Analytical Calculations of Thermodynamic Properties H. M. ROBINSON' AND HARDING BLISS2 University of Pennsylvania, Philadelphia, Penna. I T H the ever-increasing Equations for calculating entropy, enthalpy, and energy changes i n use of h i g h - p r e s s u r e terms of volume have been derived for the Van der Waals, Wohl, and methods as tools for Beattie-Bridgeman equations of state. chemical synthesis or gas separaThey have been tested for entropy and enthalpy calculations by tion, the effect of pressure on comparisons with published values determined graphically. The various thermodynamic properties becomes more and more imBeattie-Bridgeman equation is satisfactory to 5-7 per cent for entropy portant. This effect is calculated and 12 per cent for enthalpy determinations. The others are almost b y m e a n s of t h e s o - c a l l e d as good for entropy calculations but definitely inferior for enthalpy. thermodynamic identities, the terms of-which are evaluated by P-V-T d a t a or b y JouleThomson coefficients. The scarcity of the latter and the abunof state explicit in pressure should be put to more use. Since dance of the former dictate the use of the P-V-T data. These the constants for several equations of state can be evaluated are available in tabular or graphical form as the result of direct from critical constants alone, it is apparent that this method experimental measurement or in the form of an equation of can be a very simple one. The authors feel that it should be called to the attention of engineers who may be interested in state. If the thermodynamic properties are to be evaluated constructing thermodynamic charts without recourse to in terms of pressure, the quantities V and ( b V / bT), must be measured P-V-T data or graphical treatment thereof. evaluated in terms of pressure. They must then be integrated with respect to pressure. For graphical P-V-T relaA second purpose of this article is to test the three equations for their accuracy in this use. It is obvious that some actions these operations can be performed graphically, and, curacy is sacrificed by this simpler method, but the loss is not although they are tedious, they are quite accurate. For analytical equations of state the same graphical methods must great in many cases. The presentation of P-V-T data and activity coefficients be resorted to because of the fact that practically all wellfor hydrocarbons by Lewis and Luke ( I I ) , activity coefknown equations of state are explicit in pressure. The necesficients for all gases by Xewton ( I S ) , and thermodynamic sary terms cannot, therefore, be evaluated as functions of properties by Edmister (7) as functions of reduced state inpressure. dicate a trend to the sacrifice of some degree of accuracy for For exploratory work in this field it would often be desirable ease and simplicity of use. This paper is another step in that to avoid these tedious graphical methods by the use of endirection. tirely analytical methods. The purpose of this paper is to point out that, if one chooses volume as the independent Calculations variable instead of pressure, the necessary differentiations and integrations can be performed with ease. I n this case The thermodynamic identities relating the variation of the quantities P, (dP/bT)., and V(bP/bV),mustbeevaluenergy, enthalpy, entropy, etc., with volume a t a constant ated in terms of volume and integrated with respect to it. temperature are well known. They are presented, along It is apparent that these operations can be performed readily with many others, by Lewis and Randall and by Bridgman with equations of state explicit in pressure. Starting with (6),and will not be repeated here. They depend on the evaluathese quantities, expressions for isothermal A H , A S , and tion of (dP/ bT)v, P , and P ( b P / ' d V ) as previously menAL' (from which A F, A A , and A lnfcan be calculated readily tioned. The differentiations and integrations with respect by well-known relations) have been derived for three widely t o volume are matters of elementary calculus. They have used equations of state. They are presented and later tested. been performed analytically and yield the following results: The authors claim no particular originality for the use of this mathematical device, but they do feel that it has been VAN DER WAALSEQUATION unduly neglected. Lewis and Randall (IO) used it to calculate fugacity, but they did not apply it t o other properties. Beattie (1, 2 ) mentioned it, but he limited himself to the Beattie-Bridgeman equation of state. Keenan and Keyes (9) used exactly the same principle except that they correlated the P-V-T data in an equation explicit in 8. This is the ideal method because analytical treatment in terms of pressure then becomes possible. However, the authors feel (3) that the abundance of P-V-T data in the form of equations
,
1 2
Present address, 49th and Looust Streets, Philadelphia, Ponns. Present address, Yale University, New Haven, Conn.
