Hard Sphere Model with Central Attractive Forces

Hard Sphere Model with Central Attractive Forces. A great improvement over the hard sphere model physically, but one th( t is much more difficult to deal with mathematically, is the hard sphere molecule which is capable of exerting attractive forces, centrally directed (i.e., the force between two molecules depends only on the distance between them and is directed along their line of centers). The molecule is still spherical, and the closest distance of approach of two molecules is given by their mean diameter. This model can now account for the properties of condensed states. 

S3rmmetrical Molecule with Central Forces. If we discard the idea of a hard sphere and replace it by a molecule that is capable of exerting both attractive and repulsive forces but acts centrally, we have the closest approach yet to real molecules, and also the model that is most difficult to treat. Such a molecule is characterized completely by the function chosen to represent its force field. A function commonly used is the Lennard-Jones function 

---

Elastic hard spheres with superposed central attractive forces This is the so-called van der Waals model for which the equation of state is... 

Fig. 2-1 Models of intermolecular potentials, (a) Forceless mass points (b) elastic hard spheres (c) elastic hard spheres with superposed central attractive forces (d) molecules with central finite repulsive and attractive forces (e) square-well model (f) point centers of inverse-power repulsion or attraction.

If an elastic-hard-sphere model is not used, S(i y) in Eq. (2-26) must be properly expressed as a function of for the specified model. For example, for elastic hard spheres with superposed central attractive forces, the maximum value of To for contact collisions is di2. and if the attractive forces are weak,... 

A ternary collision may be conveniently pictured as a very rapid succession of two binary collisions one to form the unstable product, and the second, occurring within a period of about 10 sec or less, to stabilize the product. It is immediately obvious that it is not possible to use the elastic-hard-sphere molecular model to represent ternary collisions since two such spheres would be in collision contact for zero time, the probability of a third molecule making contact with the colliding pair would be strictly zero. It is therefore necessary to assume a potential model involving forces which are exerted over an extended range. One such model is that of point centers having either inverse-power repulsive or inverse-power attractive central forces. This potential, shown in Fig. 2-If, is represented by U r) = K/r. For the sake of convenience, we shall make several additional assumptions first, at the interaction distances of interest the intermolecular forces are weak, that is, U(r) < kT second, when the reactants A and B approach each other, they form an unstable product molecule A B when their internuclear separations are in the range b third, the unstable product is in essential... 

---