Equal moles of hydrogen and oxygen gases are placed in a container with a pin-hole through which both can escape. What fraction of the oxygen escapes in the time required for one-half of the hydrogen to escape ?

Options

A. `1//4`
B. `3//8`
C. `1//2`
D. `1//8`

Answer

To solve the problem, we will use Graham's law of effusion, which states that the rate of effusion (or diffusion) of a gas is inversely proportional to the square root of its molar mass.

### Step-by-Step Solution:

1. **Identify the Molar Masses**:
- The molar mass of hydrogen (H₂) is 2 g/mol.
- The molar mass of oxygen (O₂) is 32 g/mol.

2. **Apply Graham's Law**:
- According to Graham's law, the rate of diffusion of gas A (hydrogen) to gas B (oxygen) can be expressed as:
\[ \frac{\text{Rate of H₂}}{\text{Rate of O₂}} = \sqrt{\frac{M_{\text{O₂}}}{M_{\text{H₂}}}} \]
- Substituting the molar masses:
\[ \frac{\text{Rate of H₂}}{\text{Rate of O₂}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4 \]
- This means:
\[ \text{Rate of H₂} = 4 \times \text{Rate of O₂} \]

3. **Express the Rates**:
- Let the rate of diffusion of oxygen be \( r \). Then, the rate of diffusion of hydrogen will be \( 4r \).

4. **Determine the Time for Hydrogen to Escape**:
- If we let \( t \) be the time required for half of the hydrogen to escape, then the amount of hydrogen that escapes in time \( t \) is:
\[ \text{Amount of H₂ escaped} = \frac{1}{2} \text{(initial moles of H₂)} \]

5. **Calculate the Amount of Oxygen Escaped in the Same Time**:
- The amount of oxygen that escapes in the same time \( t \) can be calculated using the rates:
\[ \text{Amount of O₂ escaped} = r \cdot t \]
- Since we know the rate of hydrogen is \( 4r \), we can express the relationship:
\[ \text{Amount of H₂ escaped} = 4r \cdot t = \frac{1}{2} \text{(initial moles of H₂)} \]

6. **Relate the Amounts**:
- From the earlier relationship, we have:
\[ \frac{\text{Amount of O₂ escaped}}{\text{Amount of H₂ escaped}} = \frac{1}{4} \]
- Therefore, if half of the hydrogen escapes, the amount of oxygen that escapes will be:
\[ \text{Amount of O₂ escaped} = \frac{1}{4} \times \frac{1}{2} \text{(initial moles of H₂)} = \frac{1}{8} \text{(initial moles of O₂)} \]

7. **Calculate the Fraction of Oxygen Escaped**:
- The fraction of oxygen that escapes in the same time is:
\[ \text{Fraction of O₂ escaped} = \frac{\text{Amount of O₂ escaped}}{\text{Initial moles of O₂}} = \frac{1/8}{1} = \frac{1}{8} \]

### Final Answer:
The fraction of the oxygen that escapes in the time required for one-half of the hydrogen to escape is \( \frac{1}{8} \).