Low-Molecular-Weight Mixtures van Laar Model

The methods of statistical mechanics can be used to derive an equation for A conf by assuming that the molecules are placed randomly into cells that are of molecular size and are arranged in the form of a three-dimensional lattice (represented in two dimensions in Fig. 3.2(a)). The entropy S of an assembly of molecules and the total number H of distinguishable degenerate (i.e., of equal energy) arrangements of the molecules are related by the well-known Boltzmann equation  

The total number of distinguishable spatial arrangements of Na molecules of A and iVs molecules of B in a lattice of (Na + Nb) cells is equal to the number of permutations of (Na + Nb) objects consisting of Na identical objects of type A and identical objects of type B, that is. 

It is more convenient to deal with equations in terms of numbers of moles, w, and mole fractions, X. In the present case, these are given by 

Since for the formation of an ideal solution, A5nux = A5conf and A/Zmix = 0, Eq. (3.14) becomes 

Equation (3.25) provides the basis for derivation of standard thermodynamic relationships for ideal solutions (e.g., Raoult s law). To make it more generally applicable, this equation has to be modi ed for nonideality of solutions when mixing is nonathermal (i.e., AHmix 0) due to nonequivalent intermolecular interactions caused by the so-called dispersion forces. The decrease in energy due to interactions is called the interaction energy. This energy is larger for polar molecules than for nonpolar molecules. 

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