~/aT)v,n} -T-l(an/a~)T,v(aE/an)T,V, where 17 i ~/T (c) (aE/aT)v, n - (aE/aT)v, n - T-1(an/a#)T,v(aE/an)~,v >_ o. 1.23.3 Expand on the analysis of this section by examining cases where work other than mechanical (P-V) work is involved.

174

I. FUNDAMENTALS

1.23.4 The term ~ ~idnl is sometimes known as "chemical work". Is such a designation appropriate? 1.23.5 Suppose mole fractions rather than mole numbers were used as independent variables to indicate the c o m p o s i t i o n of the system. How would such a step affect the analysis? In particular, could one define a chemical potential by, say, 1.23. (a) Show under what conditions the g e n e r a l i z e d Gibbs equation TdS - dE - ZZldz i - ~ Pkdnk may be integrated to give TS - E -~Ziz i - ~Pknk . (b) Correlate these results with the conventlon~l defi{itlons for H, F, G. 1.23.7 (a) Prove that for any function of state ~ and with a set of conjugate variables (Yi,Yl), the following relation holds" Td(S/~)d(m/~) - iZ"fld(yl/~) - ~ k d ( n k / ~ ) . (b) Specialize the above to the case where ~ represents the total mass of the system, the total number of moles of the system, and the volume of the system. Take special note of the formulation for the latter case. 1.23.8 Let Z Z(T,yl,nk) be any functions of state of temperature T, deformation coordinates Yl, and mole numbers n[. Let Z ~ Z/n, with n - ~(k)nk, be the molar quantity, and let Z k " (8Z/ank)T- n be the partial molal quantity. (a) Prove that 'J' i~k with x k_- nk/n, Z - Z(k)XkZk . -- (b) Prove that d Z (8Z/OT)yl,nkdT + E(1)(aZ/ayl)T,y~r + E(k)Zkdx k. 1.23.9 ReTerring to the previous problem, let Z ~ Z/V, where V is the total volume of the system and let c k - nk/V. Prove_ that d Z (i/V){(aZ/aT)yl,nkdT + ~(aZ/ayl)z,y0#i,nk dy i +

~Zkdck }. 1.23.10 Examine t h e c o m p l e t e T a b l e o f Maxwell R e l a t i o n s b a s e d on Eqs. ( 1 . 2 3 . 3 ) - ( 1 . 2 3 . 6 ) . Discuss the utility of the various entries. 1.23.11 Discuss the question as to whether it w o u l d not have b e e n more suitable to define an increment in Gibbs free energy according to dG - - SdT + VdP + iZl ~idxl where c-i independent mole fractions appear as varia5~es rather than the c mole number n i. In fact, is there not a redundancy in the commonly used expression for dG? 1.23.12 (a) Derive the following relation (which is of interest in statistical thermodynamics) for a o n e - c o m p o n e n t system" ( 8 # / 8 n ) T , V - (V/n)Z(dP/0V)T,n . [Hint" Y o u will need several of the mathematical 'tricks' of Sec. 1.4; you need to recognize that the general equation of state interrelates T, P, V, and n; and you need to consider the fact that ~ = #(T,P,n).] (b) Show that for an ideal gas this genral result reduces to the conventional expression. 1.23.13 Show how the derivation of Eq. (1.23.7) can be h a n d l e d by use of the Planck Function introduced in Exercise 1.18.4.

THERMODYNAMICS OF OPEN SYSTEMS

I7

1.23.14 Prove that for the internal energy function the following Gibbs-Duh.em relation must hold; Ed(I/T) + Vd(P/T) -~.inld(~i/T) - 0. Show what variables, if any, must be held fixed for this relation to apply. 1.23.15 In this Section the internal energy function has been introduced in the form E - E(T,V), whereas in Section 1.18 it has been formulated as E - E(S,V). Considering that thermodynamic functions of state should be useful in deriving various intensive and extensive variables, are the two formulations equivalent? If not, which one is more fundamental? In a similar vein discuss the relation b e t w e e n H H(S,P) and H - H(T,P). 1.23.16 Consider the Grand Potential Function ~ - E - TS C -i~l~ini. (a) Derive relations for S, P, and n i in terms of appropriate derivatives involving ~. (b) Show that ~ - - PV. (c) Discuss cases where the function ~ may be of utility. 1.23.17 Prove that the Gibbs free energy is subject to the following requirements" r (ac,/aP)t,x~ ~__j n i (a-d/aP)t,xt tI1 -

I

r

(ac/aT)p.. i _tZ.1 n~ ( ac/aT)p,x, . 1.23.18 Why can one not equate Eqs. (1.23.9) and (1.23.10) with items [5] and [9] of Table 1.23.1, respectively? 1.23.19 (a) Establish relations between (aZ/an• T ena and (aZ/anl)T v,n~ for z I E,S. (b) Derive Eq. (1.23.14) using the results of ~a). (c) Prove that (1.23.13a) is equivalent to the following relation" dE I (aE/aT)v,xld T + (aE/aV)T,xtd v + T-(i)(@E/@ni) T,V,n~#idni 9 1.23.20 rntroduce the partial molal quantities Z i by=the definitions (SZ/@nl) T p n and a related set of quantities Z i (@Zl/@nl)T v n~Find 'relations for S,, in terms of ~i, Hi in terms of El, and Hi, H i in terms of E i. 1.23.21 Using the results of Exercise 1.23.20, prove that i. 23.22 Prove that the following relations are mathematically correct: dE - TdSoF - PdV + E~idn• ; dHTdSo~ + VdP + ~ • wherein dSoE - (Cv x./T)~T - (OV/OT)p,xidP and dSo~ - (dp x /T)dT + (0P/0T)vx dV'." (b) Explain very , i , i carefully why tSis _relation is not very useful or even 'dangerous'. [Hint" Hi I (aH/anl)T,e,nj ; what are the variables actually occurring in the expression for dH?]. (c) Show why it is not permissible in Eq. (1.23.15) to set ~Qr I Cv,x• [Hint" Examine the Second Law and note where the entropy enters Eq. (1.23.15) implicltly.]

176

, FUNDAMENTALS

1.24

A R E M A R K CONCERNING

SYSTEMS WITH V A R I A B L E

In See. 1.23 we introduced thermodynamic systems

of

variable

composition.

COMPOSITION

functions of state for

For

example,

the

energy

function was w r i t t e n as c dE - TdS

where

- PdV + ~ ~idnl, i-i

the

index

i

We

principle

be altered

by

the

now

runs

species.

take

occurrence

(1.24.1)

over

all

chemically

cognizance

of

the

distinguishable

fact

that

n• can

in

in the amount dn i by two mechanisms:

of

chemical

reactions

totally

(i)

within

the

system w h i c h change n i in the amount dNi, and

(ii) by transfer

across

i

boundaries,

reservoir written

of

containing

dN i moles species

in the alternative

of

i.

species

Thus,

Eq.

from

or

(1.24.1)

to

may

a be

form

c dE - TdS

Some

- PdV + ~ pi(dN i + dNi). i-i restrictions

apply:

which dN i - 0 for all i.

(1.24.2)

Consider

Then Eq.

a closed

(1.24.2)

reduces

system

for

to

c dE o - TdS

- PdV + ~ pldNi. i-I

On the other hand,

(1.24.3a)

the e x p r e s s i o n obtained in Section 1.18 for

the energy of closed systems

dE o - TdS

is given by

(1.24.3b)

- PdV.

Consistency between

(1.24.3a)

and (1.24.3b)

then demands

that

c ~idNi - 0.

(1.24.4)

i-i This

requirement

necessary

be

and sufficient

when reactions (1.24.4)

will

and

shown

in

condition

Section

2.9

to be

for e q u i l i b r i u m

take place totally w i t h i n a system. the

simplification

that

leads

from

both

a

to prevail Constraint

(1.24.3a)

to

A GENERALIZATION OF EULER'STHEOREM

I "/1

(1.24.3b) are in accord with the previously anticipated concept that changes

in internal

energy can take place

system exchanges heat, work,

only when

the

or matter with its surroundings.

Any processes occurring totally within the system do not change its internal energy E.

In view of (1.24.4)

well have been written Eq.

(1.24.1)

one could equally

for quasistatic processes

in the form C

dE - TdS - PdV + ~ #idNi. i-i However,

(1.24.5)

there is no harm in carrying along sets of terms that

ultimately vanish, and to continue using (1.24.1) with dn i - d ~ i +

dN i.

Similar

remaining

commentary

differentials

applies

of the

to

expressions

thermodynamic

for

functions

the

H,

F,

and G.

EXERCISE

1.24.1 Provide a discussion that addresses the issue as to where the driving forces leading to chemical reactions within a system 'come from', and describe 'what happens to them' as the reactions proceed.

1.25

A GENERALIZATION OF EULER'S THEOREM

(a) For later use it is important to generalize the exposition on

Euler's

theorem,

f(al,...,an;Xl,...,xm)

as

shown

below.

Consider

a

function

which meets the following requirement:

fk " f(al,'-',an;kxl,''-,kxm)

- khf(al, ...,an;xl,.-.,xm) - khf;

(1.25.1) i.e.,

f

is homogeneous

xl,...,x m. Next,

Z i-I

of

order

h

in

the

set

How do we generalize Euler's Theorem? differentiate

dai + [aaIJ aj#i,kx r

Eq.(1.25.1)

Z r-i

as follows:

d(kxr) (kxr)

ai,kxs# r

of variables

178

,.

-kh

dal + k h

Z

i-I

dxr

Z

r-i aj#i, Xr

FUNDAMENTALS

al,

Xs#r

+ hkh-lf(al,...,a=;xl,...,xm)dk.

(1.25.2)

Note that on the right side we also had to consider k as a variable because on the left k is part of the operand of the operator d. Since d(kxr) - kdx r + xrdk, we can use this relation and collect coefficients of the various da i and dx r. We obtain

n {pfk I i-~l t~al

r-I

k h @[~al]} _

m {k [-aafk ] dal +r-~l (kxr)

(k~r)'l - hkh-i de - 0.

.

k h a[~r]} d x r

(1.25.3)

Since the dal, dxr, and dk may be varied arbitrarily one can satisfy (1.25.3) only by having their coefficients vanish individually" Thus, fk ]

-

k,,

la 1 - la ] - hkh-lf. r~l Xr -

8 (kxr)

[i - l,...,n]

(1.25.4a)

[r- l,...,m]

(1.25.4b)

(1.25.4c)

On inserting (1.25.4b) into (1.25.4c) and canceling out a common factor kh-i we obtain Euler's equation in the form

m _~ Xr r i

a[~=] - hf.

(1.25.5)

A GENERALIZATIONOF EULER'ST H E O R E M

I ~9

We note the following:

(i) Only the variables

k appear in the foregoing the result

cited

in Eq.

summation; (1.4.22);

that f must be an intensive assigned to k in kx i.

(ii) if h (iii)

quantity

in

i we recover

if h -

0 this means

independent

of the value

For such functions

0

xr

homogeneous

(h-

0).

(1.25.6)

r-I

Special note should be taken of Eqs.

(1.25.5)

and (1.25.6).

(b) We now develop some general thermodynamic applicable

to solutions

with

numbers

mole

freedom. I.

Readers

nj

of c components that

involve

2

expressions

containing additional

species degrees

j of

should check each step of all derivations.

Let B represent

an extensive

thermodynamic

variable;

let Yl represent the !th additional thermodynamic variable other than mole number nj; then one can express B either as (1.25.7a)

B - B(yl,...,y,;nx,...,n=),

or as B -

B(yl,...,y,;xx,...,x=_

(1.25.7b)

x,n),

c

wlth n -

no, x 3 i no/n.

Thus,

j-I (i)

dB

-

We can now write

dy i +

i~.

', Ha

--

dn3, j --I

Yk#i n j

(i. 25.8a)

Yl,nk#j

or

dB

-

dYl +

iy

Ykr ,x3 n

c - i 018_~al j-I

dx3 +

Yl ,Xk#j ,n

Na

(1.15.8b) Yi ,xj

180

, FUNDAMENTALS then Euler' s

(ii) If all the Yl are themselves extensive, Theorem as applied to (1.25.7a) yields

BIZ

Yl

iI[

Yk#i ,ni

and thereby simplify

7~ i i

Yl ,nk#j

n.

+

If,

on

(1.25.9b)

Yl .xj

Yk#i 'xo ,n

variables, I

(1.25.9a)

nl,

(1.25.8b):

Yl

(iii)

B

Z j I[

for B as specified in (1.25.7b), we apply Eq. (1.25.6)

whereas,

B-

+

the

hand,

other

the

Yl are

then, on account of (1.25.6) Eq.

Z

~

1Yl. nk,~j

j-I

n3, so that

all

intensive

(1.25.9a) becomes

" B- ~n-B ~

0{~nj1

Y~ .nk# O

(1.25.10a)

(Yl intensive) , whereas

B- aH

(1.25.9b) now reads

n,

so that

Ha

I

nB

-- I

B -

Yi' xj

Yi' x3

(1.25.10b)

(Yl intensive). Notice

that

specified holding note

partial

only when

only

the

molal

properties

the partial

intensive

of

derivatives

variables

fixed;

B

are

are thus

properly

taken while one

should

(1.25.10b)

with

Note further that if all Yl are intensive variables

then

the

difference

between

(1.25.10a)

and

regard to the quantity labeled as B. Eq.

(1.25.10a) may be differentiated

to read:

C

_ Z ~. [an,]

aB Yi ,Xk#j ,n

s=l

[axa) Yl,Xk~j,n

(z.25.zl)

A GENERALIZATION OF EULER'STHEOREM

W i t h n, - x,n,

(an,/aXj)yl,xk#j - 0 for (J ~ s), or n (for j - s),

or - n (for j - c)

a[a_,~]

l 8 [

[explaln

this last result!],

we now find

(1.25.12)

n(B,j - Be).

I

Yl, Xk,~-1 2.

Suppose

thermodynamic (1.25.7a) apply,

next

function

that

which

or (1.25.7b);

now Eqs.

If

all

Euler r e l a t i o n

the

Yl are

(1.25.6)

(1.25.8a)

an

.intensSve

specified

extensive

by

and (1.25.8b)

variables,

Eqs. still

function.

we

use

the

c

Yl

+

Yk#i ,n~

If we switch the case

7.

in going from

Yi +

(ii) If, however, on account

of

to mole

fractions

to (1.29.9b),

then,

as was

we obtain

n.

(1.25.13b)

Yi'x3

the various

(1.25.6),

(1.25.13a)

Yl, nk#,~

(1.25.9a)

Yk#t ,x3 ,n

simplify

nj.

j-1

from mole numbers

0 -- i~.l

0 -

be

to find

'

7.

then,

represents again

but we replace B by B m B/n for the intensive

(i)

0 -

B

may

Eqs.

Yl are intensive (1.25.13a),

and

variables, (1.25.13b)

to

.,. 8[~n3] ~c

jl

n~,

(1.25.14a)

Yl ,nk#i

or to

0- Ha

n. Yl,X3

(1.25.14b)

182

, FUNDAMENTALS

Equation (1.25.14b) is of particular importance" If B is an intensive thermodynamic variable, then Eq. (1.25.8b) is given by

dB - iZ I -

dy i +

(1.25.15)

dxo,

j--

Yk#i ,x3

Yi ,Xk#j

where it is implied that n is held fixed. (iii) We take note of the following restrictions are intensive) 9

~

p'~,'nl -

lenj

,,.. ,, '[~]

J

Yi, nk~i

"

n

(all Yl

'-~:,,.,:,

- --r n

-n

Yl, nk#i

Yi, nk#j

(1.25.16a)

'[~I

.

Yi ,Xk~3

r~ . [Sxj

J

.

.

i o[~,I

-

B3 -- B c 9

n

Yi ,Xk#j

Yi ,Xk#3

(1.25.16b) Thus, we could equally well put Eqs. (1.25.8a) and (1.25.9) in the form 2 a[ ] I c Z [aylJ dyi + n Z Bdnj i-I Yk#i ,n3 j -i

dB-

_ B_ dn,

n

(1.25.17a)

and

dS -

iZ I

dy~ +

--

j=,

(Bj

- Be) dxi

Yk#i' xj

2

af

~

c- I

- iZ I ~a-~i] -

dy• + Yk#i ,xj

I

j-I

Bjdxj.

(1.25.17b)

LEGENDRETRANSFORM AND STABILITY CONDITIONS

1.26 (a)

I

THE LEGENDRE TRANSFORM AND STABILITY CONDITIONS In

this

section

we

concern

ourselves

stability of a system at equilibrium. we

8~

must

first

deal

with

the

with

problems

of

By way of preliminaries

topic

of

generalized

Legendre

transforms. Let Y(Xo,Xl,... ,xt) be a thermodynamic function of interest that

depends

solely

on the

extensive

set of variables

{xl}.

Then t

dY- i7.o_

dx i x3#i

t ~ pidxl, i-O

(1.26.1)

wherein (1.26.2) We now introduce a (partial) Legendre transform Z of Y by the definition n

Z(Po,Pl,...,Pn,Xn+1,...,xt)

- Y(xo,Xl,...,xt)

with the Pl specified by (1.26.2). a

function

replaced

in which

by

their

differential of Z.

the

first

conjugate

-

~ PlXi,(l.26. 3) s i

Equation (1.26.3) represents n variables

quantities

p 9

of

Y

have

Now

take

n

i -I

the

This yields

dZ(po,...,Pn,Xn+1,...,xt) - dY(xo,...,x t) - 7. i-O t n n t -

been

"

" 9

xldpl-

n

pidxi-

7. i-O

xldpl

n

i ~n+l pldxl - i~O x~dp~.

i-O

(1.26.4) Accordingly,

Po,

9 9 9 ,Pn,Xn+l,

9 9 9 ,Xt)

-

"

Xk

for k _< n

(1.26.5)

184

I. FUNDAMENTALS

818---~'-k}I~ From Eq.

Lax

j

-

(1.26.6)

for k > n.

(1.26.1) one obtains

(1.26.7)

ta-- J"

On a second differentiation of (1.26.5) and with respect to Pk by

use

of

(I. 26.7)

one

arrives

at

the

very

important

relationship a2z

ap~

i -

-

a2y/ax~ (k

_< n),

(1.26.8)

showing that the second derivative of Y with respect to x~ ha_____ss a

sign

oDDosite

to

that

of

the

Les

transform

Z

with

respect tO the conjugate variable pw. (b)

We

now

illustrate

applying these results

the

preceding

to the energy E -

methodology

by

E(S,V,nl,...,nt)

and

entropy S - S(E,V,nl,...,nt) , both of which depend on extensive variables.

Now, at equilibrium,

a

with

maximum

respect

includes

a demand

example,

if

transferred

we

to

S for an isolated system is at

the

for uniformity

partitioned

the

energy from one half

imposed

constraints;

of properties. system

in

two

this

Thus,

for

halves

and

to the other half,

we would

violate the equilibrium condition and the system would then be at lower total entropy than in its uniform equilibrium state. Were

this portion removed one would

and the condition of uniformity violated. i

achieve

phase

separation

of composition would

then be

Thus,

S(E-AE,V,xz,...,xt)

+ iS(E+aE,V,xz,...,xt)

_< S ( E , V , x z

.... ,xt).

(1.26.9) As AE ~ 0 one may rewrite this in the differential

form

105

LEGENDRETRANSFORM AND STABILITY CONDITIONS

S(E,V, lxi}) - (AE)S'(E,V, lxi}) + ~1 (AE) 2S "(E,V, lx~}) + ... + S(E,V,{xs}) + (AE)S'(E,V,{xs})

+ ~i (AE) 2S " ( E , V , {x i})

(1.26.10)

+... _< 2S(E,V,[x i}),

on expansion in a Taylor's series in the variable E; the primes indicate first and higher order partial derivatives. (1.26.10)

Equation

simplifies to

(82S/SE2)v,xl _< O. A derivation

(1.26.11)

similar

to

(1.26.10)

for a transfer

of volumes

shows that (82S/SV2)E,xl _< 0.

(I. 26.12)

In precisely the same manner one finds that (~2S/~n2)s,v,n3# i < O.

(1.26.13)

The foregoing

extensive

variable

shows that no matter what particular

is altered,

S is always

concave.

We now note

some

immediate consequences of those equilibrium requirements. P i Since dS - ~ dE + ~ d V

t E j V,x i

~~

dnl, it follows that

_< O, (1.26.14) -

-

- T2

V,x i

"

T2

Cv

V,z i

or

C v >_ O;

(1.26.15)

i.e., the heat capacity of constant volume of a uniform system can only be zero or positive.

In Exercise 1.26.1 the reader is

186

, FUNDAMENTALS

directed to use Eq.

(1.26.12)

in similar fashion,

and then to

explore the resulting consequences. One

can proceed

similarly

with

energy,

stability criterion is turned around. I

except

that

the

For a uniform system,

i E(S-•S ,V, {n i}) + ~ E(S-AS ,V, {n i}) >_ E(S,V,{n i}),

(1.26.16)

which, by the argument developed in conjunction with (1.26.10), leads to (a2E/aS2)v,x i >__ O.

(I. 26.17a)

Similarly, (@2E/aV2)s,x i >__ O,

(aSE/an~)s,v,nj,~i z o.

(1.26.17b,c)

But (1.26.18)

(a2E/aS2)v,x i - (aT/aS)v,xl _> O, showing that S increases with T, as it must, self-conslstency with the manner a positive

if we are to have

in which T was introduced as

integrating denominator.

Similarly,

(a2E/aV2)s,xl - - (aP/aV)s,zl >__ o, which

shows

that

in

an

adiabatic

(1.26.19) process

pressure must lead to a reduction in volume.

an

increase

in

This, of course,

is a very comforting conclusion. (c) Now we must be extraordinarily of the thermodynamic E.

careful with the rest

functions that are Legendre transforms of

First, we can use the standard procedure that was employed

in deriving Eq.

(a2H/aSZ)P,xi ~_ O,

(1.26.17)

to find (1.26.20)

LEGENDRE TRANSFORM A N D STABILITY CONDITIONS

I87

whence,

(8T/aS)p,xl

>__ O.

(1.26.21)

But notice now that H is the Legendre transform of E and hence, subject to Eq. derivative

(1.26.8).

of H with respect

respect to V.

_

Thus,

Therefore,

the sign of the second partial

to P is opposite

that of E with

contrary to naive preconceptions,

(@2E/aV2)-I S,xl _ (a2H/ae2)s,x i _< 0,

(1.26.22)

or

(av/aP)s,x i _ 0;

of.

(1.26.15).

and

with

(1.26.21) The

Eq.

(1.16.19).

One

leads to T/Cp >_ 0, whence

Helmholtz

free

energy

F is

the

Legendre transform of E with respect to S and is a minimum for constant T.

We can proceed straightforwardly

to find

(a2F/aV2)T,x i >_ O,

(1.26.24)

whence

- (aP/aV)T,xt

>_ o,

which verifies

or

(av/aP)T,xl

_ 0.

(2.2.5a)

Here S i dS/dt represents the rate of entropy production in the time

interval

negative, no

work

involve

dt

for

the

corresponding

system;

and it vanishes at equilibrium. has heat

been

performed,

alone.

It

is

and

all

We then rewrite Eq.

transfers

reasonable

dE'/dt here with the rate of heat flow, Q, boundary.

be

In the present case

energy

therefore

S cannot

to

must equate

across the internal

(2.2.5a) as S - A(I/T)Q.

Next,

define a heat flux by the relation JQ - Q/A where A is the cross sectional area of the diathermic partition. that

the

temperatures

T"

and T" are very

Moreover, nearly

suppose

constant

in

both compartments and that the changeover from T" to T" occurs only over a small distance the

latter

being

~ perpendicular

essentially

at

the

to the partition,

midpoint

over

this

distance.

Then the product A~ roughly defines a volume V over

which

temperature

the

A(I/T)VJQ/2.

changes

we

may

now

write

S

-

In the limit of small 2 the ratio A(I/T)/~ becomes

the gradient V(I/T); moreover, production

occur;

the ratio w

of entropy per unit volume,

- 8 is the rate of

an intensive

quantity

[~

2. EQUILIBRIUM IN IDEAL SYSTEMS

that is of great theoretical in rewriting Eq.

(2.2.5a)

interest.

We have thus succeeded

in the more fundamental

form

(2.2.5b)

- V(I/T)JQ ~ 0. The preceding chain of reasoning a proper derivation of Eq. Chapter 6. aspects

(2.2.5b)

We nevertheless

departure,

because

one

is obviously very crude;

for

the reader is referred to

adopt this result as our point of

can

then proceed

of the theory of Irreversible

with

some

important

thermodynamics

without

having to cope with the full machinery of Chapter 6. In conjunction with Eq. (2.2.5b), note that JQ and V(I/T) m F t may be considered as conjugate variables, quantities

in that these two

occur as the product of a flux JQ and a generalized

(thermal) force or affinity Ft, such that schematically 0 - FtJ Q >_ O.

