The ratio of rates of diffusion of `CO_2` and `SO_2` at the same temperature and pressure will be

Options

A. `4 : sqrt 11`
B. `11 : 4`
C. `1 : 4`
D. `1 : 6`

Answer

To determine the ratio of rates of diffusion of \( CO_2 \) and \( SO_2 \) at the same temperature and pressure, we can use Graham's law of effusion, which states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass.

### Step-by-Step Solution:

1. **Identify the Molar Masses:**
- Calculate the molar mass of \( CO_2 \):
- Carbon (C) = 12 g/mol
- Oxygen (O) = 16 g/mol
- Therefore, \( CO_2 = 12 + (16 \times 2) = 12 + 32 = 44 \) g/mol
- Calculate the molar mass of \( SO_2 \):
- Sulfur (S) = 32 g/mol
- Oxygen (O) = 16 g/mol
- Therefore, \( SO_2 = 32 + (16 \times 2) = 32 + 32 = 64 \) g/mol

2. **Apply Graham's Law:**
- According to Graham's law, the ratio of the rates of diffusion of two gases is given by:
\[ \frac{R_1}{R_2} = \sqrt{\frac{M_2}{M_1}} \]
- Here, \( R_1 \) is the rate of diffusion of \( CO_2 \), \( R_2 \) is the rate of diffusion of \( SO_2 \), \( M_1 \) is the molar mass of \( CO_2 \), and \( M_2 \) is the molar mass of \( SO_2 \).

3. **Substitute the Values:**
- Substitute the molar masses into the equation:
\[ \frac{R_{CO_2}}{R_{SO_2}} = \sqrt{\frac{64}{44}} \]

4. **Simplify the Ratio:**
- Calculate the square root:
\[ \frac{R_{CO_2}}{R_{SO_2}} = \sqrt{\frac{64}{44}} = \sqrt{\frac{16}{11}} = \frac{4}{\sqrt{11}} \]

5. **Final Ratio:**
- Thus, the ratio of the rates of diffusion of \( CO_2 \) to \( SO_2 \) is:
\[ R_{CO_2} : R_{SO_2} = 4 : \sqrt{11} \]

### Conclusion:
The ratio of rates of diffusion of \( CO_2 \) and \( SO_2 \) at the same temperature and pressure is \( 4 : \sqrt{11} \).