If two molecules of A and B having mass 100 amu and 64 amu respectively and rate of diffusion of A is `12 xx 10^(-3)`, then what will be the rate of diffusion of B?

Options

A. `15 xx 10^(-3)`
B. `64 xx 10^(-3)`
C. `5 xx 10^(-3)`
D. `46 xx 10^(-3)`

Answer

To solve the problem, we will use Graham's law of diffusion, which states that the rate of diffusion of two gases is inversely proportional to the square root of their molar masses. The formula can be expressed as:

\[ \frac{r_A}{r_B} = \sqrt{\frac{m_B}{m_A}} \]

Where:
- \( r_A \) and \( r_B \) are the rates of diffusion of gases A and B, respectively.
- \( m_A \) and \( m_B \) are the molar masses of gases A and B, respectively.

### Step 1: Identify the given information
- Mass of molecule A = 100 amu
- Mass of molecule B = 64 amu
- Rate of diffusion of A, \( r_A = 12 \times 10^{-3} \)

### Step 2: Calculate the molar masses of A and B
Since the problem states that the mass given is for two molecules, we need to find the mass of one molecule:
- Mass of one molecule of A = \( \frac{100 \text{ amu}}{2} = 50 \text{ amu} \)
- Mass of one molecule of B = \( \frac{64 \text{ amu}}{2} = 32 \text{ amu} \)

### Step 3: Apply Graham's law of diffusion
Using Graham's law, we can set up the equation:
\[ \frac{r_A}{r_B} = \sqrt{\frac{m_B}{m_A}} = \sqrt{\frac{32}{50}} \]

### Step 4: Simplify the equation
Calculating the right side:
\[ \frac{r_A}{r_B} = \sqrt{\frac{32}{50}} = \sqrt{\frac{32}{50}} = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

### Step 5: Rearranging to find \( r_B \)
From the equation:
\[ r_B = \frac{5}{4} r_A \]

### Step 6: Substitute the value of \( r_A \)
Now substituting the value of \( r_A \):
\[ r_B = \frac{5}{4} \times (12 \times 10^{-3}) = \frac{60}{4} \times 10^{-3} = 15 \times 10^{-3} \]

### Final Answer
Thus, the rate of diffusion of B is:
\[ r_B = 15 \times 10^{-3} \]