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

Options

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//sqrt2`

Explanation

To solve the problem regarding the mixture of one mole of hydrogen and one mole of helium in a fixed volume container at temperature T, we will follow these steps:

### Step 1: Determine the Average Energy per Mole of the Gas Mixture
The average energy per mole of an ideal gas is given by the formula:
\[ E = \frac{f}{2} RT \]
where \( f \) is the degrees of freedom of the gas.

- For hydrogen (H₂), which is a diatomic gas, \( f = 5 \) (3 translational + 2 rotational).
- For helium (He), which is a monatomic gas, \( f = 3 \).

Now, we can calculate the total energy for each gas:
- Total energy of hydrogen:
\[ E_{H_2} = \frac{5}{2} RT \]
- Total energy of helium:
\[ E_{He} = \frac{3}{2} RT \]

The total energy of the mixture is:
\[ E_{total} = E_{H_2} + E_{He} = \frac{5}{2} RT + \frac{3}{2} RT = 4RT \]

Now, to find the average energy per mole of the gas mixture (which has 2 moles in total):
\[ E_{average} = \frac{E_{total}}{2} = \frac{4RT}{2} = 2RT \]

### Step 2: Calculate the Ratio of the Speed of Sound in the Gas Mixture to Helium
The speed of sound in a gas is given by:
\[ v = \sqrt{\frac{\gamma RT}{M}} \]
where \( \gamma \) is the heat capacity ratio and \( M \) is the molar mass.

- For helium, \( \gamma_{He} = \frac{5}{3} \) and \( M_{He} = 4 \, g/mol \).
- For the mixture, we need to calculate \( \gamma_{mix} \).

Using the formula for \( \gamma \) of a mixture:
\[ \frac{1}{\gamma_{mix} - 1} = \frac{n_1}{\gamma_1 - 1} + \frac{n_2}{\gamma_2 - 1} \]
where \( n_1 = n_2 = 1 \) (1 mole of each gas).

Calculating:
\[ \frac{1}{\gamma_{mix} - 1} = \frac{1}{\frac{5}{3} - 1} + \frac{1}{\frac{7}{5} - 1} \]
This gives:
\[ \frac{1}{\gamma_{mix} - 1} = \frac{1}{\frac{2}{3}} + \frac{1}{\frac{2}{5}} = \frac{3}{2} + \frac{5}{2} = 4 \implies \gamma_{mix} = 5 \]

Now, we can find the ratio of the speed of sound:
\[ \frac{v_{mix}}{v_{He}} = \frac{\sqrt{\frac{\gamma_{mix} RT}{M_{mix}}}}{\sqrt{\frac{\gamma_{He} RT}{M_{He}}}} = \sqrt{\frac{\gamma_{mix} M_{He}}{\gamma_{He} M_{mix}}} \]
Calculating \( M_{mix} \):
\[ M_{mix} = \frac{(1 \cdot 2) + (1 \cdot 4)}{2} = 3 \, g/mol \]
Now substituting values:
\[ \frac{v_{mix}}{v_{He}} = \sqrt{\frac{5 \cdot 4}{\frac{5}{3} \cdot 3}} = \sqrt{\frac{20}{5}} = \sqrt{4} = 2 \]

### Step 3: Calculate the Ratio of RMS Speed of Helium Atom to Hydrogen Molecule
The RMS speed is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
Calculating for both gases:
- For helium:
\[ v_{rms, He} = \sqrt{\frac{3RT}{4}} \]
- For hydrogen:
\[ v_{rms, H_2} = \sqrt{\frac{3RT}{2}} \]

Now, the ratio:
\[ \frac{v_{rms, He}}{v_{rms, H_2}} = \frac{\sqrt{\frac{3RT}{4}}}{\sqrt{\frac{3RT}{2}}} = \sqrt{\frac{2}{4}} = \frac{1}{\sqrt{2}} \]

### Conclusion
The correct statements are:
1. The average energy per mole of the gas mixture is \( 2RT \).
2. The ratio of the speed of sound in the gas mixture to that in helium is \( \sqrt{\frac{6}{5}} \).
3. The ratio of the RMS speed of helium atoms to that of hydrogen molecules is \( \frac{1}{\sqrt{2}} \).