Statistical Foundations of Solution Thermodynamics

In Section 3.4a we examine a model for the second virial coefficient that is based on the concept of the excluded volume of the solute particles. A solute-solute interaction arising from the spatial extension of particles is the premise of this model. Therefore the potential exists for learning something about this extension (i.e., particle dimension) for systems for which the model is applicable. In Section 3.4b we consider a model that considers the second virial coefficient in terms of solute-solvent interaction. This approach offers a quantitative measure of such interactions through B. In both instances we only outline the pertinent statistical thermodynamics a somewhat fuller development of these ideas is given in Flory (1953). Finally, we should note that some of the ideas of this section are going to reappear in Chapter 13 in our discussions of polymer-induced forces in colloidal dispersions and of coagulation or steric stabilization (Sections 13.6 and 13.7). 

Before considering how the excluded volume affects the second virial coefficient, let us first review what we mean by excluded volume. We alluded to this concept in our model for size-exclusion chromatography in Section 1.6b.2b. The development of Equation (1.27) is based on the idea that the center of a spherical particle cannot approach the walls of a pore any closer than a distance equal to its radius. A zone of this thickness adjacent to the pore walls is a volume from which the particles —described in terms of their centers —are denied entry because of their own spatial extension. The volume of this zone is what we call the excluded volume for such a model. The van der Waals constant b in Equation (28) measures the excluded volume of gas molecules for spherical molecules it equals four times the actual volume of the sphere, as discussed in Section 10.4b, Equation (10.38). 

A point of entry for statistical considerations into thermodynamics is the Boltzmann entropy relationship 

Entropy changes can also be developed in terms of Equation (45)  

Our interest is in the entropy of mixing —that is, AS for 1 + 2 - mixture —which may be written 

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