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 in the gas (\(v\)), we can follow these steps:

### Step 1: Write the formula for \(v_{rms}\)
The rms speed of molecules in a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \(R\) is the universal gas constant,
- \(T\) is the absolute temperature,
- \(M\) is the molar mass of the gas.

### Step 2: Write the formula for the speed of sound \(v\)
The speed of sound in a gas is given by the formula:
\[ v = \sqrt{\frac{\gamma RT}{M}} \]
where:
- \(\gamma\) is the adiabatic index (ratio of specific heats).

### Step 3: Set up the ratio \(\frac{v_{rms}}{v}\)
Now, we can set up the ratio of \(v_{rms}\) to \(v\):
\[ \frac{v_{rms}}{v} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{\gamma RT}{M}}} \]

### Step 4: Simplify the ratio
We can simplify the ratio:
\[ \frac{v_{rms}}{v} = \sqrt{\frac{3RT/M}{\gamma RT/M}} = \sqrt{\frac{3}{\gamma}} \]

### Step 5: Final expression
Thus, the final expression for the ratio \(\frac{v_{rms}}{v}\) is:
\[ \frac{v_{rms}}{v} = \sqrt{\frac{3}{\gamma}} \]

### Conclusion
The ratio of the rms speed of molecules in a gas to the speed of sound in the gas is \(\sqrt{\frac{3}{\gamma}}\).