# Statistical mechanics of coupled solid solutions in the dilute limit

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## Abstract

A statistical mechanical analysis of the limiting laws for coupled solid solutions shows that the random model, in which the configurational entropy is calculated as if atoms mix randomly on each crystallographic site, is correct as a first approximation. In coupled solid solutions, since atoms of different valence substitute on the same sites, significant short-range order which reduces the entropy can be expected. A first-order correction is rigorously obtained for the entropy in dilute binary short-range ordered coupled solid solutions: where

$$\bar S^{{\text{XS}}} {\text{/R = }}Q\left( {{\text{e}}^{--H_{\text{A}} /{\text{R}}T} \left( {\frac{{H_{\text{A}} }}{{{\text{R}}T}} + 1} \right) - 1} \right)N_2^a N_4^b ,$$

*Q*is the number of positions an associated cation pair can assume per formula unit,*H*_{A}is the association energy per formula unit, and*N*_{2}^{ a }and*N*_{4}^{ b }are the site occupancy fractions for atoms*2*and*4*that are dilute on sites*a*and*b*.*S*^{XS}is the configurational entropy minus the random model entropy. Aluminous pyroxenes on the joints*diopside-jadeite*and*diopside-CaTs*are examined as examples.A generalization for dilute multiple component solutions, including possible long-range ordering variations is given by: where

$$\frac{{\bar S^{{\text{XS}}} }}{{\text{R}}}{\text{ = }}\sum\limits_i {\sum\limits_j {\sum\limits_k {Q_i } } \left( {{\text{e}}^{--H_{\text{A}}^{j{\text{ }}k{\text{, }}i} /{\text{R}}T} \left( {\frac{{H_{\text{A}}^{j{\text{ }}k{\text{, }}i} }}{{{\text{R}}T}} + 1} \right) - 1} \right)N_j^l N_k^m ,} $$

*i*labels each crystallographically distinct pair,*j*and*k*label atomic species,*l*and*m*label crystallographic sites, and the*N*'s are site occupancy fractions for the solute atoms.A total association model is examined as well as the partial association and random models. Real solution behavior must lie between the total association model and the random model. Molecular models in which the ideal activity is proportional to a mole fraction, which in itself is not always unambiguously defined, do not lie in this range and furthermore have no physical justification.

## Keywords

Entropy Solute Atom Formula Unit Random Model Component Solution
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