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 at pressure \( p \) to that at pressure \( 2p \), we can use the formula for rms speed, which is given by:

\[ c_{\text{rms}} = \sqrt{\frac{3p}{\rho}} \]

where \( p \) is the pressure and \( \rho \) is the density of the gas.

### Step 1: Write the rms speed for pressure \( p \)

For pressure \( p \):

\[ c_{\text{rms},1} = \sqrt{\frac{3p}{\rho_1}} \]

### Step 2: Write the rms speed for pressure \( 2p \)

For pressure \( 2p \):

\[ c_{\text{rms},2} = \sqrt{\frac{3(2p)}{\rho_2}} = \sqrt{\frac{6p}{\rho_2}} \]

### Step 3: Relate the densities at different pressures

Using the ideal gas law, we know that \( p = \rho RT \) (where \( R \) is the gas constant and \( T \) is the temperature). Therefore, we can express the densities in terms of pressure:

\[ \rho_1 = \frac{p}{RT} \quad \text{and} \quad \rho_2 = \frac{2p}{RT} \]

### Step 4: Substitute the densities into the rms speed formulas

Now substituting \( \rho_1 \) and \( \rho_2 \) into the rms speed equations:

\[ c_{\text{rms},1} = \sqrt{\frac{3p}{\frac{p}{RT}}} = \sqrt{3RT} \]

\[ c_{\text{rms},2} = \sqrt{\frac{6p}{\frac{2p}{RT}}} = \sqrt{3RT} \]

### Step 5: Calculate the ratio of rms speeds

Now we can find the ratio of the rms speeds:

\[ \frac{c_{\text{rms},1}}{c_{\text{rms},2}} = \frac{\sqrt{3RT}}{\sqrt{3RT}} = 1 \]

### Step 6: Conclusion

Thus, the ratio of the rms speed of an ideal gas molecules at pressure \( p \) to that at pressure \( 2p \) is:

\[ \frac{c_{\text{rms},1}}{c_{\text{rms},2}} = 1 \]