RMS speed of a particle is `V_(rms)` at pressure P. if pressure is increased to two times, then at constant temperature rms speed becomes

Options

A. `2V_(rms)`
B. `3V_(rms)`
C. `V_(rms)`
D. zero

Answer

To solve the problem of how the RMS speed of a particle changes when the pressure is doubled at constant temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding RMS Speed**: The root mean square (RMS) speed 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. **Initial Conditions**: Let's denote the initial pressure as \( P \) and the initial RMS speed as \( V_{rms} \).

3. **Effect of Pressure on Density**: According to the ideal gas law, at constant temperature, if the pressure of a gas is increased, the density of the gas also increases. The relationship can be expressed as:
\[ P \propto \rho \quad \text{(at constant T)} \]
where \( \rho \) is the density of the gas.

4. **Doubling the Pressure**: If the pressure is increased to \( 2P \), then the density will also increase to \( 2\rho \) (since pressure is directly proportional to density at constant temperature).

5. **RMS Speed with New Conditions**: The new RMS speed can be expressed using the modified density:
\[ V_{rms}' = \sqrt{\frac{3P'}{\rho'}} \]
where \( P' = 2P \) and \( \rho' = 2\rho \).

6. **Substituting New Values**: Substituting the new pressure and density into the RMS speed formula:
\[ V_{rms}' = \sqrt{\frac{3(2P)}{2\rho}} = \sqrt{\frac{3P}{\rho}} = V_{rms} \]

7. **Conclusion**: Therefore, the new RMS speed remains the same as the initial RMS speed:
\[ V_{rms}' = V_{rms} \]

### Final Answer:
The RMS speed of the particle remains unchanged, i.e., \( V_{rms}' = V_{rms} \).