Second Virial Coefficient of
Nonpolar
129
Substances
The consistency of the derived sublimation vapor pressures at various aluminum chloride pressures suggests that the CuAlCk solid phase remained at fixed composition. Data from the 11runs in which CuAlCld(s)was present are tabulated below the dotted line in Table I (supplementary material); the vapor pressure dependence on temperature is also shown in Figure 1. A least-squares treatment, assuming the simple vaporization reaction CuAlC14(~)= CuAlCld(g)
(2)
led to a predicted enthalpy of 34.3 f 1.0 kcal mo1-l and an entropy of 47.8 f 2.4 cal molm1deg-l for (2) at a mean temperature of 473 K. With an estimated ACp of -6 cal mol-l deg-' for (2), a combination of results for (1) and (2) with JANAF values for CuCl(s) and AlC13 leads to a predicted standard enthalpy of formation of -208 f 3 kcal mol-' and an absolute entropy of 44.5 i 4 cal mol-1 deg-l for CuAlCld(s) at 25O. The entropy value is not greatly different from the sum of the entropies of CuCl(s) and AlC13(s), 47 cal mol-l deg-l, as generally expected for such complexes. The temperature dependence of the saturation pressures of CuAlClr(g) when a liquid phase is present along with CuCl(s) reflects not only the variation of the vapor pressure with temperature but also the effect of the change in solubility of CuCl in the liquid complex with temperature. While the calculated pressures of CuAlC14 above 500 K (Table I, supplementary material) do appear to converge, a t a given temperature, on a limiting value presumed to characterize the saturated liquid, this upper limit could not
be reliably fixed, Le., one could not be certain in every case when a liquid phase was actually present, or if the liquid phase had reached saturation equilibrium with CuCl(s). Hence we have not attempted to calculate the properties of the liquid phase. Acknowledgment. This work was supported by a grant from the National Science Foundation, GP 37033X. Supplementary Material Available: Table I, experimental transpiration data (2 pages). Ordering information is available on any current masthead.
References and Notes (1) J. Kendall, E. D. Crittenden, and H. K. Miller, J. Am. Chem. SOC.,45, 963 (1923). (2) Eg. B. H. Johnson (Esso Research and Engineering Co.) U S . Patent 3475347 (Cl. 252-429; Bolj) Oct 28, 1969. (3) Eg. D. G. Walker (Tenneco Chemicals, Inc.) Ger. Offen. 2057162 (C1.C Olgb) June 3, 1971. (4) W. C. Laughlln and N. W. Gregory, lnorg. Chem., 14, 1263 (1975). (5) W. C. Laughlin and N. W. Gregory, lnorg. Chem., In press. (6) C-F. Shleh and N. W. Gregory, J. Phys. Chem.. 79,828 (1975). (7) D. L. Hilden and N. W. Gregory, J. Phys. Chem., 76, 1632 (1972). (8) R. R. Richards and N. W. Gregory, J. Phys. Chem., 68, 3089 (1964). (9) Perkin-Elmer Model 303. (10) R. A. J. Shelton, Trans. FaradaySoc., 57, 2113(1961). ( I 1) "JANAF Thermochemical Tables", Revised Edition, Thermal Laboratory, Dow Chemical Co., Mldland Mich. (12) G. I. Novikov and F. G. Gavryuchenkov, Russ. Chem. Rev., 36, No. 3, 156 (1967). (13) C. R. Boston in "Advances in Molten Salt Chemistry", J. Braunstein, G. Manantov, and 0. P. Smith, Ed., Plenum Press, New York, N.Y., 1971. (14) N. C. Baenziger, Acta Crysfallogr.,4, 216 (1951). (15) K. Balasubrahmanyamand L. Nanis, J. Chem. Phys., 42, 676 (1965). (16) L. Brewer, G. R. Somayajulu, and E. Brackett, Chem. Rev.. 63, 111 (1963).
The Second Virial Coefficient of Nonpolar Substances R. M. Gibbons The Brifish Gas Corporatlon, London Research Sfaflon, London SWS, Eng/and (Received April 15, 1975) Publication costs assisted by The Britlsh Gas Corporatlon
Continued fractions have been developed for the second virial coefficient for the Barker Bobetic potential which contain, without significant error, all the thermodynamic information contained in the exact expression. Constants are obtained for the BB potential for Ne, Ar, Kr, and Xe. The general problem of obtaining approximate expressions for the results of statistical mechanical theories is discussed and compared with the analogous problem of using analytical mathematical functions on digital computers.
