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Two containers of equal volume contain the same gas at the 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 a common temperature $T$ . The ratio $\frac{P}{T}$ is equal to

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$\frac{P_{1} T_{2} + P_{2} T_{1}}{T_{1} T_{2}}$

$\frac{P_{1} T_{2} + P_{2} T_{1}}{T_{1} + T_{2}}$

$\frac{1}{2} [\frac{P_{1} T_{2} + P_{2} T_{1}}{T_{1} T_{2}}]$

$[\frac{P_{1} T_{2} - P_{2} T_{1}}{T_{1} T_{2}}]$

answer is C.

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Detailed Solution

For a closed system, the total number of moles remains constant. So

$P_{1} V = n_{1} R T_{1}$ and $P_{2} V = n_{2} R T_{2}$

$∴ P (2 V) = (n_{1} + n_{2}) R T$

$∴ \frac{P}{T} = \frac{(n_{1} + n_{2})}{2 V} R = \frac{1}{2} [\frac{P_{1}}{T_{1}} + \frac{P_{2}}{T_{2}}]$ $= \frac{1}{2} [\frac{P_{1} T_{2} + P_{2} T_{1}}{T_{1} T_{2}}]$

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