Reino W. Hakala Oklahoma City University Oklahoma City, Oklahoma 73106
A Derivation of the Virial Equation Based on the Hard-Sphere Model of Gases
It is generally believed that, aside from the effect of finite molecular size, deviations from ideal gaseous behavior cannot be explained except on the basis of intermolecular forces of attraction. Such is not the case, however, although intermolecular forces certainly do contribute significantly to nonideality. It will be shown in the derivation which is given below' that the hard-sphere model of gaseous molecules, according to which the molecules occupy a definite volume which is independent of pressure and temperature, but otherwise behave as ideal gases, leads directly to the Reichsanstalt (Holborn-Otto) form of the virial equation:
or to the corresponding power series in the gas density. The usual simple kinetic theory of gases disregards the fact that molecular clustering can occur as a result of elastic collisions. When, for example, three mole-
cules collide simultaneously, one of them can carry away sufficientkinetic energy and linear momentum that the other two molecules will remain together. This is quite aside from any considerations of intermolecular forces of attraction. (The latter will, of course, enhance the formation of molecular clusters.) Though the concentration of binary clusters arising in this manner would be small in comparison with the concentration of single molecules, and the concentrations of larger clusters still smaller, the fact remains that clusters of molecules would exist in the gas phase even if intermolecular forces of attraction were entirely absent. We shall now learn what effect the existence of molecular clustering has on the form of the equation of state of gases. For mathematical convenience, we shall asume that the molecules are hard spheres. That is, they occupy a definite volume independent of pressure aud temperature, and they do not interact with one another except on collision. Since intermolecular forces of attraction are assumed to be absent, Dalton's law of partial pressures holds: P i = P , + P 1 + P 1 +. . .
This derivat,ion follows that of BAND,W. C., "Introd~lction to Quantum Statistics," D. Van Nostrand Co., Inc., New York, 1955. Cham VI. sections 6.4 to 6.6, but is preatly simplified and diffek in most oi the details
where Piis the partial pressure exerted by those clusters which contain i molecules. I n the event that all of the clusters become completely dissociated, t,he pressure increases to the value
It is not difficult to convert this to the equivalent Leiden (Onnes) form,
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P*
=
PI
+ 2p1 + 3Ps + .. .
Since the molecules of gas occupy definite volumes but do not interact except on collision, the pressure is also given by the extended ideal gas law nRT p * = --V- nb
where t,he symbols possess their usual significance. Combining these three equations, we ohtain the expression P(V - nb)- -- PI P* Pa . .. RT PI 2P1 3Ps ..
+ + + + + +
Employing the mass-action law,
Applying this result, the series P = PI
+ K,P,= + KsPt3+
.
becomes P I = P - K2Pa
+ (2K2 - Kz)PS (3K2 - 3KlKa + KdP4 +
whence
and P ( V - nb) nRT
P, = KT,'
where K t iq the equilibrium constant for the formation of clusters containing i molecules, it follows that P = PI + K S P , ~ K3Px3 . . . and P(1' - nb) = PI KaP12+ K.PL~ . . . PI + ~ K ~ P 3KgPi3 I ~ + ... nRT
+
+
+
+
+
The right-hand side of the last equation can be obtained as a single power series by carrying out the indicated division, or by applying Rlaclaurin's theorem. The result is:
On comparing the last two equations, we see that whence the compressibility factor is given by the power series This result provides a way to convert the ratio of power series in PIto a ratio of power series in P. First, we need to establish a theorem on the reversion of power series, i.e., given the power series y = f(x), how to find the power series x = g (y). I n the series y =
x-
a#
(no constant term)
n= 1
B, = b - KIRT B1 = b' - 2bKzRT 2(2Kn2- Ka)(RT)% 6b(2Ka2- K$)(RT)* BI = b"3b2KnRT - (20K2 - 18K2K1 3Kd(RTY
we make the substitution
+ +
+
and thereby obtain
Since y = y, the first coefficient equals unity while all remaining cnefficientsnecessarily vanish: ahl = 1 alb2 albr
This result is of the same form as the Reichsanst,alt virial equation. By reversion of series, the Reichsanstalt form can be converted to the Leiden form.3 The Leiden virial coefficients are thereby found to be given by
+ albia
=
0
+ 2adA + axbLa= 0
. .. from which it follows thatZ
b,
a. = -a,'
+-2a?a P
... I t may be observed that, in both forms of the virial equation, the ith virial coefficients,A,-, and Bi-I, i 22, are functions of KP, K3,. . .,KG hence depend on the formation of binary, ternary, . . ., i-nary clusters of molecules. We have thus shown that deviations from ideal gaseous behavior, apart from the effect of molecular size, are due to the occurrence of molecular clusters, which can form even in the absence of intermolecular forces of attraction. % Along list of formulas for the coefficients b, has been given by VANORSTRAND, C. E.,Phil.Mag.,19,360 (1910); ADAMS,E. A. (Editor), "Smithsanian Mathematical Formulae and Tables of Elllotic Functions." Smithsonian Institution, Washington, . ~ . dl922,p. , 116. ' This method of conversion from one form of virialequation to the other is due to EPSTEIN, L. F., J. Chem. Phys., 20,1981 (1952). His formula for E,, our A&,is incorrect as published. The correct formula. is E, = (E, - 4D,B, - 2C2
+ 10C,B,Z-5Ba4)I(RT)4
where, in our notation, B, = BI, . . ., E , = Bc
Volume 45, Number I , January 1968
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