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 by 40% at a temperature of 153°C, we will use the first-order kinetics formula. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Given Data
- The reaction is first-order.
- The rate constant \( k = 3.3 \times 10^{-4} \, \text{s}^{-1} \).
- The extent of reaction completion is 40%.

### Step 2: Define the Variables
- Let \( A \) be the initial concentration of cyclobutane. We can assume \( A = 100 \) (this is arbitrary and simplifies calculations).
- Let \( x \) be the amount of cyclobutane that has reacted. Since the reaction proceeds 40% to completion, \( x = 40 \).
- The remaining concentration of cyclobutane will be \( A - x = 100 - 40 = 60 \).

### Step 3: Use the First-Order Kinetics Formula
The formula for a first-order reaction is given by:
\[ k = \frac{2.303}{t} \log \left( \frac{A}{A - x} \right) \]
Substituting the known values into the formula:
\[ 3.3 \times 10^{-4} = \frac{2.303}{t} \log \left( \frac{100}{60} \right) \]

### Step 4: Calculate the Logarithm
Calculate the logarithm:
\[ \log \left( \frac{100}{60} \right) = \log(1.6667) \approx 0.2198 \]

### Step 5: Rearranging the Equation to Solve for Time \( t \)
Now, rearranging the equation to solve for \( t \):
\[ t = \frac{2.303 \times 0.2198}{3.3 \times 10^{-4}} \]

### Step 6: Calculate \( t \)
Calculating the numerator:
\[ 2.303 \times 0.2198 \approx 0.506 \]
Now substituting back into the equation for \( t \):
\[ t \approx \frac{0.506}{3.3 \times 10^{-4}} \approx 1538.79 \, \text{s} \]

### Step 7: Convert Seconds to Minutes
To convert seconds to minutes:
\[ t \approx \frac{1538.79}{60} \approx 25.65 \, \text{minutes} \]

### Step 8: Rounding to the Nearest Whole Number
Rounding \( 25.65 \) to the nearest whole number gives approximately \( 26 \) minutes.

### Final Answer
It would take approximately **26 minutes** for the isomerization to proceed 40% to completion at 153°C.

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