Provisional Thermodynamic Functions for Helium 4 for Temperatures from 2 to 1500 K with Pressures to 100 MH (1000 Atmospheres)

Citation preview

zyxwvutsrq zyxwvutsr zyxwvutsr

The National Bureau of Standards I was es_tablished by an act of Congress March 3, 1901. Today, in addition to serving as the Nation's central meashhment laboratory, the Bureau is a principal focal point in the Federal Government for assuring maximum application of the physical and engineering scienccz- to th,advancement of technology in industry and commerce. To this end the Bureau conducts research and provides central national services in four broad program areas. These are: ( 1 ) basic measurements and standards, ( 2 ) materials measurements and standards, ( 3 ) technological measurements and standards, and ( 4 ) transfer of technology. The Bureau comprises the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Center for Radiation Research, the Center for Computer Sciences and Technology, and the Office for Information Programs. THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consistent system of physical measurement; coordinates that system with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry, and commerce. The Institute consists of an Office of Measurement Services and the following technical divisions: and MolecApplied Mathematics-Electricity-Metrology-Mechanics-Heat-Atomic ular Physics-Radio Physics '-Radio Engineering '-Time and Frequency '-Astrophysics '-Cryogenics.' THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research leading to improved methods of measurement standards, and data on the properties of well-characterized materials needed by industry, commerce, educational institutions, and Government; develops, produces, and distributes standard reference materials; relates the physical and chemical properties of materials to their behavior and their interaction with their environments; and provides advisory and research services to other Government agencies. The Institute consists of an Office of Standard Reference Materials and the following divisions: Analytical Chemistry-Polymers-Metallurgy-Inorganic Materials-Physical Chemistry. THE INSTITUTE FOR APPLIED TECHNOLOGY provides technical services to promote the use of available technology and to facilitate technological innovation in industry and Government; cooperates with public and private organizations in the development of technological standards, and test methodologies; and provides advisory and research services for Federal, state, and local government agencies. The Institute consists of the following technical divisions and offices: Engineering Standards-Weights and Measures - Invention and Innovation - Vehicle Systems Research-Product Evaluation-Building Research-Instrument Shops-Measurement Engineering-Electronic Technology-Technical Analysis. THE CENTER FOR RADIATION RESEARCH engages in research, measurement, and application of radiation to the solution of Bureau mission problems and the problems of other agencies and institutions. The Center consists of the following divisions: Reactor Radiation-Linac Radiation-Nuclear Radiation-Applied Radiation. THE CENTER FOR COMPUTER SCIENCES AND TECHNOLOGY conducts research and provides technical services designed to aid Government agencies in the selection, acquisition, and effective use of automatic data processing equipment; and serves as the principal focus for the development of Federal standards for automatic data processing equipment, techniques, and computer languages. The Center consists of the following offices and divisions: Information Processing Standards-Computer Information - Computer Services - Systems Development-Information Processing Technology. THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and accessibility of scientific information generated within NBS and other agencies of the Federal Government; promotes the development of the National Standard Reference Data System and a system of information analysis centers dealing with the broader aspects of the National Measurement System, and provides appropriate services to ensure that the NBS staff has optimum accessibility to the scientific information of the world. The Office consists of the following organizational units: Office of Standard Reference Data-Zlearinghouse for Federal Scientific and Technical of Technical Information and Publications-Library-Office of Information .'-Office Public Information-Office of International Relations. I

zyxwv

Headquarters and Laboratories at Gaithersburp. Maryland. unless otherwise noted ; mailina address Washington, D.C. 20234.

Located at Boulder. Colorado 80302. :I Located at 5285 Port Royal Road, Springfield. Virginia 2P151. 2

zyxwv zyxw zyxwv zyxw zyxwvutsrqp zyxwvu ERRATA

R . D. McCarty: 1‘

Page

5

l?rovisional Thermodynamic Functions f o r Helium 4 for Temperatures from 2 to 1500 K with Pressures to LOO MN/m2 (1000 Atmospheres) NBS Report 9762

--

Change line

6

as follows:

lambda point pressure = .00504. MN/m2,

Pa@;e 30

Replace Table

9 with

(.0497 atm)- [3]

attached corrected Table.

ANDARDS REPORT

NATIONAL BUREAU NBS PROJECT 27502-2750422

NBS REPORT August 1, 1970

9762

zy

PROVISIONAL THERMODYNAMIC FUNCTIONS FOR HELIUM 4 FOR TEMPERATURES FROM 2 TO 1500 K WITH PRESSURES TO 100 MN/m2 (1000 ATMOSPHERES)

zyxwv zyx

R. D. McCarty Cryogenics Division Institute for Basic Standards National Bureau of Standards Boulder, Colorado 80302

zyxw zyxwv

This work has been supported by National Aeronautics and Space & w E r E i o n cq_^ Headquarters Fund Transfer No. R-06-O06-@Ke and W12-745

IMPORTANT NOTICE NATIONAL BUREAU OF STANDARDS REPORTS are usually preliminary or progress accounting documents intended for use within the Government. Before material in the reports is formally published it is subjected to additional evaluation and review. For this reason, the publication, reprinting, reproduction, or op.en-literature listing of this Report, either in whole or in part, is not authorized unless permission is obtained in writing from the Office of the Director, National Bureau of Standards, Washington, D.C. 20234. Such permission is not needed, however, by the Government agency for which the Report has been specifically prepared if that agency wishes to reproduce additional copies tor its own use.