Note that 0 > 0 means either that F t > 0, Jo > 0 (i.e., T"

> T') or that F t < 0, JQ < 0 (i.e., T" > T"); heat

flows

from

the region

lower temperature. and F t vanish.

to that

When 0 - 0, equilibrium prevails;

both JQ

to

should

introduce

be

raised

the

as

to

temperature

whether

it

is

concept

in

a

nonequillbrlum situation.

The answer is in the affirmative

the following

conditions

portions occurs

sufficiency

of the system are very

very

of

the heat flux JQ.

question

meaningful

temperature

These facts give rise to the viewpoint that the

force F t 'drives' The

of higher

in either event,

slowly.

In

this

have been met"

large, event

T"

The

and the heat and

T"

if two

transfer

are

sensibly

uniform over both regions and most of the temperature variation takes place in the immediate volume A~ of the interface. The relation between F t and JQ cannot be determined from classical further

thermodynamics information

experimental

results,

such

alone. as

It

is necessary

microscopic

or further postulates.

to

transport

supply theory,

It is generally

accepted that for a system close to equilibrium there exists a proportionality between force and flux of the form Jo -

l~Ft,

(2.2.6a)

199

ACHIEVEMENT OF EQUILIBRIUM

where L t is a p a r a m e t r i c function independent of Jo or Ft, k n o w n as

the

phenomeno!ogical

process under study.

JQ

-

L~V(I/T)

where one

~ ~

-

~F~

(et/T 2)vT

-

-

for

the

irreversible

of Fourler's

(2.2.6b)

~VT,

(I~/T 2) is the thermal

formulation

present

-

-

coefficient,

Note further that

conductivity;

L~w of heat

Eq.

(2.2.6)

conduction.

is

In the

scheme we may set

-

(2.2.7)

J~/L~,

which requires

that L t _> 0 and ~ _> 0 in order that 8 _> O.

(c) We next examine

the

case

of

an

isolated

compound

system c o n t a i n i n g a sliding p a r t i t i o n that is initially locked and provides

for adiabatic

insulation

of two compartments

at

pressures P" and P", temperatures T" and T", and volumes V" and V".

The system is allowed to relax after slowly r e l e a s i n g

lock and changes

slowly

rendering

in both compartments

relation

dS

allowed.

The constraints

-

T-I[dE

dV" - O, as before)

+

Now,

PdV],

no

other

dS

- T -I

(8S/8V)E - P/T;

dS -- (I/T')dE"

Eq.

in accord with

forms

of

work

being

It should now be clear

+ (8S/aV")E, dV"

(2.2.8)

>_ o.

[dE +

PdV}

(2.2.8)

one

finds

(OS/OE) v -

I/T

and

then becomes

+ (I/T")dE" + (P'/T')dV"

(2.2.9)

+ (P"/T")dV" >_ O.

Finally,

the

in (b),

( 8 S ' / 8 E " ) v , dE" + ( 8 S " / 8 E " ) v . d E "

from

Entropy

are dV" + dV" - 0 (rather than dV" -

adopted

+ (as/av")~.dV"

diathermic.

can now occur

and dE" + dE" - 0.

that by the procedure

dS-

the p a r t i t i o n

the

with dE" - -

dE"

and dV" - -

dV"

one obtains

'200

2. EQUILIBRIUMIN IDEAL SYSTEMS

- (dS/dt)

- (I/T"

+ (P'/T"

This led

to

introduce

setting

A(I/T)

-

(2.2.10)

>_ 0.

in c o n j u n c t i o n up

of

Eq.

I/T'

-

with

the arguments

(2.2.5b),

the fluxes JE i dE'/Adt

affinities -

- P"/T")(dV'/dt)

expression,

the

convert

-I/T")(dE'/dt)

suggests

and Jw i dV'/Adt

I/T" and

A(P/T)

-

A(P/T)/2

V(P/T).

"

We further

production

per unit volume

by

thickness

of the partition).

introduce

0 - S/V-

that

-

P"/T ~ into

- V(I/T)

identifies

conjugate

w

(~ is the

leads

V(I/T)

to

by

the

the

We then obtain

and

requirements

fluxes

then

phenomenological

in

the

coefficients discuss the

V(P/T)

and

Equilibrium

is

then

0,

which

-

P"

condition

These

T"

FT -

-

T"

relations

to

assume

a

Fp -

and

P"

between linear

quantities

appropriate

to the

in much

call

entropy p r o d u c t i o n

form,

Since we require

Lzz -> O; for w h i c h

as

as and

termed

L

are

the

situation

further

attention

(2.2.12)

to

at hand.

detail

the

phenomeno!ogical

fact

We

in Chapter that

the

shall

6. rate

For of

is given by (2.2.13)

- FTJ m + FpJ w = LzzF 2 + (Lz2 + L2z)FpF T + L22F 2 >_ 0.

sufficient

Jw

forces

L2FT + L22Fp,

Jw-

various

we

as

equations

this matter

moment

well

that

postulated

LIIFT + LI2Fp; which

as

necessary

constraints.

JE-

JE

variables.

equilibrium are

small

(2.2.11)

force/flow

characterized

and Fp

the rate of e n t r o p y

- V(I/T)J E + V(P/T)Jw, which

we

and that we

P'/T'

such that for small 2, F T - A ( I / T ) / 2

that

0 to be n o n n e g a t i v e

it is b o t h n e c e s s a r y

and

to set

4L 1zL22 -

(Lz2 + L2z)2 _> O;

a derivation

is f u r n i s h e d

(2.2.14)

L22 _> O; in S u b s e c t i o n

(e).

ACHIEVEMENT OF EQUILIBRIUM

201

(d) As a final g e n e r a l i z a t i o n subsystems

separated

and p e r m e a b l e the

two

by

to one

species

compartments.

covered

by

a

material.

a rigid

to

extension

equilibrate

the case of two

that

is d i a t h e r m i c

in d i f f e r e n t

the p a r t i t i o n

adiabatic,

This c o n s t r a i n i n g

allowed

present

Initially

stationary,

we examine

partition

impenetrable

the

small

-

(I/T"

+ (~"/T" from w h i c h

and the s y s t e m

opening.

- ~"/T" ) (dn"/dt),

it follows

in c o n s o n a n c e

with

write

dn'/Adt

as

down

well

may

case.

be

That

equations

(2.2.15)

(2.2.16)

our the

as

is,

findings set

their

suitably one

of

of Section fluxes

conjugate

Jz"

2.1.

dE'/Adt

generalized

and

forces

The various remarks made earlier paraphrased

to

postulates

linear

Jn-

apply

to

the

Jn " Fz in the

present

phenomenological

be r e w r i t t e n

(2.2.17)

L43FT + L44Fn-

The rate of entropy production,

b a s e d on Eq.

(2.2.15),

may then

as

- FrJ z + FnJ n - I~3FTz + (L34 + L43)FTF n + L44Fnz >_ O,

L33 >_ 0,

In a d d i t i o n

of the form

Jz- L33FT + Lz4Fn;

subject

an

that at e q u i l i b r i u m

and F n - V(~/T).

section

By

- I/T") (dE"/dr)

T" - T" and ~" - /J",

V(I/T)

of

to obtaln

w = (dS/dt)

may

sheet

of earlier d i s c u s s i o n we invoke the r e l a t i o n dS - T -I

[dE - ~dn]

one

in

is c o m p l e t e l y

sheet is p u n c t u r e d

through

amounts

(2.2.18)

to the requirements

4L33L44 - (L34 + L43) 2 >_ 0,

L44 >- 0.

(2.2.19)

202

2. EQUILIBRIUM IN IDEAL SYSTEMS

The partition

generalization

of

is to be handled

(e)

We

considering

justify

a related

two-component,

(d)

to

the

in Exercise

expressions problem.

(i-

such

of

as

Compositional

one-phase mixture

>_ 0

ease

a

sliding

2.2.4. (2.2.14) stability

by of a

requires

1,2)

(2.2.20)

j-1 as the criterion for uniformity of composition. this may be expressed as

(dnl dn2)

Gii Gi2

Since

[Gij - (82G/SniSnj)T,p]

> 0 .

dnil dn2J

Gai G22

dn i and dn 2 are

itself must be positive elgenvalues

W

requirement

of

the

In matrix form

(2.2.21)

arbitrary

as well, matrix

is both necessary

the

as will

are

symmetric be

positive

G matrix

the case or

if all

null;

this

and sufficient.

We are thus led to examine the associated characteristic equation with eigenvalues

Gii - W Gi2

G2i G22 - W

(GII +

(2.2.22)

-0,

which may be expanded

W 2 -- W

W,

G22 ) +

into the form (Gi2 - G2i)

(GIIG22

-

(2.2.23)

G212) ** 0.

We now attend to the third term above on the right, with

the Gibbs-Duhem

(1.22.26)

xid~ilT,p

relation

at constant

T and P:

beginning Equation

reads

+ x2d~21T,p

-- 0,

(2.2.24a)

ACHIEVEMENT OF EQUILIBRIUM

203

or

xl [(8#l/Sxl)z,pdxl

+ (a#l/Sxz)z,pdx2]

+ (a#jaxz)T.pdx2] But

since

rewrltten

a/Ox I - -

+ x z [(8#2/axl)T,pdxl

- O.

8/ax 2 and

dx I - -

(2.2.24b)

dx2,

the

above

may

be

as

2[xl(81~l/Sx2)t,p

+ x2(8t~zlSxz)t,p]

= O,

(2.2.25)

or a s

nlG12 + n2G22

(2.2.26)

--O,

in the n o t a t i o n

of Eq.

By s i m i l a r

(2.2.21).

methodologies

we find

(2.2.27)

niG11 + n2G12 - O.

From

(2.2.26)

and

(2.2.27)

we o b t a i n

- nIG11 - n2G12

(2.2.28a)

-

(2.2.28b)

nlG12 -

n2G22.

On d i v i d i n g

(2.2.28a)

by

(2.2.28b)

GllG22 -

G22,

or

GllG22 -

G22 -

O.

We

see

that

the third

term

in Eq.

thus

we then find

(2.2.29)

(2.2.23)

vanishes.

are then left w i t h

W 2 -- W ( G l l

+

G22 ) -

for w h i c h

the two roots

(2.2.30)

0,

are

We

204

2. EQUILIBRIUM IN IDEAL SYSTENS

W - 0

or

(2.2.31)

W - G11 + G22.

We now insist on having W ~_ O; then

(2.2.32)

011 + Oz2 >_ O. However, sign.

to satisfy

(2.2.29),

G11 and G22 must have

the

same

Accordingly,

O11 >_ O,

022 >_ O.

Equations (2.2.14).

The

derivation which Exercises

2.2.1

(2.2.33)

(2.2.29) and (2.2.33) form an analogue of Eqs. reader

is

invited

leads directly and 2.2.2

to

to Eq.

for a better

undertake (2.2.14),

a

similar

and to work

understanding

of

the

topics under discussion.

EXERC I S ES 2.2. I Examine the following derivations yielding conditions of stable equilibrium" (a) Suppose half the volume of a system of total entropy S, volume V, and energy E is changed so that one half has an entropy (1/2) (S + 6S) and the other half has an entropy (I/2)(S - 6S). Prove that in second order approximation the energy of the total system has increased by (1/2) (62E/6S2)v(6S)2. (b) Since for a stable system of total fixed entropy S and total volume V the energy should be a minimum, show that the preceding quantity is positive and that the condition of stability may be rewritten as (6S/6T) v > 0. (c) Interpret the result. 2.2.2 Repeat 2.2.1 except that the Helmholtz free energy of the system is to be fixed and the temperature is to be kept constant, whereas the volume of half the system is to be changed to (1/2)(V + 6V) and that of the other half to (1/2)(V - 6V). (a) Prove that (a2F/aV2) T > O, and (b) that (av/aP) T < 0 are conditions of stable equilibrium. What inequality must be set on the isothermal expansion coefficient ~? 2.2.3 On the basis of part (d) of the text, prove that, at constant temperature, material flux takes place along the direction of decreasing chemical potential. 2.2.4 Generalize part (d) of the text so as to be able to handle a system which is divided into two compartments by a partition that slides with friction.

TI-IE CLAUSIUS-CLAPEYRONEQUATION

2.3

SYSTEMS

20

OF ONE COMPONENT

AND

SEVERAL

PHASES:

THE

CLAUS IUS-CLAPEYRON EQUATION According

to

the

Gibbs

phase

rule

the number

of

degrees

of

freedom of any system is decreased by one for every additional phase that is added.

Thus,

for a one component system, f -

2,

I, or O, depending on whether the system consists of one, two, or three phases.

For example,

for the case of liquid water,

the two degrees of freedom are temperature and pressure, of which may be varied over wide state of aggregation of water. are

required

to

coexist,

independently adjustable;

limits without

However,

then

T

both

altering

the

when water and steam

and

P

are

no

longer

the pressure of the closed system is

now determined by the temperature or vice versa - one now has only one degree of freedom. pressure

on

equilibrium steam

would

Thus,

the

water-steam

value

(by means

continue

to

condense,

to the temperature

present.

The

withdrawn

of

to maintain

were

raised

of a piston,

appropriate

heat

if at fixed T < 647.2 K the

system

at

T, until

condensation

above

for example), the

vapor

the then

pressure

only liquid water must

be

is

continually

the system at the fixed temperature

T.

After condensation of all steam the pressure will rise as the piston

is

pressure

pushed

against

is slightly

water would continue

the

reduced

water.

Conversely,

(by withdrawal

to evaporate,

of

the

if

the

piston),

as long as the temperature

T is maintained, by supplying the heat of vaporization from the reservoir.

Ultimately,

only steam remains

in the system.

If it were desired to have ice, water, and steam coexist, then there

is no degree of freedom left:

maintained

at the point T - 273.16 K and P - 4.58 torr.

this reason,

For

the triple point of water serves as a convenient

thermometric reference standard These

The system must be

matters

are

well

(see Section 1.2). summarized

pictorially

in

the

phase diagram for water, Fig. 2.3.1 (not drawn to scale), which is a representative plot of pressure versus three

solid

curves

which

T and P may be

separate altered

out

three

temperature.

large

arbitrarily

regions

while

The

within

maintaining

206

2. EQUILIBRIUMIN IDEAL SYSTEMS

X 217.7 a t m H20 Solid,/

1 otto

Gas

4.58 m m (nonlinear)

I

I'~ 273.16 373.15

273.15

64Z2

T ( K)---~ (nonlinear)

FIGURE 2.3.1 The phase diagram of water

(schematic).

water

in its solid,

forms.

hand,

if

two phases

independent;

the

loci

presence

of

are

to

two

coexist,

phases.

T and

dependence

of curves

experimental

of

or gaseous

functional

indicated by means the

liquid,

For

values

example,

P are

T(P)

in the phase

(T,P)

On the other or

P(T)

diagram,

compatible if

it

no

is

longer is

now

which are with

the

desired

to

achieve coexistence of solid and gas, it is necessary to adjust T and P so that they fall somewhere on the curve OT.

Solid and

liquid can coexist only if the (T,P) coordinates lie along TX; liquid and gas are encountered simultaneously only when lles

along

earlier:

TC.

T

represents

C represents

the

the critical

triple point

point T-

(T,P)

discussed

647.2 K, P -

217.7 atm, above which the gas cannot be distinguished from the liquid. An alternative

presentation

of information

is achieved

through Fig. 2.3.2, which shows plots of P versus V (the molar volume) for various temperatures near the critical point. shaded

region

indicates

an

excluded

domain,

pressure region in which two phases coexist: 473.2 K,

gas

i.e,

The

a volume-

for instance,

at

at molar volume V a coexists with liquid at molar

THE CLAUSIUS-CLAPI::YRONEQUATION

2g 7

H20

|

217.7(Pr 6 7 5 T(K) 84.8

647.2

15.3

,.,,,,

Vc V

FIGURE

~

2.3.2 Phase behavior of water near its critical point.

volume

Vb;

the

temperature:

region As

the

Vb
x I.

is richer in

If PI > P2 then xl/(l - xl) > xl/

These matters are also illustrated in Fig.

2.6.1.

EXERCISES

2.6.1 For benzene and toluene, the vapor pressures of the pure components are 118.2 and 36.7 Hg, respectively. Prepare a graph llke that shown in Fig. 2.6.1 and indicate the approximate compositions at which there is a maximum difference in the composition of the liquid and vapor phase. 2.6.2 For the equimolar liquid mixture of benzene and toluene what is the ratio of masses of benzene to toluene in the vapor phase? 2.6.3 The vapor pressures of pure liquids A and B at 300 K are 200 and 500 mm Hg, respectively. Calculate the mole fractions in the vapor and the liquid phases of a solution of A and B when the equilibrium total vapor pressure of the binary liquid solution is 350 mm Hg at the same temperature. Assume that the liquid and vapor phases are ideal.

2.7

TEMPERATURE DEPENDENCE OF COMPOSITION OF SOLUTIONS

We have just examined the effects of varying the composition of solutions varying

at constant T and P; we now consider

the

temperature.

This

situation

the effect of

is more

complex

in

that the vapor pressure of the two species and the composition of the two phases, as well as the total pressure,

are altered.

This comes

for example,

about because

a rise

in temperature,

will favor vaporization of the more volatile component,

thereby

enriching the gas phase and depleting the liquid phase of this component. though

Thus,

the

all x i and all x i (or Pi) are altered,

overall

chemical

reactions

would be

too

impose the

system occur.

complex

condition

to be

is An of

closed, analysis

and

even

of all

immediate

use.

though

these We

even no

factors

therefore

that the total pressure P be maintained

TEMPERATUREDEPENDENCE

constant.

23 I

Experimentally

this

may

be

achieved

by

having

present in the gas phase a third, nonreactive gas that does not appreciably neutral

dissolve

gas

increments

is

in the solution.

either

(through

a

metered

in

or

semipermeable

maintain the overall P values.

As T is altered, else

membrane)

Alternatively,

the

withdrawn

in

as

to

needed

the solution is

encased in a pressure vessel with a moveable piston,

which

is

readjusted so as to maintain a constant overall pressure. We

invoke

the equilibrium

#i(g) - #i(~) "

Then ( ~

- ~P

constraint

for each

species:

of Section 2.4)

(2.7.1)

#~/RT + 2n P i - #~/RT + 2n xl, so that for the contemplated changes

[ (a/aT)(/~/RT) ]p + (a2n Pt/aT)p

[ (a/aT)(#~/RT) ]p + (a2n x~/aT)p. (2.7.2)

I

Now let xl designate the mole fraction of i in the vapor phase; then Pi - xIP. then

It follows that (8~n Pi/gT)p-

(8~n xl/OT)p.

We

find

a

-

xi)

a

_ _ - * -

RT

p

__

-

RT 2

P

RT 2

"

(2.7.3) Here

we

second

first

utilized

criterion

for

Eq.

(2.5.3)

ideal

and

gaseous

then

and

introduced

liquid

the

solutions,

whereby the molar enthalples for the pure components, H i and H~, are equal, respectively, same constituent

to the partial molal enthalpies of the

in solution

(Hi) and in the vapor phase

(Hi).

Notice that the preceding derivation can also be applied to

any

two-phase

liquids, solutions.

a

mixture

llquld-solld

such

as

system,

two or

sets two

of

types

immiscible of

solid

In these cases the dashed and undashed quantities

refer to the two respective phases.

For the special case of a

solution in equilibrium with a pure solid the dashed quantities

232

2. EQUILIBRIUM IN IDEAL SYSTEMS

x i remain

constant and

Eq. (2.7.3)

then

reduces to the form

(8~n xl/ST)p,xj - A H ~ / R T z where AH~ is the enthalpy of fusion for component

2.8

i which freezes out as a pure solid.

LOWERING OF THE FREEZING POINT AND ELEVATION OF THE BOILING POINT OF A SOLUTION

(a) One of the interesting problems in solution thermodynamics is the change

in freezing and boiling points

which a solute has been added. shown

that

liquid

A

the

addition

causes

of

to

Extensive experimentation has

a material

a lowering

of a solvent

of

the

B

that

freezing

dissolves

point

of A;

in for

example,

A and B may represent water and sugar,

respectively.

We

for

how

look

an

analytic

relation

that

shows

much

the

freezing point is depressed when a given quantity of solute is dissolved in the solvent. We consider a two-phase A (e.g.,

system consisting of pure solid

ice) in equilibrium with the solution of B dissolved

in liquid A (e.g., sugar in water).

This requires the equality

of the chemical potential ~A in both phases: ,A'

~,

(2.8.1)

~A(T,P) - ~z(T,P) + RT ~n xz, where #A and #z are the chemical solid and in the liquid state, One

should

condition

note

(2.8.1).

the At

potentials

of pure A in the

respectively.

implication

constant

P,

of a

the

equilibrium

given

temperature

corresponds to particular composition, xz, of the solution, vice versa. investigate

Thus, x z becomes this dependence

a function of T; x z - xz(T).

further it is ~ propos

and To

to subject

(2.8.1) to partial differentiation with respect to T" {a[(~

-

- ~,)/RT]/aT}p-

(~

- ~)/RT:'

,, E~/RT:' -

(a.~n x z / a T ) p .

(2.8.2) In

this

expression

alternatively,

we

have

first

invoked

Eq.

(1.23.19)

or,

(2.7.3) in different notation; H z and H A are the

233

CHANGES IN FREEZING AND BOILING POINTS

molar

enthalples

quantities

of pure

enter

respectively.

here

Next,

fusion

of pure

change

accompanying

AB.

Lastly,

A;

we

the right.

liquid A

because we have

note

that

again

and pure

solid A.

correspond

These

to #i and ~a,

defined L~ as the molar heat this

the transfer

invoked Eq.

Note

they

is not

the molar

of

enthalpy

of pure A into the solution

(2.8.1)

to obtain the quantity

that x I depends

implicitly

on T;

on see

Exercise 2.8.1. Equation i

(2.8.2) may be rewrltten as

d~n x I -

- (~.~/RTa)dT,

(P constant)

(2.8.3)

f

where

the limits

situations" is

the

on the integral

correspond

to two different

T~ is the freezing point of pure A (x I - I) and T I

freezing

point

fraction of A is x I.

of

the

solution

for

whfch

the

mole

In Exercise 2.8.1, we pose the problem as

to why a change in T should result in a change in composition. We introduce a temperature dependence of L~ in first order through use of the Kirchhoff relation, - (Cp) I - (Cp)A

Section 1.18,

(OL~/aT)p

ACp, in which ACp represents the difference in

molar heat capacity of pure liquid A and pure solid A.

Under

the assumption that the variation of ACp with T may be ignored, we find L~-

L~ + ACp(T - Tf),

(2.8.4)

where L~ is the value of Lf at T - Tf. Thus, ~:Id2n

xl--

.[TT~[E~/RT 2 + (ACe/RT2)(T- T~)]dT (P constant). (2.8.5)

Define the lowering of the freezing point by

8 f - T~- TI,

(2.8.6a)

so that

T1 -

T~[1

-

(e~/T~)].

(2.8.6b)

234

2. EQUILIBRIUM IN IDEAL SYSTEMS

Equation

(2.8.5)

may then be i n t e g r a t e d

to y i e l d

- ~n xll P - [ (L~/RTf) - (ACp/R) ] (Tf - TI)/T I - (ACp/R)2n (TI/Tf) ]p -

[(L~/RTf) - (ACp/R)] [Of/Tf(l - Of/Tf)]

In

general

(O~/Tz)llP.

(BOp/R) 2n [1 -

-

it

is an e x c e l l e n t

small powers of 0f/Tf"

We obtain

approximation

(i-

x 2) --x 2.

Equation

to e x p a n d

(2.8.7)

...].

Further,

then reduces

- 2n x I -

to

(2.8.8)

x 2 = (~.~/RTf)(ez/Tz) + [ (E~/RTf) - (n~p/2R)] (ez/Tz) 2.

The r e p l a c e m e n t o f Eq.

2n x I by x 2 does limit the a p p l i c a b i l i t y

(2.8.8) to dilute solutions,

essential

in

(I - Of/Tf) -I = I + Of/Tf + ...

and 2n (i - Of/Tf) = - [Of + O~/2T~ + -2n

(2.8.7)

of

but this l i m i t a t i o n is c l e a r l y

if the theory of ideal solutions

is to be a p p l i c a b l e

to real cases. Equation

(2.8.8)

is a q u a d r a t i c

be solved to find 8f in its d e p e n d e n c e of all of the above a p p r o x i m a t i o n s to n e g l e c t inverted

Of-

the term in (Sf/Tf) 2.

equation on x 2.

in 8f w h i c h may However,

in v i e w

it is g e n e r a l l y a p p r o p r i a t e Equation

(2.8.8)

may

then be

to read

(2.8.9)

(RT~/E~)x 2.

