Two thermally insulated vessels 1 and 2 are filled with air at temperatures $(T_{1}, T_{2})$ , volume $(V_{1}, V_{2})$ and pressure $(P_{1}, P_{2})$ respectively. If the valve joining the two vessels is opened, the temperature inside the vessel at equilibrium will be

$T_{1} + T_{2}$

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$(T_{1} + T_{2}) / 2$

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$T_{1} T_{2} (P_{1} V_{1} + P_{2} V_{2}) / (P_{1} V_{1} T_{2} + P_{2} V_{2} T_{1})$

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$T_{1} T_{2} (P_{1} V_{1} + P_{2} V_{2}) / (P_{1} V_{1} T_{1} + P_{2} V_{2} T_{2})$

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Solution

The correct option is
B

$T_{1} T_{2} (P_{1} V_{1} + P_{2} V_{2}) / (P_{1} V_{1} T_{2} + P_{2} V_{2} T_{1})$

According to the kinetic theory, the average kinetic energy(KE) per molecule of a gas = $\frac{3}{2} k T .$ Let $n_{1} a n d n_{2}$ be the number of moles of air in vessels 1 and 2 respectively.
Before mixing, the total KE of molecules in the two vessels is
$E_{1} = \frac{3}{2} n_{1} k T_{1} + \frac{3}{2} n_{2} k T_{2} = \frac{3}{2} k (n_{1} T_{1} + n_{2} T_{2})$
After mixing, the total KE of molecules is
$E_{2} = \frac{3}{2} (n_{1} + n_{2}) k T$
Where T is the temperature when equilibrium is established. Since there is no loss of energy (because the vessels are insulated), $E_{2} = E_{1} o r$
$\frac{3}{2} (n_{1} + n_{2}) k T = \frac{3}{2} k (n_{1} T_{1} + n_{2} T_{2})$
or $T = \frac{n_{1} T_{1} + n_{2} T_{2}}{(n_{1} + n_{2})}$ ~~~~~~~~~(1)
Now $P_{1} V_{1} = n_{1} R T_{1} a n d P_{2} V_{2} = n_{2} R T_{2}$ which give
$n_{1} = \frac{P_{1} V_{1}}{R T_{1}} a n d n_{2} = \frac{P_{2} V_{2}}{R T_{2}}$
Using these in Eq. (1) and simplifying, we get
$T = \frac{T_{1} T_{2} (P_{1} V_{1} + P_{2} V_{2})}{(P_{1} V_{1} T_{2} + P_{2} V_{2} T_{1})}$

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Two thermally insulated vessels $1$ and $2$ are filled with air at temperature ( $T_{1}$ , $T_{2}$ ), volume ( $V_{1}$ , $V_{2}$ ) and pressure ( $P_{1}$ , $P_{2}$ ) respectively. If the value joining the two vessels is opened, the temperature inside the vessel at equilibrium will be

Two thermally insulated vessels 1 and 2 are filled with air at temperatures $(T_{1}, T_{2})$ , volume $(V_{1}, V_{2})$ and pressure $(P_{1}, P_{2})$ respectively. If the valve joining the two vessels is opened, the temperature inside the vessel at equilibrium will be