If rate of diffusion of A is 21 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. **Understand the relationship**: According to Graham's law, the rate of diffusion (R) of two gases A and B can be expressed as:
\[ \frac{R_A}{R_B} = \frac{1}{\sqrt{\frac{d_A}{d_B}}} \]
where \(d_A\) and \(d_B\) are the densities of gases A and B, respectively.

2. **Set up the equation**: We know from the problem that the rate of diffusion of A is 21 times that of B:
\[ R_A = 21 R_B \]
Substituting this into the equation gives:
\[ \frac{21 R_B}{R_B} = \frac{1}{\sqrt{\frac{d_A}{d_B}}} \]
This simplifies to:
\[ 21 = \frac{1}{\sqrt{\frac{d_A}{d_B}}} \]

3. **Rearranging the equation**: To eliminate the square root, we take the reciprocal:
\[ \sqrt{\frac{d_A}{d_B}} = \frac{1}{21} \]

4. **Squaring both sides**: Now, we square both sides to remove the square root:
\[ \frac{d_A}{d_B} = \left(\frac{1}{21}\right)^2 \]
This gives:
\[ \frac{d_A}{d_B} = \frac{1}{441} \]

5. **Finding the density ratio**: The density ratio of A to B can be expressed as:
\[ d_A : d_B = 1 : 441 \]

### Final Answer:
The density ratio of A to B is \(1 : 441\).

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