Ordinarily, expressed

in terms

the

lowering

of

of the m o l a l i t y

the

freezing

point

m 2 of the solute.

is

We note

that m 2 - I000 n2/nIM1, where M I is the gram m o l e c u l a r w e i g h t of the solvent" MjI000

further,

for dilute

x 2 - n2/(n I + n2) = n2/n I.

solutions.

Hence,

Thus,

(2.8.10)

Of = (RT2fMjI000 ~.~)m2 - Kfm 2.

This respects.

relationship First,

in

the

is

x2/m 2 =

rather

remarkable

approximation

scheme

in used

several here

the

l o w e r i n g of the freezing point is a linear f u n c t i o n of x 2 or m 2. Second,

the p r o p o r t i o n a l i t y

factor

is d e p e n d e n t

solely on the

CHANGES IN FREEZINGAND BOILING POINTS

properties way

23

of the pure solvent

depends

on

the

(i.e.,

properties

of

on M1, Tf, L~) and in no

the

solute.

For

a

fixed

solvent Kf is thus p r e d e t e r m i n e d and is m a x i m i z e d for solvents with

high

heats but

freezing

of fusion.

there

associate

are

points, All

high

molecular

of this accords

exceptions,

arising

weights,

with most

from

cases

and

low

experiments, where

solutes

or dissociate.

(b)

The

foregoing

procedure

requires

small

changes

to

become applicable to the elevation of the boiling point for an ideal

solution

solvent. [~y - ~ ]

containing

nonvolatile

~I(T,P) + RT 2n xi(2) - ~ ( T , P ) where

2

and

Accordingly

g

refer

[note Eq.

RT

---

solutes

*

to

+ RT ~n x1(g),

the

liquid

and

-H1 m~

gaseous

0~n

phases.

[x1(g)/xl(2)]

(2 8 12)

m

,

RT 2

the

(2.8.11)

(1.23.19)],

.

P

solvent

vapor

alone

makes

up

the gas

phase.

that event xl(g) - i, and we can then set x1(~) ~ x I. (2.8.12) now reduces

(a2n xl/aT) P

-

~

8T

P

Suppose

and a volatile

The equilibrium constraint for the solvent now reads

In

Equation

to

(H~ - H -*i)/ RT2 = - ( ~ / R T 2 ),

(2.8.13)

where L v is the molar heat of v a p o r i z a t i o n of phase A from the pure

liquid

atmosphere,

into

the

gas

phase.

T - T b represents

We next

(a) rewrite

If P is m a i n t a i n e d

(C~-

introduce C~) ( T -

Kirchhoff's Tb),

one

the normal boiling point.

(2.8.13)

as (2.8.14)

- d2n Xll P - ( ~ / R T 2)dTIP , (b)

at

law

in the a p p r o x i m a t i o n

L v - L$ +

(c) introduce the elevation of the b o i l i n g

236

2. EQUILIBRIUMIN IDEAL SYSTEMS

point by the definition 8~ ~ T I - T b so that T I - Tb(l + 8~/Tb) , and

(d) expand

8b/T b .

(I + 8~/Tb) -I and ~n

(I + 8~/T b) in powers

of

This yie Ids

- ~n x I = x2 - ( ~ / R T b)(8~/T b) - [ ( ~ / R T b) - (ACp/2R)](Sb/Tb) 2.

(2.8.15) On

neglect

of

the

last

term

on

the

right,

the

resulting

equation may be inverted to read 8 b - (RT~/~) x2

(2.8.16)

and 8~ - ( R T ~ I / 1 0 0 0 ~ ) m 2 - Kbm2.

(2.8.17)

The various remarks made in conjunction with Eq. (2.8.10) also apply here with appropriate modifications.

EXERC I S ES 2.8.1 Explore the physical meaning of the relation Lf/RT 2 - (8~n xl/aT) P. Why does the composition of the solution change as the freezing point of the solution is altered? 2.8.2 (a) Determine the mole fraction of benzene in other solvents at I aim and 0~ with which it forms ideal solutions, given the fact that its normal melting point is 5.51~ and that A C p 0, H~ -H~ - 2366 cal/mol. (b) Determine the mole fraction under a total pressure of I00 aim, given the fact that V(soln) - V ( s ) - i0 cm3/mol of benzene. 2.8.3 A solution of 20.0 g of a certain solute (molecular w e i g h t 168.1) in i000 g of benzene (molecular weight78.1) has a boiling point of 80.40~ The normal boiling point of benzene is 80.I0~ (a) Predict the normal boiling point of a solution of I0.0 g of the same solute in i000 g of benzene. (b) Find the experimental value of K b for benzene. (c) Calculate the value of Kb, given that the heat of vaporization for benzene is 7.370 kcal tool-I at its boiling point. 2.8.4 An experimentalist reported a freezing point change of 1.02~ for a solution containing i0.01 at. % of sucrose in water. For pure H20 , H ~ - 1.4363 kcal/mol at 273.15

CHEMICALEQUlUBRlUM

23 7

K. Determine the molecular weight of sucrose from these data, and discuss the reason for any discrepancies compared to the molecular weight as calculated from the formula for sucrose. 2.8.5 Compare the approximation (2.8.10) with (2.8.7) by numerical computation of the value of O~ for a I molal solution of sugar in water. The heat capacities of liquid water and ice are 18 and 9 kcal/mol-deg, respectively. 2.8.6 The heat of vaporization and the heat of fusion of water are 540 and 80 cal/g respectively. (a) For a solution of 1.2 g of urea in i00 g of water, estimate (i) the boiling point elevation, (ii) the freezing point depression, (iii) the vapor pressure lowering at IO0~ Assume ideal-solution and ideal-gas behavior and assume urea to be nonvolatile. (b) Discuss the foregoing properties of a solution of 1.2 g of a nonvolatile solute of molecular weight 108 in I00 g of water.

2.9

CHEMICAL EQUILIBRIUM:

GENERAL PRINCIPLES AND APPLICATION

TO IDEAL GASES (a) One of the most important applications of thermodynamics to chemistry occurs some

general

chemical which,

in the treatment of equilibrium.

principles

reaction,

below.

Consider

vlrR I + v2rR2 +

We develop

the prototype

...= vlpP I + vgpP 2 +

for the reaction as. written,

R I, R 2,

of

...,

a in

... are different

chemical reagents, PI, P2, --- are different chemical products, and the Vlr , vlp are stoichiometry coefficients for reagents and products,

respectively.

It

is convenient

to abbreviate

above reaction by writing ~ viA i - 0 in which, the

v i are

negative

or

positive

according

species A i designate reagents or products,

the

by convention,

as

the

chemical

for the reaction as

written. We

allow

equilibrium

the reaction

to proceed

until

or to a steady state condition

it has

come

to

and then maintain

constant T and P; this latter restriction is often overlooked. The fundamental expression for the Gibbs free energy reads G = Z(1)ni~i, where

the mole numbers

are initially arbitrary. virtual displacement is given by

n i and chemical potentials

~i

The change in G which results from a

(see Section 1.16) for the above reaction

2.,~

2. EQUILIBRIUMIN IDEAL SYSTEMS

6G - ~ /~,6nl + ~ n,6,1 - ~ ,,dnl The term Z(1)n• Gibbs-Duhem An

(2.9.1)

vanishes at constant T and P by virtue of the

relation. important

displacements

constraint

-

in

carrying

O.

This

nfinitesimal,

virtual

to accord with the chemical reaction

can

virtual

be

guaranteed

by

(which also implies

introducing

6A, of the chemical

that the change

in mole number of every species

Then Eq.

(2.9.1)

an

"reversible")

of advancement, 6n i - vi6A.

out

is that the individual mole numbers n• must change

in proper synchronization wiAi

(T,P constant).

unit

reaction Z(1)viAi- 0, such i is given by

reads

6G - ~ ~i6nl - (~ wi~i)6A - (AGd)6A

(T,P constant),

(2.9.2a)

in which AGd "

(6r

~ v~#~[T.p.

(2.9.2b)

One must be very careful about the interpretation quantity

AGd:

differential reaction.

Gibbs This

advancement nearly

as indicated

preserves

calculated,

energy

accompanies

of the

participating

free

by the defining

reaction

the

in

number

the

change

an

per

of moles

T and

n i of

The

as shown in (2.9.2b),

unit

in

reversible

P, which

every

quantity

it is a

change

infinitesimal

at constant

reaction.

relation,

of the

very

species

AG d

may

i be

from the chemical potentials

#i prevailing in the reacting mixture during the infinitesimal, reversible reaction operating

advancement 2H z +

02 -

a fuel

cell

of

2HzO

the may

reaction. be

containing

carried nil2 moles

For

example,

the

out

reversibly

by

of H 2 gas

and no2

moles of 02 gas in appropriate compartments over the electrodes, and containing nil2o moles of water as the medium into which the electrodes is operated

are dipped reversibly

(see Section 4.6). by maintaining

As long as the cell

an appropriate

counter

emf, so that at the conclusion n~2 - 26A moles of H 2 gas and no2 -

6A moles of 02 gas remain in the compartments,

and nil2o + 26A

CHEMICALEQU~UBR~UM

239

moles of water are encountered, AGd(T,P)

directly.

a measurement of the emf yields

However,

if

the

reaction

is

allowed

to

proceed to the extent that vi6A becomes comparable to any of the above

n i's,

the

mole

numbers

are

no

longer

constant;

the

resulting Gibbs free energy change is then no longer identical with the quantity AG d in Eq. (2.9.2). detonation

of

calorimeter"

two moles

An extreme example is the

of H 2 and

one

mole

of 0 2 in a bomb

here the mole numbers are altered to the maximum

possible degree, T and P are no longer maintained constant, and the reaction is not carried through reversibly. Any measurement of the total free energy change AG for such a process will not even be

remotely

examples

should alert

when

different

reaction;

we

related

the reader

authors shall

to AG d as

refer

defined

to problems

to

a

consistently

free

use

earlier.

which may arise

energy

the

These

change

subscript

d

reminder of the differential nature of the quantity.

in

a

as

a

Equation

(2.9.2) shows clearly that AG d involves the chemical potential ~i for every

species participating

in the reaction under

the

prevailing steady c.ondltions which remain essentially unaltered in the virtual displacements.

Alternatively,

one may view AG d

as the change in Gibbs free energy when the reaction is carried out

such

that

6A equals

one mole

copy of the system under study. ~i can be determined one

need

not

in an essentially

Obviously,

is satisfactory

restrict

oneself

to

infinite

any method by which

for use in Eq. measurements

(2.9.2b);

carried

out

during the actual reaction. Equation

(2.9.2)

external constraints mole

number

values,

applies whether

the system

in a steady state condition in which the

n i remains

invariant

at

arbitrarily

or whether equilibrium prevails,

appropriate equilibrium values.

However,

with

be

taken

into

account;

we

thereby

the n i assuming

- 0 must

(6G/6A)T,e

obtain

condition characterizing chemical equilibrium,

(~ vi/~i),q- O.

prescribed

according to Section

1.16, in the latter instance the constraint also

is held by

an

important

namely,

(2.9.3)

240

2. EQUILIBRIUM IN IDEAL SYSTEMS

Note that if Z(i)l,,,i/,l i < 0 (> 0) the reaction as written will tend to occur spontaneously

(in the opposite direction).

(b) For an ideal homogeneous + RT s

Pi [see Eq.(2.4.15)],

gaseous system, # i -

#~P(T)

so that the equilibrium condition

(2.9.3) reads 0

- ~

,I~P(T) + RT (~i vl ~n Pi).q-

(2.9.4)

It is useful to rewrite Eq. (2.9.4) as follows" - ~ l,'i/i~tl(T)/RT Here

we

parametric

' p_.t. " ,In K, - ( ~ vl ,In Pi),q - ,In 'I' i F-] ~l,q(2.9.5a) have

function

of

T

that

alone,

characteristics

the

while

the

side

is a

right-hand

side

on

mixture.

It is therefore appropriate to introduce, a new quantity

left-hand

side

variables.

is

and

left-hand

depends (2.9.5a),

the

recognized

Kp which

actually

composition

shows

of

the

gas

as shown in

explicitly

independent

Kp is termed an equilibrium

of

that

the

composition

constant-

a highly

undesirable appellation because this quantity obviously varies with T. However,

the term "equilibrium constant"

is so firmly

entrenched that we shall continue to use it here.

A convenient

reformulation of (2.9.5a) Kp-

~

is found by taking antilogarithms:

(PVl).q. It is possible

Eq. (2.9.3). (2.4.17)] ~ i -

(2.9.5b) to provide

alternative

formulations

For an ideal gas one can set [see Eqs. ~c(T) + RT s

c i or ~ i -

for

(2.4.16),

~X(T, P) + RT s

xl, to

obtain 2n K c - - ~ ~C(T)/RT - (~ v i 2n cl).q

(2.9.6)

2n K x - - ~ v i ~ x(T,P)/RT - (~i v l 2n x i),q.

(2.9.7)

We

have thereby introduced two new

equilibrium

'constants,'

CHEMICAL EOUlUBRlUM

again

7.4 I

independent

of concentrations,

also being parametrically

dependent

the

one

(2.9.7)

on the total pressure

the system.

The problem of dimensionality

This matter

is briefly

discussed

in Eq.

later

of

arises once again:

and

fully

treated

in

Section 3.7(g). (c) The In the

evaluation

general

case,

of AG d is of considerable

when

equilibrium

mole numbers n i are arbitrary. advanced

infinitesimally

does

interest.

not prevail,

the

If the reaction Z(1)v• i - 0

and reversibly

at constant

is

T and P,

then under such condltlons -

§ RT

vl

Pi

298a

--

RT (~ v, ~n Pi).q + RT ~ v i ~n P,

(2.9.8b)

--

RT 2n Kp + RT ~ v• 2n Pi, I

(2.9.8c)

where

we have

inserted

Eq.

(2.9.5a)

on the right-hand

side.

Here and later on it is of the utmost importance to distinguish between Y~(1)vl 2n Pi, which merely from (E(1)vi 2n Pi).q, which at

this

evident

point

can

lead

has the same form as ~n Kp,

is identical with 2n Kp. Confusion to

disastrous

consequences.

that Kp and K c do depend on the units

pressures

and gas concentrations.

AGd/RT

independent

is

examlnation of Eq.

of

such

However,

choices,

as

It

chosen

is

for gas

the corresponding is

clear

from

an

(2.9.8b).

It is convenient

at this stage

to introduce

free e n e r g y change AG~ for the reaction.

This

a standard

is the value

of

AG d when the reaction is advanced infinitesimally and reversibly under

conditions

where

all

gaseous

constituents

are

at unit

partial pressure (usually one atmosphere) at the temperature of interest.

It may

not

be

possible

actually

reaction of interest under such conditions, not

detract

from

the

correctness

of

the

to

execute

the

but this fact does assertion.

The

definition is also consistent with the fact that we had earlier

4~

2. s

set AG d - Y.(t)vi~i; hence AG~ - Y.(1)ui~p.

IN IDs

Equation (2.9.5a)

SYSTs

thus

reduces to AG~ - - RT ~n Kp More generally,

Kp-

or Eq.

(2.9.8c)

exp(- AG~/RT).

(2.9.9)

reads (2.9.10)

AG d - AG~ + RT ~ v i ~n P i.

Note that the equilibrium constant Kp is directly related by Eq. (2.9.9) to the free energy change of a chemical reaction under standard conditions. under

Further,

arbitrary,

the AG d value

reversible,

for the reaction

isothermal,

and

isobaric

conditions may be written as a sum of AG~ and of a 'correction term' RT Z(1)vl 2n Pi. If

equilibrium

(2.9.10) AG~--

reduces

prevails,

AG d -

O;

in

this

(2.9.11a)

which is consistent with both remarks

made

quantity AG~ in Eqs. additional

(2.9.9) and (2.9.5).

earlier

(2.9.9),

requirement

concerning

(2.9.10),

AG d apply

and (2.9.11),

the

with the

advancement

at of

The definition (2.9.11b)

-

with ~

m ~p,

appropriate for

to

that all species must be maintained

their standard states during the infinitesimal the reaction.

Eq.

to

RT (~ v i 2n Pi).q,

The

event

the

gaseous

leaves no ambiguity" tables,

desired species

One measures,

or looks up in

the molar Gibbs free energy #~P (specified

standard

state,

usually

P-

i involved in the reaction.

I arm)

of

all

These quantities

are then to be combined as required by (2.9.11b). The

reader

the analogues

should have

of Eq.

(2.9.8c),

no difficulties namely

in constructing

2.4 3

CHEMICAL EQUlUBRlUM

AG d - - RT ~n K c + RT ~ v i ~n c i - AG~" + RT ~ v i ~n c i (2.9.12) AG a - - RT ~n K Z + RT ~ w i ~n x i It should be

clear

that

in these

again specified by (2.9.11b), or ~x.

Once more

confusion

between

corresponding

v i ~n xi.(2.9.13)

+ RT

equations

but now ~

AG~" and AG~" are

represents

either ~ c

the reader must be at great pains

to avoid

Z(t)v t

and

2n

ci

or

Ci)eq- 2n

(Z(t)u i 2n

Z(t ) u i

2n

K c or (Z(t)u i 2n

xt

Xi)eq-

the

2n K x.

The confusion easily arises because of the close similarity

of

the expressions. (d)

We

are

equilibrium pressure.

often

interested

~constant'

with

consider the definitions

variation

temperature

and

(2.9.5),

(2.9.6),

vl d - ~

R

[ - # ~ p]

aT

(2.9.14)

T

with

-

The cautionary

RT 2

applies

to

discussion

This

represents

the

total

(1.23.19)

and

Thus,

AH~ (2.9.14)

" RT 2"

is one formulation

(1886).

AH~"

~ wiH~

of

as well as (2.9.7).

(82n Kp/SP) T - (8~n Kc/SP) T - 0.

d~n Kp

Equation

the

We base our further discussion on Eq.

Obviously,

aT

in

of van't Hoff's

concerning the

equation

AG d and AG~ also

differential

enthalpy

evolution accompanying unit advancement of the reaction Z(1)v• i - 0 when the latter while

maintaining

standard

is changed reversibly

constant

states.

No

- Z(1)viH ~.

differentiation

of ~

partial molal T

under

methods

enthalpy

standard described

T and P, and all

ambiguity

definition AH~

and infinitesimally

results

species from

One should recall

in Eq.

(2.9.14)"

of species

conditions.

H~ thus represents be

the the

at temperature

determined

by

the

in Section 1.19.

One can reformulate the preceding results" and Pi - xIP,

of

that H~ arose by

i maintained

H~ may

use

in their

Since Pi - c•

244

2

2n K p -

2n Ko + ~ v i 2n R T -

EQUILIBRIUM IN IDEAL SYSTEMS

(2.9.15)

2n Kx + ~ v i 2n P.

Therefore,

d~n K o

d~n Kp

--

dT

d~n RT

AH~

~ vi

AH~ - RTAv

dT

RT 2

T

RT 2

-- ~ V i

dT

(2.9.16) W i t h AV IT,P - AvRT/P,

this becomes

(2.9.17)

d2n Kc/dT - AE~/RT 2 .

Finally,

(a2n K~/aT)p

and in v i e w of Eq.

(1.23.18),

(a2n Kx/aP) T - - ~ (vl/RT) i .

(2.9.18)

- d2n Kp/dT - AH~/RT 2,

'

- - (

v•

aP

~V~/Rr.

T

(2.9.19)

In p r a c t i c e Eq. (2.9.14) and its analogues are f r e q u e n t l y used

in

reverse.

d(#i/T)/dT

For

ideal

are identical;

gaseous

hence,

AH~-

mixtures

Alld - Z(1)viH•

Alld - RT 2 (d2n Kp/dT),

which

shows

empirically the

and so

(2.9.20)

if

the

variation

or from theoretical

gaseous

integration

2n

that

and

d(p~/T)/dT

reaction

~(1)viA i

of

analysis, -

0.

Kp

with

T

is

known

AH d may be found for

Note

further

that

by

of (2.9.20)

[K~(T2)/Kp(TI) ] -

~2

(AHd/RT2)dT,

(2.9.21)

T1 w h i c h requires

that one specify the d e p e n d e n c e

of All on T. Here

Kirchhoff's

law may be used if no other d e t a i l e d i n f o r m a t i o n

available.

Alternatively,

is

one may use the average value of AH d

CHEMICAL EOUILIBRIUM

- < A H a > , and move this constant outside

245

the integral to obtain

2n [Kp(T z)/Kp(T I) ] - (/R) (I/T I - I/Tz).

(2.9.22)

EXERCISES

2.9.1 (a) Prove that the quantities AG~, AH~, and AV~ refer not only to gases at unit pressure, but also to pure components. (b) Devise a semlpractlcal scheme for carrying out reactions in a fashion compatible with the maintenance of all gaseous components in their standard, pure states. (c) Carefully discuss the difference between AG~ - Z(1)vi~ i and AG m Z(1)nlu i -Z(1)nl, where ' and " refer to states before the start and after completion of the reaction Z(1)viA i - O. (d) O b t a i n relations among AG~, AG~', and AG~". 2.9.2 Consider the possible equilibrium n-C4H10(g ) iso-C4H10(g ) . The standard free energies of formation of normal and isobutane at 298 K are -3. 754 and -4. 296 kcal mol -I, respectively. (a) If we arbitrarily assign a value of zero to G ~ for the normal compound (G~ ) show that this means assigning a value o f - 5 4 2 cal tool-I to the iso compound (G~,o). (b) Assuming, first, that neither gas shows any tendency to convert to the other, find the free energy of a system c o n s i s t i n g of 0.800 tool of the iso and 0.400 tool of the normal in which the gases are unmixed, but each at a pressure of i atm. (c) Repeat (b) for the case in which the gases, both still supposed to be inert, are mixed at a total pressure of I atm. (d) Find the chemical potential of each gas in this mixture. (e) Suppose, now, that the isomers equilibrate readily. What w o u l d be the c o m p o s i t i o n of the equilibrium mixture? Would it depend on the total pressure? (f) What is the chemical potential of each gas in this equilibrium mixture? (g) What is the total free energy of the equilibrium mixture? 2.9.3 If Kp is 0.64 (atm) for N204(g ) - 2NO2(g ) at 318 K find the degree of dissociation under a total pressure of 2.0 arm. 2.9.4 For the reaction PCI3(g ) + Cl2(g ) _ PCls(g ), AG400o _ -855 cal. Determine the fractional conversion of PCI 3 to PCI 5 and the partial pressure of PCI 5 at 400 K, if the original reaction mixture contained I mol of PCI 3 and 3 mol of CI 2 and the total pressure is m a i n t a i n e d at I arm. 2.9.5 (a) Calculate Kp, Kc, and K x for the reaction 3H2(g) + N2(g) - 2NH3(g) at i000 K and under a total pressure of i00 O atm, given that AGd1000 - 61.890 kJ/mol. Starting with a pressure of i00 atm of NH3, what fraction decomposes into H z and

~-~6

2. EQUILIBRIUMIN IDEAL SYSTEMS

N 2 at I000 K under these circumstances? Repeat the above for the same reaction at 25~ given that A G ~ 2 9 8 - - 1 6 . 3 8 kJ. 2.9.6 For the reaction CO(g) + 2H2(g ) - C H a O H ( g ) one finds at 298.15 K (f represents the formation of compounds from their elements) ,,I

,

I,

,

, , ,

,

,

,

,,,,

,

,

,,,

,

,

AG~

s~

kcal/mol

kcal/mol

kcal/mol-deg

-26.4157

-32.8079

47.301

- 4 8 . I0

-38.70

56.8

,,,,

,,

,

,,

CO(g) CH3OH(g) **

H2(g)

31.208

(a) Determine AG~2~8, AH~298, ~ , Ko, and K x at 298.15 K. (b Taking XCH30H -- 0.8, Ptot - 1/2 aim, find XH2 , Xco at 298.15 K. (c) Derive a general expression for Kp in terms of the fraction of CO consumed, starting with n moles of CO and m moles of H 2. 2.9.7 Let n~ represent the number of moles of species i present in a system which is allowed to interact chemically so that the degree of advancement is A. Find n and x i in terms of n~, A, a n d Av - ~v i. 2 . 9 . 8 At 2 ~ . 5 ~ a n d a p r e s s u r e o f 5 8 . 7 ram, t h e d e g r e e o f dissociation o f N204 i s 0 . 4 8 3 . F i n d K= a n d Kp. Calculate the volume occupied by i (original) tool of N204. At what pressure will the degree of dissociation be 0.i? 2.9.9 A mixture containing 49~ of HCI and 51 ~ of 02 by volume was heated at a constant pressure of 723 nun to 480~ At equilibrium 76~ of the HCI had been transformed according to the equation 4HCI(g) + O2(g) - 2 C 1 2 ( g ) +2H20(g). Find the values of K c and Kp (pressures in atmospheres). 2.9.10 A certain ideal gas, A, isomerizes to B, according to the reaction A - B and forms an ideal gas mixture with equilibrium constant Kp. Starting with i mol of pure A at I atm, the gas is allowed to isomerize at constant temperature and pressure until it reaches equilibrium. The isomers are then separated and each is brought to i atm at constant temperature. Find AG d for the process. Show that AG d is not equal to AG~. Explain. HCI(g), C ~ - 6.7319 + 0.4325(I0-3)T + 3.697(I0-7)T 2 02(g), -C~- 6.0954 + 2.2533(I0-3)T- 10.171 (10-7)T 2 H2(g), C~ - 7.219 + 2,374(I0-3)T + 2.67(I0-7)T 2 Cl2(g) , C ~ - 7.5755 + 2.4244(I0-3)T- 9.650(I0-7)T 2 Find Kp at 500~ and AG~500. 2.9.11 (a) The following table gives the standard Gibbs free energy of formation of Cl(g) at several temperatures. T(K) i00 I000 300 AG ~ (kcal/mole CI) 27.531 15.547 -13.487.

