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 determine the new root mean square (RMS) speed of the gas molecules after additional mass is pumped into the closed vessel. Let's break it down step by step.

### Step 1: Understand the Initial Conditions
Initially, we have a mass \( m \) of an ideal gas in a closed vessel of fixed volume. The root mean square speed of the gas molecules is given as \( v \).

### Step 2: Recall the Formula for RMS Speed
The formula for the root mean square speed \( v_{\text{rms}} \) of an ideal gas is given by:
\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

### Step 3: Analyze the Changes After Adding Mass
When an additional mass \( m \) of the same gas is pumped into the vessel, the total mass of the gas in the vessel becomes:
\[ \text{Total mass} = m + m = 2m \]

### Step 4: Consider the Pressure Change
According to the problem, the pressure in the vessel rises to \( 2P \) while the temperature remains constant. This indicates that the number of gas molecules has increased, but the temperature is unchanged.

### Step 5: Relate Pressure, Volume, and Temperature
Using the ideal gas law, we know that:
\[ PV = nRT \]
Where \( n \) is the number of moles. The pressure has doubled, but since the volume is constant and the temperature remains the same, the increase in pressure indicates that the number of moles (and hence the mass) has also increased.

### Step 6: Analyze the RMS Speed After Adding Mass
Since the temperature \( T \) remains constant and the molar mass \( M \) of the gas does not change, we can analyze the new RMS speed:
- The new total mass is \( 2m \).
- The RMS speed formula remains the same, but we need to consider the effective mass in the equation.

However, since the temperature is constant, the root mean square speed is dependent only on the temperature and the molar mass of the gas. Thus, we can conclude:
\[ v_{\text{rms new}} = \sqrt{\frac{3RT}{M}} = v_{\text{rms initial}} \]
This means that the new root mean square speed remains the same as the initial speed \( v \).

### Final Answer
The root mean square speed of the molecules now is:
\[ v \]