A closed vessel contains a mixture of two diatomic gases `A` and `B`. Molar mass of `A` is 16 times that of `B` and mass of gas `A` contained in the vessel is 2 times that of `B`. Which of the following statements are correct ?

Options

A. Average kinetic energy per molecules of `A` is equal to that of `B`
B. Root mean square value of translational velocity of `B` is four times that of `A`
C. Pressure exerted by `B` is eight times of that exerted by `A`
D. Number of molecules of `B` in the cylinder is eight times that of `A`

Explanation

To solve the problem, we need to analyze the given information about the two diatomic gases A and B, specifically their molar masses and the relationship of their masses in the closed vessel.

### Step-by-Step Solution:

1. **Identify the Molar Masses**:
- Let the molar mass of gas B be \( M_B \).
- According to the problem, the molar mass of gas A is \( M_A = 16 \times M_B \).

2. **Identify the Masses of the Gases**:
- Let the mass of gas B be \( m_B \).
- The problem states that the mass of gas A is \( m_A = 2 \times m_B \).

3. **Calculate the Number of Moles**:
- The number of moles of gas A (\( n_A \)) can be calculated using the formula:
\[ n_A = \frac{m_A}{M_A} = \frac{2 \times m_B}{16 \times M_B} = \frac{2}{16} \cdot \frac{m_B}{M_B} = \frac{1}{8} \cdot n_B \]
- Where \( n_B \) is the number of moles of gas B:
\[ n_B = \frac{m_B}{M_B} \]

4. **Find the Ratio of Moles**:
- From the above calculation, we find:
\[ n_B = 8 \times n_A \]
- This means the number of moles of gas B is 8 times that of gas A.

5. **Calculate the Root Mean Square Speed**:
- The formula for root mean square speed (\( v_{rms} \)) is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
- For gas A:
\[ v_{rms, A} = \sqrt{\frac{3RT}{M_A}} = \sqrt{\frac{3RT}{16M_B}} \]
- For gas B:
\[ v_{rms, B} = \sqrt{\frac{3RT}{M_B}} \]

6. **Find the Ratio of \( v_{rms} \)**:
- The ratio of \( v_{rms} \) of gas B to gas A is:
\[ \frac{v_{rms, B}}{v_{rms, A}} = \frac{\sqrt{\frac{3RT}{M_B}}}{\sqrt{\frac{3RT}{16M_B}}} = \sqrt{16} = 4 \]
- This means the root mean square speed of gas B is 4 times that of gas A.

7. **Conclusion**:
- Based on the calculations, we can conclude:
- The number of moles of gas B is 8 times that of gas A.
- The root mean square speed of gas B is 4 times that of gas A.
- Therefore, the correct statement is that the number of moles of gas B is 8 times that of gas A.

### Summary of Correct Statements:
- The first, second, and third statements are incorrect.
- The fourth statement is correct.