An insulated container of gas has two chambers separated by an insulating partition. One of the cham — Thermodynamics and Thermochemistry Chemistry Question
Question
An insulated container of gas has two chambers separated by an insulating partition. One of the chambers has volume $V_1$ and contains an ideal gas at pressure $P_1$ and temperature $T_1$. The other chamber has volume $V_2$ and contains the same ideal gas at pressure $P_2$ and temperature $T_2$. If the partition is removed without doing any work on the gas, the final equilibrium temperature of the gas in the container will be
💡 Solution & Explanation
Conservation of internal energy ($\Delta U = 0$) gives $T_f = \frac{n_1 T_1 + n_2 T_2}{n_1 + n_2}$. Substituting $n = \frac{PV}{RT}$ yields $T_f = \frac{P_1 V_1/R + P_2 V_2/R}{\frac{P_1 V_1}{R T_1} + \frac{P_2 V_2}{R T_2}} = \frac{P_1 V_1 + P_2 V_2}{\frac{P_1 V_1 T_2 + P_2 V_2 T_1}{T_1 T_2}} = \frac{T_1 T_2 (P_1 V_1 + P_2 V_2)}{P_1 V_1 T_2 + P_2 V_2 T_1}$.