A diatomic gas at pressure $P_0$ and volume $V_0$ is expanded adiabatically to $32$ times its origin

Question

A diatomic gas at pressure $P_0$ and volume $V_0$ is expanded adiabatically to $32$ times its original volume. What is the final pressure of the gas?

Professor PC Thomas Classes · Answer

(c) : The equation for an adiabatic process is $P_1 V_1^\gamma = P_2 V_2^\gamma$. For a diatomic gas, the specific heat ratio is $\gamma = 7/5$. Setting up the equation: $P_0 V_0^{7/5} = P_2 (32V_0)^{7/5}$. This simplifies to $P_2 = P_0 (1/32)^{7/5}$. Since $32 = 2^5$, we have $(1/2^5)^{7/5} = 1/2^7 = 1/128$. Therefore, the final pressure is $P_2 = P_0 / 128$.