## Lowconcentration species

11.8.1 Formation of the two phases

The water-soluble polymers responsible for separating water into partially-miscible liquids possess osmotic second virial coefficients which provide the following activity coefficients in water:

7a = 1 + bAA[A] + bAB[B] ye = 1 + bBB[B] + bAB[A] (11.54)

The second terms in the above equations account for interactions between like solute molecules and bAB reflects interactions between A and B molecules in the same solution. The equilibrium distributions are given by equations analogous to Eq (8.9):

These formulas are equivalent to equating the chemical potentials of A and B in the two phases.

Finally, water equilibrium between the two phases is calculated in the same manner. The activity coefficient of water in either phase is obtained from yA and yB using the Gibbs-Duhem equation for a three-component system2:

2 Eq(11.58) is the three-component analog of Eq (7.36a), with mole fractions converted to molar concentrations by dividing by vw. Although the weight fractions of the polymers in the two phases can be as large as 0.2, their mole fractions are very much smaller because their molecular weights are in the tens of thousands

In integrated form:

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