50 mL of hydrogen diffuses out through a small hole from a vessel in 20 minutes, time needed for 40 mL of oxygen to diffuse out is: 1) 12 min 2)64 min 3)8 min 4)32 min

Options

A. 12 min
B. 64 min
C. 8 min
D. 32 min.

Explanation

To solve the problem of how long it takes for 40 mL of oxygen to diffuse out, we can 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. **Determine the rate of diffusion for hydrogen (H₂)**:
- Given that 50 mL of hydrogen diffuses out in 20 minutes, we can calculate the rate of diffusion.
\[ \text{Rate of H₂} = \frac{50 \text{ mL}}{20 \text{ min}} = 2.5 \text{ mL/min} \]

2. **Set up the equation for oxygen (O₂)**:
- We need to find the time (T) it takes for 40 mL of oxygen to diffuse out.
\[ \text{Rate of O₂} = \frac{40 \text{ mL}}{T \text{ min}} \]

3. **Apply Graham's law**:
- According to Graham's law, the ratio of the rates of diffusion of two gases is equal to the inverse ratio of the square roots of their molar masses:
\[ \frac{\text{Rate of H₂}}{\text{Rate of O₂}} = \frac{\sqrt{M_{\text{O₂}}}}{\sqrt{M_{\text{H₂}}}} \]
- The molar mass of H₂ is approximately 2 g/mol and O₂ is approximately 32 g/mol.

4. **Substituting the values into the equation**:
\[ \frac{2.5}{\frac{40}{T}} = \frac{\sqrt{32}}{\sqrt{2}} \]
- Simplifying the right side:
\[ \frac{2.5 T}{40} = \frac{\sqrt{32}}{\sqrt{2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4 \]

5. **Cross-multiply and solve for T**:
\[ 2.5 T = 40 \times 4 \]
\[ 2.5 T = 160 \]
\[ T = \frac{160}{2.5} = 64 \text{ minutes} \]

### Final Answer:
The time needed for 40 mL of oxygen to diffuse out is **64 minutes**.