If the densities of methane and oxygen are in the ratio `1 : 2`, the ratio of rate diffusion of `O_(2)` and CH_(4)` is respectively

Options

A. `1 : 2`
B. `1 : 1.414`
C. `2 : 1`
D. `1.414 : 1`

Explanation

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

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The densities of methane (CH₄) and oxygen (O₂) are in the ratio of 1:2.
- This means if the density of methane is 1 unit, the density of oxygen is 2 units.

2. **Write the Formula for Rate of Diffusion:**
- According to Graham's law, the rate of diffusion (R) is given by:
\[ R \propto \frac{1}{\sqrt{D}} \]
where D is the density of the gas.

3. **Set Up the Ratio of Rates of Diffusion:**
- We need to find the ratio of the rate of diffusion of O₂ to that of CH₄:
\[ \frac{R_{O_2}}{R_{CH_4}} = \frac{\sqrt{D_{CH_4}}}{\sqrt{D_{O_2}}} \]

4. **Substitute the Densities:**
- From the given ratio of densities:
- Let \( D_{CH_4} = 1 \) (for methane)
- Let \( D_{O_2} = 2 \) (for oxygen)
- Substitute these values into the equation:
\[ \frac{R_{O_2}}{R_{CH_4}} = \frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

5. **Calculate the Numerical Value:**
- The value of \( \sqrt{2} \) is approximately 1.414.
- Therefore,
\[ \frac{R_{O_2}}{R_{CH_4}} = \frac{1}{1.414} \approx 0.707 \]

6. **Express the Ratio:**
- The ratio can be expressed as:
\[ R_{O_2} : R_{CH_4} = 1 : \sqrt{2} \approx 1 : 1.414 \]

### Final Answer:
The ratio of the rate of diffusion of O₂ to CH₄ is approximately **1:1.414**.