Q

I need help with - Thermodynamics - JEE Main

An insulated container of gas has two chambers separated by an insulating partition. One of the chambers has volume $\dpi{100} V_{1}$ and contains ideal gas at pressure $\dpi{100} P_{1}$ and temperature $\dpi{100} T_{1}$ . The other chamber has volume $\dpi{100} V_{2}$ and contains ideal gas at pressure $\dpi{100} P_{2}$ and temperature $\dpi{100} T_{2}$ . If the partition is removed without doing any work on the gas, the final equilibrium temperature of the gas in the container will be

• Option 1)

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

• Option 2)

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

• Option 3)

$\frac{P_{1}V_{1}T_{1}+P_{2}V_{2}T_{2}}{P_{1}V_{1}+P_{2}V_{2}}$

• Option 4)

$\frac{P_{1}V_{1}T_{2}+P_{2}V_{2}T_{1}}{P_{1}V_{1}+P_{2}V_{2}}$

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As we learnt in

Change in internal energy for cyclic process -

$\Delta U= 0$

- wherein

Since in a cyclic process initial and final state is same.

$U_{f}= U_{i}$

$U_{i}\:=U_{f}$

$n_{1}C_{V1}T_{1}\:+n_{2}C_{V2}T_{2}\:=(n_{1}\:+n_{2})\:C_{Vf}\:T$

$C_{V1}\:=C_{V2}\:-C_{Vf}$

$\therefore T\:=\frac{n_{1}C_{V1}T_{1}+n_{2}C_{V2}T_{2}}{(n_{1}+n_{2})\:C_{v}}$

$=\:\frac{n_{1}T_{1}+n_{2}T_{2}}{n_{1}+n_{2}}$

$n_{1}\:=\frac{P_{1}V_{1}}{RT_{1}}\:,\:n_{2}\:=\frac{P_{2}V_{2}}{RT_{2}}$

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

Option 1)

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

This option is incorrect.

Option 2)

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

This option is correct.

Option 3)

$\frac{P_{1}V_{1}T_{1}+P_{2}V_{2}T_{2}}{P_{1}V_{1}+P_{2}V_{2}}$

This option is incorrect.

Option 4)

$\frac{P_{1}V_{1}T_{2}+P_{2}V_{2}T_{1}}{P_{1}V_{1}+P_{2}V_{2}}$

This option is incorrect.

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