GATE 2024 CH – Question 57
Ethylene obeys the truncated virial equation-of-state $\frac{PV}{RT} = 1 + \frac{BP}{RT}$ where P is the pressure, V is the molar volume, T is the absolute temperature and B is the second virial coefficient. The universal gas constant $R = 83.14$ bar cm$^3$ mol$^{-1}$K$^{-1}$. At 340 K, the slope of the compressibility factor vs. pressure curve is $-3.538 \times 10^{-3}$ bar$^{-1}$. Let $G^R$ denote the molar residual Gibbs free energy. At these conditions, the value of $\left(\frac{\partial G^R}{\partial P}\right)_T$, in cm$^3$ mol$^{-1}$, rounded off to 1 decimal place, is ___________
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Correct answer: -100.50 to -99.50
Explanation
From the equation of state, $Z = 1 + \frac{BP}{RT}$, so the slope is $\frac{B}{RT} = -3.538 \times 10^{-3}$ and $B = -3.538 \times 10^{-3} \times 83.14 \times 340 = -100.0$ cm³ mol⁻¹. For the truncated virial equation $\frac{G^R}{RT} = \frac{BP}{RT}$, so $G^R = BP$ and $\left(\frac{\partial G^R}{\partial P}\right)_T = B = -100.0$ cm³ mol⁻¹.