Elementary Chemical Thermodynamics [1 ed.]

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Elementary Chemical Thermodynamics

THE GENERAL CHEMISTRY MONOGRAPH SERIES

Russell Johnse n, Editor

Florida Stat, University

Gordon M. Barrow (Case lnstilut , of 'Tahno!~y)

THE STRUCTURE OF MOLECULES

Werner Herz (Florida

State Univmity)

THE SHAPE OF CARBON COMPOUNDS

Edward L. King (Ui1iv1:r~ siiy of Col orado)

HOW CHEMICAL REACTIONS OCCUR

Bruce H . Mahan (University of

Ca lifornia , &rkdry)

ELEMENTARY CHEMICAL THERMODYNAMICS

Gregory R. Choppin (Florida Start Univmity)

NUCLEI AND RADIO· ACTIVITY

Robin Hochstrasser (University

SPECTRA, STRUCTURE, AND PHOTOCHEMJSTR Y OF ATOMS

of P,11n5yJvania)

Elementary Chemical Thermodynamics

Bruce H. Mahan Univtrsity of Cal ifornia, Berkd,y

W. A. BENJAMIN , IN C. Menlo Park, Cal iforni a • Reading, Massachusetts London • Amsterdam • Don Mills, Ontario • Sydney

ELEMENTARY CHEMICAL THERMODYNAMICS

Copyright © 1963 by Bruce H. Mahan All rights reserved

Library of Congress Catalog Cud Number 63- 16370 Manufactured in the United States of America

'flii.s P111P11ua·ipt WllS I™t i11to proJ1mio11 on Dcurnbu 21, 1962; 1kc rolumt u.w published 011 May 15, 1963; .secon.J printing, with cormtions, Ftbruary 18, 1964

"flu pllblislicr wi.!liu to ,u:huwltdg, th, mistan,, of G~lrn Fl«:k , u:fw ,di1r.d thi 1114IIUS.ript,

~na Orm Hunt, UJM produml d,, illllStr.:itiom

Editor's Foreword

T

H.E TEACHING OF GENEI\AT, COEMJSTflY to beginning students becomes each day a more challenging and rewarding task as subj ect matter becomes more diverse and more complex and as the high school preparation of tJ1e student im proves. These challenges have evoked a number of responses ; tJ1is series of monographs for General Chemistry is one such response. It is an experiment in the teaching of chemistry which recognizes a number of the problems that plague those who select textbooks and teach chemistry. First, it recognizes that no single book can ph ysically encompass all the va rious aspects of chemistry that all instructors collectively deem important. Second, it recognizes that 110 single author is capable of \\Tiling a uthoritatively on all the topics that are included in everybody's list of what constitutes general chemist.ry. Finally, it recognizes the instructor's right to choose those topics which he considers to be .i mportant without having to apologize for having omitted large parts of an extensi~e textbook. This volume, then, is one of approximately ftl'teen i11 the General Chemistry Monograph Series, each written by one or more highly qualified persons very familiar with the current status of the subject by virtue of research in it and also conversant with the problems associated with teaching the subject matter to beginning students. Each volume dea ls broadly with one of the subdivisions of general chem istry and constitutes a complet e entity, far more comprehensive in its coverage than is permitted by the limitation of the standard one-volume text. Taken together, these volumes V

vi

Editor's For,word

provide a range or topics from wh ich the individual instructor can easily select those that will provide fo r his class an appropriate coverage of t he materia l he considers most important. Furthermore, inclusion of a numher of topics that ha ve only recen tly been considered for general chemistry courses, such as thermod ynamics, molecula r spectroscopy, aud biochemistry, is planned and these vol umes will soon be availab le. In every insta11cc a modern structura l point of view has bee n adopted with the emphasis on general principles and unify ing theory. These volumes will have other uses also: selected monographs can be used to enrich t he more conventional course of stud y by providing readily available, inexpensive supplements to standa rd texts. They should a lso prove val uable to students in other areas of the ph ysical and biological sciences needing supplementar y informa tion in any fi eld of chemistry pertinent to their own special interests. Thus, students of biology will find the monogra phs on biochemistry, organic chemistry, and reaction kinetics particularly useful. Beginning students in physics and meteorology will find the monograph 0 11 thermodynamics rewarding. T eachers of elementary science will also fi11d these volumes involu oblc a ids to bri11ging them up to date in the various branches of chemistry. Each monograph has scvero.1 features whi ch make it especially useful as a n aid to teaching. These include a large num ber of solved examples a11d problems for the student, o glossary of technical te rm s, and copious illustrations. The authors of the several monographs deserve much credit for their enthusiasm which made this experiment possible. Professor Rolfe f-l erber of Rutgers Uni versity has been of in valuable assista nce in the prepa ration of this series, having supplied editorial comment and numerous valuable suggestions on each volume. Thank s are also due to Professor M. Kasha of the Florida State U11iversity fo r many suggestio11s during the planning stages and for reading severa l of th e manuscripts. Ru SSF.1,1, JOHNSEN