(4)
396
IUDUSTHI 41, A\SD ENGINEERIKG CHEMISTRI-
MARCH, 1940
AFT, AAr, and A 1nfr can br calculated in terms of the above by these relations:
WOHLEQUATIOS p
=----RT
Ti
- b,
( 1 ,
VT(V - b,)
+A T4/3 V 3
since this is intended as a test of the accuracy of these equations for thermodynamic calculations and not as a test for P-V-T exactness. This is further justified by the fact that measured volumes are usually available, but in case they are not, they can be calculated by that equation which fits best for that particular purpose. The Van der Waals constants are taken from International Critical Tables (8) except in the case of propane, for which they were calculated by the relations given in the table of nomenclature. The Wohl constants are calculated by the relations given in the nomenclature. The critical properties, necessary for these calculations, were obtained from the International Critical Tables except in the case of propane for which those of Beattie, Kay, and Kaminsky (4)(namely, T , 96.81 O C., Pa42.01 atmospheres, and V c4.43 cc. per gram) were used. The Beattie-Bridgeman constants were those listed by the originators of this equation (3) except in the case of propane, where those of Reattie, Kay, and Kaminsky were used.
TABLEI. LIFT, AAT, and 4 l n j can ~ be evaluated from these relations as discussed under the Van der Waals equation.
BE~TTIE-BRIDGEMAS EQVATION
397
DATA^
ON
PROPANE, REFERENCE PRESSURE OF
1.7 ATMOSPHERES
Sage el al. (14)
P. V2
AH? ASE
Van der Waals Wohl Ti = 589.7; VI = 2475; T R , = 0.888 21.06 i6.74 17 18.51 18 51 18.51 - 872 -269 - 1035 -5.25 -5.84 --3.57 TI'
PI' V2 AH?,' A+
B, AH?' AS3
=
Vz " AH 2:, AS2
P3" V3"
AH,: AS3
15.18 29.63 -159 -4.49 51.7 7.49 -668 -7.45
34 9.49 1261 -7.21 40.8 6.38 - 1693 -8.09
81" =
45,lO 9.49 -520 -7.00 62.9 6.38 - 770 -7.94
-
a Units: T = R.; P = a t m . ; V mole; S = B. t. u./ib. mole, R.
TABLEIT.
17 62 1 s 51 -76.9 -5 OD
649.7; VI' = 274 3; TRl' = 0.977
13.61 29.65 -458 -4.62 34 7.49 - 1861 -8.09
TI*= 879.7; P2X
Beat tieBridgeman
DATA^
ON
13.68 29.65 - 567 -4.7s 33.0 7.49 -2185 -8.56 288; T,,"
-
= 1.022
3'3.60 9.49 - 1702 -7.85 40.2 6.38 - 2406 -9.09 CU.
13.95 29.65 - 443 -4.64 35.0 7.49 - 1803 -8.13 34.80 .49 -91398 -7.52 46.2 6 38 - 2036 -8.68
ft./lb. mole; H .= B. t. u./lb.
NITROGEN, REFERENCE PRESSURE OF 1 ATMOSPHERE
M l l a r and Sullivan ( 2 2 )
Beat tieBridgeman
30 2,885 - 754 -9.83
Van der Waals Wohl TI = 225; VI = l(i4; T R , = 0.99 23.98 30.95 2.886 2.885 -938 - 520 -9.75 -8.50
30 4.96 -327.5 -8.04 60 1.492 -923 -11.18
TI' = 261; VI' = 190.8; TR,' = 1.15 29.85 2-8.10 4.96 4.96 -298 - 496 -7.51 -8.09 62.86 53.07 1,492 1,492 -886 -1339 -12.06 -10.69
29.99 4.96 -361 -7.77 60.74 1,492 - 1098 -11.61
540; VI'' 395.5; T,," = 2.38 29.51 i9.88 13.13 13.13 -78.9 -s7.2 -G.S6 -6.89 37.13 58.47 6.7 6.7 - 156 -168 5 -s.32 -s.29
29.90 13.13 -78.6 -6.9 58.52 6.7 -149.8 -8.36
(15)
--i
A F T , AAT, and A lnfT can be calculated from thepe relations as before.
