Gaseous cyclobutane isomerizes to butadiene in a first order process which has `k` value at `153^(@)C` of `3.3 xx 10^(-4) s^(-1)`. How many minutes would it take for the isomerization to proceeds `40%` to completion at this temperature ?

Options

A. `A. 26 min`
B. `B. 52 min`
C. `C. 13 min`
D. D. None of these

Explanation

To solve the problem of how long it takes for gaseous cyclobutane to isomerize to butadiene, we will use the first-order kinetics formula. Here’s a step-by-step breakdown:

### Step 1: Understand the Reaction
The reaction is a first-order process, and we are given that cyclobutane is isomerizing to butadiene. We need to find out how long it takes for the reaction to proceed 40% to completion.

### Step 2: Define Initial and Final Concentrations
- Initial concentration of cyclobutane, \( [A_0] = 100\% \)
- After 40% completion, the remaining concentration of cyclobutane, \( [A] = 100\% - 40\% = 60\% \)

### Step 3: Use the First-Order Rate Equation
The first-order rate equation is given by:
\[ k = \frac{2.303}{t} \log\left(\frac{[A_0]}{[A]}\right) \]
Where:
- \( k = 3.3 \times 10^{-4} \, \text{s}^{-1} \)
- \( [A_0] = 100 \)
- \( [A] = 60 \)

### Step 4: Substitute Values into the Equation
Substituting the values into the equation:
\[ 3.3 \times 10^{-4} = \frac{2.303}{t} \log\left(\frac{100}{60}\right) \]

### Step 5: Calculate the Logarithm
Calculate the logarithm:
\[ \frac{100}{60} = \frac{5}{3} \]
Now, calculate \( \log\left(\frac{5}{3}\right) \):
\[ \log\left(\frac{5}{3}\right) = \log(5) - \log(3) \]
Using known logarithm values:
- \( \log(5) \approx 0.6990 \)
- \( \log(3) \approx 0.4771 \)

Thus,
\[ \log\left(\frac{5}{3}\right) \approx 0.6990 - 0.4771 = 0.2219 \]

### Step 6: Rearranging to Find Time \( t \)
Now, substitute back into the equation:
\[ 3.3 \times 10^{-4} = \frac{2.303}{t} \times 0.2219 \]
Rearranging gives:
\[ t = \frac{2.303 \times 0.2219}{3.3 \times 10^{-4}} \]

### Step 7: Calculate \( t \)
Calculating \( t \):
\[ t = \frac{0.5115}{3.3 \times 10^{-4}} \approx 1549.0 \, \text{s} \]

### Step 8: Convert Seconds to Minutes
To convert seconds to minutes:
\[ \text{Minutes} = \frac{1549.0}{60} \approx 25.8 \, \text{minutes} \]

### Final Answer
Thus, it would take approximately **26 minutes** for the isomerization to proceed 40% to completion.

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