Tallaluusee, Florida

October 1962

Preface

I

N THIS BOOK I have t ried to present thermod ynam ics in a straightforward manner, emph asizing its chemica l applica tions and its ph ysical interpretation while simplifyi ng the ma lhcma tica l development. The book was written because J feel tha t inclusio11 of thermod ynamics in the beginning course is a feasible and important improvement in the chemistry curriculum. The most successful beginning courses have tried lo teach a nd explain chemica l phenomena in terms of a growing set of "' molccula r principles." l t is my experience that this approach is strengthened considerabl y if students have learned to a nalyze a chemical process by thermo.dynamic methods. The consideration of chemica l beha vior in terms of energy a nd en tropy, followed by an explanation or how the magnitude of energy and entropy changes are dictated hy atomic properties, ma kes the descriptive chemistry taught. iii the beginning course a n interesting and coherent subject. All this materia l has been ta ught iii a fi ve-week period to students who had previously been exposed to stoichiometry, colligative properties, chemical equilibrium, electrochemistry, and molecular structure. Concurrent enrollment in a beginning CA icuius course is assumed, but the number of different mothenrntical operations a student must use is, J feel, remarka bl y small . T his treatment of thermodynamics is in tended not to supplant but to supplement the more rigorous development given in the ph ysiCf1 I chemistry course; accord ingly, some topics ha ve been deempha-

vii

Pre.fact

vi ii

sized. The Helmhol tz free energy, whicb bas only limited use in chemical problems, is developed in one of the exercises. The Carnot hea t engine is treated as an application of thermodynamics rather than ns a central (and too often dull and confusing) part of the second-law development. On the other hand : i have given considerable attention to the concept of entropy and I have emphasized the interpretation of entropy changes and chemica l behavior in terms of qualitative molecular properties. The resulting gain in familiarity with the entropy concept should aid the more formal thermod ynamic treatments given in later courses. I wou ld like to acknowledge permission to reproduce th e table I have numbered 1- 1 from Zemansky's "Heat and Thermodynamics," McGraw-Hill, New York, 1951; 0nd Fig. 3- 6 is redrawn from Daniels and Alberty's " Physical Chem istry," Wiley, New York, 1955. The thermodynamic data are taken from ''Selected Values of Chemical Thermodynamic Properties,'' edited by F. D. Rossini et al., N ational Bureau of S tandards Circular 500. BRUCE

Berkeley, California Oclaber 1962

H.

MAHAN

Contents Editor's Forewo rd

V

vii

Preface lntroducti.on

1- 1 l- 2 1- 3 l--4

Thermodynamic Systems Stat.cs and State Functions Equilibrium States Temperature

II The First Law of Thermodynami cs 2- 1 Work and Heat

Ill

l2 l2 17 21 27

2-2 Pressure-Volume Work 2-3 E nthalpy 2--4 Thermochemistr~; 2- 5 Enthalpies of Formation 2- 6 Bond Energies 2- 7 Heat Capacities 2- 8 T emperature D ependence of .o.H 2--9 Explosions and Flames 2- 10 Idea l-Gas Calculations Problems

38 40 42 45 47 56

The Second Law of Thermodynamics

60

3-1 3- 2

65 72

Entropy Calculations Equilibrium iu Isolated Systems

ix

30

Contai ts

X

Molecular lulerpretalion of Entropy Eva luation of Absolute Entropies 3--4 ::\- 5 free Energy 3-,; Free Energy a nd the Equilibrium Consta nt Determination of F ree Energy Changes 3-i 3-8 T he Electrochemica l Cell Temperature Dependence of Eq uil ibria 3-9 Problems 3-3

IV Appl ica tio n s of t h e The rmod yn a mi c Prin c ip les ,J- l

Phase Equilihria Properties of Id eal Solutions ,1,--3 Chemica l App lications ,l--4 Heat Engines ,1,--5 Conclusiou Supplementa ry RcAdings Problems 4--2

Ind e x

73

79 84 89 96 99 103 107

111

lit 11 9 133

145 150 15 1 15 1

153

Physical Constant.8 and Convers ion Factors Avogadro's number N Faraday constant \i Ice point, O"C Gas constant R I cal

6.0229 X 109 molecules/ mole 96,486 coulombs 23,060 cal;~·olt equi,·a lenL 273.15°K 0.08205 liter-atm/ mole-deg 1.9872 cal/mole-deg 4.184 joules 4.184 X 107 ergs 0.04129 liter-atm

I Introduction

T

II ERMODYNAXICS provides the most general and efficient methods for studying and understanding complex physical phenomena, and t hus it should be one of the first subjects studied by a student who has a serious interest in chemistry or any physical science. The features of the ph ysical world that are most obvious to us are not the atomic properties, which we detect only with sophisticated devices, but rather the gross properties of matter obvious to one of our fiv e senses or measurable with simple .apparatus. These are properties such as pressure, volume, temperature, and composition- properties of matter in bulk, rather than of individual isolated molecules. The properties of matter which are so obvious to us are called macroscopic properties, and they are, naturall y, the first features we use to describe a physical situation. Thermod ynamics deals only with these macroscopic quantities; the concept or a molecule need never be used in a thermodynamic argument. To the student of elementary chemistry trained to speak so casually of atoms and molecules, the absence of the atomic concept from thermodynamics may seem to be a serious weakness. The opposite is true. The great st.rength of thermod ynamics is that it makes no use of the subtle and sometimes Oimsy theories of molecular structure. The only quantities and concepts which

