Speed of sound in a gas is `v` and `rms` velocity of the gas molecules is `c`. The ratio of `v` to `c` is

Options

A. `(3)/(gamma)`
B. `(gamma)/(3)`
C. `sqrt((3)/(gamma))`
D. `sqrt((gamma)/(3))`

Explanation

To solve the problem of finding the ratio of the speed of sound in a gas (v) to the root mean square (rms) velocity of the gas molecules (c), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formulas**:
- The speed of sound in a gas is given by the formula:
\[ v = \sqrt{\frac{\gamma RT}{M}} \]
where:
- \( \gamma \) is the ratio of specific heats (\( C_p/C_v \)),
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas.

- The rms speed of the gas molecules is given by the formula:
\[ c = \sqrt{\frac{3RT}{M}} \]

2. **Set up the ratio**:
- We need to find the ratio \( \frac{v}{c} \):
\[ \frac{v}{c} = \frac{\sqrt{\frac{\gamma RT}{M}}}{\sqrt{\frac{3RT}{M}}} \]

3. **Simplify the ratio**:
- When dividing the two square roots, we can simplify:
\[ \frac{v}{c} = \sqrt{\frac{\frac{\gamma RT}{M}}{\frac{3RT}{M}}} \]
- The \( RT \) and \( M \) terms cancel out:
\[ \frac{v}{c} = \sqrt{\frac{\gamma}{3}} \]

4. **Final result**:
- Thus, the ratio of the speed of sound in a gas to the rms velocity of the gas molecules is:
\[ \frac{v}{c} = \sqrt{\frac{\gamma}{3}} \]

### Conclusion:
The final answer is \( \sqrt{\frac{\gamma}{3}} \).