Dimethyl ether decomposes as
`CH_(3)OCH_(3)(g)rarrCH_(4)(g)+CO(g)+H_(2)(g)`
When `CH_(3)OCH_(3)(g)` decomposes to 20% extent under certain conditions, what is the ratio of diffusion of pure `CH_(3)OCH_(3)(g)` with methane?

Options

A. `0.59:1`
B. `1.18:1`
C. `2.36:1`
D. `1.77:1`

Explanation

To solve the problem of finding the ratio of diffusion of pure dimethyl ether (CH₃OCH₃) with methane (CH₄) after the decomposition of dimethyl ether to 20% extent, we can follow these steps:

### Step 1: Determine the moles of reactants and products
Let’s assume we start with \( n \) moles of dimethyl ether (CH₃OCH₃). Given that it decomposes to 20% extent, the amount decomposed is:
\[ \text{Decomposed moles} = 0.2n \]
This means the remaining moles of dimethyl ether after decomposition is:
\[ \text{Remaining moles of CH₃OCH₃} = n - 0.2n = 0.8n \]

### Step 2: Calculate the moles of products formed
From the decomposition reaction:
\[ \text{CH₃OCH₃} \rightarrow \text{CH₄} + \text{CO} + \text{H₂} \]
For every mole of dimethyl ether that decomposes, 1 mole of methane (CH₄), 1 mole of carbon monoxide (CO), and 1 mole of hydrogen (H₂) are produced. Thus, the moles of methane formed from the decomposition of \( 0.2n \) moles of dimethyl ether is:
\[ \text{Moles of CH₄} = 0.2n \]

### Step 3: Set up the diffusion ratio
The rate of diffusion of a gas is inversely proportional to the square root of its molar mass (Graham's law of effusion). The molar masses are:
- Molar mass of CH₄ (methane) = 16 g/mol
- Molar mass of CH₃OCH₃ (dimethyl ether) = 46 g/mol

The ratio of the rates of diffusion of dimethyl ether to methane can be expressed as:
\[ \frac{Rate_{CH₃OCH₃}}{Rate_{CH₄}} = \frac{n_{CH₄}}{n_{CH₃OCH₃}} \cdot \frac{M_{CH₄}^{1/2}}{M_{CH₃OCH₃}^{1/2}} \]
Where:
- \( n_{CH₄} = 0.2n \)
- \( n_{CH₃OCH₃} = 0.8n \)

Substituting the values:
\[ \frac{Rate_{CH₃OCH₃}}{Rate_{CH₄}} = \frac{0.2n}{0.8n} \cdot \frac{16^{1/2}}{46^{1/2}} \]
This simplifies to:
\[ \frac{Rate_{CH₃OCH₃}}{Rate_{CH₄}} = \frac{0.2}{0.8} \cdot \frac{4}{\sqrt{46}} = \frac{1}{4} \cdot \frac{4}{\sqrt{46}} = \frac{1}{\sqrt{46}} \]

### Step 4: Calculate the numerical value
Now, we can calculate the numerical value of the ratio:
\[ \frac{1}{\sqrt{46}} \approx \frac{1}{6.78} \approx 0.147 \]

### Final Ratio
Thus, the ratio of the diffusion of dimethyl ether to methane is approximately:
\[ \text{Ratio} \approx 2.36 \]