ThermodynamicsHard

Question

An insulated container of gas has two chambers separated by an insulating partition. One of the chambers has volume V1 and contains an ideal gas at pressure P1 and temperature T1. The other chamber has volume V2 and it contains the same ideal gas at pressure P2 and temperature T2. If the partition is removed without doing any work on the gas, the final equilibrium temperature of the gas in the container will be

Options

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

Solution

Let T1 > T2. Now, heat lost by gas (1) = Heat gained by gas (2)

$\text{Or, }n_{1}.C_{m}.\left( T_{1} - T_{f} \right) = n_{2}.C_{m}.\left( T_{f} - T_{2} \right) $$${\text{Or, }\frac{P_{1}V_{1}}{RT_{1}}\left( T_{1} - T_{f} \right) = \frac{P_{2}V_{2}}{RT_{2}}\left( T_{f} - T_{2} \right) }{\Rightarrow T_{f} = \frac{T_{1}T_{2}\left( P_{1}V_{1} + P_{2}V_{2} \right)}{P_{1}V_{1}T_{2} + P_{2}V_{2}T_{1}}}$$

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