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I& ALAMOS S’CIEN-TIFIC LABORAmRY ‘--”OF THE UNIVERSITY OF CALIFORNIA o LOS ALAMOS NEWMEXICO

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-..

COMPRESSIBILITY FACTORS AND FUGAC~Y COEFFICIENTS.

CALCULATED FROM THE BEATTIE-BRIDGEMAN EQUATION

OF STATE FOR HYDROGEN, NITROGEN, OXYGEN, CARBON

DIOXIDE, AMMONIA, METHANE , AND HELIUM

-. —--- -..* z—.. .. ..- --=- —- --- ,... .’ ,. ..*. — . . .,

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PERMANENT

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LEGAL NOTICE

.*

This report was prepared as an account of Govern-ment sponsored work. Neither the United States, nor theCommission, nor any person acting on behalf of the Com-mission:

A. Makes any warranty or representation, expressor implied, with respect to the accuracy, completeness,or usefulness of the information contained in this report,or that the use of any information, apparatus, method, orprocess disclosed in this report may not infrtnge privatelyowned rights; or

B. Assumes any liabilities with respect to the useof, or for damages resulting from the use of any infor-m ation, apparatus, method, or process disclosed in thisreport.

As used in the above, ‘person acting on behalf of theCommission” includes any employee or contractor of theCommission to the extent that such employee or contrac-tor prepares, handles or distributes, or provides accessto, any information pursuant to his employment or con-tract wtth the Commission.

Printed in USA. Price $1.50. Available from the

Office of Technical ServicesU. S. Department of CommerceWashington 25, D. C.

0 ----

I

LA-2271CHEMrSTRY- -GENERALTID-4500, 14th Ed.

LOS ALAMOS SCIENTIFIC LABORA~RYOF THEUNIVERSITYOF CALIFORNIA LOSALAMOS NEW MEXICO

REPORT WRITTEN August 1958

REPORT DISTRIBUTED: March 17, 1959

COMPRESSIBILITY FACTORS AND FUGACITY COEFFICIENTS

CALCULATED FROM THE BEATTIE-BRIDGE MAN EQUATION

OF STATE FOR HYDROGEN, NITROGEN, OXYGEN, CARBON

DIOXIDE, AMMONIA, METHANE, AND HELIUM

by

C. E. Honey, Jr.W. J. WorltonR. K. Zeigler

This report expresses the opinions of the author orauthors and does not necessarily reflect the opinionsor views of the Los Alamos Scientific Laboratory.

Contract W-7405 -ENG. 36 with the U. S. Atomic Energy Commission

-1-

ABSTRACT

Compressibility factors and fugacity coefficients for hydrogen, nitrogen,

oxygen, carbon dioxide, ammonia, methane, and helium were calculated by

use of the Beattie-Bridgeman equation of state. The results are tabulated

for various pressures up to several hundred atmospheres, and temperatures

up to several hundred degrees, at sufficiently close intervals to allow for

easy interpolation.

A comparison is made of the calculated compressibility factors with

the experimental values over those ranges of temperatures and pressure

where the latter are available. From this comparison an attempt is made

to indicate the probable reliability of the fugacity coefficients by the number

of significant figures shown.

ACKNOWLEDGMENTS

The authors wish to acknowledge the advice and cooperation of Joe Fred

Lemons and Bengt G. Carlson, Group Leaders of Group CMF-2 and Group T-1,

respectively, in which this work was done. They also wish to acknowledge

the helpful conversation and advice of Roger H. Moore, Group T-1, with re-

spect to the mathematics and the calculations and the help of H. F. Franzen

in the comparison of the calculated quantities with the experimental quantities.

-3-

Introduction

In a recent study of the magnesium-hydrogen system, measurements

were made of the hydrogen pressures at equilibrium and it was necessary

to calculate the fugacities corresponding to these pressures before calculat-

ing the pertinent thermodynamic quantities. The method having been developed

and the program prepared, the calculations were extended to other tempera-

tures and pressures and to other gases. The results are presented here in

the hope that they may be of value in future investigations. Any suggestions

as to improvements will be welcome.

