A mixture of `H_(2)andO_(2)` taken in a bulb in 2 : 1 mole ratio is allowed to diffuse through a fine hole. The composition of the gases (mole ratio) coming out initially is 8 : 1. The composition of the gases after some time may be

Options

A. `8:1`
B. `9:1`
C. `7:1`
D. `4:1`

Answer

To solve the problem, we need to analyze the diffusion of the gases \( H_2 \) (hydrogen) and \( O_2 \) (oxygen) based on their mole ratio and their rates of diffusion.

### Step-by-Step Solution:

1. **Initial Mole Ratio**: We start with a mixture of \( H_2 \) and \( O_2 \) in a 2:1 mole ratio. This means for every 2 moles of hydrogen, there is 1 mole of oxygen.

2. **Molecular Mass Calculation**:
- The molecular mass of \( H_2 \) is \( 2 \, \text{g/mol} \).
- The molecular mass of \( O_2 \) is \( 32 \, \text{g/mol} \).

3. **Rate of Diffusion**: According to Graham's law of effusion, the rate of diffusion is inversely proportional to the square root of the molar mass. Thus, we can calculate the rate of diffusion for both gases:
- Rate of diffusion of \( H_2 \) is proportional to \( \frac{1}{\sqrt{2}} \).
- Rate of diffusion of \( O_2 \) is proportional to \( \frac{1}{\sqrt{32}} \).

4. **Comparing Rates**:
- The ratio of the rates of diffusion of \( H_2 \) to \( O_2 \) is:
\[ \text{Rate}_{H_2} : \text{Rate}_{O_2} = \frac{1}{\sqrt{2}} : \frac{1}{\sqrt{32}} = \sqrt{32} : \sqrt{2} = 4 : 1 \]
- This means that hydrogen diffuses 4 times faster than oxygen.

5. **Initial Composition of Diffused Gases**:
- Initially, the gases coming out are in the ratio of \( 8:1 \) (given in the problem). This indicates that more hydrogen is diffusing out compared to oxygen due to its higher rate of diffusion.

6. **Change in Composition Over Time**:
- As time progresses, the amount of hydrogen will decrease faster than the amount of oxygen because hydrogen is diffusing out more rapidly.
- Therefore, the mole ratio of \( H_2 \) to \( O_2 \) will decrease over time.

7. **Final Composition**:
- After some time, the composition of the gases will shift towards a higher proportion of oxygen compared to hydrogen. The ratio will be less than \( 8:1 \) but more than \( 4:1 \) (theoretical maximum based on diffusion rates).
- A reasonable estimate for the composition after some time could be \( 7:1 \) (as suggested in the video).

### Conclusion:
The composition of the gases after some time may be \( 7:1 \) (for \( H_2 : O_2 \)).