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(D) Number of moles in the first vessel is $\mu_1 = \frac{P_1 V}{R T_1}$.Number of moles in the second vessel is $\mu_2 = \frac{P_2 V}{R T_2}$.When bo

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$\frac{P_1}{2T_1} + \frac{P_2}{2T_2}$

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(D) Number of moles in the first vessel is $\mu_1 = \frac{P_1 V}{R T_1}$.Number of moles in the second vessel is $\mu_2 = \frac{P_2 V}{R T_2}$.When both vessels are joined,the total number of moles $\mu$ remains conserved,so $\mu = \mu_1 + \mu_2$.The total volume of the combined system is $2V$. Let the final pressure be $P$ and final temperature be $T$.Using the ideal gas equation for the combined system: $\mu = \frac{P(2V)}{RT}$.Equating the total moles: $\frac{P(2V)}{RT} = \frac{P_1 V}{R T_1} + \frac{P_2 V}{R T_2}$.Canceling $V$ and $R$ from both sides: $\frac{2P}{T} = \frac{P_1}{T_1} + \frac{P_2}{T_2}$.Therefore,the ratio $\frac{P}{T} = \frac{1}{2} \left( \frac{P_1}{T_1} + \frac{P_2}{T_2} \right) = \frac{P_1}{2T_1} + \frac{P_2}{2T_2}$.

Two containers of equal volume contain the same gas at pressure $P_1$ and $P_2$ and absolute temperature $T_1$ and $T_2$ respectively. On joining the vessels,the gas reaches a common pressure $P$ and common temperature $T$. The ratio $P/T$ is equal to

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