If rate of diffusion of A is 7 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 given that the rate of diffusion of A is 7 times that of B, 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 is expressed by Graham's law of effusion/diffusion:
\[ \frac{R_A}{R_B} = \frac{\sqrt{D_B}}{\sqrt{D_A}} \]
where \( R_A \) and \( R_B \) are the rates of diffusion of gases A and B, and \( D_A \) and \( D_B \) are their respective densities.

2. **Set Up the Equation**: We know from the problem that:
\[ R_A = 7 \times R_B \]
Substituting this into Graham's law gives:
\[ \frac{7 \times R_B}{R_B} = \frac{\sqrt{D_B}}{\sqrt{D_A}} \]

3. **Simplify the Equation**: The \( R_B \) terms cancel out, leading to:
\[ 7 = \frac{\sqrt{D_B}}{\sqrt{D_A}} \]

4. **Square Both Sides**: To eliminate the square roots, we square both sides of the equation:
\[ 7^2 = \frac{D_B}{D_A} \]
This simplifies to:
\[ 49 = \frac{D_B}{D_A} \]

5. **Find the Density Ratio**: Rearranging the equation gives us the density ratio:
\[ \frac{D_A}{D_B} = \frac{1}{49} \]
Therefore, the density ratio of A to B is:
\[ D_A : D_B = 1 : 49 \]

### Final Answer:
The density ratio of A to B is \( 1 : 49 \).