Which of the following pairs of gases corresponds to the ratio of the rates of diffusion as `sqrt2 : 1`?

Options

A. `H_(2) and He`
B. `He and CH_(4)`
C. `H_(2) and CH_(4)`
D. `CH_(4) and SO_(2)`

Answer

To solve the question regarding which pair of gases corresponds to the ratio of the rates of diffusion as \( \sqrt{2} : 1 \), we will use Graham's law of effusion/diffusion. According to Graham's law, the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. This can be expressed mathematically as:

\[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \]

Where:
- \( r_1 \) and \( r_2 \) are the rates of diffusion of gas 1 and gas 2, respectively.
- \( M_1 \) and \( M_2 \) are the molar masses of gas 1 and gas 2, respectively.

### Step-by-Step Solution:

1. **Identify the Ratio**: We need to find a pair of gases such that the ratio of their rates of diffusion is \( \sqrt{2} : 1 \). This implies:
\[ \frac{r_1}{r_2} = \sqrt{2} \]

2. **Set Up the Equation**: From Graham's law, we can set up the equation:
\[ \sqrt{\frac{M_2}{M_1}} = \sqrt{2} \]

3. **Square Both Sides**: Squaring both sides to eliminate the square root gives:
\[ \frac{M_2}{M_1} = 2 \]

4. **Interpret the Result**: This means that the molar mass of gas 2 (the heavier gas) is twice that of gas 1 (the lighter gas):
\[ M_2 = 2 \times M_1 \]

5. **Choose Gases**: Now we need to find pairs of gases where one gas has a molar mass that is double that of the other.

- For example, if we take Hydrogen (H2) with a molar mass of 2 g/mol and Helium (He) with a molar mass of 4 g/mol:
\[ M_{He} = 4 \text{ g/mol}, \quad M_{H2} = 2 \text{ g/mol} \]
Here, \( M_{He} = 2 \times M_{H2} \).

6. **Calculate the Ratio**: Using the values:
\[ \frac{r_{H2}}{r_{He}} = \sqrt{\frac{M_{He}}{M_{H2}}} = \sqrt{\frac{4}{2}} = \sqrt{2} \]

7. **Conclusion**: The pair of gases that corresponds to the ratio of the rates of diffusion as \( \sqrt{2} : 1 \) is Hydrogen (H2) and Helium (He).

### Final Answer:
The correct pair of gases is **Hydrogen (H2) and Helium (He)**.