U.S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS

T A B L E O F CONTENTS

zyxw

..................... iv v LISTOFFIGURES . . . . . . . . . . . . . . . . . . . . FORWORD . . . . . . . . . . . . . . . . . . . . . . . . . vi ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . 1 zyxwvutsrqponmlkjihgfedcbaZYX . I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . 2 NOMENCLATURE. CONVERSIONS. AND FIXED I1. POINTS FOR HELIUM . . . . . . . . . . . . . . . . . . . 3 SURVEY O F T H E LITERATURE . . . . . . . . . . . . . 6 111. 7 T H E LIQUID-VAPOR COEXISTENCE REGION . . . . . . IV . A. Vapor P r e s s u r e . . . . . . . . . . . . . . . . . . 7 B . D e n s i t i e s of the C o e x i s t i n g Liquid and V a p o r . . . 9 T H E EQUATION O F S T A T E. . . . . . . . . . . . . . . . 10 V. LIST O F TABLES

1

. VI1.

............ DISCUSSION O F ERRORS . . . . . . . . . . . . . . . . .

VI11

SUGGESTIONS FOR F U T U R E WORK

VI

.

IX X

.

.

CALCULATION O F P R O P E R T I E S

........... REFERENCES . . . . . . . . . . . . . . . . . . . . . . . APPENDIX A .Deviation P l o t s . . . . . . . . . . . . . .

iii

14 16

16 19

22

zyxw zyxwvuts zyxw LIST O F T A B U S

Page

Table Table Table Table Table Table

1.

2.

3.

4.

5.

6.

Table

7.

Table

8.

Table

9.

Table 10. Table 11.

... Constraints onEquation 2 . . . Coefficients f o r Equation 1 . . Coefficients f o r Equation 2 . .

. . . . Coefficients f o r Equations 4 and 5 . Coefficients f o r Equation 6 . . . . .

Constraints on Equation 1

. . . .

. . . .

. . . . . . . . .

10

......

31

e

. . . .

. . . .

.

. Second Virial Coefficients f o r Helium 4 . Coefficients f o r Equation 7 . . . . . . Saturation P r o p e r t i e s . . . . . . . . .

. . . .

e

. . . .

. . . .

Provisional Thermodynamic P r o p e r t i e s f o r H e l i u m 4 ( P r e s s u r e in M N / m 2 )

Provisional Thermodynamic Propertie s f o r Helium 4 ( P r e s s u r e in a t m o s p h e r e s )

iv

...

7

8

8

8

11 12

13 30

114

Figure 1 2 3

4 5

6

7

zy

zyxwvut zyx z zy LIST O F FIGURES

Page

A comparison of experimental P - V - T d a t a .

Deviation plot of the 1958 h e l i m vapor p r e s s u r e temperature scale.

..

e

......... . Deviation plot of saturation densities . . . . . . . Deviation plot of entropies of vaporization . . . e

e

23

24 25

26

Deviation plot between P-V-T d a t a by Sullivan [I41 and equation of s t a t e

27

Deviation plot between P - V - T d a t a by Canfield e t a1 [13] and equation of s t a t e .

28

........

..... Deviation plot between P - V - T d a t a by Lounasmaa [71 and equation of state. . . . . . . .

V

29

FOREWORD

This r e p o r t is i n response to many requests f o r thermodynamic property data f o r helium.

Although the data and functions given in this

r e p o r t should be t r e a t e d as preliminary, and therefore subject to change, they do r e p r e s e n t a substantial improvement over previous correlations.

Therefore, this preliminary document is n e c e s s a r y to

satisfy the interim need since the preparation and publication of the final version would otherwise entail substantial delay in providing availability of the data.

Vi

zyx

PROVISIONAL THERMODYNAMIC FUNCTIONS FOR HELIUM 4 FOR TEMPERATURES FROM 2 TO 1500 K WITH PRESSURES TO 100 MN/m2 (1000 ATMOSPHERES)

F. D. McCarty ABSTRACT The thermodynamic properties f o r helium 4 f r o m 2 to 1500 K with p r e s s u r e s to 100 MN/m2 (1000 atmospheres) a r e presented.

En-

z

tropy, enthalpy, internal energy, specific heat, speed of sound, density and t e m p e r a t u r e are tabulated for selected isobars.

F o r convenience,

these data a r e given i n both p r e s s u r e units of MN/m2 and atmospheres. An equation of state covering the e n t i r e range of p r e s s u r e and t e m p e r a -

t u r e is presented.

Equations f o r the density of the saturated liquid and

vapor are included as well as an equation which r e p r e s e n t s the 1958 helium vapor p r e s s u r e temperature scale.

Key words: Helium 4, p r e s s u r e , temperature, density, entropy, enthalpy, internal energy, specific heat, speed of sound, velocity of sound, second virial coefficient, vapor p r e s s u r e , equation of state, liquid, gaseous, coexistence.

I.

INTRODUCTION

During the course of this correlation of the thermodynamic prope r t i e s of helium, two facts began to dominate the overall character of the task.

First, it became apparent quite e a r l y in the analysis of the

data gathered f r o m the l i t e r a t u r e that even though a l a r g e body of data exists for helium, there a r e gaps t o be filled, and a r e a s where existing

zy

data a r e either of limited scope o r a r e in disagreement with other data to such an extent that m o r e experimental work is needed.