Chonical 'fhmnodynamic.s enter thermod ynamics arc the experimentally measurable macroscopic properties of matter. Thermodynamics provides a fram ework for systematizing, discussing, and relating these properties. That is one of the reasons why the subject is so important to the understanding of the physical world and why it deserves careful stud y by the beginning student. Thermodynamics does more than provide a useful scientific language; it calls attention to the two macroscopic properties which are most fundam entally responsible for the behavior of matter. The first or these quantities is energy; the second, less fam iliar, •is entropy. We say that these quantities are most funda~ me11tal because the eventual course of all physical events can be summarized by two statements: The energy of the universe is conserved. The entropy of the universe increases. These are the first and second laws of thermod ynamics. An understanding or these sta tements, and of how energy and entropy depend on the other physical properties of matter, allows us to estahlisb cri teria for predicting the exte11t to which a chemical re~ action may p,roceed under a given set of conditions. That is, thermod ynam ics can show tis how to calculate the eq uilibrium constant of a reaction which has never been run, just by using data obtained from measurements of th~ individual properties of the pure reactants and products. Furthermore, it can tell us how that equilibrium constant wiU vary as temperature is changed. We shall find that there is a wealth or other a pplications in which thermodynamic anal)·sis is used to obtain a maximum amount of information from experiments that are conveniently done ; thereby, ex perim ents that arc difficult, or practically impossible, are avoided. NatUral ly, an examination of only the macroscopic properties of matter does not completely sa tisfy anyone familiar with the concepts and theories of molecular structure and behavior. A working scientist always tries to explain his observations of macroscopic properties in terms of wbat he knows of molecular structure. While thermodynamics is not a molecular theory, it does simplify the job of understanding ph ysica l phenomena in terms of molecular properties. Thermodynamics shows clearly that energy and en-

Introduction lropy are the quan tities that control the behavior of matter. T hus, theories of molecular structure have a well-defined question to answer : How do energy and entropy depend on the structure and properties of individual molecules? In what fo llows we shall demonstrate the use of tbermodynamic concepts in solving chemical problems ; hut as opportunity allows, we shall also tr y to explain the qualitative connection between thermodynamics and molecular structure.

1- 1

THEI\MOOYNA MI C SYSTE MS

T he most effi cient way to lea rn aOOut the behavior of matter is lo conduct con trolled experiments. Naturally, at any one time we ca n make measureme11ts on on ly a small segment of the physica l universe, and it is necessa ry to defm e tJud imits of this segment. quite carefully. Tbe part of the universe under in vestigation in an experiment is called the 1,1ys tem, and all other objects which may act on the system arc called the surroundings. In the most fortunate instances, the boundaries which separate the system from its surrou ndings are experimentally well defined. For example, there is no quest.ion that for most reactions which occur in aqueous solution it is perfectl y legitima te lo claim that the glass beaker which contains Lhe solu tion is not part of the re. action system, but instead forms a rather innocuous part of the surroundings. If we were concerned with the solubili ty of a salt in waler, a few experiments with Pyrex, quartz, and porcelain containers would demonstrate that the concentration of a saturated solution is independent or' the nature of the vessel in which it is contained. It would then be proper to claim that. the solubility we had determ ined was a property of the system salt. and water, and not of the system salt. and water and beaker. As another example of the im portance of real izi ng the limi ts of a system, suppose we wish to iuvestigate the relation between the pressure and vol ume of a certain amouut of gas. If we do this experiment in a rigid steel cyliuder equjppcd with a frictionless piston , then as we increase the pressure on the gas at a constant temperature, the change in volume is determined by the properties

Chemical Cfhtrmodynamic.s of the gos alone. In this case it is correct to say that the pressurevolume properties of the system are those of a gas. On the other hand, if we had placed the same volume of gas in a rubber balloon and then varied the pres.sure on the balloon, the observed change in volume would be determined not only by the properties of the gas but also by the elastic properties of the balloon. Tn this case the system would have to be regarded as the gas and its container. Clearly, if we were interested only in the properties of the gas, we would choose to investigate it by using the steel cylinder and piston.

1- 2

STATES AND STATE F UNCTIONS

After we have defined the limits of the system which we choose to discuss or investigate, we can proceed to descr-ibe the properties of the system. The purpose of this description is to allow a-1y other scientist to reconstruct the system in all its important detail so that his replica will behave identically to the original. The description of the system therefore must be complete, but it is also desirable that it be as concise as possible. That is, it should be limited to information which actually affects the measured behavior of the system. For example, if we have the simple system of a single particle of known mass acted upon by a known force such as gravity, experience tells us that it can be completely described by giving the values of its position and velocity. In other words, we have olr served repeatedly that all single particles of mass m placed at a certain point x, y, z above the earth and given certain velocity components Vz, v~, and v. will behave the same. Therefore, we conclude·that the only information necessary to completely de~ scribe tlie condition of the particle are its positional coordinates and velocity components. All other properties of the singleparticle system, such as its kinetic or potential energy, are fixed by these six numbers. The act of completely describing a system is called specifying its state. As we have just seen, the state of a single.particle system can be specified by only six numbers. The systems which