Introduction Empirical correlations for the second virial coefficient, B , are still widely used t o d a ~ , l -even ~ though there are a number of intermolecular potentials which describe the second virial data for many substances better than any empirical c ~ r r e l a t i o n . ~ The - ~ reason for this is the complexity of the calculations to evaluate B from the intermolecular potentials, which precludes the use of these models for routine calculations of B. It is the purpose of this note to show how simplified expressions for the values of B calculated from intermolecular potentials may be obtained which are
simple enough to use routinely. Since this is a general problem with all statistical mechanical theories, we start by discussing what is required of such simplified forms before discussing the application of the method to the calculation of B for the Barker Bobetic (BB) potential? which was chosen as an example because it is one of the best potentials for argon which has yet been devised. We conclude with a short discussion of the application of the method to other statistical mechanical theories. The integral for B for a realistic potential is typical of statistical mechanical theories of fluids in that the answer is provided as an intractable integral which can only be The Journal of Physical Chemistry, Vol. 80, No. 2, 1976
130
R. M. Gibbons
evaluated by a numerical integration, It is this feature which makes all these theories so inconvenient to use. However, to use these theories it is not necessary to have the exact answer; in fact, such exact answers always contain far more information than is ever required. For example, exact answers provide exact information for all orders of derivatives for all values of the temperature and density including negative and complex values of these variables. (In practice the accuracy is limited to the round-off error of the computer even for exact answers.) Clearly, to use these theories to calculate thermodynamic properties, much of this information is not required, as we are normally only interested in the free energy, and its first few derivatives, over a restricted range of temperature, density, and composition. The initial problem then reduces to representing, sufficiently accurately, the results of numerical integrations in the statistical mechanical theories over a very restricted range of temperature, density, and composition. This problem is in many ways analogous to that faced by computer manufacturers in representing mathematical functions for use on a computer. They solved that problem by developing expressions, normally continued fractions, which represented the function over a range of values with an error about equal to the round-off error of the computer. Similar techniques can be used for statistical mechanical theories as is shown below for B. While the best possible result is to develop an expression with an error equal to the round-off error of the computer, in many cases even this is not necessary. It is frequently sufficient that the errors in the calculated values, and the first few derivatives, be much less (say a factor of 10 less) than the errors in the experimental data; the errors in the calculated values will then not be significant in making comparisons with experimental data. The problem then becomes one of obtaining such ap proximate forms. It is shown in the next section how this may be done for B for the Barker Bobetic potential using continued fractions obtained by standard methods from the results of numerical integrations of eq 1.
A Continued Fraction for B The values of B for the BB potential are obtained from the following integral: B = -2aN
JI-
(e-ulkT - 1)R2 dR
(1)
where u is the BB potential. For the reduced temperature range of 0.5-20.0, this integral can be represented by the following continued fractions:
+
B* = BH* Bs*
(2)
where B* is the reduced value of B, Bl(2arm3/3),
+
BH* = egT*(ag al/T*
+ az/T*2 +
1.01T*3)/(bo+ bl/T*
Bs* = exp(f/T)(co+ cdT* + c ~ / T + *~ 1.0/T*3)/(T*(dg+ d l + d2/T* a, = a, = a, =
+916.14919 8555.2627 +8101.6196
c, = 58147.879 c , = +8618.2141 c , = 1468.565
+ bz/T*')
+ d3/T*2))
(3)
bo = 1914.8161 b , = 13230.40 b , = 11315.04 g = -0.008
f = 0.7
d o = 35416.003 d , = 10895.233 d , = 1132.4217
The Journal of Physical Chemistry, Vol. 80, No. 2, 1976
TABLE I: Force Constants for the BB Potential
Substance
elk, K
r min, nm,
r m s error, cm3 mol-'
Neon Argon Krypton Xenon
42.00 140.58 199.60 186.06
0.3084 0.3777 0.4004 0.4468
0.5 2.6 3.8 0.4
where T* is the reduced temperature, kT/e, and E and rm are force constants for the BB potential as defined in ref 7 . The root-mean-square errors for these expressions for B*, T* dB*/dT*, T*2d2B*/dT*2are
