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Ideal Gas Equation

Two container...

Question

Two containers of equal volume contain the same gas at pressures $P_{1}$ and $P_{2}$ and absolute temperatures $T_{1}$ and $T_{2}$ , respectively. On joining the vessels, the gas reaches a common pressure $P$ and common temperature $T$ . The ratio $\frac{P}{T}$ is equal to

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

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\frac{P_{1} T_{1} + P_{2} T_{2}}{{(T_{1} + T_{2})}^{2}}

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\frac{P_{1} T_{2} + P_{2} T_{1}}{{(T_{1} + T_{2})}^{2}}

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

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Solution

The correct option is D $\frac{P_{1}}{2 T_{1}} + \frac{P_{2}}{2 T_{2}}$
$P_{1} V = n_{1} R T_{1}$
$P_{2} V = n_{2} R T_{2}$
where V is the volume of each vessel.
When the vessels are joined, $P (2 V) = (n_{1} + n_{2}) R T$
$∴ \frac{P}{T} = \frac{1}{2} \frac{(n_{1} + n_{2}) R}{V} = \frac{1}{2} (\frac{P_{1}}{T_{1}} + \frac{P_{2}}{T_{2}}) = \frac{P_{1}}{2 T_{1}} + \frac{P_{2}}{2 T_{2}}$

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