CHEMICALEOUlUBRlUM

247

For the reaction (i/2)C12(g) - e l ( g ) , find the e q u i l i b r i u m constant Kp at each of these temperatures. Find the mean AH~ and then find AS~ per unit reaction at each of the temperatures. (b) At 1700~ and i.i0 atm, i g of chlorine occupies 1.96 liters. What are Kp , K x, K c for the reaction (1/2) Cl2(g) - C l ( g ) ? Assume ideal-gas behavior. (c) At 1600 K, the degree of dissociation of chlorine is 0.071 at i atm. What are Kp, Kx, and Kr for the reaction Cl2(g) - 2Cl(g)? Assume ideal-gas behavior. 2.9.12 (a) At 448~ the degree of d i s s o c i a t i o n of HI into H 2 and 12 is found to be 0.2198. Determine Kx, Kc, and Kp. (b) If 0.05 tool of H 2 and 0.01 of 12 are brought in contact at that temperature in a volume of 3 liters, what c o n c e n t r a t i o n of HI is produced? (c) At 357~ K c - 0.01494; if 0.04 mol of HI, 0.03 tool of 12, and 0.02 tool of H 2 are e q u i l i b r a t e d at that temperature in a volume of 3 liters, calculate the c o m p o s i t i o n of the e q u i l i b r i u m mixture in terms of mole fractions, molaritles, and partial pressures. 2.9.13 Let K 1 and K 2 be the e q u i l i b r i u m constants at a temperature T for the reactions 2CO 2 - 2CO + 02 and 2H20 - 2H 2 + 02, respectively. Find K 3 at temperature T for the reaction H 2 + CO 2 - H20 + CO in terms of K I and K 2. 2.9.14 For the reaction 2SO3(g ) - 2SO2(g ) + 02(g ) one finds values of K c - 3.15 x 10 -4 , 3.54 x 10 -3 , and 2.80 x 10 -2 P llter -I at 627, 727, and 832~ respectively. (a) Determine the average enthalpy of reaction under standard conditions in the intervals 627-727~ and 727-832~ (b) Determine K c and Kp at 500, 700, and 1000~ (c) What fraction of SO 3 is d i s s o c i a t e d at these temperatures if the total pressure is 2 a t m in each case? (d) A s s u m i n g air to consist of 79% N 2 and 21% 02 by volume, what percentage of SO 2 is transformed into SO 3 if equal volumes of SO 2 and air are mixed at 727~ at a total pressure of 1 arm? 2.9.15 (a) Derive an expression showing how AS values for a chemical reaction carried out at temperature T I may be related to AS for the same reaction at a different temperature T 2. (b) For the reaction H2(g) + Cl2(g ) - 2HCl(g), the standard entropies at 298 K are 31.208, 53.288, and 44.646 eu/mol for H2(g) , Cl2(g) , and HCI(g), respectively. Also, C ~ - 6.9409 - 1.999 x 10-4T + 4.808 x 10-TT 2 cal/deg-mole (H2) C ~ - 7.5755 + 2.4244 x 1 0 - 3 T - 9.650 x 10-7T 2 cal/deg-mole (C12) ~ 0 _ 6.7319 + 0.4325 x 10-3T + 3.697 x 10-TT 2 cal/deg-mole (HCI) Calculate AS~ at 298 and 1200 K for these reactions. 2.9.16 (a) Find K for H2(g ) + 12(g ) - 2Hl(g); AG~98 = - 4 . 0 1 kcal. (b) Find Kp for 2Hl(g) - H2(g) + 12(g ). (c) Find Kp for (I/2)H2(g) + (1/2) 1 2 ( g ) HI(g). (d) For H2(g) + 12(g) O 2Hl(g), Kp - 870 at 298 K and AHd2s8 - -2.48 kcal. Find Kp at 328 K, assuming AH d is constant in this temperature range.

~-48

2. EQUILIBRIUMIN IDEAL SYSTEMS

2.10

CHEMICAL EQUILIBRIUM

(a)

The

methodology

of

applied

to homogeneous

phases.

Here Eq.

IN HOMOGENEOUS

IDEAL SOLUTIONS

the

preceding

section

ideal

solutions

which

(2.5.1) is applicable,

now

be

condensed

~I(T,P) - ~ ( T , P )

In xl, where ~i is the chemical potential value obtained when x i - x~ - i.

will

form

+ RT

of pure i, i.e.,

the

The equilibrium condition now

reads

0-

(

vi~i).q-

Again,

vi~i(T,P) + RT (

v i ~n xl).q.

(2.1o.i)

it is convenient to define an equilibrium

'constant'

as

follows"

2n K . - -

~ vllm~(T,P)/RT- (~ w,

This

the

has

same

form

as

Eq.

In x,).q. (2.9.5a)

(2.10.2) except

that

present case K Z depends parametrically on T and P.

in

the

Accordingly,

by the method of Section 2.9,

(a2n

K./aT)p -

vIHI(T,P)/RT 2 - AH$/RT 2

(a~n K./aP)T - -

vIVI(T,P)/RT - - AV$/RT,

and, by analogy with Eqs.

AG d - -

(2.10.3) (2.20.4)

(2.9.8),

RT 2n K x + RT ~ v i 2n x i

(2.10.5a)

- AG~* + RT ~ w i 2n x i. (b)

Since

frequently

(2.10.5b)

equilibrium

expressed

molallty

m i or

relation

between

constants

in terms

molarity xl,

ml,

cl,

in

of c o n c e n t r a t i o n it

and

is c i.

necessary From

the

solutions units to

are

such as

examine

definition

the of

molarlty and mole fraction we obtain

cl/x i - (nl/VL) (~nj/nl),

v L - V/lO00,

(2.10.6)

249

HOMOGENEOUS IDEAL SOLUTIONS

where

V

is

quantity ms/p,

the

volume

is related

and the mass

of

the

solution

to the density of the

solution

in

cm 3.

p of the

The

latter

solution

by V -

is given by m s - Y.(j)njMj,

where Mj is the gram atomic mass of constituent

J.

Thus,

i0009(i + n~/nz + ns/nz +...) ci/x i - I0000~ nj/~ njMj -

M I + (n2/nz)M2 + (na/nz)M3 +... (2.10.7)

Consider

now the case of a dilute

sensibly

expect

designate

the

near-ideal solvent.

niMi/n I

I

PRaoult's

i 'as'e~ 'on

.--2>> i implies that AG~ is a large negative quantity, equilibrium

the

specification

of K or Kq causes no difficulty It

(3.7.12),

evaluated

the

AG d is

fundamental

and as

of choice adopted for Kq

in

reference

state functions the

arbitrariness

units

matched

the composition

thermodynamic AHd)

It

to be

the vagaries

exactly

standard

specifying

or

Thus,

ql are

activity coefficients,

(3.7.12).

changes

71 and

(3.7.9c)

the pertinent

Thus,

reaction

has proceeded nearly to completion in the direction as written. Conversely, when

the

for values reaction

direction.

Thus,

is

of K or Kq /aP]t,.,, i -

where

ai(X)/SP]T,.i

[82n

use

[a,en a~m>/aP]t..,, i I

I

was

made

of

-

Vi/RT

differentiation

Section

3.7(a),

whereas

V i - Vi(T,I ).

Eq.

(3.5.15)

as

(3.4.5)

discussed

with

q -

c

in

and

xl

Vl/RT - V~/RT -

[a~n p/OP]T,x~ (3.10.2)

pi(T,P)/OP]T,xl,

also used Eq.

[a~n Fi(m)/aT]p,ml ffi ffi [ 0 2 n

and

P:

[a,~n a~C>/aP]T,ci - [O,~n a[X)/aP]T,xi

[a~n P/OP]T,~t -

+ lira P~l [a2n

to

Here one must be very careful

V i - Vt(T,P), to

v i to

is r e f e r r e d

to

we find at constant

where we have

(a#i/aP)T,xl-

respect

[a~n r'~o>/aP]:r.ci -

-

of

with

According (3.4.2),

and

note that #i(T,P,x•

the standard chemical potential. the

(3.1o.t)

V~/RT,

(3.5.12a)

arrive at the final result; in

-

[a.~n r~*>/aP].t..i

(3.10.1).

Similarly,

we find

[a~n ai(m)/0T]p,mt- [as r[=>/aT]p,~i

at(X)/0T]p,xi

-

-

(H i -

Hi)/RT 2

(3.10.3)

[a2n r~=>/aT]p.= i ffi [a2n a~=>/aT]p,= i ffi [a2n a~X>/aT]p,xi +

[a2n

(pt/p)/@T]p,x i

---

(H i - H i ) / R T 2

+

[0s

(pl/p)/OT]p,x i.

(3.10.4) Note

here

enthalpy from

that

of pure

~i(T,P,xl) ,

solution

Hi,

derived

i at unit refer

at pressure

P.

to

from

pressure, the

pi(T,l,xl), while

partial

is

the

molar

Vi and Hi, d e r i v e d

molar

quantities

in

3 2.4

3. CHARACTERIZATION OF NONIDEAL SYSTEMS

(b) The variation of the different equilibrium constants with

temperature

Since Kx(T), in Eq.

and

pressure

as determined

(3.7.8a),

Eq. (3.7.6a),

may be

determined

as

follows:

in terms of the chemical potentials

is identical to K(T) as specified by the ~

in

these two quantities may be considered as a unit.

One obtains d~n K(T)

d~n Kx(T ) u

dT

- ~ v,H, x~, one reverts back to a single phase,

and PI again varies with x I until

the value

is

liquid

PI - PI is

reached for x I - i. Note merge

with

further the

that as x I ~ i, all curves

one

for

which

B-

O;

this

in Fig. agrees

3.13.3

with

the

experimental fact that Raoult's Law always holds in this range. Similarly,

as

xI ~

0

consistent with Henry's with

each

solution.

one

obtains

Law; For

Raoult's Law are encountered,

here B


0

line

region

as x I ~ 0 varies deviations

from

one finds positive

354

3. CHARACTERIZATION OF NONIDEAL SYSTEMS

deviations positive

with

slopes

departure

Finally,

that

become

steeper,

we

should

note

that

in

terms

R(x I ~n x I + x 2 ~n x2) represent

which,

in the present approximation

an ideal solution. AH-

AG + TAS.

the

B

We may rewrite

w/RT

-

definitions

greater

the

Eq.

(3.13.13)

the entropy

scheme,

the

of mixing,

is the same as for

The term Bxlx 2 is then to be identified with

AG m - RT{X 1 ~n x I + x 2 ~n xz} in w h i c h

the

from ideallty.

and AH

for

mole

Eq.

(3.13.13)

as

(3.13.17)

+ wxlxz, - RTBxlx 2 - wxlx 2.

fraction

we f i n d

On introducing

[AG -

nAG -

(n 1 +

na)nC] (3.13.18)

AG m - RT(n I ~n x I + n 2 ~n x2) + w(nlnz/n ).

It is quite remarkable how many firm deductions are b a s e d on

a

single

formulation the

hypothesis.

(3.13.12)

specification

mixing phase

functions separation

fn equilibrium value

of

critical

B

in

Fig.

predicted.

of

AGm,

does occur

composition

of

Finally,

The

shown

in

by

by

Eq.

incipient the

foregoing

power of thermodynamic

is

leads directly

to

other

(3.13.3)-(3.13.7).

When

of the two phases

(3.13.14). phase

mixture

deviations

Margules All

one may construct which

the

(3.13.13).

the c o m p o s i t i o n

specified for

3.13.3

as

are then found from

is

with

for rl, thermodynamics

required

(3.13.15c).

Beginning

are

a beautiful

critical

separation

and

the

specified

by

Eq.

diagrams from

The

such as shown

Raoult's

illustration

Law

are

of

the

methodology.

EXERCISES 3.13.1 Sketch out derivations by which AG m and other mixing functions may be specified in terms of 71(T,p,cl) and 71 (T, P ,ml). 3.13.2 (a) Derive Eq. (3.13.11) and (3.13.12) in detail, proving that D I - D 2 - 0 and B I - B 2 R B as stated in the text. (b) Expand on the derivation by adding terms in x 3 and x 4 to Eqs. (3.13.9). Obtain expanded relations for ~n Vl and AG m.

MIXING IN NONIDEALSOLUTIONS

3~5

3.13.3 (a) Starting with Eq. (3.13.13) derive ex~resslons for ASm/RT, ~/RT, AVm/RT , AEm/RT , (ACp) m/RT , (ACv)m/RT on the assumption that B depends on T via B - w/RT, w being constant. (b) Make sketches showing the dependence of each of these on x z - x for B - - i , 0, 1.5, 3, at T - 300 K. (c) Comment on the nature of each of the curves. 3.13.4 (a) Determine the Henry' s Law constant in terms of B for a binary solution to which Eq. (3.13.13) applies. (b) Discuss negative departures from Raoult's Law in terms of Eq. (3.13.13). 3.13.5 The normal isotopic abundances for Li are 92.48 mole % for 7Li and 7.52 mole % for SLi. Making reasonable approximations, determine the entropy, enthalpy, and Gibbs free energy changes on mixing the pure isotopes. Discuss your results in terms of the statements made in Section 1.21 in conjunction with the Third Law of Thermodynamics. 3.13.6 For some kinds of high-polymer solutlons, the chemical potentials of solvent and solute are given by the approximate equations

~I - G~(T,P) + RT {2n 41 + (I - ~ )

2n (I - 41)[ + w(l - 41) 2, G z - G~(T,P) + RT {2n 42 - (r - I) 2n (I - 4z)'} + rw(l - 42) 2,

where r stands for the degree of polymerization and 41, for the volume fractions 41 - NI/(NI + rN2) 4 z - rNz/(NI + rNz). Here N I and N 2 are the number of solvent molecules and of solute high-polymer molecules, respectively. It is assumed that each polymer molecule consists of r monomers, and that the volume of a monomer approximately equals the volume of a solvent molecule. (a) Determine ~I and ~z in terms of x I and xz, and x I and x 2 in terms of 41 and 42. (b) Compare the entropy of mixing of this solution with that of an ideal solution. (c) Derive the relation of the enthalpy of mixing to the concentration of this solution. 3.13.7 The following expression has been proposed for the activity coefficient of component 1 in a two component solution" ~n F 1 = AV~422/RT, where V I is the molar volume of pure i, at temperature T and total pressure P, and A is a constant. The volume fraction 42 is given by 42 " n2V2/[ nlVl + n2V2 ] 9 On the basis of elementary theories one finds that A - [ (A~ap)i/V~] I12 - [ (A~ap) 2/V2 ]-* 1/2. (a)

Derive

a relation

for

2n F 2.

(b)

Prove that

the

free

energy

of mixing is given by AG m - RT[n I 2n x I + n 2 2n x2] + A(nlV I + n2Vz)4142. (c) Let the second term on the right of the expression for AG m be the excess free energy of mixing G.. Obtain expressions for

56

3. CHARACTERIZATION OF NONIDEAL SYSTEMS

S., H., V.; the final relations

should include the functions

=~- (~i)-~(a~'~/aT)p; ~ . -

(~'~)-~(a~'~/aP)~.

(d) Simplify the expressions in (c) by n e g l e c t i n g terms in (@I ~2)(=I - =2) and (~i - ~2)(~I - ~2)- Provide arguments to show why these quantities should be small. 3.13.8 It is found empirically that, for many systems, the equilibrium vapor pressure of component 2 in a nonldeal solution is given by P2 - (2x z - x~)P~. (a) Obtain an expression for 72 relative to the solvent standard state for component i and determine V2 at x 2 - 1/2. (b) Obtain an analytic expression for PI in terms of x I. (r Does one obtain positive or negative deviations from Raoult's Law? For what range is dPT/dx 2 > O? For what value of x 2 is a m a x i m u m e n c o u n t e r e d in PT, the total pressure in the system? 3.13.9 Let the ~unsymmetrlcal' Gibbs free energy of a mixture be given by the expression G - nl#~ + n2# ~ + RTIn I 2n (nl/(n I + rn2))l + RTIn 2 ~n (rn2/(~ I + rn2)) 1 + Bnln 2, where B - B(T,P), ~ #~(T,P). (a)'Obtaln the relation for the molar Gibbs free energy of the mixture solely in terms of mole fractions. (b) Obtain S, H, V, E, Cp, and C v in terms of xl, x 2 and parameters. (c) Determine the vapor pressures PI and_P2 in terms of a modified Raoult's Law. (d) Plot AS~/R, A H ~ R T , AGm/RT, P2 as functions of x 2 for r 1/2, 2, 1/5, 5; assume constant T and P. Here AG m refers to the molar free energy of mixing. Comment on the nature of the curves you obtain. 3.13.10 For the acetone-carbon disulfide system the following activities have been reported at 35.17~ (subscript 2 refers to CS2) -

X2

6.24 x 10 -2 0.1330 0.2761 0.4533 0.6161 0.828 0.935 0.9692

a2

0.216 0.405 0.631 0.770 0.835 0.908 0.960 0.970

al

0.963 0.896 0.800 0.720 0.656 0.524 0.318 0.1803

Determine AGm, AHm, TAS m for the system and plot the results. Do you detect any evidence for phase separation? 3.13.11 Let the excess molar and partial molal Gibbs free energy for a two-component mixture relative to an ideal mixture be given by AG I - x2[A - BT + C(x 2 - x I)], AG - xlx 2(A - B T _ + Cx2) , in which A, B, and C are constants. Determine H, S, G2, HI, H2, $I, S 2 for this system. 3.13.12 The following type of equation is frequently used in the empirical representation of data in a binary solution"

PHASESTABILITY

3 ~7

~n V l - AI(T)x~ + BI(T)x~ + CI(T)x~ + DI(T)x s, _where_ A I through D I are independent of composition. Determine AOm, / ~ , and AS m for the binary solution from this relation. 3.13.13 The heats of mixing have been reported for various solutions of carbon tetrachloride and acetonitrile at 45~ as summarized below: XccI 0. 128 0. 317 0. 407 0.419 0. 631 0. 821 AH(s4olen) 99 179 206 235 222 176 cal/mol solution (a) Using these data and the tabulations in Exercise 3.11.14, determine AHm, AGm, TAS m for Xcc14_- 0.I, 0.2, ...,0.9, and plot the results. (b) Assuming that AS m may be approximated by the relation that applies to an ideal solution, use these data to calculate the activity coefficients of CCI 4 at x - 0 . 1 2 8 , 0.317, 0.419, and 0.631, and compare these wlth the values that are interpolated from Exercise 3.11.14.

3.14

(a)

PHASE STABILITY" FROM IDEALITY We

had

earlier

GENERAL CONSEQUENCES OF DEVIATIONS

encountered

specific

examples

deviations

from ideal behavior of binary solutions

phenomenon

of

phase

separation.

We

now

where

led to the

introduce

several

generalizations on the basis of qualitative sketches introduced later. In

ideal

solutions

the

mixing

process

is

rendered

spontaneous through the positive entropy of mixing A S / R - ~ n x I + x 2 ~n x 2

I > 0; here AH- - 0.

{xI

The molar free energy for

ideal solutions reads

G -- RT

x I 2n x I + x 2 2n x 2

__

(3.14. la)

+ (x1# ~ + x2#2)

(3.14.1b)

T A S 0 + G*.

Qualitatively,

Eq. (3.14.1) may be represented as shown in Fig.

3.14.1(b), where energy is plotted schematically versus x 2 m x. The plot shows - TAS0 versus x as the bottom curve; the sloping baseline

(top curve)

x2~ 2 in which ~i and shown in the

middle,

for G* is obtained are held constant. ~ z

- T A S 0 + G',

from the sum Xl~ I

+

The resultant graph, is a

skewed

U shaped

.58

3. CHARACTERIZATION OF NONIDEAL SYSTEMS

A~

o[

o

0

A~ -TAS

0

•

0

1

•

(a)

1

0

x2

(b)

FIGURE 3.14.1 Free energies G of b i n a r y solutions. (b) AH = 0,

curve. (i)

exists

solutions

in general

there

an

expression

AS--x

parameter.

One

in Figs.

I ~n x I + x 2 ~n now obtains

3.14.1,

given term

by of

AH

= wx(l

Eq.

- xr);

(3.13.17)

(rx2) , w h e r e

(a) and

but

two

entropy

TAS

(c).

this r

term Se that

relation =

I.

r is a s u i t a b l e

curve

of

the

type

(ii) The e n t h a l p y

is a f u n c t i o n

when

corrections"

sum may be s i m u l a t e d by the

a skewed-

parts

m i x i n g no longer v a n i s h e s

occur

excess

must be a d d e d to ASo, the r e s u l t i n g

shown

(a) AH < 0,

(c) AH > 0.

For n o n i d e a l

There

1

(c)

of x 2 n x,

reduces On

to

such as the

introducing

corrections

one obtains a new curve as in part

and in part

(c) where w > 0.

The r e s u l t a n t s

of

last these

(a) w h e r e w < 0,

o b t a i n e d on a d d i n g

up the s l o p i n g baselines,

the entropy,

and the e n t h a l p y c o n t r i -

butions

the curve

In part

more

are

i n d i c a t e d by

negative

and more

(c) a n o n m o n o t o n i c

variation

over c e r t a i n ranges the n e g a t i v e shown

skewed

such

than

for part

a

(a) G is s i m p l y but

in part

of G w i t h x is obtained,

because

of x the large p o s i t i v e

contributions

shortly,

G.

associated

situation

(b),

All v a l u e s

with-

signals

the

TAS.

outweigh

As will

onset

of

be

phase

separation.

(b) In this c o n n e c t i o n

it is of some

h o w much of each phase must be p r e s e n t

interest

to e s t a b l i s h

to form a h e t e r o g e n e o u s

PHASE STABILITY mixture Table

359 of

average

3.16.1,

Consider i -

fraction that

the

second

mole

n~(n A +nB)

377,

fraction

x.

in d e r i v i n g

nA + n B moles

x m nA/n.

phase

numbers

p. n-

mole

Let

fraction

A

in

i - x"

, respectively.

in the " p h a s e

Eqs.

reader

Let

first

- n~/(n~

be 1 - f -

t h a t n A - n ( l - x), of

substituting

and

f -

(x-

I - f -

which

x')/(x"

(c) a given

+ nB)

fraction

If

the

numbers

solving

for

the

1 - x" -

total mole

Then

one

requires f one

and nA - nf(l

that

nA -

nA +

finds - x"). n A.

On

obtains

- x'),

as

(3.14.2b)

the L e v e r

Rule.

set of c o n d i t i o n s ,

x o for a s y s t e m were

in

such

(3.14.2a)

is h o m o g e n e o u s

alloy

and by

- x')

For a given

mole fraction

and

o f the

(nA + nB)/n.

n A - n ( l - f ) ( l - x'),

are k n o w n

alloy

formed

mole

N

mole

(x" - x ) / ( x "

results

be

phase

#

Conservation

consult

an overall

mixture

the

the

may

(3.14.2).

of A and B with

a two-phase

of

is g i v e n b y

The

simply

or

c a n one p r e d i c t

not?

Consider

for w h i c h

Fig.

a mechanical

a

3.14.2

whether

solution

of

is r e l e v a n t .

mixture

of

the p u r e

2~ r c u.l

GA

0

• X~X

2

FIGURE 3 . 1 4 . 2 D i a g r a m o f G v s . x , s h o w i n g how t h e h o m o g e n e o u s phase has a lower free energy than any biphasic mixture.

360

3. CHARACTERIZATION OF NONIDEAL SYSTEMS

components

A and B, then according to the Lever Rule

the free

energy of the system would be given by the intersection of the straight llne joining G A to G B with the vertlcal line at x - x o, labeled G* in the diagram. If instead the phase mixture involved two

solutions

of compositions

x a and xa,

then

the

same

Lever

Rule yields the free energy of the alloy designated as G a. Since Ga < ~* p

this new state

Analogous

remarks

is more

apply

stable

to another

than the orlginal

heterogeneous

one

9

alloy whose

two compositions are represented by x~ and xb; the corresponding free energy,

~,

is still lower.

