A bulb contains 1 mol of hydrogen mixed with one mole of oxygen at temperature T. The ratio of rms values of velocity of hydrogen molecules to that of oxygen molecules is

Options

A. `1 : 16 `
B. `1 :4`
C. `4 :1`
D. `16:1`

Answer

To find the ratio of the root mean square (rms) velocities of hydrogen molecules to that of oxygen molecules, we can use the formula for the rms 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 in kg/mol.

### Step 1: Write the rms velocity formulas for hydrogen and oxygen.

For hydrogen (\(H_2\)):
\[ V_{\text{rms, H2}} = \sqrt{\frac{3RT}{M_{H2}}} \]

For oxygen (\(O_2\)):
\[ V_{\text{rms, O2}} = \sqrt{\frac{3RT}{M_{O2}}} \]

### Step 2: Find the molar masses of hydrogen and oxygen.

- The molar mass of hydrogen (\(H_2\)) is approximately \(2 \, \text{g/mol} = 2 \times 10^{-3} \, \text{kg/mol}\).
- The molar mass of oxygen (\(O_2\)) is approximately \(32 \, \text{g/mol} = 32 \times 10^{-3} \, \text{kg/mol}\).

### Step 3: Write the ratio of the rms velocities.

Now, we can find the ratio of the rms velocities of hydrogen to oxygen:

\[ \frac{V_{\text{rms, H2}}}{V_{\text{rms, O2}}} = \frac{\sqrt{\frac{3RT}{M_{H2}}}}{\sqrt{\frac{3RT}{M_{O2}}}} \]

### Step 4: Simplify the ratio.

The \(3RT\) terms cancel out:

\[ \frac{V_{\text{rms, H2}}}{V_{\text{rms, O2}}} = \sqrt{\frac{M_{O2}}{M_{H2}}} \]

### Step 5: Substitute the molar masses into the equation.

Substituting the values of the molar masses:

\[ \frac{V_{\text{rms, H2}}}{V_{\text{rms, O2}}} = \sqrt{\frac{32 \times 10^{-3}}{2 \times 10^{-3}}} \]

### Step 6: Calculate the ratio.

This simplifies to:

\[ \frac{V_{\text{rms, H2}}}{V_{\text{rms, O2}}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4 \]

### Conclusion

Thus, the ratio of the rms velocities of hydrogen molecules to that of oxygen molecules is \(4:1\).