Second, in

the final stages of the correlation, it also became evident that substantial improvement over existing correlations had been achieved.

The

purpose of this r e p o r t i s to draw attention to these two observations

and to make provisional functions available in advance of the final publication of the correlation.

2

zyxwv zyxw 11. NOMENCLATURE, CONVERSIONS,

AND FIXED POINTS FOR HELIUM L i s t of Symbols

P T

absolute p r e s s u r e

-

absolute t e m p e r a t u r e

-

V P

R

Z

U

H S C

-

P

=V u)

B

G bi

Sgi

-

-

-

nki

-

i

-

Sli

f

ai

specific volume

density = 1/V

universal gas constant =

. 0820558 8 . atm/mol- K

compressibility factor = PV/RT specific internal e n e r g y specific enthalpy specific entropy

specific heat capacity at constant p r e s s u r e specific heat capacity at constant volume speed of sound second v i r i a l coefficient Gibbs function p a r a m e t e r s for the second v i r i a l coefficient expression

p a r a m e t e r s for the saturated density equation (liquid) p a r a m e t e r s f o r the equation of state

p a r a m e t e r s for the vapor p r e s s u r e equation ( 1 )

- p a r a m e t e r s for the vapor p r e s s u r e Superscripts

0

zyx zyx

p a r a m e t e r s f o r the saturated density equation (vapor)

-

ideal gas property

3

equation ( 2 )

Subs c r ipts

C

zyxwvu zyxwvut -

c r i t i c a l point

-

property at saturation

expr

-

experimentally determined property value

c alc

-

calculated property value

0

Sat At

melt 1 g

-

-

reference state property

lambda t e m p e r a t u r e

melting line property liquid state

gaseous state

Conversions and Physical Constants 1 thermochemical calorie = 4. 184 joules 0 ° C = 273. 15 K ( t r i p l e point of water = 273. 16) 1 liter-atmosphere = 101. 325 joules gas constant, R = .0820558 liter. atmospheres per moles degree Kelvin molecular weight = 4. 0026 (based on the carbon 12 scale)

1 atmosphere = 1. 01325

X

lo5 Newton per m e t e r 2

1 Mega-Newton (MN) = l o 6 Newton 1 l i t e r = 1000 cm3

4

zyxwv zyxwv zyxw zyxw Fixed Points for Helium

c r i t i c a l p r e s s u r e = .22746 MN, (2.2449 atrn) [ 11 c r i t i c a l t e m p e r a t u r e = 5. 2014 K"

c r i t i c a l density = 17.399 m o l / l , (. 06964 g/cm3) [ 2 ] normal boiling point = 4.224 K" [ 3 1 lambda t e m p e r a t u r e = 2. 177 K* [41

lambda point p r e s s u r e = .005147 MN, (. 0508 atm) [31

lambda point density (liquid) = 36. 514 mol/l (. 14615 g/cm3) [4]

r:

The t e m p e r a t u r e s given h e r e a r e based on an adjusted 1958 helium vapor p r e s s u r e t e m p e r a t u r e scale. E31 The c r i t i c a l temperature for helium on the 1958 scale is 5. 1994 K. The adjustments were made on the basis of NBS acoustical t h e r m o m e t r y experiments [51. In the case of the c r i t i c a l p a r a m e t e r s , the equation of state has been constrained to reproduce these values exactly. Since the actual fitting was accomplished, new work suggests that the value of 5.2014 K for the c r i t i c a l t e m p e r a t u r e may be a few milli-degrees ( a s much a s 4) high, however, the t e m p e r a t u r e s below 5K are unaffected by this new work.

zyxwvu 5

zyxwv zyx zyxwvuts 111, SURVEY O F THE LITERATURE

A s e a r c h of the world's scientific l i t e r a t u r e began with a com-

puterized s e a r c h of the holdings of the Cryogenic Data Center of the National Bureau of Standards at Boulder, Colorado. duced 634 r e f e r e n c e s .

This s e a r c h pro-

E a c h of these references were carefully evalu-

ated for possible contributions to the project of correlating the thermodynamic properties of helium.

The bibliography of e a c h selected r e f e r -

ence was scrutinized for possible new references. After the initial s e a r c h of the l i t e r a t u r e had been accomplished, a constant surveillance of the c u r r e n t l i t e r a t u r e (also provided by the

zy zyxw zy

personnel of the Cryogenic Data Center in the f o r m of their "Current Awareness

s e r v i c e ) continually updated the original bibliography.

T h e r e probably has been m o r e information published on helium than for any other fluid. 11

This i s largely due to its unique, so-called

superfluid" properties, but even when the superfluid l i t e r a t u r e i s ex-

cluded t h e r e is left a l a r g e number of references.

In 1968 Barieau [61

published a bibliography of all the references he could find which contained experimental P-V-T data on helium 4. The l i s t contains 163 references.

If for no other reason than the quantity of data available,

value judgements w e r e n e c e s s a r y to bring the experimental points down t o a number which could be handled by the p a r a m e t e r estimating techniques.

A l l data w e r e considered in preliminary analysis and in m o s t

c a s e s value judgements could be made on a scientific basis, however, in some c a s e s experimental points w e r e omitted f r o m the fitting procedures because t h e r e w e r e j u s t too many points in a given region.

