Q 11-12-100JEE MainJEE Main 2021 (31 Aug, Shift 1)Medium
For an ideal gas the instantaneous change in pressure $P$ with volume $V$ is given by the equation $\dfrac{dP}{dV} = -aP$. If $P = P_0$ at $V = 0$ is the given boundary condition, then the maximum temperature one mole of gas can attain is: (Here $R$ is the gas constant)
Answer: (D) $\dfrac{P_0}{aeR}$
Integrating: $P = P_0e^{-aV}$. For one mole, $T = \dfrac{PV}{R} = \dfrac{P_0Ve^{-aV}}{R}$.
$\dfrac{dT}{dV} = \dfrac{P_0}{R}e^{-aV}(1 - aV) = 0 \Rightarrow V = \dfrac1a$.
$$T_{\max} = \frac{P_0}{R}\cdot\frac1a\cdot e^{-1} = \frac{P_0}{aeR}$$
Solution by Sreeraj P, M.Sc Physics