These calculations were made usiug the IBM-704. The problem was

coded in the FORTRAN-I system, with the exception of the root-solving

routine, LA S 871, which is due to J. K. Everton. With these aids, the cal-

culation is quite simple, and can easily be extended at any computing instal-

lation.

Two general methods are available for the calculation of fugacities:

I) graphfcal and 2) analytical. The graphical method can always be used if

P-V-T data are available. The analytical method requires that the P-V-T

data be fitted to an analytical expression for the equation of state in a form

which can be integrated. Fortunately, the Beattie-Bridgeman equation1,2 is

of such a form and with the proper constants closely represents the P-V-T

data for many gases over a wide range of temperatures and pressures.

-5-

Integration of the Beattie-Bridgernan Equation

The Beattie-Bridgeman equation may be written in the form

PV2 V+BO-:)-.O(+)( )(=RT1–~3VT

(1)

where Ao, Bo, a, b, and c are empirical constants to be determined experi-

mentally for a particular gas and P, V, R, and T

This equation can be rearranged to give the

have their usual

foUowing form

PV4 – RTV3 -

(

Rc

)

RTBO – — – A. V2T2

/ RcB

( )RcbBo

+ RTbBo+—O–a.Ao V– =

T20

T2

meanings.

(2)

which, as can be seen, is a quartic equation in V. When the values of the

empirical constants are inserted and this eqyation solved for V at a given T

and P, there may be more than one real positive root. However, one of the

roots will correspond more closely to experimental observations than the

others and it can be chosen by inspection.

The fugacit y, f, is found by means of the following expressions

/9P

(3)

where

RT_vCY=—

P

Rearranging,

-6-

‘“(a=-+fp(%-’)“ =-Jp$+&[p’”(4)

The Beattie-Bridgeman equation may be solved for P to give

P.y

( )

+ RTB –-–A 10 -Z

T20V

(RcBO

)

1 RcbBo ~– RTbBo + — -aA —+—— (5)

T2 0 ‘3 T2 ‘4

Differentiating,

[(d’= –R;–

)

RTB –R~– A Lv o

T2 0 ‘3

(

RcBO

)

RcbBo ~+ RTbBo + — – 1aAo~———d’ (6)

T2 ‘4 T2 ‘5

Hence,

[(‘d’= –R#–

)

RTBo–~– A 2T2 -Z

0’

(

RcBO

)

3 RcbBo 4+ RTbBo + — 1–aAo–———————d’ (7)

T2 ‘3 T2 ‘4and

J ( Rc

)

VdP=– RTln’+ RTBO– —–A 2T2 Ov

(RcB

)

RcbBo 4- RTbBo+—O. ti ~+—__ (8)

T2 0 2’2 T2 3’3

Substituting in the expression for In (f/P) in Eq. 4, we have

-7-

(RcBO

)

3+ 1RcbBo ~ P=P(V=V)

– RTbBo + — –a. — (9)T2 0 2V2 T2 3V3 P=O(V=M)

At the limit P=O(V=w), PV = RT, so

and

()(

RTA

)

lnf=ln ~ + Bo–A3–&~T

(CB aA

)

cb, 04–b,o+~–fio ~+—— (11)

T3 2V2 T3 3V3

This is the equation used for calculating f for various values of P and T.

Compressibility Factors

For the calculation of the compressibility factors the Beattie-Bridgeman

equation was solved for V at the various values of T and P, and then PV/RT

was computed for each of the real positive roots. The values were then com-

pared with experimental values to pick out those corresponding to the correct

roots which were then tabulated. The results are shown in Tables II through

VIII. Table I lists the constants of the

these and the subsequent calculations.

-8

Beattie-Bridgeman equation used in

TABLE I

VALUES OF THE CONSTANTS OF THE BEATTIE-BRIDGEMANEQUATION OF STATE FOR SEVERAL GASES4

Units: atmospheres; liters per mole; “K; R = 0.08206

Gas A. a

‘20.1975 -0.00506

N2 1.3445 0.02617

02 1.4911 0.02562

C02 5.0065 0.07132

NH3 2.3930 0.17031

CH4 2.2769 0.01855

He 0.0216 0.05984

A comparison of the calculated

mentally observed values is made in

B.

