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Temperature of a Gas

Two identical...

Question

Two identical vessels contain the same gas at pressures $P_{1}$ and $P_{2}$ at absolute temperatures $T_{1}$ and $T_{2}$ respectively. On joining the vessels with a small tube, the gas reaches a common temperature $T$ and a common pressure $P$ . Determine the ratio $P / T$ .

A

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

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B

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

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C

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

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D

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

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Solution

The correct option is C $\frac{1}{2} (\frac{P_{1} T_{2} + P_{2} T_{1}}{T_{1} T_{2}}]$
On combining the vessels, the total number of moles remains constant
i.e
$n = n_{1} + n_{2}$
Using the ideal gas equation, we can write
$n_{1} = \frac{P_{1} V}{R T_{1}}$
$n_{2} = \frac{P_{2} V}{R T_{2}}$
and $n = \frac{P (2 V)}{R T}$
where $V$ is the volume of each vessel.
Thus,
$\frac{P (2 V)}{R T} = \frac{P_{1} V}{R T_{1}} + \frac{P_{2} V}{R T_{2}} o r \frac{P}{T} = \frac{1}{2} (\frac{P_{1}}{T_{1}} + \frac{P_{2}}{T_{2}}] = \frac{1}{2} (\frac{P_{1} T_{2} + P_{2} T_{1}}{T_{1} T_{2}}]$

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