Analysis of the Statements

Let's analyze each statement step-by-step. We are given a mixture of 1 mole of hydrogen ($H_2$) and 1 mole of helium (He) in a container of fixed volume at temperature T. Both gases are assumed to be ideal.

A: The average energy per mole of the gas mixture is 2RT

1. Internal Energy of an Ideal Gas: The internal energy (U) of $n$ moles of an ideal gas with $f$ degrees of freedom per molecule is given by the formula $U = n \frac{f}{2} RT$, where R is the universal gas constant.

2. Hydrogen ($H_2$): Hydrogen is a diatomic gas. At temperature T (assuming it's not high enough for vibrational modes to be active), it has 3 translational and 2 rotational degrees of freedom. So, $f_{H_2} = 5$. The internal energy of 1 mole of hydrogen is: $U_{H_2} = 1 \times \frac{5}{2} RT = \frac{5}{2} RT$

3. Helium (He): Helium is a monatomic gas. It has only 3 translational degrees of freedom. So, $f_{He} = 3$. The internal energy of 1 mole of helium is: $U_{He} = 1 \times \frac{3}{2} RT = \frac{3}{2} RT$

4. Total Internal Energy of the Mixture: The total internal energy of the mixture is the sum of the internal energies of its components. $U_{mix} = U_{H_2} + U_{He} = \frac{5}{2} RT + \frac{3}{2} RT = \frac{8}{2} RT = 4RT$

5. Average Energy per Mole: The mixture contains a total of $n_{total} = n_{H_2} + n_{He} = 1 + 1 = 2$ moles. The average energy per mole of the mixture is: $E_{avg, mole} = \frac{U_{mix}}{n_{total}} = \frac{4RT}{2} = 2RT$

   Thus, statement A is correct.

B: The ratio of speed of sound in the gas mixture to that in helium gas is $\sqrt {{6 \over 5}}$

1. Speed of Sound: The speed of sound in an ideal gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$, where $\gamma$ is the adiabatic index and M is the molar mass.

2. For Helium Gas:

   • Molar mass: $M_{He} = 4$ g/mol.
   • Adiabatic index for a monatomic gas: $\gamma_{He} = \frac{C_p}{C_v} = \frac{5R/2}{3R/2} = \frac{5}{3}$.
   • Speed of sound in Helium: $v_{He} = \sqrt{\frac{\gamma_{He} RT}{M_{He}}} = \sqrt{\frac{5RT}{3M_{He}}}$.

3. For the Gas Mixture:

   • Average molar mass: $M_{mix} = \frac{n_{H_2}M_{H_2} + n_{He}M_{He}}{n_{H_2} + n_{He}} = \frac{1(2) + 1(4)}{1+1} = \frac{6}{2} = 3$ g/mol.
   • Molar specific heat at constant volume for the mixture: $C_{v, mix} = \frac{n_{H_2}C_{v, H_2} + n_{He}C_{v, He}}{n_{H_2} + n_{He}} = \frac{1(5R/2) + 1(3R/2)}{2} = \frac{8R/2}{2} = 2R$.
   • Molar specific heat at constant pressure for the mixture: $C_{p, mix} = C_{v, mix} + R = 2R + R = 3R$.
   • Adiabatic index for the mixture: $\gamma_{mix} = \frac{C_{p, mix}}{C_{v, mix}} = \frac{3R}{2R} = \frac{3}{2}$.
   • Speed of sound in the mixture: $v_{mix} = \sqrt{\frac{\gamma_{mix} RT}{M_{mix}}} = \sqrt{\frac{3RT}{2M_{mix}}}$.

4. Ratio of Speeds: $\frac{v_{mix}}{v_{He}} = \frac{\sqrt{\frac{\gamma_{mix} RT}{M_{mix}}}}{\sqrt{\frac{\gamma_{He} RT}{M_{He}}}} = \sqrt{\frac{\gamma_{mix}}{M_{mix}} \cdot \frac{M_{He}}{\gamma_{He}}} = \sqrt{\frac{3/2}{3} \cdot \frac{4}{5/3}} = \sqrt{\frac{1}{2} \cdot \frac{12}{5}} = \sqrt{\frac{6}{5}}$

   Thus, statement B is correct.

C: The ratio of the rms speed of helium atoms to that of hydrogen molecules is ${1 \over 2}$ D: The ratio of the rms speed of helium atoms to that of hydrogen molecules is ${1 \over {\sqrt 2 }}$

1. RMS Speed: The root-mean-square (rms) speed of gas molecules is given by $v_{rms} = \sqrt{\frac{3RT}{M}}$.

2. Thermal Equilibrium: Since the gases are in equilibrium, they are at the same temperature T.

3. RMS Speed of Helium: $v_{rms, He} = \sqrt{\frac{3RT}{M_{He}}} = \sqrt{\frac{3RT}{4}}$.

4. RMS Speed of Hydrogen: $v_{rms, H_2} = \sqrt{\frac{3RT}{M_{H_2}}} = \sqrt{\frac{3RT}{2}}$.

5. Ratio of RMS Speeds: $\frac{v_{rms, He}}{v_{rms, H_2}} = \frac{\sqrt{\frac{3RT}{M_{He}}}}{\sqrt{\frac{3RT}{M_{H_2}}}} = \sqrt{\frac{M_{H_2}}{M_{He}}} = \sqrt{\frac{2}{4}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}$

   Thus, statement D is correct, and statement C is incorrect.

Conclusion: The correct statements are A, B, and D.