Introduction we shall study in thermodynamics are much more complicated, however. For example, a mole of gas is a typical thermodynamic system , and since six numbers are necessary to specify the state of a single particle, it would appear that 6 X 6 X 1021 numbers are required to specify the state of a mole of gas. Fortunately, this much information is not necessary. In fact., for the purposes of thermodynamics the st.ate of a mole of gas can be completely specified by giving the values of only two of the mllcroscopic properties pressure, volume, and temperature. The reason for this enormous simplification lies in the nature of the common laboratory experiments with which thermodyn1:1.mics is concerned. In these experiments we use devices such as meter sticks, thermometers, and manometers, or their more sophisticated equivalents, to examine the properties of systems. None of these instruments is sensitive to the behavior of single atomic particles. None of them gives us a record of the position and velocity of each atom or molecule. Instead, each of these devices is sensitive to a general property produced by the average behavior of all the particles in the system. In effect, these instruments are too large to respond to the behavior of the individual submicroscopic particles. The properties to which they do respond are macroscopic properties of the system as a whole. As long as we use measuring devices which are sensitive only to the macroscopic properties of matter, the detailed knowledge of ihe position and velocity of individual atoms in a system is not required in order to describe the conditions or results of oUr experiments. Thermodynamics is concerned only with the macroscopic properties of matter; therefore., the thermodynamic state of a system can be completely described in terms of a few macroscopic quantities. The macroscopic quantities that are used to specify the state of a thermodynamic system are called state variables or state functions, because their values depend only on the condition, or state, of the thermodynamic system. There are algebraic relationships between the state variables of a thermodynamic system. For instance, for a given amount of material, pressure, volume, and temperature are not all independent of each other but are connected by a mathemlltical relation called the equation of

Clumic:al 'Thmnodynamics The simplest example of such a relation is the ideal-gas eq uation of stat.e, PV = rt RT. Liquids, solids, and nonideal gases have equations or state which are generally considerably more complicated. The nature or the atoms in a system determines the algebraic form or the equation or state, and thus this equation is one of the properties which characterizes a system. It is very im portant to understand the significance of the term "state fu nction." We shall see presently that, in add ition to pressure, volume, and temperature, energy is a state function; and there arc many others. All slate functions have the following important properly: Once we specify the state of a system by giving t he values of a few or the state functio ns, the values of all other state fu nctions arc fixed. We can use the ideal-gas eq uation of state to demonstrate this fact. When the pressure and temperature of one mole of tm idea l gas are specified, the volume must assume the value V = RT/ P. All other state fu nctions such as energy also automaticall y assume definite va lues which a.re determined by the values of P and T. State functions have one other importaut property. When the state of a system is altered, the change in any state fun ction depends only on the initial andfi11al stales of the system, and not on how the change is accomplished. For example, when a gas is com pressed from an initial pressure P, to a final pressure A, the change in pressure ll.P is given by the expression' Hlatc.

ll.P

= A-A

Only the in itial and final values of the pressure determine .6.P. Any intermediate values which P may have assumed in changing from A to P 2 are immaterial. This is b y no menus a trivial property, and it is possessed only by-state functions. By way of analogy, we can say t hat any person who t ravels from Boston to San Francisco has changed his position (or state) on earth by an amount that depends only on the locat ion of hfa initial state (B08ton) and his final state (San Francisco). However, the distance he travels depends not only on 1 The symbol 4 o.lwaya means the change in a quantity, a nd the change i~ a lways computed by aubtroctif18 the initit1l value o r the qua ntity Crom iu fina l vt1lue.

Introduction

7

the location of the cities but also on the path he takes from one to the other. Thus, position is a state function, but distance traveled is not. These are the significant properties of state functions, and it is these properties which, we shall find, make state fun ctions so useful. 1- 3

EQUILIBRIUM STATES

There are some conditions of thermodynamic systems which cannot be described in terms of state fun ctions. For example, suppose our system is a gas confined in a cylinder with a movable piston. When the piston is motion less, as in Fig. 1- la, the stale of the gas can be specified by giving the values of its pressure and temperature. However, if the gas is suddenl y compressed, as in Fig. 1- lb, its stale cannot be described in terms of one pressure and temperature. While the piston is moving, the gas immediately in front of the piston is compressed an d heated, whereas the gas at the far end of the cylinder is not. There is, then, no such thing as the pressure or temperature of t he gas as a whole. Conditions in which the state variables are changing in time and space, called noncquiIibrium states, are not treated by thermodynamics. Thermodynamics deals only with equilibrium states in which the state variables have values that are uniform and constant throughout the whole system. Let us examine t he criteria for an equilibrium state more ca refull y. First, the mechanical properties of a system must be uniform and constan t. This means that there must be no unbalanced forces acting on or within a syste.m, since any unbalanced forces would cause t he volume to change continuously and we would be unable to specify the state of the system. Second, t he chemical composition of a system at equi librium must be uniform , and there must be no net chem ical reactions taking place. The occurrence of auy net chemica l change would inevitabl y change such properties of a system as its density or temperature and make specification of its state impossible. The third and final criterion of an equilibrium state is that the temperature of the system must be uniform and must be the same as the temperature of the surroundiugs. Whenever a temperature

