The root mean square speed of the molecules of an enclosed gas is 'v'. What will be the root mean square speed if the pressure is doubled, the temperature remaining the same?

Options

A. `3v`
B. `4v`
C. `v`
D. `5v`

Answer

To solve the problem, we need to analyze how the root mean square (RMS) speed of gas molecules changes when the pressure is doubled while keeping the temperature constant.

### Step-by-Step Solution:

1. **Understand the Formula for RMS Speed**:
The root mean square speed (v_rms) of gas molecules is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.

2. **Identify the Conditions**:
In this problem, we are told that the pressure of the gas is doubled while the temperature remains constant. Thus, \( R \), \( T \), and \( M \) do not change.

3. **Relate Pressure and Density**:
According to the ideal gas law:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, and \( T \) is temperature. If the temperature remains constant and the pressure is doubled, the density of the gas also changes. Since \( P \) is proportional to density (\( \rho \)), we can say:
\[ P \propto \rho \]
Therefore, if the pressure is doubled, the density will also double.

4. **Substituting into the RMS Speed Formula**:
We can express the RMS speed in terms of pressure and density:
\[ v_{rms} = \sqrt{\frac{3P}{\rho}} \]
If the pressure is doubled (let's denote the new pressure as \( P' = 2P \)) and the density is also doubled (let's denote the new density as \( \rho' = 2\rho \)), we can substitute these into the formula:
\[ v'_{rms} = \sqrt{\frac{3P'}{\rho'}} = \sqrt{\frac{3(2P)}{2\rho}} = \sqrt{\frac{3P}{\rho}} = v_{rms} \]

5. **Conclusion**:
Since the new root mean square speed \( v'_{rms} \) is equal to the original root mean square speed \( v_{rms} \), we conclude that:
\[ v'_{rms} = v \]

### Final Answer:
The root mean square speed remains the same, \( v'_{rms} = v \).