The rms velocity of hydrogen at S.T.P is `(mu) ms^(-1)`. If the gas is heated at constant pressure till its volume is three fold, what will be its final temperature and rms velocity ?

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between volume and temperature at constant pressure.
At constant pressure, the volume of a gas is directly proportional to its temperature. This can be expressed as:
\[ V \propto T \]
This means that if the volume changes, the temperature will also change proportionally.

### Step 2: Set up the relationship for the change in volume.
Given that the volume is increased to three times its original volume, we can write:
\[ V' = 3V \]
Where \( V' \) is the final volume and \( V \) is the initial volume.

### Step 3: Relate the initial and final temperatures.
Using the proportionality established in Step 1:
\[ \frac{V'}{V} = \frac{T'}{T} \]
Substituting \( V' = 3V \):
\[ \frac{3V}{V} = \frac{T'}{T} \]
This simplifies to:
\[ 3 = \frac{T'}{T} \]
Thus, we can express the final temperature \( T' \) in terms of the initial temperature \( T \):
\[ T' = 3T \]

### Step 4: Determine the initial temperature.
At standard temperature and pressure (S.T.P), the initial temperature \( T \) is:
\[ T = 273 \, \text{K} \]

### Step 5: Calculate the final temperature.
Substituting the value of \( T \) into the equation for \( T' \):
\[ T' = 3 \times 273 \, \text{K} = 819 \, \text{K} \]

### Step 6: Find the root mean square (rms) velocity.
The formula for the root mean square velocity \( V_{rms} \) is given by:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
Where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas.

### Step 7: Relate the initial and final rms velocities.
Since the molecular mass \( M \) remains constant, we can express the relationship between the initial rms velocity \( V_{rms} \) and the final rms velocity \( V'_{rms} \):
\[ \frac{V'_{rms}}{V_{rms}} = \sqrt{\frac{T'}{T}} \]
Substituting \( T' = 3T \):
\[ \frac{V'_{rms}}{V_{rms}} = \sqrt{\frac{3T}{T}} = \sqrt{3} \]
Thus:
\[ V'_{rms} = V_{rms} \sqrt{3} \]

### Step 8: Calculate the final rms velocity.
If the initial rms velocity is \( U \):
\[ V'_{rms} = U \sqrt{3} \]

### Final Answers:
- Final temperature \( T' = 819 \, \text{K} \)
- Final rms velocity \( V'_{rms} = U \sqrt{3} \, \text{m/s} \)