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

Options

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

Explanation

To solve the problem, 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. Here's the step-by-step solution:

### Step 1: Understand the relationship between the rates of diffusion and density
According to Graham's law, the rate of diffusion (R) of two gases A and B can be expressed as:
\[ \frac{R_A}{R_B} = \sqrt{\frac{d_B}{d_A}} \]
where \(d_A\) and \(d_B\) are the densities of gases A and B, respectively.

### Step 2: Set up the equation with the given information
From the problem, we know that the rate of diffusion of A is 2 times that of B:
\[ R_A = 2R_B \]
Substituting this into the Graham's law equation gives us:
\[ \frac{2R_B}{R_B} = \sqrt{\frac{d_B}{d_A}} \]
This simplifies to:
\[ 2 = \sqrt{\frac{d_B}{d_A}} \]

### Step 3: Square both sides to eliminate the square root
To eliminate the square root, we square both sides of the equation:
\[ 2^2 = \frac{d_B}{d_A} \]
This results in:
\[ 4 = \frac{d_B}{d_A} \]

### Step 4: Rearrange to find the density ratio
From the equation \(4 = \frac{d_B}{d_A}\), we can express the density ratio of B to A:
\[ \frac{d_B}{d_A} = 4 \]
This means that the density of B is 4 times the density of A.

### Step 5: Write the final answer
Thus, the density ratio of B to A is:
\[ \text{Density ratio of B to A} = 4:1 \]

### Conclusion
The correct option is that the density ratio of B to A is 4:1.

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