Question

Two containers of equal volume contain the same gas at the pressure $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 a common temperature $T$ . The ratio is equal to:

• A. $\frac{p_{1} T_{2} + p_{2} T_{1}}{T_{1} \times T_{2}}$
• B. $\frac{p_{1} T_{2} + p_{2} T_{1}}{T_{1} + T_{2}}$
• C. $\frac{1}{2} \, \left[\frac{p_{1} T_{2} + p_{2} T_{1}}{T_{1} T_{2}}\right]$
• D. $\frac{p_{1} T_{2} - p_{2} T_{1}}{T_{1} \times T_{2}}$

Answer 1

Answer: (C)

For a closed system, the total number of moles remains constant. So
$p_{1}V=n_{1}RT_{1} \, \text{and} \, p_{2}V=n_{2}RT_{2}$
$\therefore \, p\left(2 V\right)=\left(n_{1} + n_{2}\right)RT$
$\therefore \, \frac{p}{T}=\frac{\left(n_{1} + n_{2}\right)}{2 V}R=\frac{1}{2}\left[\frac{P_{1}}{T_{1}} + \frac{P_{2}}{T_{2}}\right]$
$\Rightarrow \frac{p}{T}=\frac{1}{2}\left[\frac{p_{1} T_{2} + p_{2} T_{1}}{T_{1} T_{2}}\right]$