Discussion Forum

 A container of fixed volume has a mixture  of one mole of hydrogen  and one mole  of helium  in equilibrium at temperature  T. Assuming the gases are ideal, the correct statement is/are

Answer:

Explanation:

(a)  Total  Internal energy

          $U=\frac{f_{1}}{2}nRT+\frac{f_{2}}{2}nRT$

                    $(U_{ave})_{per mole}=\frac{U}{2n}$

  =     $\frac{1}{4}\left[ 5RT+3RT\right]=2RT$

  (b)                    $\gamma_{mix}=\frac{^{n_{1}}C_{p_{1}}+^{n_{2}}C_{p_{2}}}{^{n_{2}}C_{v_{1}}+^{n_{2}}C_{v_{2}}}$

      $=\frac{(1)\frac{7R}{2}+(1)\frac{5R}{2}}{(1)\frac{5R}{2}+(1)\frac{3R}{2}}=\frac{3}{2}$

              $M_{mix}=\frac{n_{1}M_{1}+n_{2}M_{2}}{n_{1}+n_{2}}$

                   $\frac{M_{1}+M_{2}}{2}=\frac{2+4}{2}=3$

 speed of sound          $V=\sqrt{\frac{\gamma RT}{M}}$

        $\Rightarrow$                $V\propto\sqrt{\frac{Y}{M}}$

                 $\frac{V_{mix}}{V_{He}}=\sqrt{\frac{\gamma_{mix}}{\gamma_{He}}\times\frac{M_{He}}{M_{mix}}}$

                   =    $\frac{V_{mix}}{V_{He}}=\sqrt{\frac{3/2}{5/3}\times\frac{4}{3}}=\sqrt{\frac{6}{5}}$

 (d)        $V_{rms}=\sqrt{\frac{3RT}{M}}\Rightarrow V_{rms} \propto \frac{1}{\sqrt{M}}$

                    $\frac{V_{He}}{V_{H}}=\sqrt{\frac{M_{H}}{M_{He}}}=\sqrt{\frac{2}{4}}=\frac{1}{\sqrt{2}}$