6

IV. A.

zy zyxw

THE LIQUID-VAPOR COEXISTENCE REGION

Vapor P r e s s u r e

The 1958 helium vapor p r e s s u r e t e m p e r a t u r e scale [3] is r e p r e sented by the following two equations.

10 (2-i) In P = x f i T i=1

where T is i n d e g r e e s Kelvin and p r e s s u r e is in microns.

The range of

validity f o r this equation is 5. 1994 to 2. 172 K .

z zyxw

where T and P a r e in the s a m e units as equation ( 1 ) but the range of validity is 2. 172 t o 0. 5 K .

Equation (1) was f i t t o 61 1 smoothed points f r o m [31 taken e v e r y 0 . 005 K between 2. 172 and 5. 1994 K .

In addition to the 611 data points,

six constraints w e r e imposed on the l e a s t s q u a r e s e s t i m a t e of the

parameters.

These constraints a r e listed in table 1.

Table 1 Constraints on Equation 1 1.

P

=

37800. at T = 2. 7

2.

P

=

760000. at T = 4.215

3.

P

= 1718000. at T = 5. 1994

4.

dp/dt =

92960. at T = 2. 17

5.

dp/dt =

717100. at T = 4. 22

6.

dp/dt

=

1267000. at T = 5 . 2

the coefficients (f.) are given i n table 3. 1

7

1

z zyxw zyx

Equation (2) was f i t t o 333 points taken e v e r y 0.005 K f r o m 0.5 K t o

2. 172 K. C3l In addition to the 333 data points, two constraints w e r e imposed on the least s q u a r e s estimate of the parameters.

See table 2.

Table 2

Constraints on Equation 2 1. P

= 37800. at T = 2. 172

2. dp/dt = 92960. at T = 2. 17

The coefficients (a.) f o r equation (2) are given i n table 4. 1

The maximum deviation i n p r e s s u r e f o r either equation ( 1) o r ( 2 ) is

. 02%.

The maximum deviation i n t e m p e r a t u r e predicted by either

equations (1) o r (2) is 0. 0001 K .

See figure 2.

Deviations refer t o de-

viations f r o m the 1958 helium vapor p r e s s u r e s c a l e as defined by reference [3 1.

zyxwv

Table 3 Coefficients for Equation 1 f l = -3.9394635287 f =

fa = fq=

1.4127497598

a1 = -4.9510540356

X

lo2

a2

- 1. 6407741565 X l o 3

=

X

a a = -3.7075430856 x

a4 =

X

lo4

as = -3.0048545554 x

1. 6621956504 x

lo5

a6

= 4.9532267436 x

f7 = -3.2521282840 x lo5

a7 = -5.9337558548 x

fa =

fs = -2.7771806992

X

lo5

ae =

5.23111296025

X

X

lo5

as = -3.3950233134

X

f a o = 8.3395204183, X lo5

a10

= 1. 6028674003

all = -5.3541038967

X X

a l a = -1. 6146362959

X

ax4 = 9.8811553386

8

X

1. 1990301906

alg=

lo3

1.2880673491 x lo4

lo4

3.9884322750,

lo1

6.5192364170 x 10'

X

1. 1974557102

fe = -5.5283309818 fe=

Table 4 Coefficients f o r Equation 2

lo4 lo4 lo4 lo" lo4 lo4 lo3 lo" lo2

The 1958 helium t e m p e r a t u r e scale C31 has been under intensive scrutiny in r e c e n t y e a r s .

Many investigators have published experimen-

t a l r e s u l t s which indicate the 1958 helium scale is in e r r o r by a detectable amount.

Without exception, the pertinent analysis c a r r i e d out for

this correlation by the author to date indicates two things.

F i r s t , the

zy zyxw z zyxw zyxw zyxwvut

c r i t i c a l p r e s s u r e of helium 4 is about 10 m i l l i m e t e r s of m e r c u r y lower

than i s reported in the 1958 helium scale, and second, the t e m p e r a t u r e s

in the 1958 helium scale are lower than the actual thermodynamic scale. These findings a r e consistent with r e c e n t experimental work. [5, 101. On the basis of the p r e s e n t correlation

* all t e m p e r a t u r e s below

5. 2 K were adjusted according to:

T = T~~ t ,001 t . 0 0 2 ~ ~ ~

(3)

If one takes as a best e s t i m a t e , the c r i t i c a l p r e s s u r e of helium to :g

*

and solves equations ( 3 ) and ( l ) , a c r i t i c a l t e m p e r a g t u r e of 5. 2014 r e s u l t s . be 1705. 0 m m H

B. Densities of the Coexisting Liquid and Vapor

The density of the saturated vapor is given by:

The density of the saturated liquid i s given by: 6 i/3 pa = pC t ~ s ~ ~ ( ~ - T / T ~ ) i=1

(5)

where the density is in g/cm3, and the t e m p e r a t u r e i s in degrees Kelvin. and s1 a r e given in table 5. The p a r a m e t e r s for g equations (4) and (5) were determined by l e a s t s q u a r e s using the e.xperiThe p a r a m e t e r s s

mental data of [12].

The s a t u r a t e d liquid data below the lambda

The work of Plumb and co-workers [SI was given heavy weight in this determination. The work of Edwards [l] and Roach [ 111 s e r v e d a s a basis for this e stimate. :k

**

9

t e m p e r a t u r e w e r e omitted.

zyxw

The average deviation for the liquid equa-

tion was 0. 02770 with a maximum of 0. 170 ( in density) at 4 . 4 K.

