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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 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

  • 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}}

 
Answers (1)
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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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