If rate of diffusion of A is 9 times that of B, what will be the density ratio of A and B?

Options

A. 1:81
B. 81:1
C. 1:9
D. 9:1

Answer

To solve the problem of finding the density ratio of substances A and B given that the rate of diffusion of A is 9 times that of B, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**:
The rate of diffusion (R) of a gas is inversely proportional to the square root of its density (d). This relationship can be expressed mathematically as:
\[ R \propto \frac{1}{\sqrt{d}} \]
or
\[ \frac{R_A}{R_B} = \frac{\sqrt{d_B}}{\sqrt{d_A}} \]

2. **Set Up the Given Information**:
We know from the problem that:
\[ R_A = 9 R_B \]

3. **Substitute into the Equation**:
Substitute \(R_A\) into the relationship:
\[ \frac{9 R_B}{R_B} = \frac{\sqrt{d_B}}{\sqrt{d_A}} \]
This simplifies to:
\[ 9 = \frac{\sqrt{d_B}}{\sqrt{d_A}} \]

4. **Square Both Sides**:
To eliminate the square roots, we square both sides of the equation:
\[ 9^2 = \left(\frac{\sqrt{d_B}}{\sqrt{d_A}}\right)^2 \]
This gives:
\[ 81 = \frac{d_B}{d_A} \]

5. **Find the Density Ratio**:
Rearranging the equation gives us the ratio of the densities:
\[ \frac{d_A}{d_B} = \frac{1}{81} \]
Therefore, the density ratio of A to B is:
\[ d_A : d_B = 1 : 81 \]

### Final Answer:
The density ratio of A to B is \(1 : 81\).

---