The

average deviation for the vapor equation w a s 0. 170with a maximum of 0.470 ( i n density) at 2 . 2 K.

See f i g u r e 2.

The range of the liquid equa-

tion is f r o m the lambda point to the c r i t i c a l point and the range of the vapor equation is f r o m 2. 2 K to the c r i t i c a l point.

zyxwvu zyxwvu Table 5

Coefficients f o r Equations 4 and 5

Equation 5

Equation 4

= -6.9267495322 x

s

g1.

3

g5 s

1.2874326484 x 10-1

11

zyxwvuts

s = -1.2925325530 X 10” ga s . = 2.9347470712 X 10-1 €3 s = -4. 0806658212 x 10-1 g4

=

S

ge

=

3. 5809505624

X

= -4.3128217346 x 10”

S

12

=

S S

10-l

14

S

= -3.3509624489 =

3.0344215824

=

- 1.098 1289602

15

= -1. 1315580397

X

10-1

S

16

V.

1.7851911824

13

THE EQUATION O F STATE

The development and least s q u a r e s fitting of the equation of s t a t e for helium *as accomplished in two s e p a r a t e stages.

The first stage

w a s the determination of the second v i r i a l coefficient f r o m 2 to 1500 K , which i s given by:

where B is the standard second v i r i a l f r o m the expansion P = P R T (1. 1- BP t

..

* .

) and h a s the units of l i t e r s per mole.

After extensive

analysis of all the isothermal P-V-T data found in the l i t e r a t u r e f o u r s o u r c e s of P-V-T data w e r e chosen for the final least s q u a r e s f i t of the equation (6).

The data of Canfield, e t al. [I31 were chosen for the

10

zyxw zyx zy

t e m p e r a t u r e range of 133 t o 273 K. The data of Sullivan [ 141 were

chosen f o r the temperature range of 70 to 120 K. The d a t a of W h i t e , et al. [15] w e r e used f o r the temperature range of 20 to 70 K, and

finally, the d a t a of K e l l e r [161 w e r e used for t e m p e r a t u r e s f r o m 2. 154 t o 3.957 K. In the c a s e of s o u r c e s [131 and 1141 care was taken to -

I

exclude d a t a above p r e s s u r e s where higher o r d e r virials began to contribute.

In addition t o the P-V-T d a t a mentioned above, two other kinds of d a t a w e r e fitted t o equation (6) simultaneously with the P-V-T data. F o r details on the technique of simultaneous least squares estimation see Hust and McCarty [I?].

The speed of sound d a t a by Plumb and

Cataland [SI, which cover the t e m p e r a t u r e range between 2.323 and 20. 051 K , w e r e i n s e r t e d into the f i t as well as the second v i r i a l coefficient d a t a of Yntema and Schneider [ 181 which cover the temperature range of 273 to 1473 K. The p a r a m e t e r s for equation (6) are given i n table 6. Table 6. . I .

Coefficients f o r Equation 6'

&

- = -5.0815710041

X

= -9.5759219306

10''

b2 = -1. 11,68680862 X lo'*

by =

lo-'

3.9374414843

bs =

1. 1652480354

X

lo-"

be = -5. 1370239224

=

7.4474587998

X

10""

b9 =

& = -5. 3143174768

X

10-1

b4

X

2.0804456338

~

%:

The number of significant f i g u r e s l i s t e d for these and all other tabulations of coefficients f o r the various equations are i n no way indicative of the accuracy of the properties calculated f r o m the equations. However the number of significant f i g u r e s given are n e c e s s a r y to avoid arithmetic e r r o r s .

11

zyxwvu zyxwvut Table 7.

Second V i r i a l Coefficients f o r Helium 4

T emp K

B x 103 &/mol

B x 103 &/mol

Temp K

B x 10' &/mol

Temp K

2

-203.81

40

7. 62

200

12.33

4

85.83

50

9. 33

2 50

12. 16

51.20

60

10.37

300

11.95

33.88

80

11.49

400

11. 51

10

-

23.37

100

12.00

500

11.10

15

-

9.25

120

12.25

600

10. 71

20

-

2.25

140

12. 36

800

10.02

25

1. 8 5

160

12.39

1000

9.41

30

4. 49

180

12.37

1500

8. 12

6 8

zyxw

The total equation of s t a t e is given by: P = pRT[l

+ B(bi, T)p] +

8

n

i =1

3

+

L

(1 n .p6T

-.

-

ii

p3T

(1. 5

-

i/2)

i).

(7)

Where P is i n a t m o s p h e r e s , p is i n m o l e s per l i t e r , T is in d e g r e e s Kelvin, and B(b. T ) is r e p r e s e n t e d by equation (6). 1

m e t e r s f o r equation (7) a r e given in table 8.

12

The p a r a -

zyxwvu Table 8.

