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laws

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 as shown in the figure, the gas reaches a common temperature T and a common pressure P. Determine the ratio $\frac{P}{T}$ .

Moderate

A

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

$\frac{1}{2} [\frac{P_{1} T_{1} + P_{2} T_{22}}{T_{1} T_{2}}]$
C

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

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

Solution

On combining the two vessels the total number of moles remains constant, i.e., n = $n_{1} + n_{2}$
Using gas equation, we can write, $n_{1} = \frac{P_{1} V}{{RT}_{1}}; n_{2} = \frac{P_{2} V}{{RT}_{2}}$ and $n = \frac{P (2 V)}{RT}$
where V is the volume of each vessel.
Thus $\frac{P (2 V)}{RT} = \frac{P_{1} V}{{RT}_{1}} + \frac{P_{2} V}{{RT}_{2}} or \frac{P}{T} = \frac{1}{2} [\frac{P_{1}}{T_{1}} + \frac{P_{2}}{T_{2}}]$

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