A vessel contains dimethyl ether at a pressure of 0.4 atm. Dimethyl ether decomposes as `CH_(3)OCH_(3)(g)+CH_(4)(g)+CO(g)+H_(2)(g).` The rate constant of decomposition is `4.78 xx 10^(–3)" min"^(–1)`. Calculate the ratio of initial rate of diffusion to rate of diffusion after 4.5 hours of initiation of decomposition. Assume the composition of gas present and composition of gas diffusing to be same.

Explanation

To solve the problem, we need to calculate the ratio of the initial rate of diffusion to the rate of diffusion after 4.5 hours of decomposition of dimethyl ether. The decomposition reaction is given as:

\[ \text{CH}_3\text{OCH}_3(g) \rightarrow \text{CH}_4(g) + \text{CO}(g) + \text{H}_2(g) \]

### Step 1: Understand the Given Data
- Initial pressure of dimethyl ether, \( P_0 = 0.4 \, \text{atm} \)
- Rate constant, \( k = 4.78 \times 10^{-3} \, \text{min}^{-1} \)
- Time, \( t = 4.5 \, \text{hours} = 4.5 \times 60 \, \text{min} = 270 \, \text{min} \)
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