Ch,mical 'fhnmodyttamics

(,)

(b) Figu re l-l I The 11t a t c o C a !j'.as co nfm etl 11 11 i n (a) by a motion leu pis ton can he 1:1pecifit:d b y th e va l ues ofil.t!I preMure and temper ature, A gas be in g com pressed a s in (b } docs not ban~ uniform propertieit, a nd it11 11t ate cannot be s pecified by o n e pttss ure or lernper a tur-e.

difference ex ists, hea t tends t o flow until the tempera ture dilfercncc disappears. Any system i11 which heat is fl ow ing has macroscopic proper lics which Are not un iform and wh ich may be c.:hanging in time; therefore, such a sys tem ca nnot be in an equilibrium state.

l---4

TEMPERATURE

We have emphasized that state functions are of foremost importance in thermodyna mics. Some of these state fun ctio11§,.lilce pressure, volume, and chemica l composition, are familiar quantities

Introduction and need no elaborate explanation. 011 the othe.r .hand, temperature, although a common quantity , has rather subtle conceptu a l origins. The idea or a quantitati ve tempera ture was introduced because it became obvious that the results or ma ny experiments depend ed on what we qualitatively ex perieuce as " hotness." The first step in crea ting a temperature sca le is to find some convenient property or matter wh ich depends in a sim ple way on hotness. T emperature can be indica ted, for example, by the densi ty of liquid mercury, which is commonl y measured by the distance mercury expa nds from a bulb into a glass capillary tube. The centigrade temperature scale is defin ed by assigning a va lue of zero tempera ture uni ts, or degrees, to the length of !he mercury column when the thermometer is immersed in a n ice-water ba th and 100 degrees to the length of the colum n when the thermometer is in contact with water at its normal boiling poi nt. In termediate temperatures a re defined by placing 99 equa lly spaced ma rks betwee11 the two calibration poinL,;. It is clea r that the temperature indicated by this thermometer depends on the properties of the materials used in its construction. Furthermore, by dividing the length between 0° and 100° into 100 equa l units, we are rea lly say ing that temperature is something which depends linearly on t he volume of mercury. If some other liquid is used in the thermometer, the rcs11 ltii1g temperature scale is different. For example, suppose we use water as the working liquid. We mark the thermometer in the same way, noting the lengt h of the wa ter colum n at th e ice point a nd boiling point and dividing the interva l into JOO equal units. Now we pu t our mercury thermometer a nd water thermometer in a bath at 0° and slowly raise the temperature. When the mercury thermometer reads 4°, the water thermometer reads - 0.36°. This happens because, as the temperature is raised from 0° to 4° on the mercury sca le, water contracts instea d of ex pa nding. If we used the properties of wa ter to defin e temperature, we would have to say that when hotness increases, temperature sometimes goes up and sometimes goes down. A temperature sca le based on water could be used, but it would be qualitatively rather complicated. In practice, several different properties of many different materia ls a re used to measure temperature. From ou r discussion we have to ex pect that in

Chemic.al '-Thtrmcdynamics

IO

Table 1- 1 Compari80n of Thcr1no m e t crs•

Constantt'Olume

ConstantPlulinum

volume

hydrogen

air

thermometer l(P)

lhermomeler l(P )

0 20 40 60 80 100

,i.0. 00 1 59.990

20.240 40.360 60.360

79.987 100

80.2•10 100

20.008

Merr,ury resi&lance thermometer Thermocouple lhermonuler l(R ) l(emf) 1(0

0 20.150

,rn.297 60.293 80.1 47

100

0 20.091 40. 111

60.086 80.041 100

• After J\'l. W. Zcmansk)', " l [crit nnd T hermody namics," l\kG raw-Hill, New York , 195 1.

genera l t hese different thermometers will read d ifferently when they are in contact with the sa me bod y. T able 1- 1 is a comparison or five types of thermometers. Eve n though ca librated to read the same at the boiling a nd freezing poin ts of water, the thermometers deviate frorn one another at iutermcdiatc temperatu res. In order to allow differen t workers to reproduce each other's e.xpcrimcnts, we need some reliable stand ard temperature SCA.le based on a u easil y measured property of a readily ava ilable material. Experiments show that, at low densities, a ll gases a t constant volume have the sa me depemlence of pressure on temperatu re as rncosured, fo r example, by the mercury scale. Since a ll ideal gases respond iden tica ll y to a given temperature change, it is convenient to redefine temperature in terms of the properties of idea l g.ises rather than• of liq uids. T emperature, theu, is th at quan tity wh ich depends linelLrly 0 11 the pres.s;ure of an ideal gas held at eo11stant volume. T o express this a lgebra ica ll y, we write

P =Po+ Poat

Introduction

II

where l is the tempcro ture, Po is the gas pressure at zero degrees 0 11 the temperature scale, and 1/ o: is a consta nt wh ich depends on the size of the degree. When the frcc;,;ing a11d boiliug points of water (both taken al a pressure of I alm) a rc assigned the va lues 0° a nd 100°, respectively, 1/ or = 273. 15° a11d the tempera ture sca le is co iled the ce11 t igrade perfect gas scale. The ratio of two prCSsurcs P 1 nnd Pl, which correspond to the temperatures f1 a lld Ii, is

