GATE 2024 CY – Question 64

Physical Chemistry · Equilibrium: Laws of thermodynamics. Standard states. Thermochemistry. Thermodynamic functions and their relationships: Gibbs-Helmholtz and Maxwell relations, Gibbs-Duhem equation, Van’t Hoff equation. Criteria of spontaneity and equilibrium. Absolute entropy. Partial molar quantities. Thermodynamics of mixing. Chemical potential. Fugacity, Activity and Activity coefficient. Chemical equilibria. Dependence of equilibrium constant on temperature and pressure. · 2 marks · Numerical answer

2 mol of a monoatomic ideal gas initially at 5 L and 10 bar expands irreversibly and adiabatically against constant final pressure 1 bar. The final volume in L (one decimal place) is _____. Given $R=8.314\times10^{-2}$ L bar mol−1 K−1.

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Correct answer: 31.84 to 32.16

Explanation

The expansion is **adiabatic** ($q=0$) and irreversible against a constant external pressure equal to the final pressure, $P_{ext}=P_f=1$ bar. For a monatomic ideal gas, $C_v=\tfrac32R$.

**First law:** $\Delta U=-P_{ext}\,\Delta V$.
$$nC_v(T_f-T_i)=-P_{ext}(V_f-V_i).$$

Using $nRT=PV$, we have $nC_v\Delta T=\tfrac32\,(P_fV_f-P_iV_i)$:
$$\tfrac32\,(P_fV_f-P_iV_i)=-P_{ext}(V_f-V_i).$$

**Numbers.** $P_iV_i=10\times5=50$ bar L, $P_f=P_{ext}=1$ bar, $V_i=5$ L:
$$1.5\,(V_f-50)=-(V_f-5)\;\Rightarrow\;1.5V_f-75=-V_f+5\;\Rightarrow\;2.5V_f=80 .$$

$$V_f=\mathbf{32.0\ L}.$$