GATE 2021 CH – Question 50
A binary liquid mixture consists of two species 1 and 2. Let $\gamma$ and $x$ represent the activity coefficient and the mole fraction of the species, respectively. Using a molar excess Gibbs free energy model, $\ln\gamma_1$ vs. $x_1$ and $\ln\gamma_2$ vs. $x_1$ are plotted. A tangent drawn to the $\ln\gamma_1$ vs. $x_1$ curve at $x_1=0.2$ has a slope of −1.728. The slope of the tangent drawn to the $\ln\gamma_2$ vs. $x_1$ curve at the same mole fraction is _____ (correct to 3 decimal places).
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Correct answer: 0.422 to 0.442
Explanation
At fixed temperature and pressure the activity coefficients satisfy the Gibbs–Duhem equation:
$$x_1\,d\ln\gamma_1+x_2\,d\ln\gamma_2=0.$$
Divide by $dx_1$ and use $x_2=1-0.2=0.8$:
$$\frac{d\ln\gamma_2}{dx_1}=-\frac{x_1}{x_2}\frac{d\ln\gamma_1}{dx_1}=-\frac{0.2}{0.8}(-1.728)=\mathbf{0.432}.$$
The slopes have opposite signs, as required by the weighted derivative balance.