0.02096

0.05046

0.04624

0.10476

0.03415

0.05587

0.01400

b

-0.04359

-0.00691

0.004208

0.07235

0.19112

-0.01587

0

-4Cxlo

0.0504

4.20

4.80

66.00

476.87

12.83

0.0040

compressibility factors with the experi-

Tables IX through XV. The experimental

values were obtained from the following sources:

Hydrogen Woolley, Scott, and Bri&widde5 and Hilsenrath et al.6

Nitrogen Hilsenrath et aL6

Oxygen Hflsenrath et al.6

Carbon dioxide Hilsenrath et al.6

Ammonia Beattie and Lawrence7

Methane Kvalnes and Gaddy8

Helium Michels and Woutersg and Wiebe, Gaddy, and Heins10

Fugacity Coefficients

The fugacity coefficient, i e., f/P where f = fugacity and P = pressure,

is reported rather than the fugacities themselves, as this gives at a glance

a value for the relation between the two quantities.

-9-

The calculated

xVI through XXII.

values of the fugacity coefficients are shown in Tables

Errors and Significant Figures

The calculated values of the compressibility factors and the fugacity

coefficients are, of course, exact for a given set of values for the constants

in the equation. h the tables of the compressibility factors the numbers

are reported to the fifth decimal place. A comparison with the experimental

values where these are known will allow one to estimate how many significant

figures should be used in any particular case. For a discussion of the ac-

curacy of the experimental values the reader is referred to the references

cited.

In the tables of fugacit y coefficients an attempt has been made to give

some indication of how closely the numbers tabulated ought to agree with ex-

perimental values if they were available. To do this, it was noted from

Equation II above that in (f/P) = h (RT)/(PV) + correction terms) where

the correction terms are in general small. It was therefore assumed that,

to a first approximation, calculated values of f/P will deviate from experi-

mental values in about the same way as is the case for calculated values of

PV/RT. Therefore, in those regions when PV~T (talc) - PV/RT(expt) is

less than 0.0005, f/P is given to five decimal places. Where this difference

is between 0.0005 and 0.005, four decimal places are given. Where it is be-

tween 0.005 and 0.05, three decimal places are given. And where it is greater

than 0.05, only two decimal places are given. In those regions where experi-

mental values of PV/RT are not available for comparison with the calculated

values, estimates of the difference were made and the number of significant

figures adjusted accordingly. The authors hope that, if they have erred in

this, it has been in the direction of too many significant figures rather than

too few.

-1o-

References

1.

2.

3.

40

5.

6.

7.

8.

9.

10.

J. A. Beattie and O. C.~, 3133, 3151 (1928).

Bridgeman, J. Am. Chem. Sot., 49, 1665 (1927);—

J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory ofGases and Liquids, pp. 253, 254, John Wiley and Sons, Inc., New York,1954.

I. M. Klotz, Chemical Thermodynamics, p. 236, Prentice-Hall, Inc.,New York, 1950.

J. A. Beattie and W. H. Stockmayer, VI%e Thermodynamics and Statis-tical Mechanics of Real Gases, ~tChapter II in Vol. II of A Treatise onPhysical Chemistry, edited by H. S. Taylor and S. Glas stone, D. VanNostrand Company, Inc., N. Y. 1951.

H. W. Woolley, R. B. Scott, and F. G. Brickwidde, J. Research Natl.Bur. Standards, 41, 379 (1948).

J. Hilsenrath, et al., Tables of Thermal Properties of Gases, NBSCircular 564, 1955.

J. A. Beattie and C. K. Lawrence, J. Am. Chem. Sot., ~2, 6 (1930).

H. R. Kva.lnes and V. L. Gaddy, J. Am. Chem. Sot., ~3, 394 (1931).

A. Michels and H. Wouters, Physics, E, 923-39 (1941).

R. Wiebe, V. L. Gaddy, and C. Heins, J. Am. Chem. Sot., ~, 1721(1931).

-11-

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