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

Explanation

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

### Step-by-Step Solution:

1. **Identify the Molar Masses**:
- Molar mass of Hydrogen (H₂) = 2 g/mol (since each H atom has a molar mass of approximately 1 g/mol).
- Molar mass of Oxygen (O₂) = 32 g/mol (since each O atom has a molar mass of approximately 16 g/mol).

2. **Apply Graham's Law of Effusion**:
- According to Graham's law, the rate of effusion of a gas is given by:
\[ \text{Rate} \propto \frac{1}{\sqrt{\text{Molar Mass}}} \]
- Let \( r_H \) be the rate of effusion of hydrogen and \( r_O \) be the rate of effusion of oxygen.
- Therefore, we can write:
\[ \frac{r_H}{r_O} = \sqrt{\frac{M_O}{M_H}} \]
- Substituting the molar masses:
\[ \frac{r_H}{r_O} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4 \]
- This means that the rate of effusion of hydrogen is 4 times that of oxygen:
\[ r_H = 4 r_O \]

3. **Determine the Time for Effusion**:
- Let the time taken for half of the hydrogen to escape be \( t \).
- In this time, the amount of hydrogen that escapes is \( \frac{1}{2} \) of the total amount of hydrogen.
- The amount of oxygen that escapes in the same time can be calculated using the rates:
\[ \text{Amount of } H_2 \text{ escaped} = r_H \cdot t \]
\[ \text{Amount of } O_2 \text{ escaped} = r_O \cdot t \]

4. **Calculate the Fraction of Oxygen Escaped**:
- Since \( r_H = 4 r_O \), we can express the time taken for hydrogen to escape:
\[ \frac{1}{2} = r_H \cdot t \implies t = \frac{1/2}{r_H} \]
- Now substituting \( r_H \) in terms of \( r_O \):
\[ t = \frac{1/2}{4 r_O} = \frac{1}{8 r_O} \]
- Now, substituting this back to find the amount of oxygen escaped:
\[ \text{Amount of } O_2 \text{ escaped} = r_O \cdot t = r_O \cdot \frac{1}{8 r_O} = \frac{1}{8} \]

5. **Calculate the Fraction of Oxygen Escaped**:
- Since we started with equal moles of hydrogen and oxygen, the fraction of oxygen that escapes is:
\[ \text{Fraction of } O_2 \text{ escaped} = \frac{\text{Amount of } O_2 \text{ escaped}}{\text{Total amount of } O_2} = \frac{1/8}{1} = \frac{1}{8} \]

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