P: 1:' AH2
AS?
Comparison with Graphical Methods The utility and accuracy of this method can beet be demonstrated by comparison with the results of graphical computations already presented in the literature. Kitrogen, methane, and propane were chosen for comparison. The data of Millar and Sullivan (12) for nitrogen, Edmister (6) for methane, and Sage, Schaafsma, and Lacey (14) for propane were selected. Comparisons are made a t three temperatures, two of which were chosen deliberately near the critical point, since it was felt that the test would be most severe there. For each temperature the entropy and enthalpy changes in going from approximately one atmosphere to some higher pressure were calculated. One of these higher pressures was also chosen to be near the critical. The limits of integration are measured volumes, not calculated volumes,
P2 V2' AHz' AQz'
Pa V3' AH:' AS3
Ti"
P2" V2" AH::
ASz
P3"
V3" AH8 ASa"
30 13.13 -73.8 -7.27 60 6.7 -138.8 -8.69
-
-
T = ' R.; P = atm., V a Units: mole, S = B . t. u./lb. mole, R.
= cu.
29.31 2.885 732 -9.13
-
ft./lb. mole; H = B. t. u./lb.
INDUSTRIAL AND ENGINEERIXG CHE-MISTRY
398
equation fails by 70 per cent in some cases. It is apparent that the Beattie-Bridgeman equation is distinctly superior for enthalpy calculations throughout the whole range and that it can be expected to yield an accuracy of 12 per cent with isolated deviations of greater magnitude. Since the calculations of enthalpy changes from P-V-T data, as performed graphically by other investigators, involve V T ( dV/dT)p while entropy changes involve only (dV/dT),, it is t o be expected that any experimental errors which may have occurred in the original data will probably reflect more seriously in the enthalpy calculations. Therefore, the poorer agreement in enthalpy comparisons may not be entirely due to inaccuracies in the equation of state. It should be emphasized that the Wohl equation has been tested more severely than the others because in all cases its constants have been calculated from critical data alone.
TABLE111. DATAQ ON METHAKE, REFEREXCE PRESSURE OF
Edmister ( 6 )
1 ATMOSPHERE
Van der Waals
Ti = 365.7; Vi
P2
50 3.167 886 -9.47
VI AHz ASS
-
PP' V2'
= 266.8,
50.1 3.167 -961 -9.29
Wohl
43.8 3.167
TI' = 491.7; VI' = 358; T E E i= 1.436 49.1 49.7 6.34 6.34 -372 -526 -8.25 -8.65 114.3 116.9 2.278 2.275 -966 -1241 -10.76 -11.41
50 6.34 -412.5 -8.36 120 2.278 1021 -11.04
AH*' A&'
Pa: Va
-
AHs' ASS'
BeattieBridgeman
TR, = 1.066 --a
- 1893 -11.21
45.5 3.167 -965 -9.73
50.1 6.34
- 403
-8.39 118.6 2.278 - 1049 -11.11
TI"= 851.4; Vi" = 622: TRq" = 2.48 Pa"
V2" AHa" AS!"
PI"
Vdt
AHs"
ASS" a
-
Units:
mole; S
49.3 12.41
50 12.41 -83.6 -7.84 120 5.27 -271.6 -9.81
T =
O
- 149.6
-7.89 112.8 5.27 404 -9.16
-
50 12.41 -103 S -7.91 118.5 5.27 -267.5 -9.79
50.1 12.41 -135.2 -7.91 115.6 5.27 -301.5 -9.54
R . ; P = a t m . ; V = cu. ft./lb. mole; H = B. t. u./lh.
B. t. u./lb. mole,
' R.
