If rate of diffusion of A is 18 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 18 times that of B, 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: Substitute the given values
According to the problem, the rate of diffusion of A is 18 times that of B:
\[ R_A = 18 R_B \]
Substituting this into the equation gives:
\[ \frac{18 R_B}{R_B} = \sqrt{\frac{d_B}{d_A}} \]
This simplifies to:
\[ 18 = \sqrt{\frac{d_B}{d_A}} \]

### Step 3: Square both sides to eliminate the square root
To eliminate the square root, we square both sides of the equation:
\[ 18^2 = \frac{d_B}{d_A} \]
Calculating \( 18^2 \):
\[ 324 = \frac{d_B}{d_A} \]

### Step 4: Rearrange to find the density ratio
Rearranging the equation gives:
\[ \frac{d_A}{d_B} = \frac{1}{324} \]
Thus, the density ratio of A to B is:
\[ d_A : d_B = 1 : 324 \]

### Final Answer
The density ratio of A to B is \( 1 : 324 \).

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