This expression shows t hat a more con ve nient way to express temperature wo uld he to defin e a new scale by t he equutio11 T

=

~+l

=

273. 15

+l

The1, we wou ld ha ve P1

p; =

T,

r;

The quantity 'I' is ca lled the absolute, or Kelvin , temperature. From our a rgume11ts so far , it appears tha t this definition of tern pcralure a lso depends 0 11 the properties of a specific materia l, the ideal gas. However, we sha ll see later that thermod yriamics provides a way of definin g temperature wh ich is i11depemlc11 t of the properties of a ny materia l, and the resulti ng sca le is identical with that defin ed by idea l-gas proper.tics. The fact that the Kelvin temperature is independent of the detai led structure or pro1>erties of a 11 y ma terial is the reason it is so im portant in tl1crinod y11amics. The purpose of t his first chapter has been to introd uce some of the la nguage of Lhermod yna mics. With the concepts of system, state, state function , nud temperature in mind, we ca n begi11 a stud y of the first law of thermod ynam ics anchauica l sta te of the a toms C and O and the product CO2 molecules are another. The reaction is the process which takes the atoms from the initial mechani• ca l state to the final state. Since Eis a sta te fun ction, the energy change OE associated with the reaction depends on ly on the nature or the initial and final sta tes a nd does nol depend on the method or pa th by which the reaction is run. The ilE of each chemica l reaction run at constant temperature is a quantitative measure of the relative bond streugths and inter• molecular forces in reactan ts a nd products. Hefcrence to Eq. (2-2) and use of our ex pression for pressure.volume work shows how the AE or a chemica l reaction might be measured. We have 6.E

=q-

w

= q-

rv, Pu. dV

J,,

v,

If we carry out a reaction keeping the volume of the system constant, that is, if

then

rv, P•• dV -

J",,

0

aud

.t.E = q t::.E'

= qv

0

(constant volume) (2-4)

Equation (2-4 ) says that the change in interna l energy is L'{fual to the heat absorbed by the system when a proc.ess occurs al co11slanl t"Olw11e. In proctice then, t::.E for a chem ica l reaction can be measured by running the reaction in a closed vessel , or bomb, and noting the heat evolved or absorbed as the reaction occurs.

'fh, First Law of 'fhtrmodJMmics Although qv can be measured without too much difficulty, chemica l reactions are most commonly run in open vessels, not at constant volume, but at consta nt pressu re. Under conditions or constant pressure, the hea t absorbed by the system is nol equal to qv or lo !:JE. Therefore, despite the fact that /lE has a rather simple physical interpretation, it is not the most appropriate quantity for characterizing chemical situa tions . We shall now develop a new fun ction or state, called the enthalpy, which is helpful in discussiug reactious run at constant pressure. We define the enthalpy H by the equation H - E + PV

(2-5)

This definition assures us that His a f u.nclion only of lht stalt of a syskm, since E, P, and V a re all state fun ctions. Notice that H m ll8t have the units of energy. We ca n develop some or the propert ies or entha lpy by finding its differential dH from Eq. (2-5) :

r/jl - dE + d(PV) dll - dE+ PdV+ VdP

(2-6)

This equation relates a n infinitesimal change in H to similar changes in E, P, and V. To find the difference in H between two states l and 2, we integrate Eq. (2- 6):

{"' dll - J.' ' dE+ l v, PdV+

kl

B1

l''

V dP

P1

V1

t,JJ-aE+ l v' PdV + l '' vdP V,

•

Pi

Now for l!:.E let us suhstitu te the ex pres.sio11

aE -q-

l '•

PdV

f.'•'•

Vd P

v,

Then we get

aH-q+

(2-7)

T his is a general equation for computing MJ for any change of state. Suppose, however, we have a restricted process-one done at constant pressure. Then, since P, = Pi,

JP' VdP P,

0

(constant pressure)

and t:i.H = q p

(2-8)

The enthalpy change is equal lo the heat absorbed only when a process is carried out al co,uta,if pressure. Representing the heat by qp serves to emphasize this restrict.ion. Because of the equality between t:i.H and qp, the en thalpy is often called the heat content of a system. However, this name tends to invite confu sion between enthalpy and heat q, which, as Eq. (2- 7) shows, are not always equal. The enthalpy change is equal to q on ly for processes which take place at constant pressure. Notice carefully that for any process at constant pressure in which heat is ernlved by the system, qp, and thus t:i. H, is a negative number. A negative va lue for t:i.H means that the fin al state of the system has a smaller ent halpy than the initial state, since tJ.H = H , - H ;. Processes for wh ich ll.H aud qp are negative are said to be exothermic. On the other hand , if a system absorbs heat, qp and ll H are positive and the process is sa id to be endotherm ic. So far , our results show that llE = qv ilH ""' qp

(2-4)

(2-8)

In order to find the relationship between 6.E and ll.H , we return to the definition of fl, Eq. (2-5) ,

H -E+ PV aH - aE

+ d(PV) + a (P V)

t:i.J-1 = llE

+

dH - dE

P:Vi - AV1

That is, t:..J-1 and llE differ only b y the difference in the PV products or the fi!1al and initial states. For chemical reaction at

