If rate of diffusion of A is 24 times that of B, what will be the density ratio of A and B?

Answer

To solve the problem, we will 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 density.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We know that the rate of diffusion of gas A (Ra) is 24 times that of gas B (Rb).
- This can be expressed as:
\[ R_a = 24 \times R_b \]

2. **Apply Graham's Law**:
- According to Graham's law, the relationship between the rates of diffusion and densities can be expressed as:
\[ \frac{R_a}{R_b} = \sqrt{\frac{D_b}{D_a}} \]
- Here, \(D_a\) and \(D_b\) are the densities of gases A and B, respectively.

3. **Substituting the Given Values**:
- Substitute \(R_a\) with \(24 \times R_b\) in the equation:
\[ \frac{24 \times R_b}{R_b} = \sqrt{\frac{D_b}{D_a}} \]
- The \(R_b\) cancels out:
\[ 24 = \sqrt{\frac{D_b}{D_a}} \]

4. **Square Both Sides**:
- To eliminate the square root, square both sides of the equation:
\[ 24^2 = \frac{D_b}{D_a} \]
- This simplifies to:
\[ 576 = \frac{D_b}{D_a} \]

5. **Finding the Density Ratio**:
- Rearranging the equation gives us the density ratio:
\[ \frac{D_a}{D_b} = \frac{1}{576} \]
- Therefore, the density ratio of A to B is:
\[ D_a : D_b = 1 : 576 \]

### Final Answer:
The density ratio of A to B is \(1 : 576\).