If `v_(rms)` is the rms speed of molecules in a gas and v is the speed of sound waves in the gas, then the ratio `(v_(rms))/v` is

Options

A. `sqrt(3/gamma)`
B. `sqrt(gamma/3)`
C. `sqrt(3gamma)`
D. `sqrt3/gamma`

Explanation

To find the ratio of the root mean square (rms) speed of molecules in a gas, \( v_{rms} \), to the speed of sound waves in the gas, \( v \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formulas**:
- The formula for the rms speed of molecules in a gas is given by:
\[ v_{rms} = \sqrt{\frac{3P}{\rho}} \]
where \( P \) is the pressure of the gas and \( \rho \) is the density of the gas.

- The formula for the speed of sound in a gas is given by:
\[ v = \sqrt{\frac{\gamma P}{\rho}} \]
where \( \gamma \) is the specific heat ratio (also known as the adiabatic index).

2. **Set up the ratio**:
- We need to find the ratio \( \frac{v_{rms}}{v} \):
\[ \frac{v_{rms}}{v} = \frac{\sqrt{\frac{3P}{\rho}}}{\sqrt{\frac{\gamma P}{\rho}}} \]

3. **Simplify the ratio**:
- We can simplify the ratio by dividing the two square roots:
\[ \frac{v_{rms}}{v} = \sqrt{\frac{3P/\rho}{\gamma P/\rho}} = \sqrt{\frac{3}{\gamma}} \]
- Here, the \( P \) and \( \rho \) cancel out since they are common in both the numerator and denominator.

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

### Conclusion:
The ratio \( \frac{v_{rms}}{v} \) is equal to \( \sqrt{\frac{3}{\gamma}} \).