A drug with concentration 0.90 M decomposes via acid-catalysed hydrolysis with a rate constant of 0.031 M−1 h−1. What is the shelf-life of this drug in hours? Report your answer to 2 significant figures.

Question

BlackTom AI · Accepted Answer

For a second-order reaction the integrated rate law and the t90 definition are the key points. The units of k (M−1 h−1) identify second-order kinetics, and "shelf-life" in pharmaceutics normally means t90 (the time for the concentration to fall to 90% of its initial value).

Relevant equations and concepts:
- Second-order integrated law for A → products: 1/[A] − 1/[A0] = k t.
- Define t90 by [A] = 0.90[A0]. Substitute and solve for t.

Step-by-step reasoning and algebra:
1. Start with 1/[A] − 1/[A0] = k t.
2. Put [A] = 0.90[A0]: 1/(0.90[A0]) − 1/[A0] = k t.
3. Factor 1/[A0]: (1/0.90 − 1)·(1/[A0]) = k t.
4. Simplify inside parentheses: 1/0.90 = 1.111... so 1.111... − 1 = 0.111... = 1/9.
   Thus (1/9)·(1/[A0]) = k t.
5. Solve for t: t = 1 / (9 k [A0]).

Plug in the numbers (showing intermediate arithmetic):
- k = 0.031 M−1 h−1, [A0] = 0.90 M.
- k [A0] = 0.031 × 0.90 = 0.0279 h−1.
- 9 k [A0] = 9 × 0.0279 = 0.2511 h−1.
- t = 1 / 0.2511 = 3.9829... hours.

Round to two significant figures: 4.0 hours.

Why this is correct and common errors:
- Using a first-order t90 formula (t = −ln(0.90)/k) would be incorrect because k has units indicating second-order; that would give ≈3.4 h, which is wrong here.
- Using percentage loss instead of the 90% remaining definition or misidentifying the reaction order are other common mistakes.

Conclusion: The calculated shelf-life (t90) is 4.0 hours, matching the provided answer.

A drug with concentration 0.90 M decomposes via acid-catalysed hydrolysis with a rate constant of 0.031 M−1 h−1. What is the shelf-life of this drug in hours? Report your answer to 2 significant figures.Numerical

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