If rate of diffusion of A is 10 times that of B, what will be the density ratio of A and B?

Answer

To solve the problem of finding the density ratio of substances A and B given that the rate of diffusion of A is 10 times that of B, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between rate of diffusion and density**:
The rate of diffusion (R) is inversely proportional to the square root of the density (D) of the substance. This relationship can be expressed mathematically as:
\[ R \propto \frac{1}{\sqrt{D}} \]
or
\[ R = k \cdot \frac{1}{\sqrt{D}} \]
where \( k \) is a constant.

2. **Set up the equation for the rates of diffusion**:
Given that the rate of diffusion of A (Ra) is 10 times that of B (Rb), we can write:
\[ R_a = 10 R_b \]

3. **Express the rates in terms of densities**:
Using the relationship from step 1, we can express the rates of diffusion in terms of their densities:
\[ R_a = k \cdot \frac{1}{\sqrt{D_a}} \quad \text{and} \quad R_b = k \cdot \frac{1}{\sqrt{D_b}} \]

4. **Substitute the expressions for Ra and Rb**:
Substituting these expressions into the equation from step 2 gives:
\[ k \cdot \frac{1}{\sqrt{D_a}} = 10 \left( k \cdot \frac{1}{\sqrt{D_b}} \right) \]

5. **Cancel the constant k**:
Since k is a constant and is present in both terms, we can cancel it out:
\[ \frac{1}{\sqrt{D_a}} = 10 \cdot \frac{1}{\sqrt{D_b}} \]

6. **Rearrange the equation**:
Rearranging this gives:
\[ \sqrt{D_b} = 10 \cdot \sqrt{D_a} \]

7. **Square both sides**:
Squaring both sides to eliminate the square root gives:
\[ D_b = 100 \cdot D_a \]

8. **Find the density ratio**:
The density ratio of A to B can be expressed as:
\[ \frac{D_a}{D_b} = \frac{D_a}{100 \cdot D_a} = \frac{1}{100} \]

9. **Final result**:
Therefore, the density ratio of A to B is:
\[ D_a : D_b = 1 : 100 \]