Coefficients for Equation 7

1. 9381451090 x

nll =

n32

=

3. 6811167132 x lo"

n12= -4. 1496408960 x

n33 = -1.4830691828 x l o a 4

n13 = -5. 7465772899 x

nX =

n u = -4.3470945634 x l o e 3

n3s = -3. 3908190224 x

11x5

= -6. 8383888924 x

n16 = -2. 1382474225 x

2. 7106954908 x lo-'

n17

=

n18

= -1.2627967788 x

na = n22

11.41

n31

=

2. 5875753380 x l o e 3

n51=

x

n52

7.9041608815 x 10"

=

-2.8278987249 x l o m 7

zy zyxw 1. 7336410358 x lo-"

nW= -2. 5454187855 x

4. 9728101217 x 10''

= -3.8116033499

1. 9624080242 x

n43 = -1.4024724318 x 10'"

na = -1.0839788073 x lo'' na =

=

n42=

1. 5527899712 x l o m 5

= -3. 6110403503

3.0596174335 x l o m 4

x 10'"

5.9883101090 x 10"'

n61

=

n62

= -4.9653052187 x lo'?

y

= -0.0005

The p a r a m e t e r s for equation (7) w e r e determined by l e a s t

s q u a r e s estimation techniques using s e v e r a l different kinds of input data, simultaneously.

Selected P-V-T d a t a f r o m Lounasmaa [ 71.

Canfield, e t al. E131, Sullivan [14], Wiebe, e t al. [19], Glassford and Smith (91, and e l Haddi, et al. [12] w e r e included in the f i t together with the C

V

d a t a of Lounasmaa

[TI.

In addition to experimental data, the condition of the Gibbs function of the saturated liquid being equal to the Gibbs function of the saturated vapor at a common t e m p e r a t u r e w a s inserted into the f i t at 30 t e m p e r a t u r e s between 2.2 and 5 . 1 K.

Also, the entropy of vapori-

zation was put into the f i t at t h e s e s a m e t e m p e r a t u r e s .

Equations (2),

(3), (4),and ( 5 ) w e r e used t o generate the p r o p e r t i e s needed f o r the fitting equations.

The fitting equations a r e :

13

G1

zyxwv zyxwvu zyxw zyxwv -(-)dp 1 a p P ap T

- Gg

g

As a result of including the above two equations i n the actual

l e a s t s q u a r e s estimation procedure,

only one equation is needed t o

calculate a given derived property i n both t h e gaseous and liquid states. In all t h e equation of s t a t e work done a t this laboratory in the past this has not been possible.

Previously, derived p r o p e r t i e s in the compressed

liquid region ( T < Tc, p > psat

w e r e calculated in some other manner

Using which a l w a y s c r e a t e d a matching problem at T f o r a l l p > p C c' the above technique eliminates t h i s problem a s only one equation i s used over the e n t i r e region. The final l e a s t s q u a r e s f i t of t h e data described above was cons t r a i n e d at the c r i t i c a l point t o

and t o t h e s t a t e point of P = 2. 2449 atm, pc = 0. 06964 g / c m 3 , and C

T

C

= 5.2014K.

VI.

CALCULATION O F PROPERTIES

All the tabulated property values appearing i n this r e p o r t have been calculated using equation (7) and the following relationships.

14

z zyxw zyxwvu zy

S = S0 T

T ] d p + lTCP o dT T

0

(10)

0

0

+ -P -

T R T + l T C o dT P 0

0

where So = 37.068 J / m o l K, and HT = 146.328 J / m o l , C

T0

0

P

0

= 5/2 R

Co = 3 / 2 R , T = 4.22 K iLnd P = 1 . 0 Y 0 0

a2P V

V

P

In all cases t h e input v a r i a b l e p a i r was p r e s s u r e (P) and t e m p e r a t u r e (T).

This r e q u i r e s a n i t e r a t i v e solution t o equation (7).

In

the c o u r s e of preparing the tabulations it was learned that in solving equation (7) f o r density f o r all p r e s s u r e s at t e m p e r a t u r e s below 20 K, it is important to begin the iteration with a reasonable guess. r e a s o n is that a whole s e t of unwanted solutions exist in the liquidvapor coexistence region and another s e t exist at densities slightly l a r g e r than the solid densities at t e m p e r a t u r e s t o 20 K.

15

The

zyxw

DISCUSSION O F ERRORS

VII.

The equation of state is estimated to be accurate to 1. 5 percent in at least one of the two s t a t e variables of p r e s s u r e and density.

is t r u e for all regions where experimental data exist.

This

This means

that in difficult regions such as the c r i t i c a l region where the density may be a s much as 10 percent in e r r o r , the p r e s s u r e will be accurate to a few tenths of a percent.

In the compressed liquid region the in-

v e r s e relationship may be t r u e , i. e . , p r e s s u r e s may not be accurate but the densities w i l l be.

In m o s t regions, the equation is much better

than the 1.5 percent stated above, for example, for the t e m p e r a t u r e range of 20 to 500, the accuracy is estimated to be on the o r d e r of

0.5 percent,

Representative deviation plots a r e given in appendix A.

The amQunt of e r r o r p r e s e n t in the derived properties is a

zyxwvuts zyxwvut

function of many variables and is difficult to evaluate.

However, some

indication may be had from the comparison of calculated and experimental specific heats

tion of 3 to 4 percent.

["I.

This comparison shows an average devia-

These e r r o r s will be g r e a t e r when approaching

a saturation boundary o r the larnbda line.

The latent heats of vapor-

ization calculated f r o m the Clapeyron relationship a g r e e to within a few tenths of a percent with those calculated f r o m the equation of state, except above 5 K where the latent heats approach zero.

VIII.

SUGGESTIONS FOR FUTURE WORK

T h e r e a r e about t h r e e regions of P - T where new P-V-T m e a s urements a r e needed.

