The root mean square velocity of a gas is doubled when the temperature is :

Options

A. increased four times
B. increased two times
C. reduced to half
D. reduced to one fourth

Explanation

To solve the problem of determining how the root mean square (RMS) velocity of a gas changes with temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for RMS Velocity**:
The root mean square velocity (v_rms) of 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 in Kelvin,
- \( M \) is the molar mass of the gas.

2. **Set Up the Relationship**:
We want to find out how the temperature \( T \) must change in order to double the RMS velocity \( v_{rms} \). Therefore, we set up the equation:
\[ 2v_{rms} = \sqrt{\frac{3R(2T)}{M}} \]

3. **Square Both Sides**:
To eliminate the square root, we square both sides of the equation:
\[ (2v_{rms})^2 = \frac{3R(2T)}{M} \]
This simplifies to:
\[ 4v_{rms}^2 = \frac{3R(2T)}{M} \]

4. **Substitute the Original RMS Velocity**:
Substitute \( v_{rms}^2 \) from the original formula:
\[ 4\left(\frac{3RT}{M}\right) = \frac{3R(2T)}{M} \]

5. **Cancel Common Terms**:
Since \( \frac{3R}{M} \) is common on both sides, we can cancel it out:
\[ 4T = 2T \]

6. **Solve for Temperature**:
Rearranging gives us:
\[ 4T = 2T \implies 2T = 0 \implies T = 0 \]
This indicates that we need to find a new temperature \( T' \) such that:
\[ T' = 4T \]
Therefore, to double the RMS velocity, the temperature must be increased to four times its original value.

### Conclusion:
The temperature must be increased by a factor of 4 in order to double the root mean square velocity of the gas.