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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 vessles, the gas reaches a common pressure $P$ and common temperature $T$. The ratio $P/T$ is equal to
$\frac{{{P_1}}}{{{T_1}}} + \frac{{{P_2}}}{{{T_2}}}$
$\frac{{{P_1}{T_1}\, + {P_2}{T_2}\,}}{{{{({T_1} + {T_2})}^2}}}$
$\frac{{{P_1}{T_2}\, + {P_2}{T_1}\,}}{{{{({T_1} + {T_2})}^2}}}$
$\frac{{{P_1}}}{{2{T_1}}} + \frac{{{P_2}}}{{2{T_2}}}$
Solution
Number of moles in the first vessel
$\mu_{1}=\frac{P_{1} V}{R T_{1}}$
Number of moles in the second vessel
$\mu_{2}=\frac{P_{2} V}{R T_{2}}$
If both vessels are joined together, then quantity of gas remains same, i.e.,
$\mu=\mu_{1}+\mu_{2}$
$\frac{P(2 V)}{R T}=\frac{P_{1} V}{R T_{1}}+\frac{P_{2} V}{R T_{2}}$
$\frac{P}{T}=\frac{P_{1}}{2 T_{1}}+\frac{P_{2}}{2 T_{2}}$
