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JEE Main & Advanced Physics Kinetic Theory of Gases Sample Paper Topic Test - Kinetic Theory of Gases

  • question_answer
    Two thermally insulated vessels 1 and 2 are filled with air at temperature \[{{T}_{1}},\,{{T}_{2}};\] volumes \[{{V}_{1}},\,{{V}_{2}}\] and pressures \[{{P}_{1}},\,{{P}_{2}},\] respectively. If the value 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) \[\frac{{{T}_{1}}{{T}_{2}}({{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}})}{{{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}}{{T}_{1}}}\]

    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 guiding principle in this problem is that the total number of moles of the system remain the same.
    \[\frac{{{P}_{1}}{{V}_{1}}}{R{{T}_{1}}}+\frac{{{P}_{2}}{{V}_{2}}}{R{{T}_{2}}}=\frac{P({{V}_{1}}+{{V}_{2}})}{RT}\]
    or         \[T=\frac{P({{V}_{1}}+{{V}_{2}}){{T}_{1}}{{T}_{2}}}{{{P}_{1}}{{V}_{1}}{{T}_{2}}+{{P}_{2}}{{V}_{2}}{{T}_{1}}}\]
    By Boyle's law, \[P({{V}_{1}}+{{V}_{2}})={{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}}\]
    \[\therefore \]    \[P=\frac{{{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}}}{{{V}_{1}}+{{V}_{2}}}\]
    \[\therefore \]    \[T=\frac{({{P}_{1}}{{V}_{1}}+{{P}_{2}}{{V}_{2}}){{T}_{1}}{{T}_{2}}}{{{P}_{1}}{{V}_{1}}{{T}_{2}}+{{P}_{2}}{{V}_{2}}{{T}_{1}}}\]


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