A mixture of n mole of Nitrogen and 1 mole of hydrogen is kept is a container at room, temperature. The ratio of r.m.s. velocities of hydrogen and nitrogen molecules is:

Options

A. `sqrt(14)`
B. `sqrt(28)`
C. `n xx 14`
D. `n//26`

Explanation

To find the ratio of the root mean square (r.m.s.) velocities of hydrogen and nitrogen molecules, we can follow these steps:

### Step 1: Understand the formula for r.m.s. velocity
The formula for the root mean square velocity (v_rms) of a gas is given by:

\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

where:
- \( R \) = universal gas constant
- \( T \) = absolute temperature (in Kelvin)
- \( M \) = molar mass of the gas (in kg/mol)

### Step 2: Identify the molar masses
- For hydrogen (H₂), the molar mass \( M_H \) is approximately 2 g/mol, which is 0.002 kg/mol.
- For nitrogen (N₂), the molar mass \( M_N \) is approximately 28 g/mol, which is 0.028 kg/mol.

### Step 3: Write the r.m.s. velocity for both gases
- For hydrogen:

\[ v_{rms, H_2} = \sqrt{\frac{3RT}{M_H}} = \sqrt{\frac{3RT}{0.002}} \]

- For nitrogen:

\[ v_{rms, N_2} = \sqrt{\frac{3RT}{M_N}} = \sqrt{\frac{3RT}{0.028}} \]

### Step 4: Calculate the ratio of r.m.s. velocities
Now, we can find the ratio of the r.m.s. velocities of hydrogen to nitrogen:

\[ \frac{v_{rms, H_2}}{v_{rms, N_2}} = \frac{\sqrt{\frac{3RT}{0.002}}}{\sqrt{\frac{3RT}{0.028}}} \]

### Step 5: Simplify the ratio
The \( 3RT \) terms cancel out:

\[ \frac{v_{rms, H_2}}{v_{rms, N_2}} = \sqrt{\frac{0.028}{0.002}} = \sqrt{14} \]

### Final Answer
Thus, the ratio of the r.m.s. velocities of hydrogen to nitrogen is:

\[ \frac{v_{rms, H_2}}{v_{rms, N_2}} = \sqrt{14} \]