T H E GENERALIZED THREE-PARAMETER EPSTEIN EQUATION OF STATE JOSEPH JOFFE AND GHANSHYAM R . P A T E L Newark College of Engineering, Newark, N . J .
P-V-T data for four hydrocarbons, representative of gases with nonpolar and nonspherical molecules, were used to study the Epstein equation of state, a form of the virial equation containing six virial coefficients, Critical constants are used in the calculation of the fourth, fifth, and sixth virial coefficients. Epstein has recommended that the second and third virial coefficients be computed from the Lennard-Jones 6-1 2 or some similar potential function, using values of the force constants obtained from experimental P-V-T or transport property data. The present study shows that the Epstein equation reproduces the P-V-T behavior of gases with nonpolar nonspherical molecules better when the Lennard-Jones force constants are obtained from critical constants in accordance with the law of corresponding states. Accordingly, the three critical constants are the only data needed to evaluate all virial coefficients in the Epstein equation. The equation finds its most important application in the calculation of thermodynamic properties.
PSTEIN
has recently proposed a virial equation of state
E containing virial coefficients through the sixth, as follows (7) : z = 1
+ B ( T ) / V + C(T)/VZ + [D’/V3
+ E’/V4 + F ’ / V 5 ] ( T c / T ) 3 (1)
T h e second and third virial coefficients, B ( T ) and C( T),are conveniently evaluated as in the simpler Hirschfelder-BirdSpotz equation (3) z = 1
+ B ( T ) / V + C(T)/V2
(2)
from the Hirschfelder-Bird-Spotz tables ( 4 ) ,whereas the fourth, fifth, and sixth virial coefficients are calculated by Epstein from the critical constants by imposing the condition on Equation 1 that the critical isotherm in the P-V plane must pass through the critical point with zero slope and must have a point of inflection a t the critical point. I t follows from the imposed conditions that :
D’ = [+15 zC - 10 - 6 B ( T c ) / V c- 3 C ( T c ) / V c *Vc3 ] (3) E’
=
[-24
Z,
F’ = [+IO
+ 15 + 8 B ( T c ) / V c+ 3 C ( T c ) / V c 2 ] V c 4(4) Z,
-
6 - 3 B ( T c ) / V c- C(Tc)/Vc*]Vc5(5)
Since the Hirschfelder-Bird-Spotz tables are based on the Lennard-Jones intermolecular potential, values of the LennardJones force constants, b,(or ro) and e l k , are required for the calculation of the second and third virial coefficients. Values of these constants are usually obtained from P-V-T or transport property data, a procedure recommended by Epstein. Epstein showed that, whereas the Lennard-Jones form of the intermolecular potential is limited in its applicability to nonpolar spherically symmetrical molecules, the introduction of higher order virial coefficients based on critical constants into Equation 1 extends its range of application to nonpolar gases in general (7). Moreover, while Equation 2 is said to be limited to a density not in excess of 40y0 of the critical density ( 6 ) ,Equation 1 in virtue of the presence of higher order virial coefficients is valid u p to the critical density. Equation 1 is considered by Epstein to involve five independent constants, b,(or yo), c/k, T,, P,,and Vc, and is stated by him to have an accuracy and 374
l&EC FUNDAMENTALS
range of validity comparable to that of the five-constant Beattie-Bridgeman equation. While the virial equation (Equation 2) was extended to gases with nonspherical molecules by Epstein by the addition of higher order virial coefficients, Nelson and Obert have pointed out that gases consisting of nonpolar nonspherical molecules can be handled with Equation 2 and the Hirschfelder-Bird-Spotz tables, provided that the force constants, bo and e / k , of such gases are obtained from critical temperature and critical pressure by means of the law of corresponding states (6). It appears reasonable to ask whether the applicability of the Epstein equation, Equation 1, to gases with nonspherical molecules is improved if the constants bo and ~ / k are calculated from the critical constants. Such a procedure has the further advantage that it reduces the Epstein equation to a generalized three-parameter equation, a n analytical form of the three-parameter law of corresponding states. I n the investigation here reported, (7), Equations 1 and 2 were tested with P-V-T data from the literature for four hydrocarbons: ethylene, ethane, propane, and n-butane (8-10, 72). These compounds were selected because their molecules are nonpolar and nonspherical, Lennard- Jones force constants are available in the literature for these gases, and these gases have been used in the derivation of generalized charts and equations of state. The calculation procedure consisted in substituting experimental values of the volume into Equations 1 and 2 and finding the per cent deviation of the calculated compressibility factor from the experimental value over a range of temperature from 60’ to 500’ F. and
Table 1.
Compound
Ethylene Ethane Propane n-Butane
Values of Lennard-Jones Force Constants Force Constants from Force Constants from Critical Constants Second Virial Coejicients (Method A ) (Method B ) elk, bo, elk, bo, O K. cc./mole K. cc./mole 214.0 230.9 279.6 321.4
95.3
107.7 149.7 192.8
199.2 243.0 242.0 297.0
116.7 78.0 226.0 155.0
~
~~
Table II. Summary of Results
( % deviation in compressibility factor) Equation 1 Compound Ethylene Ethane Propane n-Butane
hro.of Points 170 72 141 69
Method A Av. Max. 0.57 1.08 0.59 0.75
-4.81 -4.06 -3.64 -2.72
Equation 2
Method B Av. Max. 1.09 2.91 1.47 4.09
pressures from 1 to 300 atm. Experimental data corresponding to gas densities appreciably above the critical density were not used. Values of the critical constants were obtained from the literature (5, 77). Two basic calculation procedures were employed. Method A
+11.58 -11.52 $22.41 +13.01
z = 1
Method A Av. Max. 1.41 2.84 1.02 1.70
Av. 0.92 3.13 1.86 6.34
-36.20 -33.60 -11.38 -27.10
+ B ( T ) / V + C ( T ) / V 2+
Method B Max.
