When r, p and M represent rate of diffusion, pressure and molecular mass, respectively, then the ratio of the rates of diffusion `(r_A r_B)` of two gases A and B, is given by?

Options

A. `(P_A//P_B)^(1//2) (M_A/M_B)`
B. `(P_A//P_B) (M_B//M_A)^(1//2)`
C. `(P_A//P_B)^(1//2) (M_B/M_A)`
D. `(P_A//P_B) (M_A//M_B)^(1//2)`

Explanation

To find the ratio of the rates of diffusion \( \frac{r_A}{r_B} \) of two gases A and B, we can 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 molecular mass and directly proportional to its pressure.

### Step-by-Step Solution:

1. **Understanding Graham's Law**: According to Graham's law, the rate of diffusion \( r \) of a gas is proportional to the pressure \( P \) and inversely proportional to the square root of its molecular mass \( M \). This can be expressed mathematically as:
\[ r \propto \frac{P}{\sqrt{M}} \]

2. **Expressing the Rates for Gases A and B**:
- For gas A:
\[ r_A \propto \frac{P_A}{\sqrt{M_A}} \]
- For gas B:
\[ r_B \propto \frac{P_B}{\sqrt{M_B}} \]

3. **Setting Up the Ratio**:
To find the ratio of the rates of diffusion of gases A and B, we can write:
\[ \frac{r_A}{r_B} = \frac{\frac{P_A}{\sqrt{M_A}}}{\frac{P_B}{\sqrt{M_B}}} \]

4. **Simplifying the Ratio**:
This can be simplified to:
\[ \frac{r_A}{r_B} = \frac{P_A}{P_B} \cdot \frac{\sqrt{M_B}}{\sqrt{M_A}} \]
or
\[ \frac{r_A}{r_B} = \frac{P_A}{P_B} \cdot \sqrt{\frac{M_B}{M_A}} \]

5. **Final Expression**:
Thus, the final expression for the ratio of the rates of diffusion of two gases A and B is:
\[ \frac{r_A}{r_B} = \frac{P_A}{P_B} \cdot \sqrt{\frac{M_B}{M_A}} \]