Question

Two containers of the equal volume contain the same gas at 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 common temperature $T$ . The ratio $\frac{P}{T}$ is equal to

• A. $\frac{P_{1}}{T_{1}}+\frac{P_{2}}{T_{2}}$
• B. $\frac{P_{1} T_{1} + P_{2} T_{2}}{\left(T_{1} + T_{2}\right)}$
• C. $\frac{P_{1} T_{2} + P_{2} T_{1}}{\left(T_{1} + T_{2}\right)^{2}}$
• D. $\frac{P_{1}}{2 T_{1}}+\frac{P_{2}}{2 T_{2}}$

Answer 1

Answer: (D)

Number of moles in the first vessel
$\mu _{1}=\frac{\textit{P}_{1} \textit{V}}{\textit{RT}_{1}}$
Number of moles in the second vessel
$\mu _{2}=\frac{\textit{P}_{2} \textit{V}}{\textit{RT}_{2}}$

If both vessels are joined together, then the quantity of gas remains the same, i.e., $\mu = \mu _{1} + \mu _{2}$
$\frac{\textit{P} \left(2 \textit{V}\right)}{\textit{RT}}=\frac{\left(\textit{P}\right)_{1} \textit{V}}{\left(\textit{RT}\right)_{1}}+\frac{\left(\textit{P}\right)_{2} \textit{V}}{\left(\textit{RT}\right)_{2}}$
$\frac{\textit{P}}{\textit{T}}=\frac{\textit{P}_{1}}{2 \textit{T}_{1}}+\frac{\textit{P}_{2}}{2 T_{2}}$