The ratio of rms speed of an ideal gas molecules at pressure p to that at pressure 2p is

Options

A. `1:2`
B. `2:1`
C. `1: sqrt2`
D. `sqrt2 :1`

Explanation

To find the ratio of the root mean square (rms) speed of an ideal gas molecules at pressure \( p \) to that at pressure \( 2p \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for RMS Speed**:
The rms speed (\( v_{rms} \)) of an ideal gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M_w}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M_w \) is the molecular weight of the gas.

2. **Relate RMS Speed to Pressure**:
The rms speed can also be expressed in terms of pressure (\( P \)) and density (\( \rho \)):
\[ v_{rms} = \sqrt{\frac{3P}{\rho}} \]
This shows that the rms speed is related to the pressure of the gas.

3. **Establish the Relationship Between RMS Speeds at Different Pressures**:
We need to find the ratio of rms speeds at pressure \( p \) and pressure \( 2p \):
\[ \frac{v_{rms}(p)}{v_{rms}(2p)} \]

4. **Substitute the Values**:
Using the formula for rms speed:
\[ v_{rms}(p) = \sqrt{\frac{3p}{\rho_1}} \quad \text{and} \quad v_{rms}(2p) = \sqrt{\frac{3(2p)}{\rho_2}} \]
Assuming the density changes proportionally with pressure (for an ideal gas at constant temperature), we can say \( \rho_2 = 2\rho_1 \).

5. **Calculate the Ratio**:
Now substituting the expressions:
\[ \frac{v_{rms}(p)}{v_{rms}(2p)} = \frac{\sqrt{\frac{3p}{\rho_1}}}{\sqrt{\frac{3(2p)}{2\rho_1}}} \]
Simplifying this gives:
\[ = \frac{\sqrt{3p}}{\sqrt{3(2p)/2}} = \frac{\sqrt{3p}}{\sqrt{3p}} \cdot \sqrt{\frac{2}{1}} = \frac{1}{\sqrt{2}} \]

6. **Final Answer**:
Thus, the ratio of the rms speed of an ideal gas molecules at pressure \( p \) to that at pressure \( 2p \) is:
\[ \frac{v_{rms}(p)}{v_{rms}(2p)} = \frac{1}{\sqrt{2}} \]

### Conclusion:
The correct answer is \( \frac{1}{\sqrt{2}} \).