Conclusions The results shown in Tables I, 11, and I11 can best be compared at similar conditions of reduced temperature. When such a comparison is made for the entropy, it is apparent that at or below the critical, the Beattie-Bridgeman equation is distinctly superior and fits to within 0.5 per cent with the exception of one point which deviates by 7 per cent. The Wohl equation agrees to 6 per cent, the Van der Waals to 13 per cent. At temperatures slightly above the critical both the Beattie-Bridgeman and Van der Waals equations fit within 7 per cent, but the Wohl equation deviates as much as 18 per cent. At temperatures well above the critical, all three fit within 4.5 per cent (the Beattie-Bridgeman within 4 per cent) with the exception of one Van der Waals point which deviates by 7 per cent. Although no attempt was made to test these equations at frequent intervals, i t is true that the conditions chosen were deliberately severe. It seems, therefore, that the Beattie-Bridgeman equation should be accurate to 5-7 per cent for entropy calculations and the others to 10-15 per cent. This rather good agreement is understandable, when i t is observed that the first term of the entropy equation is substantially the same for all three equations of state and that this term is of predominant importance. When a similar comparison is made for the enthalpy, the agreement is much less striking. At temperatures a t or below the critical, the Beattie-Bridgeman equation fits within 12 per cent, the Wohl within 25 per cent, and the Van der Waals within 70 per cent. The last is in part due to the method of evaluating the constants for propane, where the worst discrepancies were observed. At temperatures slightly above the critical, the Beattie-Bridgeman equation fits within 11 per cent except for one point which deviates by 20 per cent. The Wohl equation is very poor in this range, deviating over 100 per cent in one case, although its usual disagreement was about 45 per cent. The Van der Waals equation agrees within 10 per cent on nitrogen and methane but deviates by 55 per cent on propane, which is in part due to the constants as explained above. At temperatures well above the critical the Beattie-Bridgeman equation retains its superiority, falling for the most part within 11 per cent. There is one bad deviation of about 60 per cent. The others agree to about 25 per cent, although the Van der T a a l s
VOL. 32, NO. 3
Acknowledgment The authors wish to acknowledge the helpful comments of Barnett F. Dodge and N. W. Krase in reviewing the manuscript.
Komenclature pressure temperature molar volume = universal gas constant a, = constant in Van der Waals equation; units V2P Determinable from critical constants thus: a, = 3P,V,2 bo = constant in Van der Waals equation; units V :
P T V R
= = =
a,
=
h.
=
V./.?
conytant in-iTiohl equation; units V2PT:
a, = 6VC2P,T, b, = constant in Wohl equation: units V ; b, = Vd/4 cur = constant in Wohl equation; units V'T'/aP: c.. = 4V,aP,T,'/a
ab = constant in Beattie-Bridgeman equation; units V A D = constant in Beattie-Bridgeman equation; units V2P b b = constant in Beattie-Bridgeman equation; units V BO = constant in Beattie-Bridgeman equation; units V Cb = constant in Beattie-Bridgeman equation; units T3V S = molar entropy H = molar enthalpy F = molar free enthalpy, Gibbs free energy; F = H - T S A = molar Helmholtz free energy; A = - TS
= fugacity Subscript c = critical
f
Literature Cited (1) Beattie, J. A.,Phys. Rev., 31, 680 (1928). (2) Ibid., 32, 691 (1928). (3) Beattie, J. A., and Bridgeman, 0. C., J . Am. Chem. Soc., 50, 3133 (1928). (4) Beattie,'J.-A:, Kay, W.C., and Kaminsky, J., Ibid., 57, 1589 (1937). (5) Bridgman, P., "Condensed Collection of Thermodynamic Formulas", Cambridge, Mass., Harvard Univ. Press, 1925. (6) Edmister, W. C., IND. EKG.CHEM.,28, 1112 (1936). (7)Ibid., 30,352 (1938). (8) International Critical Tables, Vol. I, p. 42, New York, McGrawHill Book Co., 1933. (9) Keenan, J. H., and Keyes, F. G., "Thermodynamic Properties of Steam", New York, John Wiley &- Sons, 1936. (10) Lewis, G. N., and Randall, M., "Thermodynamics", New York, McGraw-Hill Book Co.,1923. (11) Lewis, St'. K., and Luke, C. D., Trans. Am. SOC.Mech. Engra., Petroleum illech. Eng., 54,8 (1932). (12) Millar, R. W., and Sullivan, J. D., U. S. Bur. Mines, Tech. Paper 424 (1928). (13) Newton, R. H., IND. EKG.CHEM.,27, 302 (1935). (14) Sage, B. H., Schaafsma. J. G., and Lacey, W.N., I M d . , 26, 1218 (1934).
