Gasesous cyclobutane isomerizes to butadiene following first order process which has half life of `150.5` minute at certain temperature. How long will take for the process to occur to the extent of `40%` at the same temperature ?

Options

A. 103 minutes
B. 121 minutes
C. 111 minutes
D. None of these

Explanation

To solve the problem of how long it will take for gaseous cyclobutane to isomerize to butadiene to the extent of 40% at a certain temperature, we can follow these steps:

### Step 1: Understand the Reaction and Given Data
We know that the isomerization of cyclobutane to butadiene follows first-order kinetics. The half-life (t₁/₂) of the reaction is given as 150.5 minutes. We need to find the time required for the reaction to proceed to 40% completion.

### Step 2: Calculate the Rate Constant (k)
For a first-order reaction, the rate constant (k) can be calculated using the formula:
\[ k = \frac{0.693}{t_{1/2}} \]

Substituting the given half-life:
\[ k = \frac{0.693}{150.5} \]

Calculating this gives:
\[ k \approx 0.0046 \, \text{min}^{-1} \]

### Step 3: Determine the Initial and Final Concentrations
If we assume the initial concentration of cyclobutane is 100%, then after 40% of the reaction has occurred, the remaining concentration of cyclobutane will be:
\[ \text{Final Concentration} = 100\% - 40\% = 60\% \]

### Step 4: Use the Integrated Rate Law
The integrated rate law for a first-order reaction is given by:
\[ \ln\left(\frac{[A]_0}{[A]}\right) = kt \]

Where:
- \([A]_0\) = initial concentration (100%)
- \([A]\) = final concentration (60%)
- \(t\) = time
- \(k\) = rate constant

Rearranging the equation, we can express it as:
\[ t = \frac{1}{k} \ln\left(\frac{[A]_0}{[A]}\right) \]

### Step 5: Substitute the Values
Substituting the values into the equation:
\[ t = \frac{1}{0.0046} \ln\left(\frac{100}{60}\right) \]

Calculating the natural logarithm:
\[ \ln\left(\frac{100}{60}\right) = \ln(1.6667) \approx 0.511 \]

Now substituting this back into the equation:
\[ t = \frac{1}{0.0046} \times 0.511 \]

Calculating this gives:
\[ t \approx 111.0 \, \text{minutes} \]

### Conclusion
Thus, the time required for the process to occur to the extent of 40% is approximately **111 minutes**.