Three closed vessels A, B and C are at the same temperature T and contain gasses which obey the Maxwellian distribution of velocities. Vessel A contain only `O_2 and N_2`. If the average speed of the `O_2` molecules in vessel A is `v_1` that of the `N_2` molecules in vessel B us `v_2,` the average speed of the `O_2` molecules in vessel C is

Options

A. `((v_1 + v_2))/(2)`
B. `V_1`
C. `sqrt(v_1 v_2)`
D. None of these

Explanation

To solve the problem, we need to understand how the average speed of gas molecules is determined in different vessels. The average speed of gas molecules can be derived from the kinetic theory of gases, which states that the average speed of gas molecules is given by the formula:

\[ v_{avg} = \sqrt{\frac{8RT}{\pi m}} \]

where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( m \) is the mass of a gas molecule.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have three vessels (A, B, and C) at the same temperature \( T \).
- Vessel A contains \( O_2 \) and \( N_2 \).
- Vessel B contains \( N_2 \) with an average speed \( v_2 \).
- We need to find the average speed of \( O_2 \) in vessel C.

2. **Average Speed of Gases**:
- The average speed of gas molecules depends on their mass and the temperature of the gas.
- For \( O_2 \) in vessel A, the average speed is given as \( v_1 \).
- The average speed of \( N_2 \) in vessel B is given as \( v_2 \).

3. **Independence of Average Speed**:
- The average speed of a specific gas (like \( O_2 \)) is independent of the presence of other gases in the same vessel.
- Therefore, the average speed of \( O_2 \) in vessel C will be the same as in vessel A, provided that the temperature remains constant.

4. **Conclusion**:
- Since the average speed of \( O_2 \) in vessel A is \( v_1 \), the average speed of \( O_2 \) in vessel C will also be \( v_1 \).

Thus, the average speed of the \( O_2 \) molecules in vessel C is:

\[ \text{Average speed of } O_2 \text{ in vessel C} = v_1 \]

### Answer:
The average speed of \( O_2 \) molecules in vessel C is \( v_1 \).

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