1.

In the o r d e r of their importance they are:

zyxw

The a r e a of the P - V - T surface f r o m about 2 . 2 K to 20 K with

p r e s s u r e s to at least 350 atmospheres.

At the p r e s e n t time only one

source of experimental data exists which covers even p a r t of this range and which i s extensive enough to be of value. 171 Unfortunately these

16

zy zyx

d a t a have some s e r i o u s shortcomings which have hampered the present

First these data do not include the saturated liquid and

correlation.

the saturated vapor, which means that saturation data must be added f r o m some other source. problems.

This immediately introduces consistency

Second, when these data a r e compared to other experimen-

tal d a t a at 100 atm, the two d i s a g r e e by about 1. 3% in density. corresponds to a 7y0 e r r o r i n p r e s s u r e .

This

See figure 1. Since no other

s o u r c e exists with which to compare the two sets of data, one h a s no choice but to assign a 1 . 5 % uncertainty i n density t o the two s e t s and therefore the sa.me uncertainty to the calculated values f r o m the equation of state.

zyxwv zyxwv

It is difficult t o t e l l f r o m Lounasmaas' thesis [7] exactly what his t e m p e r a t u r e s c a l e was between 4.2 and 12 K, and conversion to what is

now the b e s t e s t i m a t e of absolute t e m p e r a t u r e s i n this range of t e m p e r a t u r e i s impossible. 2.

Low p r e s s u r e , i s o t h e r m a l m e a s u r e m e n t s f r o m 4 . 2 to 7 0

would be v e r y helpful.

If these measurements a r e to be of value, they

m u s t be of high accuracy.

The m e a s u r e m e n t s a r e needed to check the

WBS accous t i c a1 measurements 3.

K

.

P - V - T measurements a r e needed in the 20 to 7 0 K region

from low p r e s s u r e s to

1 0 0 0 atm.

Some high p r e s s u r e measurements

exist in this region [8], but t h e i r accuracy is poor, making them of little use to the c o r r e l a t o r . 4.

In addition to the P-V-T m e a s u r e m e n t s , specific heat and

speed of sound m e a s u r e m e n t s should be made Over the s a m e regions of p r e s s u r e and t e m p e r a t u r e .

Ideally, all of the measurements should

be made a t the same laboratory i n one continuous experimental p r o -

gram.

17

zyxwv zyxwvu zy

The correlation work just completed indicates improvement over the present equation o f state can be accomplished by a better functional form

~ Q the T

representation of the high density fluid.

The

final version of this correlation will reflect further effoyts tQwards a better functional representation of the high density fluid,

18

zyx z zyxwvuts IX.

Edwards, M.

mFERENCES

E , The coexistence c u r v e of 4-He above 0. 98 Tc,

(1966), Proc. 10th Internat'l. Conf. Low Temp. Phys. , MOSCOW

t o be published, 5 pp.

Kierstead, H. A., P r e s s u r e s on the c r i t i c a l isochore of He4,

Argonne Natl. Lab., Argonne, 111.

[3 1

60439.

Brickwedde, F. G., The 1958 He4 scale of t e m p e r a t u r e s , Part 1;

van a j k , H., Durieux, M . , Clement, J. R , and Logan, J. K.

Part 2; J. Res. Nat. Bur. Stand.

(u. s. 1,

64A (PhYS. and Chem- )s

No. 1 (Jan. -Feb. 1969).

[4 3

Kierstead, H. A., Lambda c u r v e of liquid He4, Phys. Rev., Vol. 162, NO. 1, pp. 153-161 (Oct. 5, 1967).

[5 1

zy

Cataland, G. and Plumb, H. , Isotherms determined by the

National Bureau of Standards acoustical t h e r m o m e t e r in liquid helium t e m p e r a t u r e range, J. Res. Nat. Bur. Stand. (U. S. (Phys. and Chem, ),No. 6, (Nov. -Dec.

E6 1

1965).

1, 69A

Barieau, R E , , Helium-4 experimental P-V-T references: 1895 to 1968, Bur. of Mines Info. Circ. 8388 (Aug. 1968).

l-71

Lounasmaa, 0. V., Specific heats at low t e m p e r a t u r e s , Ph. D. Thesis, Univ. of Oxford (1958). Available through University Microfilms, Inc., Ann Arbor, Mich.

zy

Dobrovol'skii, 0. A. and Golubev, I. F., Measuring t h e density of helium, Gas. P r o m . , Vol. 10, No. 7, pp. 53-54 (1965) (In Russian).

19

[9]

zyxw zyxwv zyxwvut zyxwvu zy zyx z

Glassford, A. Pe M. and Smith, A. J.

, Jr. , Pressure-volume-

t e m p e r a t u r e and internal energy data f o r helium f r o m 4.2 t o 20 K between 100 and 1300 a t m . , Cryogenics, Vo1. 6, No. 4, pp. 193206 (Aug. 1966). [lo]

Rogers, J. S.

, Tainsh, R J. , Anderson, M. S. and Swenson,

C. A. , Comparison between g a s t h e r m o m e t e r , acoustic, and platinum r e s i s t a n c e t e m p e r a t u r e s c a l e s 'between 2 and 20 K, Metrologia, Vol. 4, No, 2, pp. 47-59 (April 1968). [11 ]

Roach, P. R.

