If rate of diffusion of A is 8 times that of B, what will be the density ratio of A and B?

Options

A. 64:1
B. 1:64
C. 8:1
D. 1:8

Answer

To solve the problem of finding the density ratio of gases A and B given that the rate of diffusion of A is 8 times that of B, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between rate of diffusion and density**:
The rate of diffusion of a gas is inversely proportional to the square root of its density. This relationship can be expressed mathematically as:
\[ \frac{r_A}{r_B} = \sqrt{\frac{d_B}{d_A}} \]
where \( r_A \) and \( r_B \) are the rates of diffusion of gases A and B, respectively, and \( d_A \) and \( d_B \) are their densities.

2. **Set up the equation using the given information**:
According to the problem, the rate of diffusion of A is 8 times that of B:
\[ r_A = 8 \cdot r_B \]
Substituting this into the equation gives:
\[ \frac{8 \cdot r_B}{r_B} = \sqrt{\frac{d_B}{d_A}} \]
Simplifying this, we have:
\[ 8 = \sqrt{\frac{d_B}{d_A}} \]

3. **Square both sides to eliminate the square root**:
Squaring both sides of the equation results in:
\[ 8^2 = \frac{d_B}{d_A} \]
This simplifies to:
\[ 64 = \frac{d_B}{d_A} \]

4. **Rearrange to find the density ratio**:
To find the density ratio of A to B, we can rearrange the equation:
\[ \frac{d_A}{d_B} = \frac{1}{64} \]
This means that the density of A is 1/64th that of B.

5. **Express the final answer**:
Therefore, the density ratio of A to B is:
\[ d_A : d_B = 1 : 64 \]

### Final Answer:
The density ratio of A to B is \( 1 : 64 \).