GATE 2024 CH – Question 32
Consider a binary mixture of components $A$ and $B$ at temperature $T$ and pressure $P$. Let $\bar{V}_A$ and $\bar{V}_B$ be the partial molar volumes of $A$ and $B$, respectively. At a certain mole fraction of $A$, $x_A$: $\left(\frac{\partial\bar{V}_A}{\partial x_A}\right)_{T,P} = 22$ cm$^3$ mol$^{-1}$ and $\left(\frac{\partial\bar{V}_B}{\partial x_A}\right)_{T,P} = -18$ cm$^3$ mol$^{-1}$
The value of $x_A$, rounded off to 2 decimal places, is ________
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Correct answer: 0.44 to 0.46
Explanation
The Gibbs-Duhem equation at constant $T$ and $P$ is $x_Ad\bar{V}_A + x_Bd\bar{V}_B = 0$. Dividing by $dx_A$: $x_A(22) + (1 - x_A)(-18) = 0$, so $40x_A = 18$ and $x_A = 0.45$.