GATE 2025 CY – Question 29

Correct answer: (B) $\Delta S>0$ at constant U and V; (D) $\Delta(U-TS)<0$ at constant T and V

Explanation

Thermodynamic criteria for a spontaneous process depend on which variables are held constant.

• **A. $\Delta U>0$ at constant S and V.** False. At constant $S$ and $V$ a spontaneous process has $\Delta U<0$.
• **B. $\Delta S>0$ at constant U and V.** True. For an isolated system (fixed $U$, $V$) the entropy increases. ✓
• **C. $\Delta(H-TS)>0$ at constant T and P.** False. The criterion is $\Delta G<0$, with $G=H-TS$.
• **D. $\Delta(U-TS)<0$ at constant T and V.** True. The Helmholtz energy $A=U-TS$ decreases at constant $T$ and $V$. ✓

Answer **B and D**.