GATE 2021 CH – Question 56
A distillation column handling a binary mixture of A and B is operating at total reflux. It has two ideal stages including the reboiler. The mole fraction of the more volatile component in the residue $(x_W)$ is 0.1. The average relative volatility $\alpha_{AB}$ is 4. The mole fraction of A in the distillate $(x_D)$ is _____ (round off to 2 decimal places).
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Correct answer: 0.63 to 0.65
Explanation
At total reflux, the Fenske equation relates composition odds across N ideal equilibrium stages:
$$\frac{x_D/(1-x_D)}{x_W/(1-x_W)}=\alpha^N.$$
The stated N = 2 **already includes the reboiler**, so do not add another stage. With $x_W=0.1$ and $\alpha=4$, $x_D/(1-x_D)=16(0.1/0.9)=16/9$.
Solving gives $x_D=16/(16+9)=\mathbf{0.64}$. The increase from 0.1 is consistent with enrichment of the more volatile A.