The GATE Grind

GATE 2022 CH – Question 15

Thermodynamics · Phase equilibria and vapour-liquid equilibrium · 1 mark · Multiple choice

For a single component system at vapor-liquid equilibrium, the extensive variables $A$, $V$, $S$ and $N$ denote the Helmholtz free energy, volume, entropy, and number of moles, respectively, in a given phase. If superscripts ($v$) and ($l$) denote the vapor and liquid phase, respectively, the relation that is NOT CORRECT is

  1. $\left(\frac{\partial A^{(l)}}{\partial V^{(l)}}\right)_{T,N^{(l)}} = \left(\frac{\partial A^{(v)}}{\partial V^{(v)}}\right)_{T,N^{(v)}}$
  2. $\left(\frac{\partial A^{(l)}}{\partial N^{(l)}}\right)_{T,V^{(l)}} = \left(\frac{\partial A^{(v)}}{\partial N^{(v)}}\right)_{T,V^{(v)}}$
  3. $\left(\frac{A + PV}{N}\right)^{(l)} = \left(\frac{A + PV}{N}\right)^{(v)}$
  4. $\left(\frac{A + TS}{N}\right)^{(l)} = \left(\frac{A + TS}{N}\right)^{(v)}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (D) $\left(\frac{A + TS}{N}\right)^{(l)} = \left(\frac{A + TS}{N}\right)^{(v)}$

Explanation

At equilibrium the two phases have the same $T$, $P$ and chemical potential $\mu$. The derivative $\left(\frac{\partial A}{\partial V}\right)_{T,N} = -P$ is equal in both (A). $\left(\frac{\partial A}{\partial N}\right)_{T,V} = \mu$ is equal (B). $\frac{A + PV}{N} = \frac{G}{N} = \mu$ is equal (C). But $A + TS = U$, and $\frac{U}{N}$ (the molar internal energy) is different in the liquid and the vapour, so D is not correct.