Three closed vessels `A, B` and `C` are at the same temperature T and contain gases which obey the Maxwellian distribution of velocities. Vessel A contains only `O_(2), B` only `N_(2)` and `C` a mixture of equal quantities of `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` is `V_(2)`, the average speed of the `O_(2)` molecules in vessel `C` is (where `M` is the mass of an oxygen molecules)

Options

A. `(V_(1) + V_(2))/2`
B. `V_(1)`
C. `(V_(1) V_(2))^(1//2)`
D. `sqrt(3kT//M)`

Explanation

To solve the problem, we need to analyze the average speeds of the gas molecules in the three vessels based on the principles of the kinetic theory of gases and the Maxwellian distribution of velocities.

### Step-by-Step Solution:

1. **Understanding Average Speed Formula**:
The average speed \( V \) of gas molecules can be expressed using the formula:
\[ V = \sqrt{\frac{8kT}{\pi m}} \]
where \( k \) is the Boltzmann constant, \( T \) is the absolute temperature, and \( m \) is the mass of a gas molecule.

2. **Identifying Variables for Each Vessel**:
- For vessel A (containing only \( O_2 \)):
- Average speed of \( O_2 \) molecules: \( V_1 = \sqrt{\frac{8kT}{\pi m_{O_2}}} \)
- For vessel B (containing only \( N_2 \)):
- Average speed of \( N_2 \) molecules: \( V_2 = \sqrt{\frac{8kT}{\pi m_{N_2}}} \)
- For vessel C (containing a mixture of equal quantities of \( O_2 \) and \( N_2 \)):
- Average speed of \( O_2 \) molecules in vessel C: \( V_{C(O_2)} = \sqrt{\frac{8kT}{\pi m_{O_2}}} \)

3. **Temperature and Mass Considerations**:
Since all vessels are at the same temperature \( T \), the temperature factor \( T \) is constant across all vessels. The average speed depends on the mass of the gas molecules.

4. **Comparing Average Speeds**:
- The average speed of \( O_2 \) in vessel A is \( V_1 \).
- The average speed of \( O_2 \) in vessel C is also derived from the same formula as for vessel A, since the temperature is the same and the mass of \( O_2 \) is constant.

5. **Conclusion**:
Therefore, the average speed of the \( O_2 \) molecules in vessel C is equal to the average speed of the \( O_2 \) molecules in vessel A:
\[ V_{C(O_2)} = V_1 \]

### Final Answer:
The average speed of the \( O_2 \) molecules in vessel C is \( V_1 \).

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