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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
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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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