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JEE Main & Advanced Physics Kinetic Theory of Gases Question Bank Topic Test - Kinetic Theory Of Gases-2

  • question_answer
    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 valve joining the two vessels is opened, the temperature inside the vessel at equilibrium will be         

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

    B) \[({{T}_{1}}+{{T}_{2}})/2\]

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

    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: C

    Solution :

    [c] The number of moles of the system remains same,
    \[\frac{{{P}_{1}}{{V}_{1}}}{R{{T}_{1}}}+\frac{{{P}_{2}}{{V}_{2}}}{R{{T}_{2}}}=\frac{P({{V}_{1}}+{{V}_{2}})}{RT}\] Þ \[T=\frac{P({{V}_{1}}+{{V}_{2}}){{T}_{1}}{{T}_{2}}}{({{P}_{1}}{{V}_{1}}{{T}_{2}}+{{P}_{2}}{{V}_{2}}{{T}_{1}})}\]
    According to Boyle’s law,
    \[{{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}}=P({{V}_{1}}+{{V}_{2}})\]       \ \[T=\frac{({{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}}){{T}_{1}}{{T}_{2}}}{({{P}_{1}}{{V}_{1}}{{T}_{2}}+{{P}_{2}}{{V}_{2}}{{T}_{1}})}\]


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