The departure of a binary solution from ideal solution

Chapter 11, Problem 11.128

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The departure of a binary solution from ideal solution behavior is gauged by the activity coefficient, \(\gamma_{i}=a_{\mathrm{i}} / y_{\mathrm{i}}\), where \(a_{i}\) is the activity of component i and \(y_{i}\) is its mole fraction in the solution (i = 1, 2). Introducing Eq. 11.140, the activity coefficient can be expressed alternatively as \(\gamma_{i}=\bar{f}_{i} / y_{i} f_{i}^{\circ}\). Using this expression together with the Gibbs–Duhem equation, derive the following relation among the activity coefficients and the mole fractions for a solution at temperature T and pressure p:

\(\left(y_{1} \frac{\partial \ln \gamma_{1}}{\partial y_{1}}\right)_{p, T}=\left(y_{2} \frac{\partial \ln \gamma_{2}}{\partial y_{2}}\right)_{p, T}\)

How might this expression be used?