If rate of diffusion of A is 4 times that of B, what will be the density ratio of B and A?

Answer

To solve the problem, we need to understand the relationship between the rate of diffusion and the density of the substances involved. Here’s a step-by-step solution:

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

### Step 2: Set up the equation for the two substances
Given that the rate of diffusion of A (rA) is 4 times that of B (rB), we can write:
\[ r_A = 4 \cdot r_B \]

### Step 3: Express the rates of diffusion in terms of densities
From the relationship established in Step 1, we can express the rates of diffusion in terms of densities:
\[ r_A = k \cdot \frac{1}{\sqrt{d_A}} \]
\[ r_B = k \cdot \frac{1}{\sqrt{d_B}} \]

### Step 4: Substitute the expression for rA into the equation
Substituting the expression for rA into our earlier equation gives:
\[ k \cdot \frac{1}{\sqrt{d_A}} = 4 \cdot \left(k \cdot \frac{1}{\sqrt{d_B}}\right) \]

### Step 5: Cancel out the constant k
Since k is a common factor, we can cancel it out from both sides:
\[ \frac{1}{\sqrt{d_A}} = 4 \cdot \frac{1}{\sqrt{d_B}} \]

### Step 6: Rearrange the equation
Rearranging the equation gives us:
\[ \sqrt{d_B} = 4 \cdot \sqrt{d_A} \]

### Step 7: Square both sides to eliminate the square root
Squaring both sides results in:
\[ d_B = 16 \cdot d_A \]

### Step 8: Find the density ratio
The density ratio of B to A can be expressed as:
\[ \frac{d_B}{d_A} = 16 \]

Thus, the density ratio of B to A is:
\[ \text{Density ratio } (d_B : d_A) = 16 : 1 \]

### Final Answer
The density ratio of B to A is 16:1.

---