At a given temperature the root mean square velocities of oxygen and hydrogen molecules are in the ratio

Options

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

Explanation

To find the ratio of the root mean square velocities of oxygen and hydrogen molecules at a given temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Root Mean Square Velocity (Vrms)**:
The root mean square velocity (Vrms) of a gas is given by the formula:
\[ V_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature, and \( M \) is the molar mass of the gas.

2. **Identify the Molar Masses**:
- The molar mass of oxygen (O₂) is \( 32 \, \text{g/mol} \) (since each oxygen atom has an atomic mass of approximately 16 g/mol and there are 2 atoms in an O₂ molecule).
- The molar mass of hydrogen (H₂) is \( 2 \, \text{g/mol} \) (since each hydrogen atom has an atomic mass of approximately 1 g/mol and there are 2 atoms in an H₂ molecule).

3. **Calculate Vrms for Oxygen**:
Using the formula for Vrms:
\[ V_{\text{rms, O}} = \sqrt{\frac{3RT}{M_{\text{O}}}} = \sqrt{\frac{3RT}{32}} \]

4. **Calculate Vrms for Hydrogen**:
Similarly, for hydrogen:
\[ V_{\text{rms, H}} = \sqrt{\frac{3RT}{M_{\text{H}}}} = \sqrt{\frac{3RT}{2}} \]

5. **Find the Ratio of Vrms for Oxygen to Hydrogen**:
Now, we can find the ratio:
\[ \frac{V_{\text{rms, O}}}{V_{\text{rms, H}}} = \frac{\sqrt{\frac{3RT}{32}}}{\sqrt{\frac{3RT}{2}}} \]

6. **Simplify the Ratio**:
This simplifies to:
\[ \frac{V_{\text{rms, O}}}{V_{\text{rms, H}}} = \sqrt{\frac{3RT}{32}} \cdot \sqrt{\frac{2}{3RT}} = \sqrt{\frac{2}{32}} = \sqrt{\frac{1}{16}} = \frac{1}{4} \]

7. **Final Result**:
Therefore, the ratio of the root mean square velocities of oxygen to hydrogen is:
\[ V_{\text{rms, O}} : V_{\text{rms, H}} = 1 : 4 \]

### Conclusion:
The ratio of the root mean square velocities of oxygen and hydrogen molecules at a given temperature is \( 1 : 4 \).

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