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 a common temperature $T$. The ratio $\frac{P}{T}$ is

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

Answer 1

Answer: (B)

For a closed system, the total mass of gas or the number of moles remains constant. Let $n_{1}$ and $n_{2}$ be number of moles of gas in container 1 and container 2 respectively.
$P_{1} V=n_{1} R T_{1}$
or $n_{1}=\frac{P_{1} V}{R T_{1}} \dots$(i)
$P_{2} V=n_{2} R T_{2}$
or $n_{2}=\frac{P_{2} V}{R T_{2}} \dots$(ii)
$P(2 V)=\left(n_{1}+n_{2}\right) R T \dots$(iii)
Substituting the values of $n_{1}$ and $n_{2}$ in equation (iii), we get
$P(2 V)=\left(\frac{P_{1} V}{R T_{1}}+\frac{P_{2} V}{R T_{2}}\right) R T$ or
$\frac{P}{T}=\frac{1}{2}\left(\frac{P_{1}}{T_{1}}+\frac{P_{2}}{T_{2}}\right)$