If rate of diffusion of A is 15 times that of B, what will be the density ratio of A and B?

Answer

To solve the problem, we need to 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. **Understanding the Given Information**:
- We know that the rate of diffusion of gas A (ra) is 15 times the rate of diffusion of gas B (rb).
- Mathematically, we can express this as:
\[ r_a = 15 \times r_b \]

2. **Applying Graham's Law**:
- According to Graham's law, the relationship between the rates of diffusion and densities of two gases 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 Known Values**:
- We can substitute the expression for \(r_a\) into Graham's law:
\[ \frac{15 \times r_b}{r_b} = \sqrt{\frac{d_b}{d_a}} \]
- This simplifies to:
\[ 15 = \sqrt{\frac{d_b}{d_a}} \]

4. **Squaring Both Sides**:
- To eliminate the square root, we square both sides of the equation:
\[ 15^2 = \frac{d_b}{d_a} \]
- This gives us:
\[ 225 = \frac{d_b}{d_a} \]

5. **Finding the Density Ratio**:
- Rearranging the equation gives us the density ratio of A to B:
\[ \frac{d_a}{d_b} = \frac{1}{225} \]
- Therefore, the density ratio of A to B is:
\[ d_a : d_b = 1 : 225 \]

### Final Answer:
The density ratio of A to B is 1:225.