In a mixture of gases, the average number of degrees of freedom per molecule is 6. the rms speed of the molecules of the gas is C. the velocity of sound in the gas is

Options

A. C
B. `(2C)/(3)`
C. `(3C)/(4)`
D. `(2C)/(sqrt(3))`

Explanation

To solve the problem, we need to find the velocity of sound in a gas mixture where the average number of degrees of freedom per molecule is 6, and the root mean square (rms) speed of the molecules is given as \( C \).

### Step-by-Step Solution:

1. **Identify Degrees of Freedom**:
The average number of degrees of freedom \( f \) for the gas molecules is given as 6.

2. **Calculate Gamma (\( \gamma \))**:
The value of \( \gamma \) (the ratio of specific heats) can be calculated using the formula:
\[ \gamma = 1 + \frac{2}{f} \]
Substituting \( f = 6 \):
\[ \gamma = 1 + \frac{2}{6} = 1 + \frac{1}{3} = \frac{4}{3} \]

3. **Velocity of Sound Formula**:
The velocity of sound \( V \) in a gas is given by the formula:
\[ V = \sqrt{\frac{\gamma RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature, and \( M \) is the molar mass of the gas.

4. **Relate RMS Speed to Temperature**:
The root mean square speed \( V_{rms} \) is given by:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
We know \( V_{rms} = C \), hence:
\[ C = \sqrt{\frac{3RT}{M}} \]

5. **Express Velocity of Sound in Terms of RMS Speed**:
To find the relationship between the velocity of sound and the rms speed, we can express \( V \) in terms of \( C \):
\[ \frac{V}{C} = \sqrt{\frac{\gamma RT/M}{3RT/M}} = \sqrt{\frac{\gamma}{3}} \]
Substituting \( \gamma = \frac{4}{3} \):
\[ \frac{V}{C} = \sqrt{\frac{4/3}{3}} = \sqrt{\frac{4}{9}} = \frac{2}{3} \]
Therefore:
\[ V = \frac{2}{3} C \]

6. **Conclusion**:
The velocity of sound in the gas is:
\[ V = \frac{2}{3} C \]

### Final Answer:
The velocity of sound in the gas is \( \frac{2}{3} C \).