If rate of diffusion of A is 5 times that of B what will be the density ratio of A and B?

Options

A. `1:25`
B. `1:5`
C. `25:1`
D. `5:1`

Answer

To solve the problem of finding the density ratio of gases A and B given that the rate of diffusion of A is 5 times that of B, we can follow these steps:

### Step 1: Understand the relationship between diffusion rates and densities
The rate of diffusion of a gas is inversely proportional to the square root of its density. This relationship can be expressed mathematically as:
\[ R \propto \frac{1}{\sqrt{d}} \]
where \( R \) is the rate of diffusion and \( d \) is the density of the gas.

### Step 2: Set up the equation based on the given information
Given that the rate of diffusion of A is 5 times that of B, we can write:
\[ R_A = 5 R_B \]

### Step 3: Use the relationship between rates and densities
From the relationship established in Step 1, we can express the rates of diffusion in terms of densities:
\[ \frac{R_A}{R_B} = \sqrt{\frac{d_B}{d_A}} \]

### Step 4: Substitute the known values into the equation
Substituting \( R_A = 5 R_B \) into the equation gives:
\[ 5 = \sqrt{\frac{d_B}{d_A}} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides of the equation results in:
\[ 25 = \frac{d_B}{d_A} \]

### Step 6: Rearrange to find the density ratio
Rearranging the equation gives:
\[ \frac{d_A}{d_B} = \frac{1}{25} \]

### Step 7: Express the final answer as a ratio
Thus, the density ratio of A to B can be expressed as:
\[ d_A : d_B = 1 : 25 \]

### Final Answer
The density ratio of A to B is \( 1 : 25 \).

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