Question #42174

A cylinder contains one mole of a rigid diatomic ideal gas at equilibrium pressure $P_0$ and volume $V_0$. A piston drives the volume to oscillate as $V(t) = V_0 - \Delta V \cos(\omega t)$, where $\Delta V \ll V_0$. The translational degrees of freedom equilibrate instantaneously, but the rotational degrees of freedom have a relaxation time $\tau$. The effective molar heat capacity at constant volume is given by $C_V(\omega) = C_{V,trans} + \frac{C_{V,rot}}{1 + \omega^2 \tau^2}$. Assuming the process follows an effective adiabatic relation $P V^{\gamma_{eff}} = \text{constant}$ with $\gamma_{eff} = 1 + \frac{R}{C_V(\omega)}$, what is the amplitude of the pressure oscillations $\Delta P_0$ when the piston is driven at a frequency $\omega = \frac{1}{\tau}$?