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GATE 2023 CH – Question 41

Thermodynamics · Properties of mixtures: partial molar properties, fugacity, excess properties, activity coefficients · 2 marks · Multiple choice

The enthalpy ($H$, in J mol$^{-1}$) of a binary liquid system at constant temperature and pressure is given as $H = 40x_1 + 60x_2 + x_1x_2(4x_1 + 2x_2)$, where $x_1$ and $x_2$ represent the mole fractions of species 1 and 2 in the liquid, respectively. Which one of the following is the CORRECT value of the partial molar enthalpy of species 1 at infinite dilution, $\bar{H}_1^\infty$ (in J mol$^{-1}$) ?

  1. 100
  2. 42
  3. 64
  4. 40

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Correct answer: (B) 42

Explanation

With $x_2 = 1 - x_1$, $H = 60 - 20x_1 + x_1(1 - x_1)(2 + 2x_1)$. The partial molar enthalpy is $\bar{H}_1 = H + (1 - x_1)\frac{dH}{dx_1}$. At $x_1 = 0$: $H = 60$ and $\frac{dH}{dx_1} = -20 + 2 = -18$. So $\bar{H}_1^\infty = 60 - 18 = 42$ J/mol.