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 2RT
B. The ratio of speed of sound in the gas mixture to tah tin 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 analyze the statements one by one.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a mixture of gases: 1 mole of hydrogen (H₂) and 1 mole of helium (He).
- The gases are assumed to be ideal and are in equilibrium at temperature T.

2. **Calculating Total Energy of the Gases**:
- For hydrogen (H₂), which is a diatomic gas, the total energy (U) is given by:
\[ U_{H_2} = \frac{5}{2} nRT = \frac{5}{2} \cdot 1 \cdot RT = \frac{5}{2} RT \]
- For helium (He), which is a monatomic gas, the total energy is:
\[ U_{He} = \frac{3}{2} nRT = \frac{3}{2} \cdot 1 \cdot RT = \frac{3}{2} RT \]
- Therefore, the total energy of the mixture is:
\[ U_{total} = U_{H_2} + U_{He} = \frac{5}{2} RT + \frac{3}{2} RT = 4RT \]

3. **Calculating Average Energy per Mole**:
- The average energy per mole of the gas mixture is calculated as:
\[ \text{Average Energy} = \frac{U_{total}}{\text{Total moles}} = \frac{4RT}{2} = 2RT \]
- Thus, the first statement is correct: **The average energy per mole of the gas mixture is 2RT**.

4. **Calculating the Ratio of Speed of Sound**:
- The speed of sound in a gas is given by:
\[ v = \sqrt{\frac{\gamma RT}{M}} \]
- Here, we need to find the ratio of the speed of sound in the gas mixture to that in helium.
- For helium (monatomic gas), \(\gamma_{He} = \frac{5}{3}\).
- For the mixture, we need to calculate \(\gamma_{mix}\):
\[ \frac{N_{total}}{\gamma_{mix} - 1} = \frac{N_{H_2}}{\gamma_{H_2} - 1} + \frac{N_{He}}{\gamma_{He} - 1} \]
\[ \frac{2}{\gamma_{mix} - 1} = \frac{1}{\frac{7}{5} - 1} + \frac{1}{\frac{5}{3} - 1} \]
- Solving gives \(\gamma_{mix} = \frac{3}{2}\).
- Now, substituting into the speed of sound formula:
\[ \frac{v_{mix}}{v_{He}} = \sqrt{\frac{\gamma_{mix}}{\gamma_{He}} \cdot \frac{M_{He}}{M_{mix}}} \]
- The molar mass of the mixture \(M_{mix} = \frac{2 + 4}{2} = 3\).
- Therefore:
\[ \frac{v_{mix}}{v_{He}} = \sqrt{\frac{\frac{3}{2}}{\frac{5}{3}} \cdot \frac{4}{3}} = \sqrt{\frac{6}{5}} \]
- Thus, the second statement is correct.

5. **Calculating the Ratio of RMS Speeds**:
- The RMS speed for helium and hydrogen is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
- The ratio of RMS speeds is:
\[ \frac{v_{rms, He}}{v_{rms, H_2}} = \sqrt{\frac{M_{H_2}}{M_{He}}} = \sqrt{\frac{2}{4}} = \frac{1}{\sqrt{2}} \]
- The statement that the ratio is \( \frac{1}{2} \) is incorrect.

### Conclusion:
- The correct statements are:
- A: The average energy per mole of the gas mixture is \(2RT\).
- B: The ratio of the speed of sound in the gas mixture to that in helium gas is correct.
- C: The ratio of RMS speed of helium atoms to that of hydrogen molecules is incorrect.