The root mean square velocity of gas molecules is `10km//s`. The gas is heated till its pressure becomes four times. The velocity of gas molecules will be

Options

A. `10Km//s`
B. `20Km//s`
C. `40Km//s`
D. `80Km//s`

Explanation

To solve the problem, we need to determine the new root mean square (RMS) velocity of gas molecules when the pressure is increased to four times its original value. We can use the kinetic theory of gases to derive the relationship between pressure and RMS velocity.

### Step-by-Step Solution:

1. **Understand the Relationship**: According to the kinetic theory of gases, the root mean square velocity (VRMS) of gas molecules is related to the pressure (P) and density (ρ) of the gas by the formula:
\[ V_{RMS} = \sqrt{\frac{3P}{\rho}} \]

2. **Initial Conditions**: We are given that the initial root mean square velocity (VRMS1) is 10 km/s. Therefore:
\[ V_{RMS1} = 10 \text{ km/s} \]

3. **Change in Pressure**: The problem states that the pressure is increased to four times its original value:
\[ P_2 = 4P_1 \]

4. **New RMS Velocity**: We need to find the new root mean square velocity (VRMS2) when the pressure is increased. Using the relationship derived from the kinetic theory:
\[ V_{RMS2} = \sqrt{\frac{3P_2}{\rho}} = \sqrt{\frac{3(4P_1)}{\rho}} = \sqrt{4 \cdot \frac{3P_1}{\rho}} = \sqrt{4} \cdot \sqrt{\frac{3P_1}{\rho}} \]

5. **Substituting the Initial Velocity**: Since we know that:
\[ \sqrt{\frac{3P_1}{\rho}} = V_{RMS1} = 10 \text{ km/s} \]
We can substitute this into our equation for VRMS2:
\[ V_{RMS2} = \sqrt{4} \cdot 10 \text{ km/s} = 2 \cdot 10 \text{ km/s} = 20 \text{ km/s} \]

6. **Final Answer**: The new root mean square velocity of the gas molecules after the pressure is increased to four times is:
\[ V_{RMS2} = 20 \text{ km/s} \]

### Summary:
The root mean square velocity of the gas molecules after the pressure is increased to four times is **20 km/s**.