A closed vessel of fixed volume contains a mass m of an ideal gas, the root mean square speed being v. Additional mass m of the same gas is pumped into the vessel and the pressure rises to 2P, the temperature remaining the same as before. The root mean square speed of the molecules now is :

Options

A. `V/sqrt2`
B. `v sqrt2`
C. `2 v`
D. `v`

Explanation

To solve the problem, we need to analyze the situation step by step, focusing on the relevant formulas and concepts related to the root mean square (RMS) speed of gas molecules.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- We start with a closed vessel containing a mass \( m \) of an ideal gas.
- The initial root mean square speed of the gas molecules is given as \( v \).
- The formula for the root mean square speed (\( v_{rms} \)) is:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the gas constant, \( T \) is the temperature, and \( M \) is the molar mass of the gas.

2. **Adding More Gas**:
- An additional mass \( m \) of the same gas is pumped into the vessel.
- Therefore, the total mass of the gas in the vessel becomes \( 2m \).

3. **Pressure Change**:
- The pressure in the vessel rises to \( 2P \) after adding the additional mass of gas.
- It is important to note that the temperature remains constant throughout this process.

4. **Analyzing the RMS Speed**:
- The molar mass \( M \) of the gas does not change, as it is a characteristic of the specific gas being used.
- Since the temperature \( T \) is constant and the molar mass \( M \) remains the same, we can analyze the RMS speed again using the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
- The key point is that the RMS speed depends only on the temperature and the molar mass of the gas. Since both of these quantities remain unchanged, the RMS speed will also remain unchanged.

5. **Conclusion**:
- Therefore, the new root mean square speed of the gas molecules after adding the additional mass is still \( v \).

### Final Answer:
The root mean square speed of the molecules now is \( v \).

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