If rate of diffusion of A is 16 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 relationship between the rates of diffusion and densities**:
According to Graham's law, the rate of diffusion (r) is inversely proportional to the square root of the density (d):
\[ r \propto \frac{1}{\sqrt{d}} \]
This can be expressed as:
\[ \frac{r_A}{r_B} = \sqrt{\frac{d_B}{d_A}} \]

2. **Set up the equation based on the given information**:
We are given that the rate of diffusion of A (r_A) is 16 times that of B (r_B):
\[ r_A = 16 \cdot r_B \]
Substituting this into the equation from Graham's law gives:
\[ \frac{16 \cdot r_B}{r_B} = \sqrt{\frac{d_B}{d_A}} \]
This simplifies to:
\[ 16 = \sqrt{\frac{d_B}{d_A}} \]

3. **Square both sides to eliminate the square root**:
Squaring both sides results in:
\[ 16^2 = \frac{d_B}{d_A} \]
Which simplifies to:
\[ 256 = \frac{d_B}{d_A} \]

4. **Express the density ratio**:
Rearranging the equation gives us the density ratio of A to B:
\[ \frac{d_A}{d_B} = \frac{1}{256} \]
Therefore, the density ratio of A to B is:
\[ d_A : d_B = 1 : 256 \]

### Final Answer:
The density ratio of A to B is \( 1 : 256 \).