1821
XOTES
various protons near a polar group are assumed to be ascribable to a single complexing benzene molecule. The condition will be fulfilled only if the attraction force is almost saturated by the interaction with a single benzene molecule. This is not valid in the case of van der Waals forces. If one insists on the 1 : 1 complex model, then it is very likely that he may be led to the idea of forces and conformations of the complex which may not be close to the true nature of molecular interactions in liquid media. Certainly, this is not a desirable situation, even though the model is proposed only to explain successfully the nmr solvent effect. Here, the present author suggests that the benzeneinduced shifts should be generally explained as being due to the benzene molecules a t several association sites around a solute molecule. The detail of this clustering model, such as the number and the conformations of these representative benzene molecules, are to be worked out further in future.
in which p is the shortest distance between cores, and the nonorientational part of the potential has a minimum value of E and is equal to zero when p = po. The angles &,02, 41, and c $ ~ are as defined by Buckingham and P ~ p l e . ~ Following Kihara,2 the second virial coefficient for the potential is given by
B(T) =
N
+ Apdp + Bdp)
el del X
- lm(pzdp
87r
s,’“
dh
sin 0 2 d&
s,””
d& X
+
whereA = M o / T ,B = (1/27r) [So (1/4n)Mo2],and V o , Mo, and SOare the volume, curvature, and surface area, respectively, of the core. For simplicity, and also because it is the core most commonly used, we will now consider the core to be a sphere with radius a, so that V O= 4 / / 3 na3, M O= 4na, and So = 4 d . If eq 1 is substituted into eq 2 and the term l / ( p 2a) is expanded in a binomial series, the expression can be intergrated by the method of Buckingham and P ~ p l e . The ~ final result (valid if po > 2a) is
+
On the Kihara Core Model for Polar Molecules by T. S. Storvick
2
Department of Chemical Engineering, University of Missouri, Columbia, Missouri
B(T) =
d=O
bd[?l-’(HlZ+d(Y) -
‘/ZH6+d(Y))
-
and T. H. Spurling Department of Chemistry, University of Tasmania, Hobart, Tasmania, Australia (Received September 65, 1967)
Suh and Storvick’ recently demonstrated that the Kihara2 core potential, modified by the addition of a point dipole acting at the center of the core, correlated the second virial coefficients of a number of polar gases more successfully than the simpler Stockmayer3 model. Certain features of their derivation of the expression for the second virial coefficient for the model were unsatisfactory, and here we present a corrected derivation. Assume that a permanent point dipole of strength p is embedded at the geometric center of the core of the molecule. The orientation of the dipole within the core does not depend on the shape of the core but is assumed to be fixed for any molecular species. The core can be any convex body with a nonzero volume and we define a radius, a , which is the radius of a sphere having a volume equal to the volume of the convex body. The potential function may now be written
u = 4.[(;)12
-
(91+
( 2 cos el cos e2
P2
(P
+ 2aI3 X
+ sin el sin
82
cos (41
+ 4J)
(1)
This expression reduces to the expression for the second virial coefficient for the Lennard-Jones5 12-6 potential if a = 0, p = 0; for the Kihara2core potential if p = 0; and for the Stockmayer3 potential if a = 0. Furthermore, the equation given by Suh and Storvick’ (their eq 13) is this expression with the n series truncated a t the H6(Y) term. (1) K. W. Suh and T. S. Storvick, J . Phys. Chem., 71, 1450 (1967). (2) T. Kihara, Rev. Mod. Phys., 25, 831 (1953). (3) W. H. Stockmayer, J . Chem. Phys., 9, 398 (1941). (4) A. D. Buckingham and J. A. Pople, Trans. Faraday SOC., 51, 1173 (1955). ( 5 ) J. E. Lennard-Jones, Proc. Roy. SOC. (London), A196, 463 (1924).
Volume 78, Number 6 M a y 1068
