Chapter 15, Problem 15.7P

Pure liquid species 2 and 3 are for practical purposes immiscible in one another. Liquid species 1 is soluble in both liquid 2 and liquid 3. One mole each of liquids 1,2, and 3 are shaken together to form an equilibrium mixture of two liquid phases: an $\alpha$ -phase containing species 1 and 2, and a $\beta$ -phase containing species 1 and 3. What are the mole fractions of species 1 in the a and $\beta$ phases, if at the temperature of the experiment. the excess Gibbs energies of the phases are given by:

$\frac{{(G^E)}^a}{RT}=0.4x_1^a{x}_2^a$ and $\frac{{(G^E)}^{\beta }}{RT}=0.8x_1^{\beta }{x}_2^{\beta }$