Calculate relative rates of effusion of `SO_(2)` to `CH_(4)` in the mixture obtained by effusing out a mixture with initial molar ratio`(n_(SO_(2)))/(n_(CH_(4)))=(8)/(1)` for three effusing steps.

Options

A. `2:1`
B. `1:4`
C. `1:2`
D. none of these

Explanation

To calculate the relative rates of effusion of \( SO_2 \) to \( CH_4 \) in a mixture with an initial molar ratio of \( \frac{n_{SO_2}}{n_{CH_4}} = \frac{8}{1} \) after three effusing steps, we can follow these steps:

### Step 1: Understand the relationship between effusion rates and moles
The rate of effusion of a gas is inversely proportional to the square root of its molar mass, according to Graham's law. The formula can be expressed as:
\[ \frac{R_{SO_2}}{R_{CH_4}} = \frac{n_{SO_2}}{n_{CH_4}} \cdot \sqrt{\frac{M_{CH_4}}{M_{SO_2}}} \]
where \( R \) is the rate of effusion, \( n \) is the number of moles, and \( M \) is the molar mass of the gases.

### Step 2: Define the initial conditions
Given the initial molar ratio:
\[ \frac{n_{SO_2}}{n_{CH_4}} = \frac{8}{1} \]
Let \( n_{SO_2} = 8n_0 \) and \( n_{CH_4} = n_0 \).

### Step 3: Calculate the molar masses
The molar masses are:
- Molar mass of \( CH_4 \) (methane) = 16 g/mol
- Molar mass of \( SO_2 \) (sulfur dioxide) = 64 g/mol

### Step 4: Substitute values into the effusion rate equation
Substituting the values into the equation:
\[ \frac{R_{SO_2}}{R_{CH_4}} = \frac{8n_0}{n_0} \cdot \sqrt{\frac{16}{64}} \]
This simplifies to:
\[ \frac{R_{SO_2}}{R_{CH_4}} = 8 \cdot \sqrt{\frac{1}{4}} = 8 \cdot \frac{1}{2} = 4 \]

### Step 5: Adjust for three effusion steps
Since the question states that this occurs over three effusion steps, we need to consider how the number of moles changes after each step. After each effusion step, the number of moles of each gas will decrease proportionally to their rates.

Let’s denote the number of moles after three steps as \( n_{SO_2}^{(3)} \) and \( n_{CH_4}^{(3)} \). The ratio will remain the same but will be adjusted for the number of steps:
\[ \frac{n_{SO_2}^{(3)}}{n_{CH_4}^{(3)}} = \frac{8}{1} \cdot \left(\frac{R_{SO_2}}{R_{CH_4}}\right)^3 \]
Thus:
\[ \frac{n_{SO_2}^{(3)}}{n_{CH_4}^{(3)}} = \frac{8}{1} \cdot 4^3 = \frac{8}{1} \cdot 64 = \frac{512}{1} \]

### Step 6: Final ratio of effusion rates
Now, we can calculate the final ratio of the rates of effusion after three steps:
\[ \frac{R_{SO_2}}{R_{CH_4}} = \frac{512}{1} \cdot \sqrt{\frac{16}{64}} = 512 \cdot \frac{1}{2} = 256 \]

### Conclusion
The relative rates of effusion of \( SO_2 \) to \( CH_4 \) after three effusion steps is \( 256:1 \).