A box contains a mixture of H, and He gases. Which of the following statements are correct?

Options

A. The average translational kinetic energies of H, molecules and He atoms are same.
B. The average energies of `H_2` molecules and He atoms are same.
C. `H_2` molecules have greater average energy than that of He atoms
D. The average speed of `H_2` molecules and He atoms is same.

Answer

To solve the problem regarding the mixture of hydrogen (H₂) and helium (He) gases, we will analyze the statements based on the principles of the kinetic theory of gases.

### Step-by-Step Solution:

1. **Understanding the Mixture**:
- The box contains a homogeneous mixture of hydrogen and helium gases. A homogeneous mixture means that the composition is uniform throughout.

2. **Temperature of the Gases**:
- In a mixture of gases at equilibrium, all components share the same temperature. Therefore, the temperature of hydrogen (H₂) will be equal to the temperature of helium (He).

3. **Average Kinetic Energy**:
- The average kinetic energy of gas molecules is directly proportional to the temperature of the gas. Since both gases are at the same temperature, the average kinetic energy of hydrogen molecules will equal that of helium atoms.

4. **Molecular Weights**:
- The molecular weight of hydrogen (H₂) is approximately 2 g/mol, while that of helium (He) is approximately 4 g/mol.

5. **Root Mean Square Speed (Vrms)**:
- The root mean square speed (Vrms) of gas molecules is given by the formula:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.
- Since \( M \) for hydrogen is less than that for helium, at constant temperature, the Vrms of hydrogen will be greater than that of helium.

6. **Comparing Average Energies**:
- The average kinetic energy of gas molecules can also be expressed as:
\[ KE = \frac{3}{2} k T \]
where \( k \) is the Boltzmann constant. Since both gases are at the same temperature, their average kinetic energies will be equal.

7. **Conclusion**:
- Based on the analysis:
- **Option A**: Average translational kinetic energy of hydrogen molecules equals that of helium atoms - **Correct**.
- **Option B**: Average energy of hydrogen molecules is greater than that of helium atoms - **Incorrect** (they are equal).
- **Option C**: Hydrogen molecules have greater average energy than helium atoms - **Correct** (due to higher Vrms).

### Final Answer:
The correct statements are **Option A** and **Option C**.