If rate of diffusion of A is 2 times that of B, what will be the density ratio of A and B?

Options

A. 4:1
B. 1:4
C. 2:1
D. 1:2

Explanation

To solve the problem, we need to find the density ratio of gases A and B given that the rate of diffusion of A is 2 times that of B. 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 rates of diffusion and density**:
According to Graham's law, the rate of diffusion (r) of a gas is inversely proportional to the square root of its density (d). Mathematically, 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 know that the rate of diffusion of gas A (r_A) is 2 times that of gas B (r_B):
\[ r_A = 2r_B \]
Substituting this into the equation from Graham's law gives:
\[ \frac{2r_B}{r_B} = \sqrt{\frac{d_B}{d_A}} \]

3. **Simplify the equation**:
The \( r_B \) terms cancel out:
\[ 2 = \sqrt{\frac{d_B}{d_A}} \]

4. **Square both sides to eliminate the square root**:
Squaring both sides results in:
\[ 4 = \frac{d_B}{d_A} \]

5. **Rearrange to find the density ratio**:
This can be rearranged to express the density ratio of A to B:
\[ \frac{d_A}{d_B} = \frac{1}{4} \]

6. **Express the final density ratio**:
Therefore, the density ratio of A to B can be written as:
\[ d_A : d_B = 1 : 4 \]

### Final Answer:
The density ratio of A to B is 1:4.

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