Question

Figure shows two vessels A and B with rigid walls containing ideal gases. The pressure, temperature and the volume are p_A, T_A, V in the vessel A and p_B, T_B, V in the vessel B. The vessels are now connected through a small tube. Show that the pressure p and the temperature T satisfy $\frac{p}{T}=\frac{1}{2}\left(\frac{p_A}{T_A}+\frac{p_B}{T_B}\right)$
when equilibrium is achieved.

Answer 1

$\begin{array}{l}\\ {\text{Let the partial pressure of the gas in chamber A and B be P}}_A^{\text{'}}{\text{and P}}_B^{\text{'}}\text{, respectively.}\\ \\ \text{Applying equation of state for gas A, we get}\\ \frac{P_{A}V}{T_A}=\frac{P_A^{\text{'}}2V}{T}\\ \Rightarrow P_A^{\text{'}}=\frac{P_{A}T}{2T_A}\\ \text{Similarly, for gas B:}\\ P_B^{\text{'}}=\frac{P_{B}T}{2T_B}\\ \text{Total pressure is the sum of the partial pressures. It is given by}\\ P=P_A^{\text{'}}+P_B^{\text{'}}\\ =\frac{P_{A}T}{2T_A}+\frac{P_{B}T}{2T_B}\\ \Rightarrow P=\frac{T}{2}(\frac{P_A}{T_A}+\frac{P_B}{T_B})\\ \Rightarrow \frac{P}{T}=\frac{1}{2}(\frac{P_A}{T_A}+\frac{P_B}{T_B})\end{array}$