'Tht First Law of 'fktrmodynamics

25

constant pressure in which only solids and liquids are im'olved, AYt ~ Pi Vi, and therefore 11H and il.E are nea rly equal. However, if gases are produced or consumed in a chemical reaction, 11( P V) may be quite appreciable a nd 11H a nd ~ may differ not.iceably. For example, consider a reaction which involves only ideal gases and takes place at a cons tant temperature T: aA(g)

+ bB(g) ~ cC(g) + dD(g)

B y Dalton's law of partial pressures, PV for the products is PV(p,oducts)

~ (c + d) RT

and PV for the reactants is PV(reactauts) = (a+ b)RT Therefore, d. (PV) is given the expressiou 6(PV)

~ [(,+a)

A(PV)

~ 6 n RT

- (a+ b))RT

The total number of moles of gaseous products minus the number of moles of gaseous reactants is defined as 11n. In general, then, we have 6H t,£ + 6(PV) 6E + 6n RT (2-9)

~

~

Jn deriving Eq. (2-9) we have used the ideal-gas equation of stnte, PV = nRT. Since the quantity PV has the units of energy, so must ,tRT. This means that R has the units of energy per mole-degree. Natura lly, the numerical value of R depends 011 which energy units are used, and Table 2-l gives the value of /l

Table 2-1 The Gas Constant R

R - 0.~206 R

R

= 8.314 X 101

= 8.314

R - l.986

litcr-atm/mole-deg ergs/ mole-deg joules/ mole-deg cal/mole-deg

Chemical 'Thumodynamic.s for several of the possible e nergy units. The m ost useful of these units is the calorie, and it is very easy to reme mber tha t R has the approximate value of 2 ca.I/ mole-deg. We shall ill ustra t e the use of Eq. (2- 9) by t wo examples. EXAMPLE 2- 1

When 1 mole of ice melts at 0°C and a constant pressure of 1 atm, 1440 cal of heut is ubsorbed by the system. The molar volumes of ice and water are 0.0196 and 0.01 80 liters. respectively. Calculate All and t.E. Since t. H - qr, we have t..H • 1440 cal

+

T o find t.E by the expression 11H - t. E t..( Plr} , we must evaluate t.( P V). Since P - l a tm, we have

V1) - (1)(0.0180 - 0.0196) - - l.6 X 10-• Jiter-ntm - - 0.039 eel

t.( P V) - P t.. V - P(V1

-

Since 11H - H 40 cal, t he difference between t:.11 and 11E is trivial, a nd we can say t.E - 1440 cal. E XA a.t PL E 2-2

For the decomposition of MgCO, by t he reaction MgCO,(,) - Mg()(,)

+ CO,(g)

all - 26,000 ca l a t 900°K and I atm pressure. If the molar ,·olume of MgCO, is 0.028 liter a nd that of l\1g0 is 0.011 liter, find t:.E. Let us divide A(PV ) into two tenns: that. due to the volume change of the solids and that due to t he appearance of gas:

+

.:l.(P V) - 11(P V) ... 11 V 1, the entropy of t he universe as a resul t of the irreversib le expa nsion is greater tha n zero, as i~ consisten t with the second law of thermodynamics. These examples show how essential it is to consider the total entropy change or the system and il.8 surroundinb'S before attempting to conclude whether or not a process is spontaneous.

Chemical 'Thermodynamics

68

Tlie General I solliermal Process Although we have demonstrated that IJ.S or the system and its surroundings is positive for a particular irreversible isothermal expansion of an ideal gas, it is useful to proceed further and show t hat this is true for any irreversible isothermal expansion of any substance. Our argumen t involves a comparison bet ween the work done by the system in the reversible and irreversible expansions. If we write P 1ra for the equilibri um pressure of the system, in the case of a reversible expansion we have P,. - P; 0 ,

w,.. - J P,. dV - J P;

0 ,

dV

But when the expa nsion is irreversible, we can onl y write

w;.,,.

~ J P,. dV < J P ;"' dV -

w.,.

Therefore (3--3)

That is, since P~~ in the irreversible case is always lower than Pu (or P ;~ ,) in the reversible case, the work done by the system in the irreversible expansion must always be less than that done in the reversible case. The application of these ideas to the special case of an isothermal expansion of an ideal gas is shown in F ig. 3-2. However, Eq. (3- 3) is quite general for all isothermal processes a nd is not just restricted to ideal gases. Now we must compare the heat fo r the two types of processes. From the first law we have q..... Q irr u

=

6.E+ w,_.

= 6.E

+ W;.,,..,

Subtracting, we find

But since Wirr ev

< Wre v

'1hermodynamics-Smmd Law

m

P.

w_

~ w_

Figure 3-2 Cmnpari80n or the work done in reveniLle and i.rn:vcnible expa1111ioDJJ between the 1Jame a laleti, The area under the reversible path i.e. alway& greater than

under the ilreve.nible path.

we can write quv -

q;., •., = q,e... >

Wr ev -

Wier ev

>

0

q 1,uv

The heat absorbed by the system in an isothermal reversible process is always greater than in any corresponding irreversible process. Since the entropy change in the system is always q,ev/ T and the entropy change of the surroundings is numerically equal to -q 1.,ev/ T, we find for irreversible isothermal processes

~Stoia.I

=

7 - qiT~" > 0

(irreversible)

while for reversible processes the entropy change of th e surroundings is -q,u/T, so ~ Stot al

=~-

7

=0

(reversible)

Clu1n ical 'Thtrmodynamic.s

70

Th is comple tes our d c rno11stratio11 of the 8(,>t:ot 1d law for iwt.hermal processes.

