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 molecule of `A` is equal to that of `B`
B. Root mean square value of translational velocity of `B` is four time that of `A`
C. Pressure exerted by `B` is eight time of the exerted by `A`.
D. Number of moleucles of `B`, in the cylinder, is eight times that of `A`

Explanation

To solve the problem step by step, let's analyze the information given and derive the necessary conclusions.

### Step 1: Understand the Given Information
We have two diatomic gases, A and B. The molar mass of gas A is 16 times that of gas B, and the mass of gas A is 2 times that of gas B.

Let:
- Molar mass of gas B = \( M_B \)
- Molar mass of gas A = \( M_A = 16 M_B \)
- Mass of gas B = \( m_B \)
- Mass of gas A = \( m_A = 2 m_B \)

### Step 2: Calculate the Number of Moles of Each Gas
The number of moles \( n \) of a gas can be calculated using the formula:
\[ n = \frac{m}{M} \]
where \( m \) is the mass of the gas and \( M \) is its molar mass.

For gas A:
\[ n_A = \frac{m_A}{M_A} = \frac{2 m_B}{16 M_B} = \frac{2}{16} \cdot \frac{m_B}{M_B} = \frac{1}{8} \cdot n_B \]

For gas B:
\[ n_B = \frac{m_B}{M_B} \]

### Step 3: Determine the Ratio of Moles
From the calculations above, we find:
\[ n_A : n_B = \frac{1}{8} : 1 = 1 : 8 \]
This means that the number of moles of gas B is 8 times that of gas A.

### Step 4: Analyze Average Kinetic Energy
The average kinetic energy per molecule for a diatomic gas is given by:
\[ KE = \frac{5}{2} k T \]
where \( k \) is Boltzmann's constant and \( T \) is the temperature. Since both gases are at the same temperature, the average kinetic energy per molecule is the same for both gases.

### Step 5: Calculate Root Mean Square Velocity
The root mean square (rms) velocity \( v_{rms} \) is given by:
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
Since \( v_{rms} \) is inversely proportional to the square root of the molar mass:
\[ \frac{v_{rms, B}}{v_{rms, A}} = \sqrt{\frac{M_A}{M_B}} = \sqrt{\frac{16 M_B}{M_B}} = \sqrt{16} = 4 \]
Thus, the rms velocity of gas B is 4 times that of gas A.

### Step 6: Calculate Pressure Ratio
The pressure exerted by a gas in a closed vessel is proportional to the number of moles:
\[ P \propto n \]
From our earlier calculation:
\[ n_A : n_B = 1 : 8 \]
Thus, the pressure of gas B is 8 times that of gas A:
\[ P_B = 8 P_A \]

### Conclusion
Based on the analysis:
1. The average kinetic energy per molecule of A is equal to that of B.
2. The root mean square velocity of B is 4 times that of A.
3. The pressure exerted by gas B is 8 times that of gas A.

### Summary of Correct Statements
1. Average kinetic energy per molecule of A is equal to that of B.
2. Root mean square velocity of B is 4 times that of A.
3. Pressure of B is 8 times that of A.