Thermodynamic Processes question from JEE Main 2020 (7 Jan, Shift 2)

Answer: (a)

1. Mean collision time $\tau = \frac{\lambda}{v_{rms}}$, with $\lambda \propto V$ (fixed number of molecules) and $v_{rms} \propto \sqrt{T}$, so $\tau \propto \frac{V}{\sqrt{T}}$.
2. Adiabatic: $T \propto V^{1-\gamma}$, so $\tau \propto V \cdot V^{\frac{\gamma-1}{2}} = V^{\frac{\gamma+1}{2}}$.
3. Volume doubles: $\frac{\tau_2}{\tau_1} = 2^{\frac{1+\gamma}{2}}$
4. So the answer is (a).