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American Journal of Science, Vol. 303, October 2003, P.708-722

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On the mathematics of associated solutions

Victor C. Kress

Department of Earth and Space Sciences, University of Washington, Seattle, Washington 98195; kress{at}u.washington.edu

Many "non-ideal" or "excess" mixing properties in solid or liquid solutions arise from homogeneous order-disorder or speciation reactions. Simple Taylor expansions, such as the commonly used Margules expansion can never exactly match the enthalpy and entropy effects associated with such homogeneous reactions. Solution modeling based on associated solution theory produces what can be a more phenomenologically sound representation of the energetics of mixing in such solutions, as well as allowing incorporation of constraints from x-ray crystallography and a wide range of spectroscopic methods. Additional non-ideal interactions between species can be incorporated by combining regular- and associated-solution theory.


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  		TABLE 1 Variables used in text. Matrices and vectors are subscripted with their dimension. Subscripted lower case will generally refer to an individual element of the corresponding matrix or vector. Subscripts on partial derivatives indicate that differentiation is carried out with the subscripted variable held constant.

 
We present a detailed description of robust algorithms allowing calculation of homogeneous equilibrium, estimation of model properties and calculation of thermodynamic properties. Matrix algorithms are presented to allow calculation of species standard state and mixing properties in non-ideal associated solutions. Species stability and phase equilibrium calculations in associated solutions require the second derivative of Gibbs free energy with respect to the component vector (the Gibbs Hessian). A general algorithm is presented to calculate the Gibbs Hessian in ideal and non-ideal associated solutions. These algorithms can be applied in any number of components and species. Species can be of arbitrary stoichiometry.






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