GATE 2021 CH – Question 49
A gaseous mixture at 1 bar and 300 K consists of 20 mol% $\mathrm{CO_2}$ and 80 mol% inert gas. Assume the gases to be ideal. Take R = 8.314 J/(mol K). The magnitude of minimum work required to separate 100 mol of this mixture at 1 bar and 300 K into pure $\mathrm{CO_2}$ and inert gas at the same temperature and pressure is _____ kJ (round off to nearest integer).
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Correct answer: 125
Explanation
Reversible isothermal separation requires work equal to the increase in Gibbs free energy. For an ideal-gas mixture, $\Delta G_{mix}=nRT\sum_i y_i\ln y_i$; separating it reverses the sign.
Thus
$$W_{min}=-100(8.314)(300)[0.2\ln0.2+0.8\ln0.8].$$
The bracket is $-0.5004024$, giving $W_{min}=124810.37$ J = 124.810 kJ. The requested magnitude is **125 kJ**. It is positive because unmixing reduces mixing entropy.