GATE 2022 CH – Question 48
The molar excess Gibbs free energy ($g^E$) of a liquid mixture of A and B is given by $\frac{g^E}{RT} = x_Ax_B[C_1 + C_2(x_A - x_B)]$ where $x_A$ and $x_B$ are the mole fraction of A and B, respectively, the universal gas constant, $R = 8.314$ J K$^{-1}$ mol$^{-1}$, $T$ is the temperature in K, and $C_1$, $C_2$ are temperature-dependent parameters. At 300 K, $C_1 = 0.45$ and $C_2 = -0.018$. If $\gamma_A$ and $\gamma_B$ are the activity coefficients of A and B, respectively, the value of $\int_0^1\ln\left(\frac{\gamma_A}{\gamma_B}\right)dx_A$ at 300 K and 1 bar is _________ (rounded off to the nearest integer).
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Correct answer: 0
Explanation
For a binary mixture, $\ln\gamma_A - \ln\gamma_B = \frac{d(g^E/RT)}{dx_A}$. So the integral is the change of $g^E/RT$ from $x_A = 0$ to $x_A = 1$, and $g^E = 0$ for both pure components. The value is $0 - 0 = 0$.