If rate of diffusion of A is 13 times that of B, what will be the density ratio of A and B?

Answer

To solve the problem of finding the density ratio of gases A and B based on their rates of diffusion, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**: 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. **Given Information**: From the problem, we know that:
\[ R_a = 13 \times R_b \]
This means that the rate of diffusion of gas A is 13 times that of gas B.

3. **Substituting the Values**: Substitute the value of \( R_a \) into the diffusion relationship:
\[ 13 \times R_b = \sqrt{\frac{D_b}{D_a}} \]

4. **Rearranging the Equation**: To eliminate \( R_b \), we can square both sides of the equation:
\[ (13 \times R_b)^2 = \frac{D_b}{D_a} \]
This simplifies to:
\[ 169 \times R_b^2 = \frac{D_b}{D_a} \]

5. **Expressing the Density Ratio**: Rearranging the equation gives us:
\[ \frac{D_a}{D_b} = \frac{1}{169} \]
Therefore, the density ratio of A to B is:
\[ D_a : D_b = 1 : 169 \]

6. **Final Answer**: The density ratio of A to B is \( 1 : 169 \).