M. RIGBY
2014
A van der Waals Equation for Nonspherical Molecules by M. Rigby Chemistry Department, Queen Elizabeth College, University of London, London W8 7 A H , England (Received January IS, 1078) Publication costs borne completely by The Journal of Physical Chemistry
An equation of state of the van der Waals type is proposed for systems of nonspherical molecules. The hardsphere repulsive term is replaced by a term appropriate to nonspherical convex hard molecules. The effects of this change are found to be similar to many of the experimentally observed effects of changes in molecular shape.
The effects of molecular shape on the bulk properties of fluids have been the subject of many investigations1t2 and the experimental pattern has been well established. The work of Pitzer and his coworkers’ has shown that variations from the corresponding states behavior characteristic of simple spherical molecules may be accurately correlated using a single additional parameter. This parameter, called by Pitzer the acentric factor, is determined by a number of factors and may include contributions from the effects of nonspherical attractive and repulsive intermolecular forces and from variations in the spherically symmetric force field. Attempts to relate the observed behavior to the presence of various types of nonspherical intermolecular interaction have tended t o concentrate on the effects of contributions from nonspherical attractive terms, which may be treated as perturbations of a spherically symmetric reference potential. The effects of the presence of dipoles, quadrupoles, and higher multipoles have been investigated in this way and are now quite well understood.2 The treatment of nonspherical repulsive forces is less easy. The energy perturbations associated with the nonspherical repulsive terms are much larger than those for attractive perturbations, and it seems desirable to seek an alternative approach. The success of recent extensions of the van der Waals type of model of the liquid has suggested one possible treatment of this problem. Perturbation theories, whose physical basis is similar to that of the van der Waals equation, have been applied with great success to the description of simple f l ~ i d s . ~ -This ~ work has shown that the principal factor determining the geometric structure of a fluid is the magnitude of the repulsive forces and that these may be adequately approximated using a hard sphere model. The structure of a real liquid is thus very similar to that of a hard-sphere fluid at an appropriate density, and the attractive intermolecular forces have little effect on the structure. Equations of state embodying these principles may be written in the generalized form5 The Journal of Physical Chemistry, Vol. 76, N o . 14, 1072
PV = NkT
@($)- -(-) a N kT V
where a and p are functions only of the density, N / V . The function @ describes the equation of state of the hard-sphere fluid, and the correction due to attractive intermolecular forces is given by the function a. The simple approximate form of a used by van der Waals appears to be fairly adequate for many systems. f
f
=
aN VZ -
This form of a implies that the configurational internal energy of the fluid is directly proportional to the density. The original van der Waals form of /3 is less satisfactory, and recent worka has shown the advantages of using an accurate hard-sphere equation of state. One of the most convenient analytic representations of this is that derived from scaled particle theory,’ which reproduces the data from Monte Carlo and molecular dynamics simulations well. Since the hard-sphere model provides a good basis for the description of systems of real spherical molecules, it seemed interesting to investigate the adequacy of a similar approach to the study of systems of nonspherical molecules. This has been done by studying the properties of an equation of state of the general form of eq 1, retaining the van der Waals form of a, but utilizing a hard core term, p, which is applicable to convex hard molecules of generalized shapes. Using such an equation one may expect to establish the effects of nonspherical repulsive forces, separated from those (1) K. S. Pitzer, D. Z. Lippmann, R. F. Curl, Jr., C. M.Huggins, and D. E. Petersen, J. Amer. Chem. Soc., 77, 3433 (1955). (2) J. 9. Rowlinson, “Liquids and Liquid Mixtures,” 2nd ed, Butterworths, London, 1969, Chapter 8. (3) J. A. Barker and D. Henderson, J . Chem. Phys., 47, 4714 (1967). (4) B. TVidom, Science, 157, 375 (1967). (5) M. Rigby, Quart. Rev. Chem. SOC.,24, 416 (1970). (6) E. A. Guggenheim, Mol. Phys., 9, 199 (1965). (7) H. Reiss, H. L. Frisch, and J. L. Lebowitz, J . Chem. Phys., 31, 369 (1959).
A VAN DER WAALSEQUATION
FOR
NONSPHERICAL MOLECULES
arising from nonspherical attractive terms. A suitable equation of state for convex hard cores has been derived by Gibbons,* using scaled particle theory, and may be written
p = - -PI.' NkT
3 -
+ d(37 - 6) + d2(3 - 3 7 -I- 7 ' ) 3(1 - d ) 3
(3)
where y. is a single parameter determined by the shape of the hard core. This is defined in terms of the volume, v, mean radius, R , and surface area, S, of the hard core.
