Q. 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 $p/T$ 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) $\frac{1}{2}\ \left\lbrack \frac{p_{1}T_{2} + p_{2}T_{1}}{T_{1}T_{2}} \right\rbrack$
D) $\frac{p_{1}T_{2} - p_{2}T_{1}}{T_{1} \times T_{2}}$
Explanation
(c) For a closed system, the total number of moles remains constant. So
$p_{1}V = n_{1}RT_{1}\ and\ p_{2}V = n_{2}RT_{2}$
$\therefore\ p(2V) = \left( n_{1} + n_{2} \right)RT$
$\therefore\ \frac{p}{T}\ \ = \ \frac{\left( n_{1} + n_{2} \right)}{2}R = \frac{1}{2}\left\lbrack \frac{P_{1}}{T_{1}} + \frac{P_{2}}{T_{2}} \right\rbrack$
= $\frac{1}{2}\left\lbrack \frac{p_{1}T_{2} + p_{2}T_{1}}{T_{1}T_{2}} \right\rbrack$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.