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 (r.m.s.) velocity of gas molecules at standard temperature and pressure (STP), we can use the relationship between the velocity of sound in a gas and the r.m.s. velocity of its molecules.

### Step-by-Step Solution:

1. **Understand the relationship**: The velocity of sound \( v \) in a gas is given by the formula:
\[ v = \sqrt{\frac{\gamma RT}{M}} \]
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. **RMS velocity formula**: The r.m.s. velocity \( v_{\text{rms}} \) of gas molecules is given by:
\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]

3. **Relate \( v_{\text{rms}} \) to the velocity of sound**: We can express \( v_{\text{rms}} \) in terms of the velocity of sound:
\[ v_{\text{rms}} = \sqrt{\frac{3}{\gamma}} \cdot v \]
This is derived from the fact that:
\[ \frac{RT}{M} = \frac{v^2}{\gamma} \]
Therefore, substituting this into the r.m.s. velocity formula gives us the relationship.

4. **Substitute the values**: Given that the velocity of sound \( v = 600 \, \text{m/s} \) and \( \gamma = 1.5 \):
\[ v_{\text{rms}} = \sqrt{\frac{3}{1.5}} \cdot 600 \]

5. **Calculate \( \sqrt{\frac{3}{1.5}} \)**:
\[ \sqrt{\frac{3}{1.5}} = \sqrt{2} \approx 1.414 \]

6. **Final calculation**:
\[ v_{\text{rms}} = 600 \cdot \sqrt{2} \approx 600 \cdot 1.414 \approx 848.5 \, \text{m/s} \]

### Final Answer:
The r.m.s. velocity of the gas molecules at STP is approximately \( 848.5 \, \text{m/s} \).