Continuing this process

it is

found that at the composition x where the two phases merge into a slngle homogeneous solutlon of composition Xo, the free energy attains

its

lowest

configuration. 3.14.2,

For the type

value, of free

82G/ax 2 > 0 for all x; hence,

two points

on the

these points. U-shaped

possible

curve

always

lies

G O . This energy

is

the

displayed

stable in Fig.

any straight llne joining above

the

curve

between

When the free energy versus composition curve is

the homogeneous

solution

always

has

the

lowest

free

energy. By contrast, of

composition

curve G in Fig.

consider as

the free energy curve

sketched

3.14.1(c).

in Fig.

3.14.3,

as a function

which

reproduces

If an alloy of composition x o were

I

I

I I I

0

x'

I I 1

Xo

x"

1

X

FIGURE 3.14.3 Diagram of g versus x, illustrating a condition in which a biphasic mixture of relative composition x" and x" with respect to the second component is the most stable configuration for a solution; here x" _< x _< x". For O _< x < x " and for x" < x _< I, a homogeneous phase is stable.

PHASESTABILITY

to exist

36 I

as a

homogeneous

solution,

its

be given by the point G O on the diagram.

free energy would

On the other hand,

if

9

N

the alloy were a heterogeneous mixture of composition x a and x a for component 2, the free energy of the system would be lowered to G a.

By choosing x" and x" to be more widely

composition minimal

one progresslvely

value

G

is

lowers

reached

separated

in

the free energy until

when

the

two

phases

are

a of

composition x" and x"; here the straight line forms a tangent to the two curves near the local minima.

Any attempt to spread

the composition of the two phases further will lead to a rise N

in

free

energy,

as

is

illustrated

for

xb

and

x b,

with

a

is

thus

a

corresponding value for C~. The

stable

heterogeneous x",

state

of

mixture

the

system

under

study

of composition x" and x".

For x" < x
x n only Thus,

in the

heterogeneous region the compositions of the two phases of the mechanical mixture remain constant, but the relative amounts of material

in each phase changes with alterations

in x.

(e) The presentation may readily be generalized to a system

36~-

3. CHARACTERIZATION OF NONIDEAL SYSTEMS

Comp.B

Comp. A

1-f

X'

X" x~ x B

FIGURE 3.14.4 V a r i a t i o n of the makeup of a b i n a r y system w i t h relative c o m p o s i t i o n (mole fraction), shown as a plot of I - f and of f versus x.

in which

more

shown i n F i g .

than

two

3.14.5;

phases

here

appear.

A

typical

t h e homogeneous

example

is

and h e t e r o g e n e o u s

c o m p o s i t i o n ranges are delineated by use of an imaginary string B'

/ A~

I

T+5 i;r

P

i

B+T

l I I I

O x

FIGURE 3 . 1 4 . 5

Gibbs

free

energy

as a function

mixtures of intermediates are formed. indicated

by

cross-hatching

on the

of x when several

Stable single phases are

x scale.

PHASESTABILITY

363

that is tightly wound around the curves between points A' and B"

in

the

diagram

constructions.

to It

exhibit is

all

possible

customary

to

.common

tangent

designate

phases

consecutively by Greek lowercase letters in alphabetical order. The various phases and their composition ranges are indicated on the diagram.

In general,

if AH is small, as is likely to be

the case for homogeneous solutions, a

function

of

composition ranges

curve,

and

the

phases

are

stable

formed,

deviations

remain very small. the from

free the

energy

as

of

are

forms

compound

does

broad, over

However,

from the appropriate

shallow which

when

a

U-shaped

the

single

compound

stoichiometric

is

ratio

This is reflected in the very sharp rise of the composition

depicted in Fig. 3.14.6; over

a

composition

large.

stoichiometric ratio.

is stable

the free energy plotted as

not

even

slightly

A situation of this type is

it is seen that the composition x - x i

only a narrow

stoichiometric value;

is changed

range.

necessarily for example,

The composition

coincide

with

of the

an

ideal

the compound CuAI 2 does not

/

7"

J

+

~YAx

0

x

1

FIGURE 3.14.6 Gibbs free energy as a function of mole fraction for a binary mixture involving formation of a compound corresponding to the phase with stoichiometry Ai_xiBxl. Crosshatched composition ranges indicate stable monophasic regions.

364

3. CHARACTERIZATION OF NONIDEAL SYSTEMS

exist,

but

a compound

on the Al-rich

side

of this value

is

stable.

EXERCISES

3.14.1 Discuss in terms of chemical potentials why phase equilibrium requires the tangent constructions described in this section. 3.14.2 Prove that the common tangent construction is equivalent to the equality of chemical potentials of the phases whose compositions are given by the points of tangency.

3.15 The

DISCUSSION OF SEVERAL TYPES OF PHASE DIAGRAMS general

correlated

features with

of standard phase

the

free

3.14.

For

simplicity

where

the

liquid

energy

we

curves

immediately

phase

is

diagrams

will now be

depicted

in

Section

specialize

to

systems

homogeneous

throughout

its

composition range; the corresponding free energy curve is then U-shaped,

as depicted

in Fig.

3.14.2.

In our

first

example,

Fig. 3.15.1, the solid state also exists only as a homogeneous solution.

Quite generally,

as the temperature

is lowered the

free energy curve of the solid moves past that of the liquid, and

the

shapes)

shape

of

will

also

each be

curve

(i.e.,

altered.

the

skewness

Thus,

with

of

the

U

diminishing

temperature the two free energy curves will intersect and give rise

to

situations

where

the

common

tangent

indicates the existence of biphasic mixtures. the temperature energy

curve

T I is sufficiently high

for the

liquid,

Gi(x),

construction

In Fig. 3.15.1a

that the entire

lies below

that

for

free the

solid, Gs(x) ; at any composition x, the system is stable in the liquid state.

As the temperature is lowered to a value T A (part

(b)), the two free energy curves touch at the composition x 0; solid and liquid now coexist for the pure phase, the melting

point

of pure

A.

However,

for x >

and T A is

0 the

free

energy of the liquid remains below that of the solid. As T is reduced further to the value T 2 (part (c)), the GI(x) and Gs(x )

TYPESOF PHASED I A G R A M S

36S

(a)

T~

(b)

TA

(c)

I

T2

I

t

B

A

XB

X B -----~.-

(d)

{(e)

TB

XB,L-~

IA(I.f). . . . § . . . . . . . . . . . . .I". . ! .... t ..... ! ......... TA

T3

I

xh

r2

Liquid Solid

I A

B

x B-~,.-

B

A XB

x s or

intersect, xs
T^ > T 2 > T B > T 3.

such that in the composition

is stable;

""\

X B ~

FIGURE 3.15.1 D e r i v a t i o n of a phase curves of liquid and solid phases. curves

1,,/

A

~

0 _< x
0 for this case,

less

stable

than

the solid s o l u t i o n

a mixture

of phases;

indicated by the dotted curve at the b o t t o m

TA

(a)

i

Te,

this

is

of the diagram.

.

T1

(b)

(c)

s

I

I ~+i

b;~i XB ~

B (d)

A a I

s

t

b

XB---~ T3

B (e)

A

TA a .,~

I

~

J.. Y

L

d Xa-~ L

'

IL+s e

1 f

s

B (f) l

TB t I-1

.........

~ E ~

I

XB

c

b_....................

y

A

1L+s

T2

T~ //" A + B \\\

B

A

XB

B

A

T

X B ..______ ~

I B

FIGURE 3.15.2 D e r i v a t i o n of a phase diagram with a minimum, obtained from free energy curves of liquid and solid phases. TA > T B > T 1 > T z > T 3 > T c .

TYPES OF PHASE DIAGRAMS

36 7

SolMd Md Sol

lquld

/

To

(o)

(e) (b)

To

Tc

E

(d) Liqu,d

I

~ Tc

I Td I

5O

o A

100 B

Mole Percent B 0 A

I B XB~

FIGURE 3.15.3 Gibbs free energy curves for liquid and solid mixtures which result in a phase diagram with a maximum.

A second case which no

frequently

the components

intermediate

encountered

are only p a r t i a l l y miscible

phases

are

curves are shown in Fig.

formed.

3.15.4.

somewhat curves

composition

below

range.

the melting

for solid

liquid phase

of pure

intersect,

By the common tangent construction, the h o m o g e n e o u s

relevant

A,

free

energy

T - T I, G t (x) is stable over

At a p a r t i c u l a r

point

and liquid

The

a solid for and in w h i c h

At temperature

< Gs(x ) for all x; the homogeneous the entire

involves

value

the

as shown

free

T - T 2, energy

in part

(b).

we see that for 0 < x < x"

solid alloy is stable;

for x" < x < x" solid of

c o m p o s i t i o n x" is in equilibrium with liquid of c o m p o s i t i o n x"; for x > x" the homogeneous

liquid phase is stable.

Part (c) is

typical of a temperature T 3 at which G, and G e intersect both at the

A-rich

homogeneous x




N

Electrostatic gradient

Electr,c field

Ct > Cr

~-~-r

-G

--

~r

~r > ~I, ' •r >121,

~Cp ~

potential

Cui r e w

~'L > ~r

1

r

+

Elect rode A

wire Cu

Elect r o d e C

FIGURE 4.6.2 Schematic diagram illustrating the directions of electric field, electrostatic potential gradients, electrochemical potential gradient, and emf for a representative cell under open circuit conditions, and set up in accord with Conventions i and 2, developed later.

- RT d~n a 9_ potential

VF~, points -"

to the left and the electrochemical

itself is governed by the inequality ~e_(2) > ~e_(r).

We now modify slightly the approach in Section 2.9, where we

had

shown

temperature

that

at

equilibrium

and pressure)

(subject

AG d -- ~(i)v•

we

deal

expression equilibrium

with

(4.5.3),

charged

species;

employing

condition,

in

constant

- 0, c o r r e s p o n d i n g

the schematic chemical reaction ~(1)viAi - 0. case

to

we

In our particular

therefore

~(1)vi~i - 0 as conformity

to

with

the

adopt

the

appropriate the

earlier

discussion. As

applied

to

the

Daniell

cell

we

again

write

Zn(s)

-

Zn2+(m,) + 2e-(~) for the oxidation step on the left and Cu2+(mr) + 2e-(r) ~ Cu(s) for the reduction step on the right. then

to

be

These are

combined into the net reaction schematized by Eq.

4 J4

4. THERMODYNAMIC PROPERTIESOF ELECTROLYTES

(4.6.1), that

in w h i c h we have also included the c o m p e n s a t i n g

preserve

electroneutrality

in

the

two

anions

compartments"

We

write

Zn(s)

+ Cu2+(mz) + SO~(m z) + 2e-(r)

- Cu(s) + Zn 2+(m,) + SO~(m,) + 2e-(2). Note

that

we

have

constituents; the

two

drives

note

not

canceled

they are present

electrodes.

It

is

out

the

(4.6.1)

important

at different precisely

electronic

concentrations

this

difference

the cell o p e r a t i o n w h e n the open circuit

(4.6.1)

that for u n c h a r g e d

to

[ - ~.

This

that

is closed.

We next apply the c o n d i t i o n 7.(i)ui~i = 0 to Eq. species

at

leads

and

~u(s) + ~nSO4(s) + RT 2n a~(m,) + F~zn2+(m,) - F~so~(m,) + 2[e-(2)

-

~Zn(,) + ~cuso4(,) + RT ~n a~(m=)

+ F4cu2+(m r) - F4so~(m r) + 2[.-(r)] The ionic constituents at

the

same

corresponding is

true

standard

of

Zn 2+ and SO~ in the left c o m p a r t m e n t

electrostatic

potential

~(~),

terms in 4 cancel out from Eq.

Cu 2+ and

conditions

equilibrium

(4.6.2)

- 0.

of

SO~ on

the

right.

the

terms

in ~

can

constant-

(4.6.2)

whence

(4.6.2).

grouped

RT ~n Kq, as in Section 3.7.

solve

Eq.

2F~"

[[ _(2) - fe_(r)] - RT 2n Kq - RT 2n{a~2z(m= ) ,

the

The same

Furthermore, be

are

into

for an

We can then

for

(m,)} (4.6.3)

with RT 2n Kq -~ ~Zn(s) + ~CuSO4 - ~Cu(s) - ~ZnSO4 9

Here we have also

reintroduced

the e l e c t r o m o t i v e

force or emf,

-

the earlier

namely

by

[~ _(~) - ~ _(r)]IF. e

definition,

e

(4.6.4)

4 I5

GALVANIC AND ELECTROLYSIS CELLS

Note the following" circuit voltage' this case, is

the

that must be multiplied

difference

thermodynamic

function

state

for

F

in

which use

at

More precisely, ~ is directly related to the at the left and right

(ii) We have set up the circuit of Fig. 4.6.2 and

corresponding

inequalities

by

reversible electron flow from ~ to r. that K > 0 corresponds fact

of

in electrochemical potential

terminals.

The

by a charge,

to be compatible with the Gibbs free energy,

proper

constant T and P.

the

(i) The emf, K, is in a sense an ~open

that

the preceding

(iv)

The

emf

spontaneous,

Since ~i > [r it follows

to such a spontaneous discussion

transfers means that F represents transport.

assuming

transfer.

is based

the magnitude

(4.6.4)

is

on electron

of the charge

governed

spontaneous net chemical reaction abstracted

(iii)

solely

by

the

from (4.6.1);

can therefore introduce our prior definition AG d

-- ~ Z n 2 +

+

we

~Cu-

[~zn + ~cu2+] that involves solely the chemical potentials of the various species involved in the battery operation.

-

-

Thus,

AGd/2F

(4.6.5)

for the net chemical

reaction ~viA i - 0 as written;

AG d < 0, as expected. explicit reference

(v) In th~ final expressions there is no

to electron participation.

these quantities have their

effect

is

potentiometer

contrary

does

not

clearly

in to

measure

potential but differences electrodes.

the

(4.6.1) ; rather,

definition

what

is

often

differences

in

in electrochemical

for

a most undesirable

appellation.

~.

stated,

a

electrostatic

potential

(vi) The use of the term electromotive

nomenclature

of the

force

Unfortunately,

is

this

is now so firmly rooted that it is unlikely to be

displaced by a more appropriate The

It is not that

'canceled out' from Eq. subsumed

Correspondingly,

moreover,

preceding

discussion

terminology. presents

an

obviously

very

schematic representation of the actual processes that occur in the

operation

essentials

for

of a

the

Daniell

thermodynamic

cell.

Nevertheless,

analysis

Although we have based our discussion

have

been

on a specific

all

the

included. example,

4 [6

the

4. THERMODYNAMICPROPERTIESOF ELECTROLYTES

preceding

discussion

generalization,

to

which

is we

now

capable

of

immediate

devote

the

subsequent

presentation.

EXERCISES

4.6.1 Provide evidence to show that the emf for the combination Zn[M[Cu is the same as for the Zu[Cu couple; here M is any desired metal. 4.6.2 Provide a physical model which, on a microscopic basis, accounts for the fact that an emf is set up at the phase boundary between dissimilar substances, which permit electron transfer to occur.

4.7 (a)

GALVANIC CELLS : When

a

cell

GENERAL TREATMENT

is

functioning

reversibly,

work

is

being

performed on or by the system in transferring electric charge through

any

operation

cross

of

the

section cell

of

the

according

circuit.

to

the

reversible

generalized

equation Zce~vtAI - 0 is carried out only extent,

The

chemical

to an infinitesimal

so that it does not alter either the concentration

chemical

species

or

difference).

Then,

process

Section

electrons

(see

the

a

unit 2.9)

emf

(open

advancement involves

6A

the

through the external circuit,

of equivalents,

circuit in

the

transfer

of

potential chemical of

nF6A

where n is the number

determined by the nature

of the process,

and

where F, the Faraday (i.e. , the charge for Avogadro's number of electrons),

is 96,487

coulombs.

The work

done by

the

cell,

when the opposing emf is infinitesimally less that required for maintaining static conditions, 1.16,

for reversible processes,

other than mechanical

is nF~6~.

As argued in Section

at constant T and P, the work

is given by 6W" - -

6G.

In the present

case the cell reaction must occur spontaneously,

hence

for an

infinitesimal virtual advancement 6~, + nF~6~ = -

(6G/6~)T,e6~,

or

nF~ - - AGd,

(4.7.1)

GALVANIC CELLS

where

4 I7

AG d is

the

chemical

cell

This

in

is

increase

reaction

in

free

energy

~(1)veAI - 0 per

agreement

with

deducing Eq. (4.6.5).

the

accompanying

unit

special

the

of advancement.

case

considered

in

We had shown in Section 4.6 why one can

use the expression for the net chemical equilibrium in dealing with electrochemical processes. (b) The subsequent standard

states.

formulation

We shall

depends

on the choice

follow convention by adopting

hybrid system discussed in Section 3.7(c)"

the

(i) For pure solids

or liquids that participate in the electrochemical process, standard

state

subjected

to

is a

that

total

at

activity

for

such

(note

again

that

the

pressure

temperature of interest. vs(T,P,q:)a,q(T,P),

which

of

isolated one

substances

atmosphere

liquid,

is

at

the

As shown in Eq. (3.6.3b) the relative

species

in which

is q

given

by

represents

a s (T, P, qs)

either

x,

q/q* - I for pure materials); forming

the

component

c,

the

homogeneous

or gaseous phases,

mixtures

in

--

or

m

relative

activity so defined is almost invariably close to unity. For

of

the

(ii) solid,

the standard state is chosen to be

that in which each substance in the mixture is present at unit activity and subjected to one atmosphere at the temperature of interest.

of

these

species is specified by ao(T,P,qo) --7o(T,P,qj)aoq(T,P)qo.

The

summation

As

used

in

Eq.

(3.5.21)

the

activity

for j runs over all ionic species,

as well as over

any dissolved but un-ionized species, which participate electrochemical ao,

and

qj

for

quantities 7• illustrated

reaction. the

a•

in

individual

and q•

later

It is customary ions

by

to replace

Also,

the 7j,

corresponding

as discussed in Section 4.1,

sections.

in the

we

frequently

atmospheric pressure as the operating condition,

mean and as adopt

in which case

alq(T,l) m I for all species (~ m s,j), as may be ascertained by examination of Eqs. (3.4.10b), in conjunction with Eqs.

(3.4.12),

(3.7.15),

(3.5.17), and (3.5.20)

(3.7.16).

The equation for AG d consistent with the above choice Eq.

(3.7.12);

is

on introducing this relation in (4.7.1) we find

4 J8

4. THERMODYNAMIC PROPERTIESOF ELECTROLYTES

RT .---- ~ n

--

Kq

v, ~n a,(T,P,qs) + ~ vO ~n ao(T,P,qo)

nF

,

j

(4.7.2a) which under operating conditions of one atmosphere,

RT

RT

where

~n K q - -

Eq.

~ vj ~n (7jqj)

the same cell, (4.7.2a)

(4.7.2b)

to designate by ~

the standard emf of

in which all species are at unit activity;

from

it is seen that

RT ~o _ __nF ~n Kq

so that Eq.

AG ~

--,nF

(4.7.3)

(4.7.2a) becomes

- --

v, 2n a,(T,P,q,)

is known

depends

(P = i atm),

has been used.

(3.5.21)

It is conventional

which

reduces to

on

as

the

the

+ ~ vj ~n aj (T,P,qj) j

Nernst

choice

of

(1889).

concentration

(4.7.4), i.e. ~, does not.

(4.7.4)

Clearly,

units,

but

the

~o sum

One immediately sees the tremendous

utility of ,go measurements:

For any cell which can be set up

to simulate ionic,

solid state,

interest,

a

such

equation

,

liquid or gaseous reactions of

determination

directly

yields

the

corresponding equilibrium constant Kq. (c) It is helpful at this state to introduce a systematic nomenclature

and

analysis

cell

of

several

performance.

of

molality

in

unified is

Let

Pt

example: ml,

a

procedure

surrounded by hydrogen gas at pressure solution

particular

general

electrode,

HC~

a

permit

in

aqueous

of

The

that

explained an

terms

conventions

a

P, dip into

contact

with

a

similar solution at molality m 2, and saturated with respect to

GALVANICCELLS

AgC~,

4 I9

into which

is immersed a silver electrode.

such descriptive

verbiage

one

represents

such

To save on

a cell

by

Pt,

H2(P)] HC2 (m I, sat by H2)[HC2 (m2, sat by AgC2)IAgC2(s), Ag(s), where vertical bars

separate

different

portions

of the cell.

Pt is used here as an inert electrode material which does not participate in any oxidation processes in aqueous solution but which provides an interface for H +, Hz, e- interactions. We now introduce the

oxidative

process

reductive

process

electrons

are

external

Convention

on

given

circuit

to

always the

i:

For the cell as written

occurs

right.

off

at

the

the

right,

on

the

According left where

left to

and

move

they

are

and

this

the

scheme

through taken

the

up

by

species in solution, as illustrated in Fig. 4.6.1; conventional current flows in the opposite direction. The overall process is best visualized in terms of the two half reactions at the anode and cathode respectively. follows"

In the present scheme these read as

anodic ~1 H 2 ( P ) -

e- + AgC2(s)

-Ag(s)

overall process

H +

+

(ml) + e - , on the left,9

C~-(mz) , on

the

right;

cathodic

so

that

the

1

is representable as EHz(P) + AgC~(s) - A g ( s )

H+(m~) + C2-(mi), accompanied,

9

+

in accord with Convention (i), by

the electron flow through the external circuit from the Pt to the Ag electrode and by H + transfer from left to right. By Convention 2 the emf developed by the cell as written, under open circuit conditions,

~" -

~'l

-

-

~r

([l

-

is taken as (4.7.5)

~r)l F,

where 2 and r refer to the left and right electrodes cell as written under Convention potential

introduced

for the

I; [ is the electrochemical

in Section 4.6.

step reflects

the

process of splitting the cell operation into an oxidative

and

a reductive half reaction.

This

Convention 2 is consistent with our

prior discussion, as was shown for the general cell depicted by Fig. 4.6.1. A further check for self-consistency cell

for

infinity

which and

an

placed

additional on

the

electron

negative

is the following: is

brought

terminal,

is

up

A

from

at higher

42,0

4. THERMODYNAMIC PROPERTIESOF ELECTROLYTES

energy than the same cell for which the additional placed

on the positive

electron from the

terminal.

Hence,

electron

the movement

is

of

an

negative to the positive terminal lowers the

energy and thus the free energy,

of the cell.

The spontaneous

advancement of the cell reaction by dX requires the concomitant transfer left

of nFdX

(-)

to

potential

fi - r

right

under

concomitant

electrons (+)

through

past

near-open

change

a

difference

circuit

in

the

(4.5.3),

this

(aG/aX)TopdX pressure.

leads

a

proceeds

the

overall

reactions.

By

potential.

-

fashion by

advantage

a

to

-

Eq.

dG

-

temperature

-

and

as before. according

AGd/NF

first

dealing

to

with

is taken of the prior step in

is written

Convention

decrease

nF~,

of ~ and of

reaction

is

in the amount

According

at constant

that AG d - -

in an orderly Here,

There

yielding an overall change of

energy

determination

from

electrochemical

potential

corresponding

free

It now follows

the quantity ~o. which

to

in Gibbs

(d) The (4.7.4)

electrochemical

of

circuit

conditions.

in electrochemical

per transferred electron,

nF~SX

the external

3

one

as

a sum

of

associates

the

with

half each

half-reaction a standard oxidation potential. ~o, which is to be computed as a special case of (4.7.5): ~o _ ~

_ ~o,

where ~

(4.7.6)

and ~o are the standard oxidation

potentials

for the

half reactions at the left and right electrodes

for the cell as

written.

