If rate of diffusion of A is 20 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 20 times that of B, we can use Graham's law of effusion. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand Graham's Law
Graham's law states that the rate of diffusion of a gas is inversely proportional to the square root of its density. Mathematically, this can be expressed 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, and \( d_A \) and \( d_B \) are their respective densities.

### Step 2: Set Up the Equation
From the problem, we know that:
\[ r_A = 20 \times r_B \]
Substituting this into Graham's law gives:
\[ \frac{20 \cdot r_B}{r_B} = \sqrt{\frac{d_B}{d_A}} \]
This simplifies to:
\[ 20 = \sqrt{\frac{d_B}{d_A}} \]

### Step 3: Square Both Sides
To eliminate the square root, we square both sides of the equation:
\[ 20^2 = \frac{d_B}{d_A} \]
This results in:
\[ 400 = \frac{d_B}{d_A} \]

### Step 4: Rearrange the Equation
Rearranging the equation gives us the ratio of the densities:
\[ \frac{d_A}{d_B} = \frac{1}{400} \]

### Step 5: State the Density Ratio
Thus, the density ratio of A to B is:
\[ d_A : d_B = 1 : 400 \]

### Final Answer
The density ratio of A to B is \( 1 : 400 \).

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