Q 11-11-008JEE MainEAMCET 2012 (Engineering)Hard
The work done to increase the temperature of one mole of an ideal gas by $30°$C, if it is expanding under the condition $V \propto T^{2/3}$, is ($R = 8.314$ J/mol K)
Answer: (A) $166.3$ J
Concept. Recast as a polytropic process $PV^x = $ constant; then $W = \dfrac{nR\Delta T}{1 - x}$.
Step 1. Index. $V \propto T^{2/3}$ and $PV \propto T$ give $PV \propto V^{3/2}$, i.e. $P \propto V^{1/2}$: $x = -\dfrac12$.
Step 2. Work.
$$W = \frac{8.314 \times 30}{1.5} \approx 166.3\ \text{J}$$
Answer. $166.3$ J (Option A).
Solution by Sreeraj P, M.Sc Physics