Extensive

Table

excerpted,

are available for specifying the ~o values;

that these quantities All

half

reactions

processes,

tabulations,

themselves in

the

from

may be positive

table

are

the sign of its emf value. (4.7.6)

necessitates

around,

change

is

take note

or negative. as

oxidative

is attended

thereby changing

That is, the operation required by that

one

take

standard values for the half reactions sign

written

4.7.1

so that the complete cell reaction must be obtained

by turning one of the half reactions Eq.

which

to by

the

for

~

and

~o

cited in the table;

formulation

(4.7.6).

the the One

GALVANIC CELLS

notes

42. I

that

(4.7.5)

indeterminacies

and

(4.7.6)

and

can

arise

from

the

equally well be w r i t t e n as ~0 _ ( ~ is any arbitrary

constant.

from

fact

sign

that

problems

(4.7.6)

_ cl ) _ (~o _ ci),

The first matter

in

could

where c 1

is dealt with by

C o n v e n t i o n 4: The emf is regarded as positive for the o p e r a t i n g cell

when

the

electrode

electrons

written

The second

on

tend

the

spontaneously

left

to

to

flow

that w r i t t e n

on

from

the

the

right.

to by Convention 5: The standard I electrode potential for the half reaction EH2(P-- I am .) = H + (a+

-

item is attended

I) + e- is arbitrarily It now

values

set at zero" E~

should be very

can

be

clear

constructed:

appropriate half cells

how

H+

=

0

Table

One

4.7.1

for

successively

to the half cell

in w h i c h

the ~o couples

the r e a c t i o n

2H2 (PH2 - i arm) is carried out on the e- + H + (all+ I) right; in the half cell to be tested all d i s s o l v e d species must -

- -

be

at

unit

pressure. negative

activity The

and

all

resulting

according

reacting

emf has

to whether

gases,

a sign

at

that

the electrode

atmospheric

is positive

on the left

or

is at

a lower or higher electrostatic potential than the electrode on the right; under

this emf represents the value of ~o for the half cell

study,

since

fo for

zero by convention. completed,

the

be

standards With

half

Their physical

the chemical

respective

can

cross

which

terminals.

operation

is

occurs.

are a measure

generate

has b e e n used

checks

in the use of Eqs.

interpretation

as

a

self

That

electrons.

thus ~

half

driven

and

in the extent

can give up electrons

~wins,'

reaction

which

to their

renders

its

in the sense that the opposing

by

and ~o are both of the

(4.7.5)

is that the emf arises

the two half reactions

entities

electrode more negative

oxidation

be

is

for other half cells whose ~o is

one is forced to undergo a reduction process. cell

electrode

tabulation can thus be constructed.

as a c o m p e t i t i o n b e t w e e n to w h i c h

cells

appropriate

One must be very cautious (4.7.6).

hydrogen

Once a set of such m e a s u r e m e n t s

determined.

consistent

standard

corresponding

secondary reference to

the

tendency

the

electrode

standard for

The spontaneous at

oxidation

the half

which emfs

reaction

to

422

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

Table

4.7.1

SHORT TABLE

OF S T A N D A R D

EMF V A L U E S

B a l B a 2+

Ba-Ba

MglMg 2+

M g = Mg 2+ + 2e-

+2.363

AI[ AI3+

AI = AI3+ + 3e-

+1.662

Zn[ Zn 2+

Zn-Zn

2+ + 2e-

+0. 7628

FelFe 2+

Fe-Fe

z+ + 2e-

+0.4402

Call Cd 2+

Cd = Cd 2+ + 2r

+0.4029

pt [Ti2+,Ti 3+ Pb ]PbSO4 JSO~

2+ + 2r

(IN VOLTS)

+2.906

Ti 2+ - Ti 3+ + ePb + SO~ = PbSO 4 + 2e-

CulCuI I I -

Cu + I- - Cul + e-

Pt[HzIH +

H 2 = 2H § + 2e-

AgIAgBrJBrP t l C u +, Cu 2+ A g [AgC~ iC r Pt IHgl HgzC~2 ]C2PtllalIPt[Fe 2+, Fe 3+ AgIAg +

Cu +

=

A g B r + e-

Cu 2+ + e-

A g + C~- = AgC~

+ r

2C2- + 2Hg = Hg2C22 + 2e3I- - I~ + 2eFe 2 + -

Fe 3+ + e-

A g = Ag + + e-

PtlT2+,T23+ PtlC221C2Pt IMnZ+, MnO4Pt ISO~,

Ag + Br-=

SO~

T2 + - T~ 3+ + 2e2Or

= C~ 2 + 2e-

+0.3588 +0. 1852 0. 0000 -0. 0713 -0. 153

- 0 . 2225 - 0 . 2676 -0. 536 -0. 771 -0. 7991 -1.25 -1.3595

M n 2+ + 4H20 = MnO~ + 8H + + 5e-

-1.51

SO~ + 20H- = SO~ + HzO + 2e-

+0.93

PtJHzlOH-

H 2 + 2OH--

Pt I021OH-

4OH- = 02 + 2H20 + 4e-

Pt ]M n O 2 1 M n O ~

+ 0 . 369

2HzO + 2e-

MnO 2 +4OH- = M n O T +

2H20 +3e-

+0. 8281 -0.

401

- 0 . 588

GALVANIC CELLS

423

There

is

obviously

nothing

standard oxidation emfs.

compelling

in

the

One could equally well

use

of

deal with

tabulation in which all the half reactions are reversed.

a

These

data are compiled by coupling the appropriate half cell to the hydrogen cell operating according to the scheme ~H 2 (P~2 - I a t m ) -

H + (all+- i) + e-; all species must be present under standard

conditions. values

It

are

reversed

Table 4.7.1; the

half

in

reactions

reduction

be

clear

sign

that

the

relative

to

corresponding those

compiled

emf in

these new values are a measure of the tendency of

Correspondingly,

to

remove

electrons

one would write ~o _ ~

emfs.

formulation; the

should

Many

writers

from _ ~

the

electrode.

for the standard

prefer

this

particular

when used self-consistently one naturally obtains

same numerical

values

for ~o as by

the method

involving

standard oxidation emfs. (e) We provide a very elementary example to illustrate the preceding remarks. Pt, C ~ z ( P whose

operation,

Consider the cell I atm) IHC2 ( a • in

accord

with

i)

IAgC~(s), Ag(s),

Convention

I,

is

given

by

(25~ 2C~-(a_-

i) - C 2 2

(P-

I atm) + 2e-

~

2AgC~(s) + 2e- - 2Ag(s) + 2C~-(a_ = i)

= -1.3595 V =

+0.2225

v

=

-1.1370

V

,,

~o

2AgC~(s) - C~2(g ) + 2Ag(s)

(4.7.7)

- ~

One should carefully note how Convention 3 is applied:

~o

- ~;

However,

in

equation,

we

writing

~

the

may be read off from Table 4.7.1. lower half

reaction

as a reduction

need a sign reversal in converting ~o for an oxidative process to ~o for

a

reductive

process.

The

contributions

cancel in the algebraic addition process because is the same on both sides.

from

C2-

the activity

47-4

4. THERMODYNAMIC PROPERTIESOF ELECTROLYTES

One can now compute C22(P

-

i

arm)

+

-2(96,487)(-1.1370) reaction proceeds reader

should

statements"

AG~ for the net reaction

2Ag(s)

according

to

AG~

J/tool- 219,420 J/mol.

2AgC2(s)

=

-

2F~ ~

able

to

verify

=

Since AG~ > 0 the

spontaneously in the reverse direction.

be

-

readily

the

The

following

(i) If the half reactions had been written as C~-

- 21-C~z + e- and e- + AgC2 = Ag + C~-, the ~o values for the half reactions

and

however,

n-

the

total

~o

would

have

I in this case and A G ~ -

remained

unaltered;

109,710 J/mol.

(ii) If

the cell and cell reactions had been reversed according to the scheme Ag(s),AgC2(s)]HC2(a•

- I) lC22(P = I atm),Pt,

with half

reactions

2Ag(s) + 2 C ~ - ( a _ - I) = 2AgC~(s) + 2e-, 2e- + C~2(P --

i atm)

2C2-(a_-

-

I),

then

one would

1.1370 V and AG~ = - 2 1 9 , 4 2 0 J/mol. principle runs

that if Conventions

spontaneously

have

chloride,

but

reaction

can

with

under be

chlorine

standard

reversed

potential exceeds 1.1370 V.

+

This illustrates the general

in the direction

spontaneously

~o_

i and 2 yield a reaction which opposite

down, then one finds ~o < 0 and AG~ > 0. react

obtained

by

to that written

(iii) Silver tends to

gas

to

form

conditions

this

electrolysis

when

solid

silver

spontaneous the

applied

(iv) If the operation of the cell

is altered to read Pt,C~z(P )lHC~(a•

one obtains

a net reaction 2AgC2(s) - C22(P ) + 2Ag(s).

The Nernst equation

now reads

RT 2n [aA82 ] - --RT 2n ac~ - -1.1370 - ~-~ [a~c I 2F 2

-

-I.1370

RT - ~-~

in which the ratio

~n

a[a_~ct 1

RT 2F ~n ac~ 2 '

aAs/aAsCl differs

only slightly from unity and

acl 2 may ordinarily be replaced with Pc, 2. convince himself

that precisely

(4.7.8)

(v) The reader should

the same result,

Eq.

(4.7.7),

would be obtained if the net reaction had been written form AgC~(s) of

how

the

- 89 C~2(P ) + Ag(s).

in the

Thus,

the emf is independent

net reaction is balanced,

which is a physically

TYPESOF ELECTRODES

sensible

425

result.

On

the

other

hand,

AG~

does

depend

on n,

which again is a physically sensible result. The

reader

oxidation

-

experimental

should

reduction

carefully

study

potentials

the

are

manner

in which

handled

investigation of electrochemical

in

every

cells.

EXERCISES

4.7.1 Do the two celis Cu(s)ICu++l ICu+ICu(s) and PblCu+, Cu++ll Cu+ICu(s) correspond to the same reaction and do they have the same value ~o? 4.7.2 The emf of the cell ZnlZnC22(m) IAgC~(s)IAg, with various molalities m of zinc chloride, was found to be as follows at 25 ~ m (volts) m ~ (volts) , ;

",I

,'

:: ""', .

.

.

.

.

.

0.002941

1.1983

0.04242

1.10897

1.16502

0.09048

1.08435

0.01236

1.14951

0.2211

1.05559

0.02144

1.13101

0.4499

1.03279

,

0.007814 ,

,

,,

)etermine the standard potentlal 0f t h e Zn, Zn *T electrode. 4.7.3 Why does the electron concentration in the wire enter the Nernst equation? 4.7.4 Discuss the difference in operation and in between the two cells Pt, H2(P) IHC~(ml)IHC~(mz)IAgC2,Ag and H2(P )IHC2(m)IAgC~,Ag. Illustrate the difference in terms of Nernst equation.

4.8

not emf Pt, the

TYPES OF ELECTRODES

The following types of electrodes (a) pressure

'Gas'

Electrodes"

In

are in common use" this

scheme,

P is forced over an immersed,

and is bubbled through the solution. half reaction

=

H+

+ e-.

at

a

fixed

electrode

An example of this cell

is given by PtlH2(P atm), H+(c mol/liter), i ~H2(g )

gas

inert metal

corresponding to the

4~-~

4. THERMODYNAMIC PROPERTIESOF ELECTROLYTES

(b) Oxidation-Reductlon

Electrodes"

dips into a solution containing states.

An

example

corresponding

is

Here,

Electrode"

cell

(Q), as may

corresponding

(d)

PtlFe2+(cl),

Fe3+(c2),

This

is a specialization

an equimolar mixture of hydroquinone obtained from

contact with an inert metallic half

by

to the half reaction Fe 2+ - Fe 3+ + e-.

(b)"

and quinone

an inert metal

ions in two distinct oxidation

furnished

(c) The Ouinhydrone of case

Here,

be

quinhydrone

electrode.

represented

here

Metal-Metal

lon

(H2Q-Q) , is in

Schematically

by

to the half reaction H z Q -

(H2Q)

PtlH2Q ,

Q,

this

H+(cl),

Q + 2H + + 2e-.

Electrodes:

Here

metal

ions

in

solution are equilibrated with a metallic electrode of the same material. dipping

An into

corresponding which

react

example a

furnished

solution

to A g wlth

is

of

silver

Ag + + e-.

water

by

a

silver

nitrate:

AglAg+(cl),

Very active metals

obviously

cannot

be

electrode

such as Na

employed

in this

manne r.

(e) Amalgam equilibrated

with

Electrodes: its

ion

In this

setup

is dissolved

the metal

in a pool

to be

of mercury

into which is dipped a wire made of a noble metal.

This setup

is used for active metals which normally would react directly with water; mercury does not participate example

in the reactions.

is given by PtlHg-Na(cl) INa+(c2),

corresponding

An

to the

reaction Na(cl) - Na+(c2) + e-.

(f) Meta!-Insolub!e

Salt Electrodes"

contact with an insoluble

Here,

salt of the metal,

which

equilibrated with a solution containing the anion. is

illustrated

corresponding

by

the

half

to the reactions

cell

a metal

is in

in turn is This scheme

AglAgC~(s )ICe-(c),

Ag - Ag + + e- and Ag + + C~- --

AgC~, with a net half reaction Ag + C~- - AgC~ + e-.

(g) The Calomel Electrode" the widely used calomel

A special case of the above is

electrode,

where

a Pt wire

dips

into

427

LIQUID JUNCTION POTENTIALS

mercury in contact with a Hg2C~ 2 paste, which is in contact with a KC~ solution

saturated with HgzC~ 2.

Here the reactions

Hg -

Hg + + e - and Hg + + C~- _ EI HgzC~z occur

yielding

a net half

reaction

electrode

frequently

serves

Hg + C~- - 89 Hg2C~ 2 + e-.

as

electrode

a reference for

a

standard

calibration

This

because

use

standard

is

of

the

hydrogen

usually

quite

inconvenient.

EXERCISES

4.8.1 (a) The "dry cell" or Leclanche cell operates by oxidation of Zn(s) to Zn2+(aq), and by reduction of MnOz(s) to Mn2Oa(s) in the presence of NH4C~(aq), forming NH4OH(aq). Write out the half reactions and the complete reaction for this process. (b) Inclusive of a 50~ safety factor, what are the minimum masses of Zn and MnO 2 required to guarantee generation of a i0 ma current for I00 hours? 4.8.2 Devise a cell in which the half reactions Fe(CN) 64Fe(CN)6 a- and Mn 2+ - MnO 4- can be carried out. 4.8.3 Write down the half reactions and complete cell reactions corresponding to the following galvanic cell diagrams" (a) K(Hg) llKC~(a q) IAgC~(s)IAg (b) PtlFeC~2(aq,m2) llFeC~2(aq,m 3) IFe (c) Hg, HgO(s)INaOH(aq)IZn(OH)2(s), Zn (d) Pt,C~2(g)IKC~(aq)llBr-(a q) IBr2(~),et (e) Hg IHg2S04 (s ), Na2SO 4 (aq, m, ) I INa2S04 (aq, m r) ,Hg2SO 4 (s ) IHg. 4.8.4 Devise galvanic cells in which the following reactions can in principle be carried out" (a) (b) (c) (d) (e) (f)

H2(g) + ~~(Bgr) - H20(~) Zn(s) + (s) - Z n B r 2 ( a q) + 2Ag(s) 2Hg(~) + C~2(g) - Hg2C~2(s) Ag(s) + z 02(g ) _ AgO(s) Ag+(cz) ~ l-(c 2) - A g l ( s ) Pb + 2AgC~(s) - PbC~ 2(s) + 2Ag.

4.9 So

LIQUID JUNCTION far

we

juxtaposition anodic

have

POTENTIALS

bypassed

the

of two dissimilar

compartments

problems solutions

of the electrochemical

arising

from

in the cathodic cell.

As

the and

already

4~,~

4. THERMODYNAMIC PROPERTIESOF ELECTROLYTES

indicated

in the

discussion

of Section

4.6,

contact

between

such solutions produces a contribution to the overall cell emf which

ought

to be

taken

into

account.

Nevertheless,

it

is

generally dismissed as being small. On

a microscopic

because

two

dissimilar

equilibrium. if allowed

level,

the

liquid

solutions

junction

are

obviously

Interdiffusion of the various to proceed

indefinitely,

renders both solutions identical.

emf

this

arises not

in

ions takes place; process

ultimately

If the diffusion process can

be maintained at a sufficiently slow pace that the constitution of the solution surrounding the electrodes

is not appreciably

altered during the operation of the cell the junction potential problem may be ignored. surrounding

the

In these circumstances

electrodes

will

remain

the solutions

homogeneous;

no

appreciable concentration of foreign ions will be generated in either compartment,

and thus the chemical reactions associated

with

of the cell

the operation

are

strictly

reversible.

To

approximate this ideal condition one allows the two homogeneous electrolytes

to intermingle over a considerable

location well removed from the electrodes. zone

each

differential

infinitesimally infinitesimal out

from

layer the

advancement

without

of

distance

at a

Within the boundary

solution

neighboring

will

differ

layers,

and

only an

of the cell reaction can be carried

appreciably

violating

the

conditions

for

reversibility. Different

ions

in a given solution are characterized by

distinct diffusion coefficients that govern their net rates of diffusion

in

the

direction

opposite

to

the

concentration

gradient.

Since any electrolyte solution contains at least two

ions of opposite charge, a net charge separation will occur in a process,

whereby the faster ions move ahead of their slower

counterparts. the

faster

ions

state thus

sets

The resulting and

internal

accelerates

in, whereby both

the

electric slower

field

ones.

types of ions begin

retards A

steady to move

across the junction at comparable rates under the influence of a fairly steady electric field in the region of the junction. Suppose the

boundary region is large in extent,

so that

LIQUIDJUNCTION POTENTIALS

429

the transfer of an infinitesimal quantity of charge across the boundary layers does not appreciably alter the composition of each differential layer.

Charges are carried by different ions

in proportion to their transport numbers.

Suppose for a flow

of one Faraday in an infinite copy of the system from left to right a fraction t~ of the transferred charge is carried by the ith cation of valence z~, and a fraction t~ is carried by the ith anion of valence z ~ - - ] z ~ ] ; numbers.

Necessarily,

the t's are termed transference

Y.(j)t~ + Y.(j)t~ - i. At each location in

the bridging layer between solutions a flow of t• +

+i equivalents

of cation i occurs from right to left, and a flow o f equivalents

t~/Iz~[

of anions j takes place from left to right.

This

produces a net increase in Gibbs free energy of dG - 7. (t~/z~)d~ - 7. (t3/Iz3])d~3

i

(4.9.1)

j

for each layer in the junction region.

The overall change in

G and the concomitant emf for the entire junction solution is then -F[A~I

t~ d2n a ~ -- AG J - RT ~. ~A ZZ I

Rr 7. . 3

J'BA]z.~] t3 --

d2n a3 . (4.9.2)

One notes that in general the t's are themselves the composition or activity of the solution.

functions

of

One defines mean

tr.ansference numbers as

t-I -

~A t~ d~n a~

(4.9.3)

2n a~(B) - 2n a~(A) ' m

and similarly for t~; then

RT

A~,~ _ 7

[a~(B)]

t~

[i ~ Iz~l u

2n

laq (A)

t-~ - ~-

j z~

[a~(B) 2n

II

La~(A)

(4.9.4)

By appropriate regrouping of terms for cations and anions that belong together one obtains

(2 designates such a group)

430

4. THERMODYNAMIC PROPERTIESOF ELECTROLYTES

A~.~ _ ~

~

z;

i z_, I_

RT +-F

t~ , Izql

2n c,(A)

'

+ }

vI(B)

2n

_ j~ t] 2n ~,j+(B) . z]

"y:(A)

~j(A)

(4.9.5)

For very dilute solutions only the first term need be retained. It

is

clear

that

the

Junction

potentials

cannot

be

unambiguously determined because they involve the activities of individual

ions which cannot be measured

experimentally.

On

the other hand, if activities can be calculated successfully by the Debye-H(ickel limit

of

dilute

concentrations. species

relation solution

the

evaluated.

activities

are

In the

replaced

by

It is seen that at room temperature each ionic

contributes

[a~(B)/a~(A)]

then ~j may be

volts

of

the

order

of +

0.0592

(t•177

to ~j, and that the portions

log

arising

from

cations and anions tend to cancel.

Thus, unless ai(B)/ai(A ) is

of

is

order

I0

or

more

and

t•

of

order

unity

for

a

particular species,

the value of ~ at room temperature remains

well below 0.06 V.

Equation

be minimized

by

taking

(4.9.4) shows further that ~j can

several

precautions:

One

should

use

salts of the same valence type having a common cation or anion such

as

KC2,

KBr

or

concentration

of

electrolyte on both sides of the junction should be equal;

and

the transport numbers

Na2S04,

for the various

comparable.

In many practical

be

to

reduced

experimental

K2SO 4;

+0.003

V,

the

ionic species

cases of interest,

which

is

within

the

should be

~j can then range

of

error of many measurements.

As a practical matter,

liquid junction potential

are frequently minimized by use of salt bridges,

effects

involving

separate region connecting the two electrode compartments. solution in the salt bridge consists of high concentrations a

salt

in

mobilities.

which

the

cations

Interdiffusion

minimized by use of parchment,

and of

the

anions

have

various

collodion,

a

The of

comparable

solutions

or agar-agar gels.

is

4~ I

EMF CONCENTRATION AND ACTIVITY DEPENDENCE

EXERCISES

4.9. I For the cell with liquid junction Hg(2)IHg2SO4(s)IK2SO4(aq,m I) llK2SO4(aq,m2) IHg2SO4(s)IHg(~) show that under reversible operating conditions m2

~j

--

(3RT/2F) |

tK+ d2n (m-y+),

Jm

1

where tK+ is the transference number for the K + ion. 4.9.2 Derive an expression for the liquid junction potential of a concentration cell of the type AIA,+B,_(c, )lAv+B~-(cr)IA. 4.9.3 Determine the liquid junction emf for the cell PblH2(P - I atm) HC~(m - 0.01) IHC~(m - 0.1) IH2(P - i atm) IPt, given that t + - 5/6. 4.9.4 The emf of the cell AglAgNO3(a , - 10-3) IAgNO3(a= 10 -2)lAg is 63.1 mV at 25~ The two solutions are separated by a porous plug. Determine the average transference number of Ag + in the solution.

4.10

CONCENTRATION AND ACTIVITY DEPENDENCE OF THE EMF

The dependence of ~ on the activity of all gaseous or dissolved species

is manifested by Eq.

(4.7.2).

At 25~

for which the

standard electrode potentials are generally reported, one finds

_

~o

_

0.05915

uj ~ aj.q

log

a,us

(25 o C, E in volts), (4.10.1)

where the numerical value cited includes not only the

factor

2.303

for conversion

from natural

RT/F but also

(~n)

to common

(log) logarithms. To calculate ~ it is thus necessary to know both ~o and the various

activities

aj

and

as.

One

should

recall

that

j

enumerates the activities of all dissolved or gaseous species, whereas

s

phases.

refers The

to

components

standard

emfs

were

present dealt

as

with

pure

condensed

in Section

4.7;

activities for solutes were discussed in Sections 4.1 and 4.2, and the fugacities procedures

outlined

for gases may be computed according in

Section

3.1.

When

the

to the

Debye-H~ckel

432

4. THERMODYNAMIC PROPERTIESOF ELECTROLYTES

limiting

law may be applied

one has

an alternative

right-hand

side

the limiting

method

of Eq.

for c o m p u t i n g

(4.10.1).

At

room

in S e c t i o n

4.2,

activities

on the

temperature

(250C)

law reads

log 7• - - 0.5092 where

in the form shown

z+Iz_14~,

(4.10.2)

S - ~ ~(j)z~cj. As an i l l u s t r a t i o n

of this p r o c e d u r e

H2(P) IHC~(m)IAgC2(s)IAg(s),

for

which

consider

the

half

the cell Pt,

reactions

given by the pairs ~H 2(g) - H + + e- and e- + AgC2 - C~- + Ag, a

net

reaction

89 z(g)

+

AgC2(s)

-

Ag(s)

+

H+

+

are for C~-

Correspondingly,

RT -

(as+) (ac, -) (a~) 2n

-

a1~2

Considerable

.

(4.10.3)

simplification

is a c h i e v e d at one a t m o s p h e r e

and 300 K.

In that event a ~ c t ~ a ~

it

an

is

then

Finally,

excellent

_ ~o _ (2RT/F)

enables

of m o l a l i t y however,

a measurement in

solution

+ (2RT/F)

We (4.2.5),

next

then

using

We first

to

set

for H 2 gas

aH2 -- P H 2 -

, whence

of the mean molal activity of HC~ in the

tabulation

the ~ value

of s t a n d a r d

for the s c h e m a t i z e d

is used in reverse;

the mean activity

coefficients

be

The

determined.

the present

~n m~ - ~o _ (2RT/F)

the

in the following

cell.

i.e., from

procedure Eq.

2n (7• (m)).

extended form

emfs

for is

ions now

example.

set a• (m) - 7 • (m) m~ and rewrite

utilize

I.

(4.10.4)

the procedure

of emfs may

illustrated,

-

m and

one to calculate

Normally,

(a• (m) )2

Further,

~n(a• (m)).

One notes that a knowledge a solution

approximation

aH+ (m )ace_(m)

we write

- I.

Debye-Hfickel

(4.10.4)

as

(4.10.5) equation,

Eq.

EMF CONCENTRATION AND ACTIVITY DEPENDENCE

43

2v,v_

0. 5092z+ Iz_ 14~ log

On

V ~(m) -

-

+

Cm~.

(4.10.6)

v++v_

setting

4.2.2 ),

z+-

converting

introducing one finds

-

Iz_l - u + Eq.

(4.10.6),

v_-

i, ~ -

(4. I0.5 )

to

~

(see

common

Exercise

logarithms,

and using appropriate numerical

factors

(T = 25~

+0.11833

-i+0 . 5 0 9 2 ~ ] J

iog

_ ~o _ 0 I1833Cm~. (4.10.7)

Measurements of E are then taken for a variety of m~ values very

dilute

solutions,

yielding

a set of L values.

