If at STP , velocity of sound in a gas `( gamma = 1.5)` is 600 m/s , the r.m.s. velocity of the gas molecules at STP will be

Options

A. 400 m/s
B. 600 m/s
C. `600 sqrt(2) m//s `
D. `300sqrt(2) m//s`

Answer

To find the root mean square (RMS) velocity of gas molecules at standard temperature and pressure (STP), we can use the relationship between the speed of sound in the gas and the RMS velocity of the gas molecules.

### Step-by-Step Solution:

1. **Understand the formula for the speed of sound in a gas:**
The speed of sound \( V_{\text{sound}} \) in a gas is given by the formula:
\[ V_{\text{sound}} = \sqrt{\frac{\gamma R T}{M}} \]
where:
- \( \gamma \) is the adiabatic index (given as 1.5),
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

2. **Square the speed of sound formula:**
To eliminate the square root, we square both sides:
\[ V_{\text{sound}}^2 = \frac{\gamma R T}{M} \]

3. **Rearrange to find \( \frac{RT}{M} \):**
We can rearrange the equation to express \( \frac{RT}{M} \):
\[ \frac{RT}{M} = \frac{V_{\text{sound}}^2}{\gamma} \]

4. **Use the formula for RMS velocity:**
The RMS velocity \( V_{\text{rms}} \) is given by:
\[ V_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]

5. **Substitute \( \frac{RT}{M} \) into the RMS formula:**
Now we can substitute \( \frac{RT}{M} \) from step 3 into the RMS formula:
\[ V_{\text{rms}} = \sqrt{3 \cdot \frac{V_{\text{sound}}^2}{\gamma}} \]

6. **Factor out \( V_{\text{sound}} \):**
This can be simplified to:
\[ V_{\text{rms}} = V_{\text{sound}} \cdot \sqrt{\frac{3}{\gamma}} \]

7. **Substitute the known values:**
We know \( V_{\text{sound}} = 600 \, \text{m/s} \) and \( \gamma = 1.5 \):
\[ V_{\text{rms}} = 600 \cdot \sqrt{\frac{3}{1.5}} \]

8. **Simplify the expression:**
Simplifying \( \sqrt{\frac{3}{1.5}} \):
\[ \sqrt{\frac{3}{1.5}} = \sqrt{2} \]
Therefore,
\[ V_{\text{rms}} = 600 \cdot \sqrt{2} \]

9. **Final answer:**
Thus, the RMS velocity of the gas molecules at STP is:
\[ V_{\text{rms}} = 600 \sqrt{2} \, \text{m/s} \]