Ratio of the rate of diffusion of He to `H_(2)` at `0^(@)C` is same in the case :

Options

A. When temperature is changed to `100^(@)C`
B. When O2 and CH4 are taken instead of He and `H_(2)`
C. When volume of the flask is doubled
D. All the above are correct

Explanation

To solve the problem of finding the ratio of the rate of diffusion of Helium (He) to Hydrogen (H₂) at 0°C, we will use Graham's law of diffusion. Here’s a step-by-step solution:

### Step 1: Understand Graham's Law of Diffusion
Graham's law states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, it can be expressed as:

\[ \text{Rate of diffusion} \propto \frac{1}{\sqrt{M}} \]

where \( M \) is the molar mass of the gas.

### Step 2: Set Up the Ratio
To find the ratio of the rates of diffusion of He to H₂, we can write:

\[ \frac{\text{Rate of diffusion of He}}{\text{Rate of diffusion of H₂}} = \frac{\sqrt{M_{H_2}}}{\sqrt{M_{He}}} \]

### Step 3: Find the Molar Masses
The molar mass of Hydrogen (H₂) is approximately 2 g/mol (1 g/mol for each H atom), and the molar mass of Helium (He) is approximately 4 g/mol.

### Step 4: Substitute the Molar Masses into the Ratio
Now, substituting the values into the equation:

\[ \frac{\text{Rate of diffusion of He}}{\text{Rate of diffusion of H₂}} = \frac{\sqrt{2}}{\sqrt{4}} = \frac{\sqrt{2}}{2} \]

### Step 5: Analyze the Conditions
According to Graham's law, the ratio will remain the same under different conditions as long as the gases involved are the same. Therefore, we can analyze the provided options:

1. **Changing the temperature to 100°C**: The ratio remains the same as temperature does not affect the molar mass.
2. **Using O₂ and Methane instead of He and H₂**: We can calculate the ratio for these gases similarly, and it will yield a consistent result.
3. **Doubling the volume of the flask**: Volume does not affect the diffusion rates directly, so the ratio remains unchanged.
4. **All of the above are correct**: Since all previous conditions maintain the ratio, this option is valid.

### Conclusion
The correct answer is that the ratio of the rate of diffusion of He to H₂ at 0°C is the same in all the given cases. Therefore, the answer is **option 4: all of the above are correct**.

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