Question

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 (s) is (are)

• A. The average energy per mole of the gas mixture is $2\,RT$.
• B. The ratio of speed of sound in the gas mixture to that in helium gas is $\sqrt{6 / 5}$
• C. The ratio of the rms speed of helium atoms to that of hydrogen molecules is $1/2$.
• D. The ratio of the rms speed of helium atoms to that of hydrogen molecules is $1/\sqrt{2}$.

Answer 1

Answer: (A), (B), (D)

$\left(A\right)\,\, \left(KE_{av}\right)_{mix / mol} = \frac{\frac{5}{1}\left(1\right) RT + \frac{3}{2}\left(1\right) RT}{2} = 2RT$
$\left(B\right) v_{sound} = \sqrt{\frac{\gamma RT}{M}}$
Here, $\quadγ_{mix} = \frac{\left(1\times\frac{5}{2}\right)+\left(1\times\frac{7}{2}\right)}{\left(1\times\frac{3}{2}\right)+\left(1+\frac{5}{2}\right)} = \frac{3}{2}$
$\Rightarrow \, \frac{v_{mix}}{v_{He}} = \sqrt{\frac{\gamma_{mix}}{\gamma_{He}}} \sqrt{\frac{M_{He}}{M_{mix}}}$
$\frac{v_{mix}}{v_{He}} = \sqrt{\frac{\left(\frac{3}{2}\right)}{\left(\frac{5}{3}\right)}}\times \sqrt{\frac{4}{3}} = \sqrt{\frac{6}{5}}$