As

in

seen

from the expression on the right, a plot of the left-hand side, L, versus m~ should produce molalities where Eq.

a straight

(4.10.6)

line

in that range

is found to be valid.

slope one may determine the value of C appropriate

of

From the to the HC~

solution under study.

Extrapolation of the straight line back

to

as

m~-

0

yields

~o

the

intercept.

This

provides

a

convenient alternative method for determining the standard emf with respect to molality;

in the present case ~o _ 0.22234 V at

25~ Next, should

be

one returns noted,

to Eq.

without

(4.10.5), which was derived,

recourse

to

the

Debye-H(ickel

as

Law.

This relation may be rewritten as

log

- - 0.11833

With ~o known, m~ to

obtain

the

- log % .

(4. I0.8)

one now measures ~ at any desired value of corresponding

value

of

7•

Very

determinations of 7• are obtained in this manner. that the extended Debye-H~ckel very good approximation -0.I

Law,

as given

accurate

It turns out

in (4.10.6),

to the actual 7• (m) up to values

is

of m~

molal, but for greater HC~ concentrations it is necessary

to determine 7• (m) experimentally using Eq.

(4.10.8).

4:34

4. THERMODYNAMIC PROPERTIESOF ELECTROLYTES

One must recall that whereas Eq. (4.10.6) is unrestricted, Eqs.

(4.10.7)

uni-unlvalent

and

(4.10.8)

electrolyte

are at

must be suitably generalized

specialized

room

to

the

temperature

and

to be applicable

case

of

a

therefore

to other cases.

EXERCISES

4.10.1 (a) Devise a galvanic cell for which the reaction H2(P-I arm) + AgBr(s) - HBr(m) + Ag(s) may be carried out. (b) easurements show that at 298 K ~ o _ 0.07103 V and ~ - 0.27855 V for m - 0.02 molal. Determine v• and compare this with the value calculated on the basis of the Debye-H~ckel Theory. 4.10.2 In the operation of the cell Ag(s)IAgBr(x2) in molten LiBrlBr2(l atm) the emf at 500~ was found to be 0.7865 V when AgBr is the electrolyte, and to be 0.8085 V at a mole fraction x 2 - 0 . 5 9 3 7 . Determine the mean activity coefficient for AgBr. 4.10.3 At 5000C and I atmosphere the emf for the cell Ag(s)IAgC~(xl) in molten LiC~IC~2(I atm) was reported as follows: xI 1.000 0.690 0.469 0.136 ~(V) 0.9001 0.9156 0.9249 0.9629 (a) Determine the activity coefficient of AgC2 at these mole fractions. (b) Calculate the Gibbs free energy of transfer of one mole of AgC2 from its pure state to a solution at 500~ in which its mole fraction is 0.469. 4. i0.4 In the operation of the cell Pt, H2 (P-I atm) IHC~(c)IAgC~(s), Ag(s), Pt. at 300 K the following emf measurements have been reported: c (M) i0 -I 5x10 -2 10 .2 i0 -3 I0 -4 10 .5 I0 -6 ~(V) 0.3598 0.3892 0.4650 0.5791 0.6961 0.8140 0.93 (a) Write out the half reaction and net reaction. (b) Determine V• corresponding to c - 10 .2 , 5 x 10 -2 , 10 -I M; compare the values so determined with those calculated from the Debye -H(icke I Law. 4.10.5 Consider the cell schematized as Ag(s) IAg,m3(s)l NaC~(aq,ml) I Na(Hg)INaC~(aq,mz)INa3PO4(aq,m3)IAgC2(s). Taking account of nonideallty, derive an expression for the cell emf in terms of ml, mz, and m 3. What is the function of the Na3PO 4 in the operation of the cell? 4.10.6 Consider the cell schematically indicated as: H2(PI) ,Pt[C2HsOH , CH3NH3C2(cl), CH3NH2(c2) [AgC~(s)[Ag(s), where C2H5OH is the solvent for CH3NH3C~ which is in a saturated solution formed with solid CHsNH3C2. C2H5OH is also the solvent for CH3NH2, which is in equilibrium with gaseous CH3NH 2 that is

435

TYPES OF OPERATING CELLS

m a i n t a i n e d at pressure P2. At 25~ the emf of a cell, o p e r a t i n g under conditions where Pz - 0.983 arm and Pz - 4.15 x 10 .3 arm, is 0.697 V. (a) What is the emf of the cell w h e n P z - P 2 - i atm? (b) What is the overall reaction for the o p e r a t i o n of this cell? (c) Determine the e q u i l i b r i u m constant for the cell reaction. 4. i0.7 For the 'CaC22 cell' operating at 25 ~C, Ca(Hg) ICaC~2(m)IAgC2[Ag, one finds the following data" m (molal) 0.0500 0.0563 0.0690 0.1159 0.1194 0.1305 0.1538 0.2373 (V) 2. 0453 2. 0418 2. 0348 2. 0205 2. 0175 2. 0147 2.0094 1.9953 Determine ~ for the cell reaction and evaluate 7• (m) for CaC~ 2 in w a t e r w h e n m - 0.02, 0.2 M. 4.10.8 The results reported below pertain to the operation of the cell at 250C 9 PtlH2(latm) IHBr(aq,m)l AgBr (s)IAg. m (xl0 -4 molal) 1.262 1.775 4. 172 i0.994 18.50 37.18 (V) 0. 53300 0. 51618 0.47211 0.42280 0.39667 0.36172 m (molal) 0.001 0.005 0.01 0.02 0.05 (V) 0.42770 0. 34695 0. 31262 0. 27855 0. 23396 m (molal) 0. i0 0.20 0.50 0. 20043 0. 16625 0. 11880 Determine the activity coefficient for HBr when m 2 x 10 -4 , i0 -3 I0 -2 0 05 0 I0 0 50 4.10.9 For a cell in which the half reaction Fe(CN)~- Fe(CN)~- + e- and Mn 2+ + 4H20 - MnO% + 8H + + 5e- are to be c a r r i e d out, indicate the dependence of ~ on the mean activities of all species p a r t i c i p a t i n g in the cell reaction. 4.10.10 The following data are reported for o p e r a t i o n of the cell at 25~ 9 ZnlZnC22(aq,m) IAgC2(s)IAg(s)" m (molal) 2. 0941xi0 -3 7. 814xi0 -3 i. 236xi0 -2 2. 144xi0 -2 ~(V) I. 1983 I. 16502 i. 14951 i. 13101 m (molal) 4. 242xi0 -2 9. 048xi0 -2 0. 2211 0. 4499 ~(V) i. 10987 1.08435 1.05559 i. 03279 Determine 7• for ZnC~ 2 at m 10 -3 , 10 -2 , 10 -I molal. 9

4.11

P

"

9

9

P

9

9

TYPES OF OPERATING CELLS

The following

types of operating cells are in common use"

(a) Chemical provided

Cells"

in Section 4.7.

An example

of this type of cell was

We list a second example"

4~6

4. THERMODYNAMIC PROPERTIES OF ELECTROLYTES

ZnlZnC~2(m), reaction

AgC2(s)IAg(s),

represented

2C2-(2m).

By

the

by

which corresponds

Zn(s)

standard

+ 2AgC2(s)

to a net c h e m i c a l

= 2Ag(s)

methodology

+ ZnZ+(m)

described

earlier,

+ we

obtain

_ ~o

3RT _

~o

Here

.

2F

_

~o

_

__

2n

(azn++

a•

-

~o

it is a s s u m e d

_

~n

2F

that

[(41/S)mT•

the o p e r a t i o n

of one atm,

that a d e t e r m i n a t i o n

(4 " ii " I)

is c a r r i e d

so that a• - i.

of ~, c o r r e s p o n d i n g

the c o m p u t a t i o n

of 7• (m).

out c l o s e

in the p r e c e d i n g

(b) E l e c t r o d e made

up

of

to a g i v e n v a l u e of m,

The c o n t r i b u t i o n s

concentration reactive

of

gases

The

electrodes electrode

over

first

case

is

at c o n c e n t r a t i o n s

electrodes. Znx(cl)

The

Cells: that

In this case a cell is differ

materials

or

net

illustrated

in

which

by

involve

solely the

reactions

for

the

the

example of

equilibrium

cell

as

is e s t a b l i s h e d

when

For this

n-

Zn

written

Evidently,

a t = at,

in w h i c h

of

Zn, in

are:

leading for this case Kq

cell

RT ae - - - - ~n---, nF ar

amalgam.

the

left and r i g h t h a n d

- x Z n 2+(c) + 2xe- and x Z n 2+(c) + 2xe- - Znx(C r),

- a l / a r - I, and ,go = 0.

where

in

pressure

amalgams

c I and c r on the

to the c o m b i n e d r e a c t i o n Znx(ct) - Znx(Cr). case

have been

the e l e c t r o d e s .

Hg(c e) IZnS04(c)IZn,Hg(cr), mercury

of A g + and

analysis.

Concentration

two

to

Once a g a i n we see

C~- ions a r i s i n g from the s l i g h t s o l u b i l i t y of AgC~(s) neglected

a~t-)

2F

3RT ~n

a total p r e s s u r e

permits

RT

RT a~azncl RT - - 2n z _ ~ . ___ ~n aznct 2F a~c, azn 2F 2

-

2x if Zn x is the m o l e c u l a r

(4.11.2)

aggregation

of Zn in the

TYPES OF OPERATING CELLS

The

437

second

case

H2(P e)IHC~(c)IH2(Pr),

is

illustrated

by

the

Pt, for w h i c h the p r o c e s s e s

ce ii

read"

Pt,

Hz(P I) -

2H+(c) + 2e-, 2H+(c) + 2e- = H2(Pr) to y i e l d a net r e a c t i o n H2(PI) -H2(Pr).

Once again the e q u i l i b r i u m

constant

is unity,

~o_

0,

and therefore

RT P~ - ~ 2n --.

(4.11.3)

Pr As

expected,

whether then

~ >

AG d
Po. Third, simplify.

eliminate This yields

c

in

(6.1.15b) (6.1.9)

by

use

of

(6.1.13)

and

SHOCK PHENOMENA

mt -

529

[Vo/(V o - v)

Fourth,

- l]up

eliminate

- v p u / ( v o - v)

m via

(6.1.I0)

- u / ( v o - v).

and

solve

the

(6.1.16a)

resultant

for

u2 -

(P-

Po)(Vo- v).

Finally,

e-

eo -

which

(P

use

+

(6.1.16b)

(6.1.16b)

Po)(Vo-

is k n o w n

h

Po)(V

-

(P

(c)

We

perfect

gas

PV - n R T -

-

now

to o b t a i n

(6.1.17)

as H u g o n i o t ' s we m a y w r i t e

ho

(6.1.15a)

v)/2,

h e - e o + PVo,

-

in

+

equation.

If we

set h - e + Pv,

(6.1.18)

Vo)/2.

specialize

as a w o r k i n g

considerably

substance.

by

dealing

with

the

Then

(m/M)RT,

(6.1.19)

or

Pv - R T / M

(6.1.20)

and w.

e - cvT + c o n s t a n t

-

(Cv/M)T

w h e r e M is the m o l e c u l a r at c o n s t a n t Use

+ constant,

weight,

(6.1.21)

a n d Cv the m o l a r

heat

capacity

the

right

volume.

(6.1.21)

on

the

left

and

(6.1.20)

on

of

(6.1.17)" -

IR

(Cv/M) (T - To) - ~. ~ ( P + Po)[ (To/Po) -

(T/P) ].

(6.1.22)

30

6. IRREVERSIBLETHERMODYNAMICS

Note

the m a n n e r We

P/Po,

in w h i c h

rewrite

in terms

of w h i c h

Cv(T - To) - R ( n

Now collect

e has b e e n e l i m i n a t e d

the above

+ I)

terms

T(2C v + R + R/n)

by d e f i n i n g

Eq.

(T O

(6.1.22)

a shock

in favor strength

becomes (6.1.23)

T/n)/2.

-

of C v. by ~ -

in T and in T O 9

- (2~

(6.1.24)

+ R + IIR) To,

or

T

2Cv + R + fIR

To

2C v + R + R/If

(6.1.25)

For II >> R this

T/To ~

relation

reduces

(6.1.26)

[R/(2Cv + R)Ill.

The factor on the right appears for

to

it a new

symbol,

so f r e q u e n t l y

that we introduce

~, - R/(2C v + R) - R/(Cp + Cv).

We

then

obtain

T/T o -

(i + p,H)/(l

(6.1.27a)

+ ~,/H)

~,H

for II >> I.

At h i g h

T, Cv ~ 3R/2

for a m o n a t o m i c

diatomic

gas.

diatomic

gases respectively.

information

Hence,

on

the

(6.1.27b)

T/To

rise

~

n/4 Note

in

(d)

and diatomic

On

considerable

the

basis

n u m b e r of

or

and Cv ~ 5R/2

H/6

for

for a

monatomic

or

the route we took to o b t a i n

temperature

s h o c k e d and note that the a s y m p t o t i c monatomlc

gas,

when

limits

an

ideal

gas

for T/To d i f f e r

is for

gases.

of

the

above

interrelations

we

can

now

establish

using various

a

algebraic

531

SHOCK PHENOMENA

manipulations.

For instance"

(1) We can find the ratio

P/Po"

P/Po

from

(P/Po)(To/T) - 9(1 + #,/~)/(I + ~,ff)

- (n + ,.)/(z + ,.n) i/#, Thus, 6

(6.1.28a)

for n >> i.

(6.1.28b)

P/Po,

there exists a distinct upper limit on

for

monatomic

and

diatomic

gases,

for

very

of 4 and of large

shock

strengths. (li) Information on the mass flow velocity first using

e-

(6.1.21)

is obtained by

to determine

(6.1.29)

e o - (Cv/M)(T - To) ,

and then using this result in (6.1.15a), Eq.

(6.1.27),

and reintroducing H - P/Po.

eliminating T through This yields

u2 2~MC-~v]To[ (H- I)2] mm

(6.1.30)

~S'

H+~,

which

shows

that

there

exists

strength and mass flow velocity

a

connection

between

in a perfect gas.

(ill) We may eliminate To for the u n d i s t u r b e d medium Eq. (6.1.30) by recalling

shock from

(6.1.8) and noting that ~s - R/Cv(I +

~); on carrying out the indicated operation and taking square roots of the resultant we find"

u___I co

2 ~(~ + i)

(n - z)~l ~2 ~ + ~,

(6.1.31a)

(6.1.31b) 0.716 4~

monatomic

gas, H >> I

(6.1.31c)

32

5

6. IRREVERSIBLETHERMODYNAMICS

0.890 ~

diatomic

gas, H >> I.

(6.1.31d)

For 11 sufficiently large, u/c o > I; i.e. , the mass flow v e l o c i t y becomes

supersonic.

(iv)

Let us

examine

the

ratio

c/c o next.

We b e g i n

with

(6.1.13)

C m

U

(l

-

- (n

po/p)Co

t-

II+#a

[-.~j

(6.1.32).

-

~

,

(n

+oi

(6.1.28a).

.

.

Eliminate

u/c o in (6.1.32)

from

This yields

-

,, .

+ ~,,)]

(6.1.32)

II+p~

c

Co

[(t + ~,,n)l(n

- i) (i - ~,,1

where we had used

-

'

i

Co

1)(z

~,)

~(~ + 1)

+

1)(1

-

(n-

1) z] z/2

(n + # ,

)]

i12

2(II + #s)

.

-y(~

2 -

(6.1.33)

#,)z

This relation may be simplified by noting

from the d e f i n i t i o n

of ~, and V that ~, - (V - l)/(V + I) and i - Ps - 2/(V + I). Then

C/Co-

r

+ 1)/2v](n

+ ~,)

(6. I. 34a)

for H >> I

0.895 ~ ,

monatomic

0.926 ~ ,

diatomic

(6.1.34b)

gas, H >> i

(6. i. 34c)

gas, II >> I.

(6.1.34d)

A comparison of (6.1.34) with

(6.1.31)

establishes

that c > u;

the shock wave will always outrun the mass v e l o c i t y of the gas. c/c o - M, is called the MaGh number.

533

FIRST AND SECOND LAW IN LOCAL FORM

(d) We can write a shock equation of state by defining

-vo/v"

n-

P/Po

I'/. Then (6.1.28c) may be rearranged to read

(,7 - . , ) / ( I - ~,.).

(6.1.35)

Compare this to the case of the reversible, of state ~ - 7 7

adiabatic equation

and to the reduced isothermal equation of state

EXERCISES 6.1.1 Calculate the fractional rise in temperature for an ideal monatomic and dlatomic gas subject to adiabatic shock strengths H - I0, i00, i000. Compare with the fractional rise obtained under similar conditions for reversible adiabatic compressions. 6.1.2 The velocity of sound in water at 30 ~ is 1.528 m/sec. Find the compressibility K (l/p) (dp/dP) at that temperature. 6.1.3 (a) Assuming adiabatic conditions to apply, derive an appropriate equation for the sound velocity c in terms of T for a gas at relatively low pressure. (b) Taking V - 1.41, and an average molecular weight M - 28.9 g/tool, calculate the sound velocity in air at room temperature and the change in sound velocity in air with temperature at 273 K. 6.1.4 Prove the following relations involving shocked materials : m 2 - PPo ( P - Po)/(P - Po) u2 - (P - Po)(P0 - P)/PPo ~,- (vi ) / ( ~ + z).

6.2

IRREVERSIBLE

THERMODYNAMICS :

INTRODUCTORY COMMENTS - THE

FIRST AND SECOND LAWS IN LOCAL FORM (a)

In

the

phenomena

concluding

of

flow

part

of

of matter

and

examination of thermodynamics flow

necessarily

different

portions

involves of

a

our

study

energy,

we thus

deal

initiating

in its literal sense. nonequilibrium

given

system

with

states generally

the an

Any such in

which display

534

6. IRREVERSIBLETHERMODYNAMICS

different physical properties.

To deal with this situation we

subdivide the system into many subunits; volume

of

each

subunlt

tends

to

in the limit when the

zero,

every

intensive

and

extensive property will have been specified as a function of position. uniform

Each system

field 4s(r)

of

the

will

n

have

defined

thermodynamic been

everywhere

properties

replaced inside

by

an

41

of

a

instantaneous

the boundaries

of

the

scrutiny

of

total system. We

render

such

thermodynamics which makes

a

system

by establishing

two assertions:

subject

to

the

the Principle

of Local

(i) The instantaneous

State,

values

of

all thermodynamic quantities 4i at any given point satisfy the same

general

thermodynamic

principles

and

relations

as

the

corresponding quantities for a large copy of that small region at that

instant of time.

thermodynamics case.

of

This will permit

equilibrium

us to extend

configurations

to

the

the

present

(il) The local, instantaneous gradients in 41, and their

rates of change, do not enter the description of the states of each local system.

This point addresses the fact that at any

point r all relevant parameters are likely to be characterized by

different

gradient.

values

in contiguous

regions,

and hence,

by

a

Nevertheless, as long as the variation of ~i from one

region to the next is 'sufficiently small'

(what sufficiently

small means must be decided by experiment),

it may be left out

of account.

This represents

an assertion

that is verifiable

only by appeal to experiment. In general ~i will also depend on time; in such a case one must specify not only 41(r,t), but also a corresponding velocity function v(r,t) where

one

to describe a process.

wishes

to

restricted

interval

properties

of the

treat of

the

time.

subsystems

There do exist cases

evolution If

it

so

of

a

system

happens

remain unaltered,

in

a

that

the

and only

the

characteristics of the surroundings change, then the various ~i remain independent of t in the time interval under study and the system is said to have reached steady state conditions. more

precise

Section 6.4.

specification

of

this

state

is

furnished

A in

535

FIRST AND SECOND LAW IN LOCAL FORM

(b) In discussing properties is conventional

to represent

of inhomogeneous

extensive variables

systems

it

in terms of

specific q u a n t i t i e s -

that is, quantities per unit mass rather

than per unit volume.

When such quantities are multiplied by

the density and integrated over the volume obtains the total extensive variable. quantity element

per

unit

mass

whose

of the system one

Let 4(r,t)

distribution

over

represent a the

volume

dSr is governed by the density function p(r,t).

The

extensive variable for the entire system is then given by @(t) - ~v

P(r't)~(r't)d3r' (t)

where

-

-

(6.2.1)

"

V is the volume

enclosed by

the boundary.

One should

note that V in general may be a time variable quantity. To determine take

into

the rate of change of @ (d@/dt - @ )

account

that

not

only

the

integration limits change with time.

integrand but Reference

shows that the system whose initial boundary indicated by whose

the solid

boundaries

general

this

involves

center-of-mass may proceed from

the

are

curve

passes

schematized

by

a deformation

flow with velocity v.

in three

central

steps.

First,

of

the

also

to Fig.

the

6.2.1

is schematically

in time the

we must

dt

to a system

jagged system

curve; due

in

to a

The evaluation of d@/dt there

is a contribution

region V" encompassing the two overlapping

(v...t FIGURE 6.2.1 Change in volume of a barycentric flow. A" and A"" are boundaries adjacent to V" and V'".

;. system subjected to a the original and new

536

6. IRREVERSIBLE T H E R M O D Y N A M I C S

volumes,

drawn

as

a

shaded

region

in

contributes a quantity dt~v, (8~p/at)dar

Fig.

6.2.1.

to ~dt.

This

In the limit as

dt ~ 0, and when keeping only first order terms, one may ignore the difference between due to a

V"

and V.

Next, there is a contribution

V" that is newly

volume

occupied in the deformation

process.

This is composed of the e l e m e n t s -

p~(v.n)d2rdt, where

n is

Outer

element

the

unit

original boundary,

normal

to

the

surface

d2r a t

the

and v is in the direction of the velocity of

motion of the particles

as

they

cross

f r o m V" i n t o

V'.

It

is

transfer

of

A

clear

that

material

p~(v.n)d2r

represents

(more p r e c i s e l y ,

o f p~)

the

rate

across

the boundary

direction normal to the element of area. one

obtains

a

volume

element

of

d2r i n a

When multiplied by dt

containing

the

material

transferred in time dt across the boundary along the direction of flow.

The minus

sign arises because

n is the outer unit

normal, whereas ~dt represents the _increase in ~ in the system. (Thus,

when

v

is precisely

oppositely

directed

to n

then

a

positive contribution to ~ results in the amount dtp~vd2r. ) The overall contribution to ~dt due to the transfer just described isV"

dt L - p~v.nd2r, where A" is the bounding surface separating ^ from V'.

A

similar

from the volume V " two -

terms

may

- dt L .p~v.nd2r, arises ^ in the transfer. The latter

relinquished

now

dt~A(p~)v.nd2r,

contribution,

be

for

combined

which n

into a

differ

in

sign

magnitude as one passes from region V" to V "

- -

~A

(t)

(p~)v-nd2r + A -

and when Gauss'

-

-

single

integral,

as well

as

in

. It follows that

r ~V( t ) 8(p~)d3 "'at -'

(6.2.2)

theorem is applied (see Table 1.4.2, line (j))

one obtains

I

~V

{[a(p~)/at]

- v.(~pv)} d3r.

(t)

Equation

-

(6.2.3a)

transport equation.

is

-

often

(6.2.3a)

--

referred

to

as

the

Reynolds

FIRSTAND SECOND LAW IN LOCAL FORM

537

$

Note that if we write

Jv(t)(dp~/dt)dSr then Eq. (6.2.3a)

I

can be cast in the local form

d(p~)

a(p~)_ v.(p,~v)

I

dt

at

(6 2 3b)

-

" "

for any specific ~ which is an extensive quantity per unit mass of the system. extensive

It should be evident that if ~ is chosen as an

quantity per unit volume

rather

than per unit mass

then the density factor p may be dropped from Eq. Equation

(6.2.3b)

is

typlcal

of

whenever conservation laws apply. rate

8R/at;

i.e.,

of production this

term

form

is governed by two terms:

(or disappearance)

can

be

encountered

The overall rate of change

of the quantity p~ i R, i.e., dR/dr, (1) The

the

(6.2.3b).

traced

to

the

of R

locally,

occurrence

of

processes totally within the system, without referring to flows across

boundaries.

(il)

The balance

outflow to the surroundings,

between

influx

from or

as expressed in the divergence of

the flux vector _V'~, with J~R i Rv,_ which was earlier related to flows or transport across boundaries,

as specified by Stokes'

Law.

