A mixture of two diatomic gases exists in a closed cylinder. The volumes and velocities of sound in the two gases are `V_1, V_2, c_1 `and `c_2` respectively. Determine the velocity of sound in the gaseous mixture. (Pressure of gas remains constant),

Options

A. `c_1c_2 sqrt((V_1 + V_2)/(V_1c_2^2 + V_2c_1^2) )`
B. `c_2c_1sqrt((V_2^2 + V_1^2)/(V_1c_2^2 + V_2c_1^2))`
C. `c_2c_1 sqrt((V_2 - V_1)/(V_1c_2^2 - V_2c_1^2))`
D. `c_1c_2 sqrt((V_2 + V_1)/(V_1c_1^2 - V_2 c_2^2))`

Answer

To determine the velocity of sound in a mixture of two diatomic gases in a closed cylinder, we can use the following steps:

### Step-by-Step Solution:

1. **Understanding the Variables**:
- Let \( V_1 \) and \( V_2 \) be the volumes of the two gases.
- Let \( c_1 \) and \( c_2 \) be the velocities of sound in the two gases.

2. **Using the Formula for Velocity of Sound in a Mixture**:
- The velocity of sound in a mixture of two gases can be calculated using the formula:
\[ c = \sqrt{\frac{V_1 c_1^2 + V_2 c_2^2}{V_1 + V_2}} \]
- This formula is derived from the principles of sound propagation in different media and takes into account the contributions of both gases based on their volumes and sound velocities.

3. **Substituting the Values**:
- Substitute \( V_1 \), \( V_2 \), \( c_1 \), and \( c_2 \) into the formula:
\[ c = \sqrt{\frac{V_1 c_1^2 + V_2 c_2^2}{V_1 + V_2}} \]

4. **Final Expression**:
- This expression gives the velocity of sound in the gaseous mixture under the condition that the pressure remains constant.

### Final Answer:
The velocity of sound in the gaseous mixture is given by:
\[ c = \sqrt{\frac{V_1 c_1^2 + V_2 c_2^2}{V_1 + V_2}} \]