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Step-by-Step Solution
Step 1: Identify the Reaction Order
The question explicitly states that the isomerization of cyclobutene to butadiene is a first-order reaction.
Step 2: Recognize the Relevant First-Order Formula
For a first-order reaction, if the fraction converted (reacted) is x , then the fraction remaining is (1 - x) . The integrated rate law for a first-order reaction can be written as:
\displaystyle \ln\!\bigl(\frac{[A]_0}{[A]}\bigr) = kt,
where [A]_0 is the initial concentration and [A] is the concentration at time t . In terms of the fraction remaining, this becomes:
\displaystyle \ln\!\Bigl(\frac{1}{1 - x}\Bigr) = kt.
Step 3: Plug In the Known Values
• Fraction reacted, x = 0.40 (i.e. 40% completion)
• Therefore, fraction remaining, 1 - x = 1 - 0.40 = 0.60
• Rate constant, k = 3.3 \times 10^{-4}\,\text{s}^{-1}
Step 4: Calculate the Time in Seconds
Use the formula for first-order kinetics:
\displaystyle t = \frac{1}{k}\,\ln\!\Bigl(\frac{1}{1 - x}\Bigr).
Substituting the values, we get:
\displaystyle t \;=\; \frac{1}{3.3 \times 10^{-4}}\;\ln\!\Bigl(\frac{1}{0.60}\Bigr).
Evaluate the logarithm:
\displaystyle \ln\!\Bigl(\frac{1}{0.60}\Bigr) = \ln\!(1.6667)\approx 0.51.
Hence,
\displaystyle t \;=\; \frac{0.51}{3.3 \times 10^{-4}} \approx 1545\,\text{s}.
Step 5: Convert Time to Minutes and Round Off
Since 1\,\text{minute} = 60\,\text{seconds} , convert the time from seconds to minutes:
\displaystyle t_{\text{minutes}} = \frac{1545}{60} \approx 25.8 \,\text{minutes}.
Rounded to the nearest integer, this is approximately 26 minutes.
Answer
The time required for 40% completion of the isomerization is about 26 minutes.