Temperature Dependence of Entropy Now let us turn to t he problem of the calculation of entropy diffcre11ees between states of difl'ere11 t temperature. Again , we must imagine ourselves ta king u sys tem fro m iLs iuitial slate lo its fm a l sta te by a reversible path. A re\'Crsih le change in tempera ture can be effected hy slowly cha11ging the tempera ture of th e surroundinb'S, so t.hat the temperature of the surroundi ngs differs from the temperature of t he system b y uo more tha n a11 infinitesimal amount dT. This infinites ima l temperature difference causes a n i11fir1iLesimal amou11t of hea t dq ..,,, to be absorbed by the system at each of the temperatures between the initial a11d fina l va lues. F'or a change Laking place at constant pressure, combination of the definition s of entropy and heal capacity yields

dS - d~;," n Cp

=

if.

(consta nt pressure)

d~

~!9~

T

>

T

> 373

That is, !lG is negati ve a nd t he evapora tion to fo rm vapor at l atm pressure is spontaneous if the temperature is abotie the normal boiling point of waler. Ou the other hand , LiG will be positive if T OS < Off

This will be true if

T ~ pressure of benzene is 100 mm at 300°K. 5. Equ ation (4-18) is the expression for the solubility of a solute in an ideal solution. Evaluate the constant of il)-tegration for a solution of naphthalene in benzene from the fa cts that for pure naphthalene dissolving in itself (that is, melting), x 2 = 1 at T = 80°C nnd aHf... = 4600 cal/ mole. Use this constant to calculate the solubility of napht haleue in beuzene al 25"C. 6. The reactions listed below con be combined to form an a lternate path for the reaction HX (aq) = ff +(aq) + X-(aq), where X stands for For I. From the data given calculate aH0 for the ionization of aqueous HF and HI. By com pari ng the data for each step of the ionization process, decide why the ionization of HJ is more exothermic than that of HF.

HX(oq) - HX(g) HX (g) - H (g) X(g) H(g) X(g) - H+(g) X-(g) H • (g) x - ~ H+(oq) x-(oq)

+ +

+ + +

aH(HF),

Ml(HI),

kcal

kcal

-11.5

-5.5

134.6 232.8 -381.9

239.3 - 330.3

71.4

Index Boiling-point elevatioo, 120-124 Bond energies, 38, 135, 138 Butane, isomerization of, 134 1-Bute.ne, decomposition of, 136

Carnot engine, 145-146 Cells, electrochemical, 99-103 Clapeyron equation, ll5, 118

DlSOl'der and entropy, 75- 77 Dissociation energy, 36

Energy, conservation of, 12-13, 60 heat capacity and changes of, 4041 internal, 14-16 isothermal expansion of ideal gases and constancy of, 52-54 molecular interpretation of, 16-

17 9v and, 22 units of, 20

Enthalpy, 23 chemical reactions and changes of, 27- 28 variation with temperature, 4:HS of formation, 30 table of, (Of' various materials, 31-32 heat capacity and changes of, 40--41 and beat content, 24 q,. and, 24 relation to internal energy, 24-25 sign convention for, 24 standard state for, 30 Entropy, 63 change as a criterion for spontaneity, 64 chemical reaction and changes of, 84 computation of changes, 6-1 dependence on temperature, 7072, 76,103 disorder and, 75-76 of formation, 84 heat capacity and, 70, 79- 81

lndu

1 54

of ionization, 139- 142

molecular interpretation of, 7374

molecular properties and, 81-83, 135-136, 138-139 phase changes and, 7i- 78 second law of t hermodynamics and, 63 t hird law of thermodynamics a nd absolute values of, 79 table of, for various materials, 82-83 Equation of state, 5-6 Equilibri um, bet ween phases, 11 1128

entropy cha nge as a criterion for, 72-73 free energy and position of, 95-96 free ene rgy change as ra criterion for, 86

mechanica l. t hermal, a nd chemical, 7-8, 63 Equilibrium cons tant, 93- 9-1calculation of, 95, 97, IOI

determined by enthalpy a nd entropy, 131, 137, 141 - 143 standard cell potential and. 102 temperature dependence of, 103106, 133-138 E;1:plosions., maximum tempera ture reached in, 45-47

criterion for equilibrium as a minimum or, 86 cri terion for reveniihility, 86 criterion for spontaneity, 87- 89 electrochemical determination or cha nges or, 100 and equilibriwn constant, 89-96 or format-ion, 89 tables of, for various ma terials, 90 isothermal processes and changes of, 85 pressure dependence or, 92 Free energy, Helmholtz, llO Freezing-point depression, 125-129 relation to solubilit y, 128-129

Gas consta nt, 25 Gases, idea l, changes of state in, 48-56, 65-67 11 E consequent to, 50--52 11 G consequent to, 92