Partition function regular solution

The structure of a simple mixture is dominated by the repulsive forces between the molecules [15]. Any model of a liquid mixture and, a fortiori of a polymer solution, should therefore take proper account of the configurational entropy of the mixture [16-18]. In the standard lattice model of a polymer solution, it is assumed that polymers live on a regular lattice of n sites with coordination number q. If there are n2 polymer chains, each occupying r consecutive sites, then the remaining m single sites are occupied by the solvent. The total volume of the incompressible solution is n = m + m2. In the case r = 1, the combinatorial contribution of two kinds of molecules to the partition function is... 

In the derivation of the regular solution model the vibrational contribution to the excess properties has been neglected. However, as a first approximation the vibrational contribution can be taken as independent of the interaction between the different atoms, and this contribution can be factored out of the exponential and taken into account explicitly. The partition function of the solution is then given as... 

While a random distribution of atoms is assumed in the regular solution case, a random distribution of pairs of atoms is assumed in the quasi-chemical approximation. It is not possible to obtain analytical equations for the Gibbs energy from the partition function without making approximations. We will not go into detail, but only give and analyze the resulting analytical expressions. 

The partition function for the regular ionic solution model is now... 

The partition function of a regular solution is the sum of the Boltzmann factor over all possible configurations. 

Simplifying and, in order to be more precise, taking into account the molecular partition function, we write for the regular solution partition function... 

FIG. 11 Evolution of log P ct as a function of (a) log and (b) log Pdbe. The correlation lines are obtained with a well-distributed series of model compounds [171,220] that must be divided into H-bond donor solutes (O) and non-H-bond donor solutes ( ) with DCE. The partition coefficient values obtained for a series of P-blockers ( ) are added to both graphs, showing that only acebu-tolol (1), atenolol (3), carazolol (5), and pindolol (11) behave as regular H-bond donors in water-DCE, the other compounds deviating due to the presence of intramolecular H-bonds, whereas only atenolol (for unknown reasons) is an outlier in water-DBE. (Reprinted from Ref. 290.)... 

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