The rms velocity of hydrogen gas molecules at `NTP` is `V rms^(-1)`. The gas is heated at constant volume till the pressure becomes four times. The final rms velocity is

Options

A. `V//2`
B. `V`
C. `2 V`
D. `4 V`

Explanation

To solve the problem, we need to understand the relationship between the root mean square (RMS) velocity of gas molecules, temperature, and pressure. Here's a step-by-step solution:

### Step 1: Understand the RMS Velocity Formula
The RMS velocity (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,
- \( M \) is the molar mass of the gas.

### Step 2: Initial Conditions at NTP
At Normal Temperature and Pressure (NTP), we are given that the RMS velocity of hydrogen gas is \( V_{rms} = V \).

### Step 3: Heating the Gas at Constant Volume
The problem states that the gas is heated at constant volume until the pressure becomes four times the original pressure. According to the ideal gas law:
\[ PV = nRT \]
If the volume (V) is constant and the pressure (P) increases to four times its original value, the temperature must also increase to maintain the equation.

### Step 4: Relating Pressure and Temperature
If the initial pressure is \( P \) and the initial temperature is \( T \), after heating, the new pressure \( P' \) is:
\[ P' = 4P \]
Using the ideal gas law, we can relate the initial and final states:
\[ \frac{P'}{P} = \frac{T'}{T} \]
Thus:
\[ \frac{4P}{P} = \frac{T'}{T} \]
This simplifies to:
\[ T' = 4T \]

### Step 5: Finding the Final RMS Velocity
Now that we have the final temperature \( T' = 4T \), we can substitute this back into the RMS velocity formula:
\[ V'_{rms} = \sqrt{\frac{3R(4T)}{M}} \]
This can be simplified to:
\[ V'_{rms} = \sqrt{4} \cdot \sqrt{\frac{3RT}{M}} \]
\[ V'_{rms} = 2 \cdot V_{rms} \]
Since \( V_{rms} = V \), we have:
\[ V'_{rms} = 2V \]

### Conclusion
The final RMS velocity after heating the gas at constant volume until the pressure becomes four times is:
\[ V'_{rms} = 2V \]