Excess Gibbs energy equivalent activity coefficients

Equation 9.9-3 does not give values tor the mixture parameters a and b separately, but only for their sum. A second equation comes from requiring that the excess Gibbs energy predicted from an equation of state at liquidlike densities be equivalent to that from excess Gibbs energy or activity coefficient models discussed in Secs. 9.5 and 9.6. Since, from an equation of state, as F —> oo, V b and Ymix mix. so that liquid densities are obtained, the second equation that is used is... 

Many mixtures of interest in the chemical indu.stry exhibit strong nonidealities that can not be described by the EOS with any form of the van der Waals mixing rules. Mixing rules that combine equations of state with liquid excess Gibbs free-energy (or, equivalently, activity coefficient) models are more suitable for the thermodynamic... 

So there is complete equivalence between the use of the activity coefficient and the excess Gibbs energy. In a sense there is hardly any difference at all. If the correction for nonideality is in a logarithmic form (7 Tin 7) it can be combined with the ideal term (RT In x) as a correction to x. If it does not have a logarithmic form (wx ) it becomes a correction to the AG or A/i term. 

Relation [A 1.33] forms the link between the description of the solutions by the Lewis activity coefficients method, and that by die excess Gibbs energy. The two methods, therefore, are equivalent. 

The relation [3.87] hnks the solutions described by the Lewis method of coefficients of activity and the excess Gibbs energy method. Both methods are therefore equivalent. 

Still with the aim of having mathematical expressions for the representation of the solution, Redlich and Kister offered a representation that provides an expansion of the excess Gibbs energy, a pure-substance reference in the same state of segregation as the solution (reference (I)), the equivalent of ihe Margules expansion for the activity coefficients. For a two-component solution, the pol5momial expansion up to order m is written ... 

Clearly these two forms are ultimately equivalent via a simple expression, whieh we will estabhsh later (expression [3.83]), which links the activity coefficients of different components of the solution and their excess partial molar Gibbs energies. 

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