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JEE Main & Advanced AIEEE Solved Paper-2008

  • question_answer
    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       AIEEE  Solved  Paper-2007

    A) \[\frac{{{P}_{1}}{{V}_{1}}{{T}_{2}}+{{P}_{2}}{{V}_{2}}{{T}_{1}}}{{{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}}}\]

    B)        \[\frac{{{T}_{1}}{{T}_{2}}\left( {{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}} \right)}{{{P}_{1}}{{V}_{1}}{{T}_{1}}+{{P}_{2}}{{V}_{2}}{{T}_{2}}}\]

    C)                \[\frac{{{T}_{1}}{{T}_{2}}\left( {{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}} \right)}{{{P}_{1}}{{V}_{1}}{{T}_{2}}+{{P}_{2}}{{V}_{2}}{{T}_{1}}}\]             

    D)         \[\frac{{{P}_{1}}{{V}_{1}}{{T}_{1}}+{{P}_{2}}{{V}_{2}}{{T}_{2}}}{{{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}}}\]

    Correct Answer: C

    Solution :

                    Internal energy of the system will remain conserved. \[\left( {{n}_{1}}+{{n}_{2}} \right){{C}_{v}}T={{n}_{1}}{{C}_{V}}{{T}_{1}}+{{n}_{2}}{{C}_{V}}.\,{{T}_{2}}\] \[\left( \frac{{{P}_{1}}{{V}_{1}}}{R{{T}_{1}}}+\frac{{{P}_{2}}{{V}_{2}}}{R{{T}_{2}}} \right)T=\frac{{{P}_{1}}{{V}_{1}}}{R}+\frac{{{P}_{2}}{{V}_{1}}}{R}+\frac{{{P}_{2}}{{V}_{2}}}{R}\,\,;\]\[T=\frac{{{T}_{1}}{{T}_{2}}\left( {{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}} \right)}{{{P}_{1}}{{V}_{1}}{{T}_{2}}+{{P}_{2}}{{V}_{2}}{{T}_{1}}}\]        


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