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JEE Main & Advanced Physics Kinetic Theory of Gases Question Bank Self Evaluation Test - Kinetic Theory

  • question_answer
    For a gas sample with Np number of molecules, function N(V) is given by: \[N\left( V \right)=\frac{dN}{dV}=\left[ \frac{3{{F}_{0}}}{V_{0}^{3}} \right]{{V}^{2}}\] for \[0\le V\le {{V}_{0}}\]and \[N\left( V \right)=0\]for \[V>{{V}_{0}}\] Where \[dN\] is number of molecules in speed range V to \[V+dV.\]The rms speed of the gas molecule is

    A) \[\sqrt{\frac{2}{5}}{{V}_{0}}\]

    B) \[\sqrt{\frac{3}{5}}{{V}_{0}}\]

    C) \[\sqrt{2}{{V}_{0}}\]    

    D)  \[\sqrt{3}{{V}_{0}}\]

    Correct Answer: B

    Solution :

    [b] \[V_{rms}^{2}=\,\,<{{V}^{2}}>\,=\frac{V_{1}^{2}+V_{2}^{2}+V_{3}^{2}+.......}{N}\] \[=\frac{\int_{{}}^{{}}{{{V}^{2}}dN}}{\int_{{}}^{{}}{dN}}\text{ here }\frac{dN}{dV}=N\left( V \right)\] \[V_{rms}^{2}=\frac{1}{N}\int\limits_{0}^{\infty }{N\left( V \right){{V}^{2}}dV}\] \[=\frac{1}{N}\int\limits_{0}^{{{V}_{0}}}{\left[ \frac{3N}{V_{0}^{3}}{{V}^{2}} \right]{{V}^{2}}dV=\frac{3}{5}V_{0}^{2}}\] \[\Rightarrow {{V}_{rms}}=\sqrt{\frac{3}{5}}{{V}_{0}}\]


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