One should note that (6.2.3b) or its equivalent

d

-

/dt

( aR/at

)

-

(6.2.3c)

v.

is much more restrictive

than (6.2.3a),

balance

outflow,

between

dissipation,

influx,

not

polnt-by-polnt,

only local

for

the

basis.

and

in that one demands rate

system A

as

of

generation

a whole,

relation

of

but

this

a or

on

a

form

is

R

is

designated as an equation of continuity. In

certain

physical

situations

indestructible or uncreatable: destroyed

locally.

the

quantity

It can neither be generated nor

In such circumstances

8R/at,

the rate of

local generation or annihilation of R, must necessarily vanish. For this special case

dR/dt --

V-JR,

(6.2.3d)

53~

6. IRREVERSIBLETHERMODYNAMICS

which

implies

solely

by

outflow,

a

that net

local

change

changes in

in R

the

can be

balance

brought

between

about

influx

as expressed in divergence term - ['JR_. Eq.

and

(6.2.3d)

is a conservation equatlon.. (c) We next Second

Laws

in

turn

to

local

the

establishment

form.

This

will

of

be

the

done

First

and

under

the

important restriction that there be no motion of the center of mass of the local system" volume

deformations

v-

need be

O.

In these circumstances

considered"

dV/dt

= O,

no

thereby

greatly simplifying the analysis, while not unduly restricting its applicability.

Reference to a generalized treatment of the

problem with v ~ 01 shows that the present restriction involves the dropping of a P(dV/dt) the

First

Law,

and

of

term from the local formulation of a

tensorial

pressure-volume

term

appropriate to anisotropic media from the local formulation of the First and Second Laws.

However,

since we will continue to

confine our studies to isotropic media, and since in the steady state dV/dt vanishes, all results cited later are attainable by the

present,

more

restricted

approach.

We

divide

the

subsequent discussion into several subunits. (i) Consider a system in which n different chemical species k (k-

I, 2,...,n) are subject to r distinct chemical reactions

(~ - 1,2,...,r).

In an extension of the procedure of Section

2.9 we now represent the ~th chemical reaction by X(k)VktAk -- O, where

the Vke are

the

appropriate

stoichiometry

coefficients

matched to the various chemical species A k in reaction ~. quantities may have either sign, Then,

if AI represents

designate dAi/dt. or

the

each

of

as discussed advancement

advancement

of

this

of

in S e c t i o n

2.9.

reaction

~ we

reaction

by

~i -

The product PkVki~i then represents the mass production

depletion

chemical

rate

a unit

These

rate

per

reaction.

species

may be

unit

volume

Furthermore, subject

to

of

we

species

allow

any

for

k

in

the

conservative

the

fact

~th that

external

IS. R. de Groot and P. Mazur, Nonequilibrium Thermodynamics (North-Holland, Amsterdam, 1962).

FIRSTAND SECOND LAW IN LOCAL FORM

force per

unit

S3 9

mass fk --- _V~k, where ~k is an appropriate

time-independent potential per unit mass. that while v -

0, individual particle

It should be noted

flows do not vanish;

in

fact, we write the mass flux vector for species k as Jk m PkVk, where Pk is the density, unit mass,

of the kth species.

ECk)PkV_k/P -- I - (it)

Note that in the present case

O.

We now t u r n

of mass.

and v k the average drift velocity per

to an analysis

of

the

law of conservation

The time rate of change of mass for a given species

within a fixed volume is equal to the net rate of influx across the bounding surfaces plus the net rate of generation of k in chemical reactions.

Thus,

r

(dPk/dt)d3r ---

Pk[k "nd2r +

X ~--I

(6.2.4)

PkVkeWtd3r,

where the minus sign occurs because n is the unit normal vector directed toward the outside. the

last

negative;

term

in

(6.2.4)

Observe that for some species k

is positive

for

others

it

is

this balances out in such a manner that there is no

net formation or disappearance that occur locally. 1.4.2),

and

Eq.

of total mass

in the reactions

On introducing Gauss' Law (line (j), Table

(6.2.4) reduces to

r

dPk/dt

1-

V.J k +

(k-

..., n).

(6.2.5)

expressions for energy

balance.

~ PkVk,~,,

i, 2,

s

(ill) We

next

construct

Since by assumption v of

the

total

center

of mass

potential

energy

0, the kinetic energy term for motion vanishes. density

The

rate

of change

p~ m Z(k)Pk~,

so

long

of

the

as

one

considers only time-independent potentials ~k, may be determined from

(6.2.5)

by

multiplication

subsequently summing over k"

of

both

sides

with

~k

and

S40

6. IRREVERSIBLETHERMODYNAMICS

n

--z-(

dp~ dt

+ k-I

r I

+

x k-I

n X Pk~"k,', - -

(n V.

I ~_Jk

i-i k-i

Here

we

noted

n

-

have

written

V.(~Jk)

(6.2.6)

I Jk'fk k-I

k-i

-- ~ V . J k

+ Jk'V~

the fact that Z(k)Pk~Vkl--0 because

and

we

have

the total potential

energy of the system is conserved in every chemical reaction. In the preceding discussion and in what

follows below we

have introduced to the bar symbol to emphasize that we refer to a specific mass

(i.e.

per unit mass)

flux vector;

quantity

although ~ J k

is dimensionally

quantity.

Note

that Jk is a

has a strange appearance

correct"

Jk-

PkVk, whence

this

~Jk

-

#k~/k iS the rate of transport of potential energy density. (d)

As

a

Thermodynamics

further which

step

we

requires

introduce that

local system be strictly conserved.

the

the

total

First

Law

energy

of

of the

This means that any change

of total energy U can occur solely through influx or outflow of energy across boundaries.

This flux will be characterized by

the flux vector Ju, which corresponds to the energy density pu; m

Ju

-

The

puv.

total

energy

balance

equation

thus

reads

..Iv{(dpu)/dt}d3r - - JAJu-{~dZr; with the aid of Gauss' Theorem we then obtain

the

strictly

local

equation

for

conservation

of

total energy density:

(6.2.7)

d(p~)/dt - - V.J u. (iv) specific

We

now

introduce

energy u and the

relation u -

the

distinction

internal

specific

between energy

total

e by

the

e + @; if the center of mass of the system were in

motion then an additional kinetic energy term would have to be ~ .

added.

The

readily

grasped by an example.

a

a m

distinction between u and e may perhaps

gravitational

potential"

Consider Changes

in

be most

a system placed

in

its energy are then

541

FIRST AND SECOND LAW IN LOCAL FORM

specified by d(Mgz)

- gzdM + Mgdz.

The quantity on the left

refers to the overall change of energy du. the

right

corresponds

transported location

z;

to

a

from

infinity

the

internal,

case

de.

The

displacement

of

the

to the

thermodynamic

second

system

coordinates

leading to a change term,

entire

The first term on additional

and accreted

system have now been altered, energy,

where

Mgdz,

system

mass

at the of

the

the

in internal

corresponds

in

is

to

a

gravitational

potential #s; thus the potential energy of the entire system is uniformly

altered,

but

the

system remain unchanged. by the quantity d~s. + Mgdz corresponds

thermodynamic

coordinates

of

the

This latter alteration is symbolized

In thls elementary example d(Mgz) - gzdM

to d(pu) - d(pe) + d(p~s).

(v) The First Law of Thermodynamics

is frequently written

out in terms of the quantity pe rather than pu, and this is the path we shall follow here.

We specify the total energy flux as

n

(6.2.8) k-I

The reason for the JQ nomenclature

is that we have consistently

regarded the net energy flux density V-J u as arising from the performance

of work,

and heat flow. the

term

V-~Ck)~Jk;

identified

as

corresponding Let

us

the effects of potential

The first a

the net

two contributions

remaining heat

energy

transfer

energy changes, are contained

flux

_ V.JQ,

must

thus

involving

in be the

flux vector JQ. subtract

(6.2.6)

from

(6.2.7);

this

yields

an

expression for the rate of change of internal energy density" n

d(p~) - _ dt

V-JQ + -

~ Jk-fk. k=l

Equation (6.2.9) represents

(6 2 9) " " the local formulation of the First

Law of Thermodynamics when there is no motion of the center of mass.

Note again that Eq. (6.2.7) and (6.2.9) are not averaged

542,

6. IRREVERSIBLE T H E R M O D Y N A M I C S

over a finite volume but must be

obeyed

locally.

Also,

Eq.

(6.2.9) does not satisfy the conservation law except when fk m 0 for all k. energy

This situation arises because e is not the total

density

unless

the

system

is

free

from

all

external

forces. (e) The

Second

Law

is handled

by

recourse

equation (1.18.34) in the form (for d V - 0 ) Set dS - d(sV) - Vd(p~),

~k

~kMk,

-

nk

-

n~/Mk

-

dE - Vd(p~),

TdS-dE

to

the

Gibbs

+ ~(k)~kdnk.

and write (6.2.10)

pkVlMk

Then

T d (p~) d (p~) dt + ~ ~k dt k

Finally,

introduce

dPk dt

"

(6.2.5)

(6.2.11)

and

(6.2.9)

into

(6.2.11);

this

yields

n

-

where A e - -

+ z k

r

+ x k-i

+ z ~=i

~(k)PkVki~k is called the chemical affinity.

that 3 k and ~e refer to specific quantities.

However,

Note

the above

quantities occur exclusively in combinations such as 3.fk, ~kJk, and ~tNe; this permits the following alternative interpretation" 3 k is the flux in moles time,

~k is the

of k past unit cross

usual partial molal

Gibbs

section

in unit

free energy,

fk =

- - V ~ is the negative of the potential energy gradient per mole, Vke~ ~ is the rate of molar concentration change of species k in reaction ~, and A t the corresponding affinity. will be used interchangeably. We now rewrite

(6.2.12)

as

These quantities

543

FIRSTAND SECOND L~W IN LOCAL FORM

n d(pg)_dt

) j ~kJk /T

V-[(JQ-

n - (I/T 2) JQ.VT + (I/T 2)

~. ~kJk .VT k-I

k-I n

-

(I/T)

r

~. JTk.(~T~k - fk) + ( l / T ) k-i

~. ~,A,. ~-i

(6.2.13)

This relation can be split into two types of contributions: first

term

in

(6.2.13)

T-I(_JQ - ~.(k)~kJk). clearly

makes

involves

the

divergence

In the context of Eq.

sense

to define

of

(6.2.13)

an entropy

The

the

flux

it therefore

flux vector

by

the

following relation:

n

Js " (I/T){Jo -

The

remaining

~- ~kJk} 9 k-I

terms

(6.2.14)

on the right

of

(6.2.13)

must

represent

source terms if Eq. (6.2.13) is to be interpreted as an entropy balance equation -

V.J s we

generation

d(p~)/dt--

can

express

0

V.J s + 0. as

the

Having thus identified

rate

of

n

- - (l/T) Js'_VT - (l/T)

recognizing

that

molar

ge.neralized

the

r

_Vk~k -- _fk -- V(~k + ~k) -- V~k,

specific

potential

chemical

great fundamental Note

~

chemical

can be

potential

interest,

that we have

~'k"

potential

combined Equation

succeeded

in setting

into

the Second Law,

- - V-J s + 0,

which should be contrasted with (6.2.7).

the

a specific

(6.2.15) up

thus

P-k and

as will be demonstrated

equation for entropy density, [d(ps)/dt]

density

~. Jk'_V~k + (l/T) ~. wtA, _> 0, k-i ~=i (6.2.15)

in which we have written external

entropy

locally as follows"

is

of

later.

a continuity

in local form, (6.2.16)

We have also achieved

S44

6. IRREVERSIBLETHERMODYNAMICS

an important

separation.

The

term-

V-J s specifies

the net

transfer of entropy density across the boundaries of the local system,

whereas

generation

due

8 to

refers

to

the

(irreversible)

rate

of

processes

within the local volume element.

entropy

density

occurring

totally

In subsequent

sections

this

latter quantity will play a cardinal role. (f) A reformulation eliminating JQ between

of (6.2.15) may be achieved by first

(6.2.8) and (6.2.14);

one obtains

n

Ju -

-

(6.2.17)

I rk_J -

k-1

In the case of a single species

(k-

considered

transported

as the total

energy

Js/J1

species i, UI, likewise,

unit mass of species I, S I. U I-* - T S ~

TS - Pl.

Ju/J1

may be

per unit mass

of

is the total entropy carried per Thus Eq.

- [i, or to its equivalent,

analog of H -

I) the ratio

(6.2.17)

-E *l -

Now substitute

TS~-

(6.2.17)

specializes

to

~i , which is an into (6.2.15);

then a slight rearrangement yields n

- - (I/T 2) Ju'VT + +

r ~ (I/T)~,A, ~-I

-Ju'V(I/T)

-

Z

n

Z

(rk/T2)Jk'VT -- (I/T)

k-I

Jk'V[k

k-i

n r 7. Jk'V(rk/T) + ~ (I/T)~,A, _> 0. k-i ~-i

The form of (6.2.15)

and (6.2.18)

(6.2.18)

is highly significant.

In each case the rate of local entropy density generation, to

irreversible

processes

occurring

totally

within

a

due

local

volume element, may be written as a sum of terms of the general form 8 - E(j)Jj-X_~ >_ O, wherein the Jj represent either general fluxes or reaction velocities, forces.

and the X_~ represent generalized

As already explained in Section 2.2, this nomenclature

arises because 8 - 0 can only occur when equilibrium prevails,

LINEAR

PHENOMENOLOGICALEQUATIONS

545

m

at which

point both X j and Jj go to zero

sense Jj is a response

to the imposition

external

Of

force

development forces

and

-VT),

Such

thermodynamics pairwise

pairs

said

are

(wt, Ae/T);

(Jk/T, ~r~k/T), and (wl,

that no unique of a

on the system of an

importance

that occur

to

In this

to

our

future

are the particular

in the expression constitute

in Eq.

A,/T).

(6.2.18)they

(J,/T,

are

It is therefore

set of such pairs may be set up;

8 -

conjugate

In Eq. (6.2.15) these pairs are respectively

(Jk/T,-~k),

-VT-I),

great

of irreversible fluxes

~.(j)Jj-X_j. variables.

X_j.

for all J.

(Ju,

obvious

the selection

set as a starting point for further development

is then

simply a matter of convenience. Since

a

corresponding

flux

may

be

considered

to

be

~driven'

by

a

force, no flux can occur without a force field,

in which case all irreversible phenomena cease; 8 now vanishes, and Eq.

(6.2.16) becomes a conservation condition for entropy.

EXERCISES

6.2.1 Derive the equation of continuity for a system of constant mass in the form dp/dt + V . p v - O. 6.2.2 Explain why for reactions occurring totally within a system ~ does not contain terms in reaction velocities or affinities [see Eq. (6.2.9)], whereas ~ does: see Eq. (6.2.13). Hint: Consider the possible sources for reaction energies, and how such reaction energies would be dissipated. 6.2.3 Derive the equation r dpNk/dt - - l'_Jk + Z PkUk~k k-1

and explain its relation to the equation of continuity. 6.2.4 In Eqs. (6.2.8) and (6.2.9) JQ was identified as a heat flux vector, yet this quantity corresponds to e, the internal energy density. Consult Sections 1.8 and 1.16 and explain again why this particular designation is appropriate.

6.3 THE LINEAR PHENOMENOLOGICAL

EQUATIONS,

AND THE ONSAGER

RECIPROCITY CONDITIONS (a) We had earlier derived an equation for the local rate of entropy density production when no volume changes occur:

546

6. IRREVERSIBLETHERMODYNAMICS

- (Sps"/ST) - -

T-IJs.VT - T -I

n

r

~ Jk.V[k +

~ ~,(A,/T) _> O.

k-I In circumstances

where

no

2-1

chemical

(6.3.1)

reactions

take

where no particle fluxes occur 8 - - T-IJs.VT _> 0. - 0 only if equilibrium prevails, attain a value

of zero.

place

and

One can have

in which case Js and VT both

In the absence

of any particle

flux

the quantity T-IJs represents the heat flux JQ/T 2, as is evident from

(6.2.14).

JQ.V(I/T).

Equation

According

to

(6.3.1) our

'driven' by a gradient in I/T. problem one postulates the

linear

now

reads

standard

interpretation

JQ

is

In the simplest approach to the

that JQ is linear

relationship

8 - - T-2JQ-VT -

must

be

in V(1/T) ; moreover,

homogeneous,

so

that

no

additive constants prevent JQ from vanishing simultaneously with V(1/T).

The postulated relation

JQ- I..V(1/T) should

(6.3.2a)

hold

under

equilibrium.

Here

temperature quantity, Equation

and

conditions L-

L(T,p)

density;

one

not is

a

too

far

scalar

cannot

removed

function

introduce

a

have

of

the

tensorial

as this would generate a set of preferred directions. (6.3.2a) may be rewritten as

JQ - - (L/T 2)VT - - ~VT,

which

from

represents thus

correctness

Fourier's

recovered of

the

(6.3.2b)

Law of Heat

a well-known procedural

law,

methods

Conduction which

(1818).

attests

adopted

here.

to

We the The

quantity ~ is known as the thermal conductivity of the medium. The same argument may now be repeated

for the case where

again no reactions occur, the temperature is held constant, but now a particle flux of one type is permitted. reduces

=-

Equation (6.3.1)

to

_> o .

(6.3.3)

LINEAR PHENOMENOLOGICAL EQUATIONS

54"/

We now revert to the use of molar quantities

for which Ji.V~i -

(ml/V)v.V(Gi/m i) - (nl/V)v.V(Gi/n i) -Ji.V~i.

We shall omit the

tilde for ease of notation. V~i

is

a

driving

conjugate

molar

is invoked,

force

The form of (6.3.3) to

which

flux for species

T-iJi

i.

is

Again,

suggests

the

responding

a linear

relation

of the form

Ji - L" (T,p)V~i , which

that

connects

(6.3.4a)

the

force

and flux.

The

been absorbed in the definition of L'.

extra

factor

Equation

the form of Fickes Law (1856) for diffusion,

T -I has

(6.3.4a)

is in

usually w r i t t e n in

the form

J - - D Vc,

(6.3.4b)

where D is the diffusion of particles. for

in the

particle

The

absence

flows

(RT/c)Vc,

Eqs.

(6.3.4a) ~ -

Eq.

~

+

are

or when RT

~n

equivalent;

only u n c h a r g e d

c.

Thus,

V~

potential

and Je is the charge

- V(~/e)

and ~ the prevailing

back

from

molar

6.3.2.

concentration

are charged,

-

then one

in the form Je = L"(T,p)V(~/e),

charge,

Now write V(~/e)

Exercise

field, -

if the particles

(6.3.4a)

is the electronic

shift

(6.3.4b)

so that D - - (L'RT/c).

may recast

- eJ.

~

and c, the c o n c e n t r a t i o n

and

of electric

occur,

On the other hand,

-e

coefficient

flux;

Je-

- V~, where ~ is the chemical

electrostatic

to atomic

Suppose

where

potential;

quantities

further

that

the

in the system remains uniform.

is

such a

considered charge

in

carrier

In that event one

finds

_fl'e -

where

L"(T,p)(-

E

V~)

-

L"(T,p)E

is the electrostatic

formulation conductivity.

of

Ohm' s

Law

-

aE,

field. (1826) ;

(6.3.4c) Equation a

is

(6.3.4c) the

is one

electrical

~48

6. IRREVERSIBLETHERMODYNAMICS

(b) These examples concept

of

a

set

suffice to illustrate

of processes

for

which

the more general

the

rate

of

local

entropy density production has the form 8 - ~(J)JJ'X_0, where the Jj are all forces.

the relevant

Pairs

fluxes

of variables

and the Xj,

the corresponding

Jj, X_j, satisfying

this particular

form for 8 are said to be conjugate.

For every such pair one

postulates

of

a

direct

proportionality

the

form

J•

-

LIjX_0

between the various conjugate Ji, Xj pairs.

The validity of the

llnearlty

a

principle

experimental

data

thermodynamics. microscopic

ultimately

and

rests

in no way

However,

on

invokes

comparison

with

a new principle

of

in lowest order of approximation,

a

theory of heat and of mass flow does lead precisely

to such linear relationships. When more than one force at a time acts on a local system, a corresponding number of flows must occur simultaneously. this event

one enlarges

down simultaneous Jl-

LIIXI

JnThese

equations

LIzX2

+

2x2

relations

equations, -

+

on the original

+

"'"

+

+

...

+

postulate

In

by writing

of the form

Lln_Xn

(6.3.5)

are known

as phenomenological

or macroscopic

in which the various Jj, Xj satisfy the relationship

E(j)Jj-X_0 for

phenomenological

i 0;

I) then ~A > O, so

If two processes

thus,

for

occur

example,

it

is

possible to have ~IAI < 0 if ~2A2 > I~IAII. (b) Consider If

the

third

elementary

two reactions

process

reactions

A is

=

C

the

of the type A = B and B = C.

is

not

same

as

feasible the

the

number

linearly

reaction equations;

uncoupled.

If A = C is a feasible reaction the three processes

coupled;

generally,

coupling

occurs

are

of

of

independent

are

the reactions

number

whenever

said to be

there

is a

redundancy in the number of reaction steps. For situations not far removed from equilibrium (what this implies usual

will

linear

be

fully

relations

present context

5R. Haase, (Addison Wesley,

documented between

later), fluxes

one postulates

and

forces.

In

the the

(A, should not be confused with A)

Thermodynamics Reading, Mass.,

of Irreversible 1968).

Processes

580

6. IRREVERSIBLETHERMODYNAMICS

R ~r == X a r , A ,

(r

-

(6.11.1)

I, 2, . . . , R ) .

s-I

Coupled

equations

coefficients:

are c h a r a c t e r i z e d

a=, ~ 0 for r ~ s.

by nonvanlshing

The d i s s i p a t i o n

cross

function

is

given by

R

R

(6.11.2)

~r--~ s--~ ar'ArA" >- 0.

(c)

It

reactions

is

instructive

to

specialize

to

the

case

of

two

(R - 2):

~I = a11A1 + a12A2 a21

-

a12

(6.11.3)

(~2- a21A1 + a22A2. Then

(6.11.4)

-- aliA21 + 2aI2AIA 2 + a22A ~ >_ 0,

which

requires

Where

there

a11 >_ 0,

and a 1 1 a 1 2 -

is no coupling,

a22 >_ 0 (see

Section

2.2).

a12 - a21 - 0; in that event,

~IAI -

a11A12 >_ 0 and ~2A2 - a22A 2 >_ 0. We n e x t

inquire

approximation.

For

characterized general

~

-

eliminate

as to the range

by

n

this

purpose

I

deformation

+

~o(xl, .... ,xn+l) and

xn+ I

of v a l i d i t y

between

the

note

A

two

that

-

But

well,

the

zero

measure

of

conditions

the

deviation

deviation

at w h i c h

of

of A from the

the x i a s s u m e

x~. It is t h e r e f o r e

reasonable

in A w h i l e

all x i - x~;

lowest

setting

order,

one o b t a i n s

a

linear

system

xl,

is

then

in

Ao(xl,...,xn+l) ; one

functions

a n d A - A(xl,...,xn+l).

that

if

coordinates

~(xl,...,Xn,A) so

of the

system

to

obtain

may ~

-

as A ~ 0, ~ ~ 0 as may

from

be

the

taken

to e x p a n d ~ as a T a y l o r ' s only

a

equilibrium

their e q u i l i b r i u m v a l u e s

on r e t a i n i n g

as

the

xI -

series term

of

CHEMICAL PROCESSES

-

S8 I

(a~/aA)~r~o

On writing ~ -

a

I

A

~

+

(i-

...

1,2,...,

n).

(6.11.5a)

aA, one finds that

(ao~/aA)=i.o i

identifies

(6. ll.5b)

the coefficient

a.

To check on the adequacy of the linear approximation we now introduce

~

==

~

the law of mass action in the form

C i

- -

corresponding

C,j~',i ,

SO"

(6. l l . 6 a )

to the schematic

reaction

where, as usual, the v's are stolchlometry coefficients and the A's are reacting species; for

the

forward

(6.11.6b).

=

=

reverse

Now rewrite Eq.

process

(6.11.6a)

as

written

in

Eq.

as

- k~c,',),

u~(1

in which ~ k

and

K and m" are reaction rate constants

(6.11.6c)

is the rate of the forward reaction,

m~c~

i, and

J L

~'/~.

Referring back to Section 2.14 one notes that the affinity may be reformulated

A

-

-

Z

v lpl

"

as

RT ~n K -

RT ~

ce

,

(6.11.7)

!

where K is the equilibrium constant appropriate to the reaction (6.11.6b)

when

~ is referred to the standard chemical poten-

5 8'2

6. IRREVERSIBLETHERNODYNANICS

It now follows that Ke -A/R7 -

tlal.

is introduced ~f(l

-

in (6.11.6c)

c e ; w h e n this e x p r e s s i o n

one flnds

kKe--AIRT).

- -

Now at equilibrium, that k K -

(6.11.S)

~ - A - O; according to (6.11.8)

i, so that one obtains

the final

this means

expression

in the

form

~f(1

~0 -

-- e --A/RT),

which exhibits

(6.11.9)

an exponential

It is now clear

of w on A.

that the p o s t u l a t e d

linear

dependence

in

(6.11.5) obtains only if IA/RTI