If temperature of the gas is increased to three times , then its root mean square velocity becomes :

Options

A. 3 times
B. 9 times
C. ` (1)/(2)` times
D. ` sqrt3 ` times

Explanation

To solve the problem, we need to understand the relationship between the root mean square (RMS) velocity of a gas and its temperature. The RMS velocity is given by the formula:

\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]

where:
- \( V_{rms} \) = root mean square velocity,
- \( R \) = universal gas constant,
- \( T \) = absolute temperature of the gas,
- \( M \) = molar mass of the gas.

### Step-by-Step Solution:

1. **Identify the Initial RMS Velocity**:
The initial RMS velocity of the gas at temperature \( T \) is given by:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]

2. **Increase the Temperature**:
If the temperature is increased to three times its original value, the new temperature \( T' \) will be:
\[ T' = 3T \]

3. **Calculate the New RMS Velocity**:
Substitute the new temperature \( T' \) into the RMS velocity formula:
\[ V_{rms}' = \sqrt{\frac{3R(3T)}{M}} = \sqrt{\frac{9RT}{M}} \]

4. **Simplify the Expression**:
We can simplify the expression for the new RMS velocity:
\[ V_{rms}' = \sqrt{9} \cdot \sqrt{\frac{3RT}{M}} = 3 \cdot V_{rms} \]

5. **Conclusion**:
Therefore, when the temperature of the gas is increased to three times its original value, the new root mean square velocity becomes:
\[ V_{rms}' = 3 \cdot V_{rms} \]

### Answer:
The root mean square velocity becomes **3 times** the original velocity.