There are two vessels A and B with rigid walls containing ideal...

Using the ideal gas equation, the initial number of moles in vessel A is:

$n_A = \dfrac{P_A V}{R T_A}$

The initial number of moles in vessel B is:

$n_B = \dfrac{P_B V}{R T_B}$

When the vessels are connected, the total volume becomes $V_{total} = V + V = 2V$. Let the equilibrium pressure be $P$ and the equilibrium temperature be $T$. The total number of moles in the system remains conserved:

$n_{total} = n_A + n_B$

$\dfrac{P(2V)}{RT} = \dfrac{P_A V}{RT_A} + \dfrac{P_B V}{RT_B}$

Dividing the entire equation by $\dfrac{V}{R}$:

$\dfrac{2P}{T} = \dfrac{P_A}{T_A} + \dfrac{P_B}{T_B}$

$\dfrac{P}{T} = \dfrac{1}{2}\left(\dfrac{P_A}{T_A} + \dfrac{P_B}{T_B}\right)$

Answer: \dfrac{1}{2}\left(\dfrac{P_A}{T_A}+\dfrac{P_B}{T_B}\right)