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 use the formula for r.m.s. velocity:

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

where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

### Step 1: Write the formula for r.m.s. velocity for both gases

For hydrogen (H₂):
\[ v_{\text{rms, H₂}} = \sqrt{\frac{3RT}{M_{\text{H₂}}}} \]

For nitrogen (N₂):
\[ v_{\text{rms, N₂}} = \sqrt{\frac{3RT}{M_{\text{N₂}}}} \]

### Step 2: Identify the molar masses

The molar mass of hydrogen (H₂) is approximately 2 g/mol, and the molar mass of nitrogen (N₂) is approximately 28 g/mol.

### Step 3: Substitute the molar masses into the r.m.s. velocity formulas

Substituting the molar masses into the formulas, we have:

For hydrogen:
\[ v_{\text{rms, H₂}} = \sqrt{\frac{3RT}{2}} \]

For nitrogen:
\[ v_{\text{rms, N₂}} = \sqrt{\frac{3RT}{28}} \]

### Step 4: Find the ratio of r.m.s. velocities

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

\[ \frac{v_{\text{rms, H₂}}}{v_{\text{rms, N₂}}} = \frac{\sqrt{\frac{3RT}{2}}}{\sqrt{\frac{3RT}{28}}} \]

### Step 5: Simplify the ratio

The \( 3RT \) terms cancel out:

\[ \frac{v_{\text{rms, H₂}}}{v_{\text{rms, N₂}}} = \sqrt{\frac{28}{2}} = \sqrt{14} \]

### Final Answer

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

\[ \frac{v_{\text{rms, H₂}}}{v_{\text{rms, N₂}}} = \sqrt{14} \]