-
9.22 -11.45 +31.23 f34.73
+ E ( T ) / V 4+ F ( T ) / V 6
D(T)/V3
(8)
it may be shown, starting with basic thermodynamic relations given elsewhere (4, 7), that the enthalpy departure and entropy departure of a gas from the ideal gaseous state are given by :
Values of the Lennard-Jones force constants were calculated from the critical constants by means of the relations recommended by Nelson and Obert (6),
e / k = 0.756 T, and
T dF 6, = 17.0 T J P ,
(7)
and used to calculate second and third virial coefficients from the Hirschfelder-Bird-Spotz tables (4). Higher order virial coefficients appearing in Equation 1 were calculated from Equations 3, 4, and 5.
and
Method B
VaIues of the Lennard-Jones force constants, derived from experimental second virial coefficients, were taken from the Hirschfelder, Curtiss, and Bird tabulation (4). These values, along with those used in Method A, are shown in Table I. Except for the values of the force constants, the calculation procedure of Method B is identical with that of Method A. Calculations were carried out with the help of a digital computer. Details of all calculations are shown elsewhere (7). Table I1 lists for each hydrocarbon and each method the number of points tested, the average absolute value of the per cent deviation, and the maximum value of the per cent deviation in the calculated compressibility factor. At each temperature the largest deviations were observed to occur a t or close to the highest pressure tested, regardless of the method or equation used. An examination of the results makes it evident that Equation 1 in combination with Method A yields the most reliable results and therefore should constitute the preferred procedure. This is fortunate, since Method A requires fewer independent data than Method B when used in conjunction with the Epstein equation. In effect, Method A, involving only the critical constants P,, V,, and T,, reduces the Epstein equation to a generalized three-parameter equation, comparable in accuracy with the five-parameter Beattie-Bridgeman equation. Equation 2 in combination with Method A yields surprisingly good results, and is greatly in error only a t densities approaching or exceeding the critical density. As pointed out by Nelson and Obert (6), the advantage of a virial equation, such as the Epstein equation, lies not so much in its ability to predict compressibility factors as in its usefulness in calculating thermodynamic properties. If the Epstein equation is written as
T h e derivatives of the second and third virial coefficients,
d B / d T and dC/dT, are obtained from the Hirschfelder-BirdSpotz tables (4,while the derivatives of the fourth, fifth, and sixth virial coefficients, d D / d T , d E / d T , and dF/dT, are obtained by differentiating the expressions for these virial coefficients given by Epstein (7, 7) :
dD-- D ’ ( - 3 T C 3 / T 4 ) dT
(1 1)
dE - -- E ’ ( - 3 T c 3 / T 4 ) dT dF - -- F‘(-3TT,3/T4) dT After the work reported here had been completed, a new generalized virial equation was published by Gyorog and Obert ( Z ) , representing a modification and extension of the Hirschfelder-Bird-Spotz tables to include a fourth virial coefficient and its derivatives. T h e authors state that the inclusion of the fourth virial coefficient increases the range of the tables to a maximum density which is o n the average 2.8 times the Hirschfelder-Bird-Spotz limit. Their work is based on data for gases with spherically symmetrical nonpolar molecules. I t would be of interest to determine whether the Gyorog-Obert tables can be used with gases whose molecules are nonpolar but possess no spherical symmetry, provided that the Lennard-Jones force constants are computed from critical temperature and critical pressure as in Equations 6 and 7. VOL. 4
NO. 4
NOVEMBER 1 9 6 5
375
Should this procedure be valid, the Gyorog-Obert virial equation would in effect be a two-parameter form of the la!\. of corresponding states in contrast to the Epstein equation, which is a three-parameter form. Nomenclature
second virial coefficient, a function of temperature C( T ) = third virial coefficient, a function of temperature D ( T ) .E ( T ) .F(T) = fourth, fifth, and sixth virial coefficients D ;, constants in Epstein equation related to fourth, fifth, and sixth virial coefficients enthalpy of real gas H enthalpy of gas in ideal gaseous state at same H* temperature as H pressure P critical Dressure pc R gas constant entropy of real gas S S* entropy of gas in ideal gaseous state a t Same temperature and pressure as S T temperature critical temmrature Tc molar volume V critical volume compressibility factor = PV/RT Z critical compressibility factor Lennard-Jones force constant 3k =
B(T)
2
bo To
= =
Lennard-Jones force constant equilibrium intermolecular distance
Literature Cited
(1) Epstein, L. F., IND. ENO. CHEM.FUNDAMENTALS 1, 123 (1962). (2) Gyorog, D. A., Obert, E. F., A.I.Ch.E. J . 10, 625 (1964). (3) Hirschfelder, J. O., Bird, R. B., Spotz, E. L., Trans. A m . SOC. Mech. Engrs. 71, 921 (1949). (4) Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” pp. 1112, 1114, 1116, IViley, New York, 1954. (5) McIntosh, R. L., Dacey, J. R., Maass, O., Can. J . Res. 17, 241 (1939). (6) Nelson. L.. Obert. E. F.. A.I.Ch.E. J . 1. 74 11955). (7) Patel, G. R., M.S. thesis, Newark Cbllege of’Engineering, Newark, N. J., 1964. (8) Prengle, H. I V . , Jr., Greenhaus, L. R., York, R., Jr., Chem. En
