This equation can be used to fit activities in two component mixtures over the
entire concentration range, and to calculate the activities of both components.
It is used *E-AIM* for aqueous solutions of single organic compounds,
and is an alternative to UNIFAC for those compounds for which the necessary activity
data exist.

The expansion is described by Prausnitz et al. (1986), and is also covered in other chemical engineering textbooks. McGlashan (1963) gives examples of the use of the equation to illustrate the types of activity curves that are observed for different two component mixtures.

In *E-AIM* the Redlich-Kister expansion can be used with up to 10
fitted parameters. Examples of the use of the equation, to represent the properties
of aqueous solutions of dicarboxylic acids at 298.15 K, are given by Clegg and Seinfeld (2006).
These acids are present in the public database of compounds, on the Available Compounds selection page,
and can be included in model calculations.

Expressions for solute (S) and water (W) mole fraction activity coefficients are given below, for
terms including the first 5 fitted parameters (C_{1} to C_{5}).

ln(*f*_{S}) = g^{e}/RT + [(1 - *x*S) × d(g^{e}/RT)/d(*x*S)]

ln(*f*_{W}) = g^{e}/RT - [*x*S × d(g^{e}/RT)/d(*x*S)]

where the excess Gibbs energy per mol of substance, g^{e}, is given by:

g^{e}/RT =
*x*S(1 - *x*S)[C_{1} + C_{2}(1 - 2*x*S) + C_{3}(1 - 2*x*S)^{2} + C_{4}(1 - 2*x*S)^{3} + C_{5}(1 - 2*x*S)^{4} ...]

and its differential with respect to *x*S, d(g^{e}/RT)/d(*x*S), is given by:

d(g^{e}/RT)/d(*x*S) =
C_{1}(1 - 2*x*S)
+ C_{2}[-2*x*S(1 - *x*S) + (1 - 2*x*S)^{2}]

+ C_{3}[-4*x*S(1 - *x*S) + (1 - 2*x*S)^{2}](1 - 2*x*S)
+ C_{4}[-6*x*S(1 - *x*S) + (1 - 2*x*S)^{2}](1 - 2*x*S)^{2}

+ C_{5}[-8*x*S(1 - *x*S) + (1 - 2*x*S)^{2}](1 - 2*x*S)^{3} ...

where C_{i} are the fitted parameters. Both activity coefficients are relative to
a reference state of the pure liquid. The logarithm of the activity coefficient of the solute can be adjusted to a reference state of
infinite dilution in water (*f*_{S}^{*}) by subtracting the value obtained with the equation above for *x*S = 0, yielding:

ln(*f*_{S}^{*}) = ln(*f*_{S})
- (g^{e}/RT + d(g^{e}/RT)/d(*x*S))

This expression is valid for any concentration but, as stated above, the final term in g^{e}/RT and its
differential is calculated for *x*S = 0.

References

S. L. Clegg and J. H. Seinfeld (2006) Thermodynamic models of aqueous
solutions containing inorganic electrolytes and dicarboxylic acids at 298.15 K. 1. The acids as
nondissociating components. *J. Phys. Chem. A* **110**, 5692-5717.

M. L. McGlashan (1963) Deviations from Raoult's law. *J. Chem. Educ.* **40**, 516-518.

J. M Prausnitz, R. N. Lichtenthaler, and E. Gomes de Azevedo (1986) *Molecular Thermodynamics of
Fluid Phase Equilibria*, 2nd. Edn., Prentiss-Hall.