If rate of diffusion of A is 25 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. **Understanding the Given Information**:
- We are given that the rate of diffusion of gas A (Ra) is 25 times that of gas B (Rb).
- This can be expressed mathematically as:
\[ R_a = 25 R_b \]

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**:
- From our first step, we substitute \(R_a = 25 R_b\) into the equation:
\[ \frac{25 R_b}{R_b} = \sqrt{\frac{d_b}{d_a}} \]
- The \(R_b\) cancels out:
\[ 25 = \sqrt{\frac{d_b}{d_a}} \]

4. **Squaring Both Sides**:
- To eliminate the square root, we square both sides of the equation:
\[ 25^2 = \frac{d_b}{d_a} \]
- This simplifies to:
\[ 625 = \frac{d_b}{d_a} \]

5. **Finding the Density Ratio**:
- Rearranging the equation gives us the density ratio:
\[ \frac{d_a}{d_b} = \frac{1}{625} \]
- Therefore, the density ratio of A to B is:
\[ d_a : d_b = 1 : 625 \]

### Final Answer:
The density ratio of A to B is 1:625.

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