The half-life for the first-order conversion of cyclobutene to ethylene is 22.7 s at a given temperature. How many seconds are needed for the partial pressure of cyclobutene to decrease from 100 mmHg to 10 mmHg?

• A. 0.101 s
• B. 52.0 s
• C. 75.5 s
• D. 5233 s

Answer: (C)

First-order decay means the amount of cyclobutene decreases exponentially over time, with a constant half-life. The time to reach a certain fraction of the initial amount relates to how many half-lives have passed. To go from 100 mmHg to 10 mmHg is a reduction by a factor of 10, which corresponds to n = log2(10) ≈ 3.322 half-lives. Each half-life is 22.7 s, so the total time is t = n × t1/2 ≈ 3.322 × 22.7 s ≈ 75.5 s.

Equivalently, using the decay equation and k = ln 2 / t1/2 gives t = (ln([A]0/[A])) / k = t1/2 × (ln 10 / ln 2) ≈ 75.5 s.