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31. (AIEEE 2008 )

An insulated container of gas has two chambers separated by an insulating partition. One of the chambers has volume $V_{1}$ and contains ideal gas at pressure $P_{1}$ and temperature $T_{1}$ . The other chamber has volume $V_{2}$ and contains ideal gas at pressure $P_{2}$ and temperature $T_{2}$ . If the partition is removed without doing any work on the gas, the final equilibrium temperature of the gas in the container will be

(A) $\frac{T_{1} T_{2} (P_{1} V_{1} + P_{2} V_{2})}{P_{1} V_{1} T_{2} + P_{2} V_{2} T_{1}}$

(B) $\frac{P_{1} V_{1} T_{1} + P_{2} V_{2} T_{2}}{P_{1} V_{1} + P_{2} V_{2}}$

(D) $\frac{T_{1} T_{2} (P_{1} V_{1} + P_{2} V_{2})}{P_{1} V_{1} T_{1} + P_{2} V_{2} T_{2}}$

Correct answer is (A)

Since, no work is done and system is thermally insulated from surrounding. Therefore, total internal energy is constant, that is, $U = U_{1} + U_{2}$ Assuming gas have same degree of freedom, we have

$(n_{1} + n_{2}) C_{V} T = n_{1} C_{V} T_{1} + n_{2} C_{V} T_{2}$

Therefore,

$T = \frac{n_{1} R T_{1} + n_{2} R T_{2}}{R n_{1} + n_{2} R} = \frac{(P_{1} V_{1} + P_{2} V_{2})}{\frac{P_{1} V_{1}}{T_{1}} + \frac{P_{2} V_{2}}{T_{2}}} = \frac{(P_{1} V_{1} + P_{2} V_{2}) T_{1} T_{2}}{(P_{1} V_{1} T_{2} + P_{2} V_{2} T_{1})}$

32. (AIEEE 2004 )

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 value joining the two vessels is opened, the temperature inside the vessel at equilibrium will be

(A) $\frac{T_{1} T_{2} (P_{1} V_{1} + P_{2} V_{2})}{(P_{1} V_{1} T_{2} + P_{2} V_{2} T_{1})}$

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

(D) $\frac{T_{1} T_{2} (P_{1} V_{1} + P_{2} V_{2})}{(P_{1} V_{1} T_{1} + P_{2} V_{2} T_{2})}$

Correct answer is (A)

Here $Q = 0$ and $W = 0.$ Therefore from first law of thermodynamics $Δ U = Q + W = 0$
$∴$ Internal energy of the system with partition $=$ Internal energy of the system without partition.
$n_{1} C_{v} T_{1} + n_{2} C_{v} T_{2} = (n_{1} + n_{2}) C_{v} T$
$∴$ $T = \frac{n_{1} T_{1} + n_{2} T_{2}}{n_{1} + n_{2}}$
But $n_{1} = \frac{P_{1} V_{1}}{R T_{1}}$ and $n_{2} = \frac{P_{2} V_{2}}{R T_{2}}$
$∴$ $T = \frac{\frac{P_{1} V_{1}}{R T_{1}} \times T_{1} + \frac{P_{2} V_{2}}{R T_{2}} \times T_{2}}{\frac{P_{1} V_{1}}{R T_{1}} + \frac{P_{2} V_{2}}{R T_{2}}}$
$= \frac{T_{1} T_{2} (P_{1} V_{1} + P_{2} V_{2})}{P_{1} V_{1} T_{2} + P_{2} V_{2} T_{1}}$

Kinetic Theory of Gases