RS
7 = -
V
2015 ~
Table I: Critical Data for Nonspherical van der Waals Model Y
R/o
3 4 5 6 7 8 9 10 15
0
(PV / N k T ) c
(4) A
The density, d, is defined by
The equation of state shows the general characteristics observed with the van der Waals equation, and the critical point may be readily located in the usual manner. Because of the rather greater complexity of the equation, the solutions for several values of the shapc factor y were obtained numerically. The critical compressibility factor, ( P V I N k T ) , , was evaluated for a wide range of values of y, and the results are shown in Table I. In order to give a readily comprehended idea of the molccular shape associated with a given value of y, the dimensions of the appropriate prolate spherocylinder (see Figure 1) are given also. This shapc is sclocted for the purpose of illustration only, since a given value of y is compatible with a wide range of shapes. For each value of y studied, the vapor pressure was determined over a temperature range from about 0.5T, to T,. The method of calculation used was a numerical one, based on tho n4axwt:ll equal area rule. The den-
I
I
I
I
Method
I
- - - - - - - _ _1-
I
1-
I
_--------a -A' Q I
I
This equation of st'ate (3) appears to provide a good representat'ion of the (rather meager) data presently available for nonspherical hard mole~ules.~ The complete van der Waals type equation using the forms of p and cr defined in eq 3 and 2 is an example of a throe-parameter corresponding states relationship, and it is of interest to see to what extent such an equation can reflect the experimentally observed deviations from simple (two-parameter) corresponding states behavior. These havc been summarized by Pitzer] and Rowlinson.2 Certain general trends are evident with increasing departure from spherical symmetry. These include variations in the reduced vapor pressure as a function of reduced temperature, increases in the slope of the rectilinear diameter line, decreases in the critical compressibility factor, and increases in the configurational heat capacities. Most of these changes are reproduced qualitatively by the present model.
0.3599 0.3574 0.3555 0.3539 0 3525 0 3514 0 3503 0.3494 0.3461
1.457 2.826 3.562 4.582 5.596 6.606 7.613 12.633
R
-71
Figure 1. The prolate spherocylinder model. AA' is the axis of rotation.
sities of the coexisting phases were determined at the same time, and the rectilinear diameter curve was established.
Results Before considering in detail the results obtained, it must be emphasized that qualitative agreement with experimental trends is all that can be expected from such a model. Since the repulsive forces in real molecules are not infinite, the appropriate hard-core dimensions should presumably be temperature dependent, in analogy with the similar hard-sphere cases3 The omission of this adjustment must modify the quantitative conclusions to some extent. The critical compressibility factor is reported in Table I for a range of molecular shapes from spherical ( y = 3) to spherocylinders of length R = 1 2 . 6 ~(see Figure 1). The value of the critical compressibility factor for the spherical case is rather different from the experimental value for simple spherical molecules of around 0.28. This is clearly to some extent a comment on the simple modified van der Waals equation of state. As the shape of the molecules deviates increasingly from the spherical, the critical compressibility factor is seen to decrease slightly. This behavior is observed experimentally, when a series such as the alkanes is considered. However, the experimentally observed changes are quite large, e.g., a change from 0.290 to 0.266 on passing from CH4 to n-CsHl2. This may be contrasted with the variation from 0.360 to 0.353 on changing from the spherical case, R = 0, to the case R = 4u, a change which might be thought naively to ( 8 ) R. M. Gibbons, Mol. Phys., 17, 18 (19G9). (9) M. Rigby, J. Chem. Phys., 53, 1021 (1970).
The Journal of Physical Chemistry, Vol. 76, N o . 14, 1978
M. RIGBY
2016
It is seen that the van der Waals model reproduces
Figure 2. The reduced vapor pressure curves. A = In (P/P,),
- In (P/P&s
Experimental data (relative to argon) are shown as dashed lines for C3Heand n-CgH12.
be of the correct magnitude for the van der Waals model. The results of the vapor pressure calculations are shown in Figure 2. Rowlinson2 has illustrated the deviations of the reduced vapor pressure from the simple spherical molecule pattern and has shown how this may be correlated with nonspherical attractive terms. Thcse deviations are conveniently displayed by plotting the difference in the logarithm of the reduced vapor prcssurc, P/P,, for systems of nonspherical and spherical molecules, as a function of the reduced temperature, TIT,. The typical experimental trend is shown in Figure 2, together with the results based on the modified van der Waals equation for several values of 7.
The Journal of Physical Chemistry, Val. 76, No. 1.A 1979
qualitatively the observed effects of changing shape. The effects of moderate changes in the core shape are clearly significant, and any comprehensive treatment of shape must evidently include the effects of both attractive and repulsive contributions. The study of the slope of the rectilinear diameter lines does not lead t o any clear conclusion. For the spherical case, the van der Waals model gives a line whose slope greatly exceeds the appropriate experimental value. For nonspherical cores, this slope is increased, in agreement with experimental observation, but in view of the large discrepancy for the case of spherical molecules this agreement may be fortuitous. The van der Waals model cannot adequately describe the heat capacity of a fluid, as the assumption of a uniform attractive energy field implies that the heat capacities have the ideal gas values, with zero configurational contributions. This is of course partly an artefact of the hard-core model, and repulsive contributions would be expected if a soft repulsive model were used for the core.
Conclusion A simple equation of state for systems of nonspherical molecules is found to be able t o reproduce many of the changes in behavior found experimentally as the shape deviates increasingly from the spherical. It is suggested that the contributions of repulsive nonspherical forces are significant and should not be neglected. However, the effects of both attractive and repulsive terms are qualitatively similar for many properties and it will probably prove difficult t o separate the effects arising from different causes, from a study of experimental results.