,

P r e s s u r e - d e n s i t y - t e m p e r a t u r e s u r f a c e of He"

n e a r t h e c r i t i c a l point, Cal Review, Vol. 170, No. 1 (June 5,1968).

[ 121

El Haddi, Z. E. H, A. , .Durieux, M,

,

and Van Dijk, H.

The density

of liquid 4He under its saturated vapour p r e s s u r e ; and El Haddi,

Z. E. H. A. and Rurieux, M., The density of t h e saturated vapour of "He, Physica 41, 289-304 (1969). m3*

[13]

Canfield, F. B.

, Leland, T. W. , and Kobayashi, R. , Compressi-

bility factors f o r helium-nitrogen mixtures, J. Chem. and Eng. Data, Vol, 10, No, 2, pp. 92-96 (April 1965), [14]

Sullivan, J. A.

, P.-V-T data f o r neon and helium at t e m p e r a t u r e s

f r o m 70 K t o 120 K and p r e s s u r e s t o 690 atm, Ph. D. Thesis, Univ. Mich.

, 146 pp. (1966). A v a i l a b l e f r o m u n i v e r s i t y Microfilms,

Inc., Ann Arbor, Mich. , O r d e r No. 66-14,603. [151

White, D.

, Rubin, T. , Camky, P. , and Johnston, H. L. , The

virial coefficients of helium f r o m 20 t o 300 K, J. Phys. Chem., Vol. 64, NO, 11, pp. 1607-1612 (Nov. [16]

1960).

Keller, W i l l i a m E. , P r e s s u r e - v o l u m e i s o t h e r m s of He" below 4. 2 K, Phys. Rev,

, Vole 97, No,

Phys. Rev., Vol, 100, No. 6,

20

1, pp. 1-8 (Jan. 1, 1955) and

p. 1790 (Dee. 15, 1955).

[17,]

zyxw zyxwv zyxwvut

Hust, J. G. and McCarty, R D. , Curve-fitting techniques and

applications t o thermodynamics, Cryogenics, p. 200 (Aug. 1967). [18]

Yntema, J. L. and Schneider, W. G., Compressibility of g a s e s

at high temperatures.

111. The second virial coefficient of helium

in the t e m p e r a t u r e r a n g e 6OOOC t o 1200°C, J. Chem. Phys.,

Vol.

18, No. 5, pp. 441-646 (May 1950).

(191

Wiebe, R , Gaddy, V. L., and Heins, C., Jr., The c o m p r e s s i -

bility i s o t h e r m s of helium at t e m p e r a t u r e s f r o m -70 to 200 d e g r e e s

at p r e s s u r e t o 1000 atm., J. Am. Chem. SOC., Vo1. 53, No. 5, 11. 1721-1725 (May 1931).

21

X.

zyxwv zyxw APPENDIX A

Deviation Plots

22

zyxwvut zyxwvu

zyxwvut

zyxwvutsrqp 8

too

Glorrfotd. sai t h

Lounasmaa

zyxwvuts

2

I-

4:

L 7

w

Ym

m w

a a

*O

60

40 1

zyxwvu DENSITY IN W C C

OT/~S/TP

Figure 1.

do-’

A Comparison of Experimental P - V - T Data

The data of Lounasmaa [‘TI and Glassford [91 have been c r o s s plotted to obtain i s o t h e r m s of the same value. The lower most line in each s e t is the 4.5 K i s o t h e r m and each succeeding line is 25 K higher than the one below it. e

23

Y

zyxwvu

zyxwvutsrqpo zyxwvutsrq zyxwvutsrqp zyxwvu zyxwvutsrqponmlkjihgfed zyxw

C. 8.":

..

.. ... .' ... ..'... . . . 4.

W

'h

'.

'

T ,

'

.-.,.;

4.

.

...._ .... ....

... ...

.,.,%.

. ....

:

.

"

.'""

zyxwvutsr c _

,f

.....

...

....,...*. .._...'

.. .

.

re*r,

ci. 4

L

.:.

, ,

1.2

0:e

7

PRESSURE, M I C R O N S

:-/2-/-:

F i g u r e 2.

Deviation Plot of the 1958 Helium Vapor P r e q s u r e T e m p e r a t u r e Scale.

The difference between t e m p e r a t u r e s calculated f r o m eqwations (1) and (2) and the 1958 helium vapor p r e s s u r e temperature s c a l e [31.

24

Y

"106

0

zy zyxwv zyxw zy zyxwvutsrqponmlkjih zyxwvu 0

0

8

8

0

R

8

0

El

El

n

n

IJ

u

B f l

0

0

0

0

8

0

Saturatr4 Vapor Saturate4 L i e u i d

2. e

3.6

0

8

B

a13

8-

8

e

0

8

0

0

a

0

4.4

TEMPERATURE, K

0'/01/'C

Figure 3 .

Deviation P l o t of Saturation Densities.

Deviation plot of experimental d a t a by el Hadi [12] and values calculated from equations (4)and (5).

25

2

0.4

zyxwvutsrq

zyxwvutsrqponm Q

zyxwvutsrqponmlk zyxwvu zyxwvutsrqponmlkjihgfedcba zyxwvuts zyxwvutsrq zyxwvuts v

B

Q

Q

Q

6

0.0

B

M

t-

Q

4

U

3

w

0

v

f Lrl u

3n

Q

Q

v

0

B

-0.4

Q

v

Q B

Q

v V Q

-z. a