If rate of diffusion of A is 3 times that of B, what will be the density ratio of B and A?

Answer

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

### Step 1: Understand the relationship between diffusion rate and density
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.

### Step 2: Set up the equation with the given information
According to the problem, the rate of diffusion of A is 3 times that of B:
\[ r_A = 3 r_B \]
Substituting this into the equation from Step 1 gives:
\[ \frac{3 r_B}{r_B} = \sqrt{\frac{d_B}{d_A}} \]

### Step 3: Simplify the equation
Cancelling \( r_B \) from both sides results in:
\[ 3 = \sqrt{\frac{d_B}{d_A}} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides of the equation yields:
\[ 3^2 = \frac{d_B}{d_A} \]
which simplifies to:
\[ 9 = \frac{d_B}{d_A} \]

### Step 5: Express the density ratio
From the equation \( \frac{d_B}{d_A} = 9 \), we can express the density ratio as:
\[ d_B : d_A = 9 : 1 \]

### Final Answer
The density ratio of B to A is \( 9:1 \).

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