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`

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
- Molar mass of Oxygen (O₂) = 32 g/mol

2. **Apply Graham's Law**:
- According to Graham's law, the rate of effusion (R) is given by:
\[ R \propto \frac{1}{\sqrt{M}} \]
- Where M is the molar mass of the gas.

3. **Set Up the Ratios**:
- Let \( R_{H_2} \) be the rate of effusion of hydrogen and \( R_{O_2} \) be the rate of effusion of oxygen.
- According to Graham's law:
\[ \frac{R_{H_2}}{R_{O_2}} = \frac{\sqrt{M_{O_2}}}{\sqrt{M_{H_2}}} \]
- Substituting the molar masses:
\[ \frac{R_{H_2}}{R_{O_2}} = \frac{\sqrt{32}}{\sqrt{2}} = \frac{4}{1} = 4 \]
- This means that hydrogen effuses 4 times faster than oxygen.

4. **Determine the Time for Hydrogen to Escape**:
- Let’s assume that in a certain time \( t \), half of the hydrogen escapes. Therefore, if we denote the initial moles of hydrogen as 1 mole, then:
\[ \text{Moles of } H_2 \text{ escaped} = \frac{1}{2} \text{ mole} \]

5. **Calculate Moles of Oxygen Escaped in the Same Time**:
- Since the rate of hydrogen is 4 times that of oxygen, if \( x \) moles of oxygen escape in the same time \( t \), we can set up the equation:
\[ \frac{1/2}{x} = 4 \implies x = \frac{1/2}{4} = \frac{1}{8} \text{ moles} \]

6. **Calculate the Fraction of Oxygen Escaped**:
- The fraction of oxygen that escapes is given by:
\[ \text{Fraction of } O_2 \text{ escaped} = \frac{\text{Moles of } O_2 \text{ escaped}}{\text{Initial moles of } O_2} = \frac{1/8}{1} = \frac{1}{8} \]

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