51. A first order reaction is 87.5% complete at the end of 30 minutes. The half-life of the reaction is minute(s).
First Order Reaction is 87.5% Complete in 30 Minutes – Calculate the Half-Life
In this question, we are told that a first order reaction is 87.5% complete in 30 minutes. The objective is to determine the half-life of the reaction. Although the calculation appears lengthy, it becomes very simple once we understand how the first-order rate equation works and how the percentage completion is converted into the fraction of reactant remaining.
Understanding the Given Information
The reaction is said to be 87.5% complete. This means that 87.5% of the original reactant has already reacted, while only the remaining portion is left unreacted.
Therefore, the percentage of reactant remaining is:
100 − 87.5 = 12.5%
Expressing this as a fraction of the initial concentration:
12.5% = 12.5/100 = 0.125 = 1/8
Thus, after 30 minutes, only one-eighth of the original reactant remains.
First Order Rate Equation
The integrated rate law for a first-order reaction is
k = (2.303/t) log([A]₀/[A]t)
Substituting the given values:
t = 30 minutes
[A]t/[A]₀ = 1/8
Hence,
k = (2.303/30) × log(8)
Since
log 8 = 0.9031
Therefore,
k = (2.303 × 0.9031)/30
k ≈ 0.0693 min⁻¹
Calculating the Half-Life
For every first-order reaction, the half-life is independent of the initial concentration and is given by
t1/2 = 0.693/k
Substituting the calculated value of the rate constant:
t1/2 = 0.693/0.0693
t1/2 = 10 minutes
Alternative Conceptual Method
There is an even faster way to solve this question without using logarithms.
After one half-life, the reactant remaining becomes:
1 → 1/2
After two half-lives:
1/2 → 1/4
After three half-lives:
1/4 → 1/8
The question states that after 30 minutes, only 1/8 of the reactant remains. Since reaching one-eighth requires exactly three half-lives, we have
3 × t1/2 = 30 minutes
Therefore,
t1/2 = 30/3 = 10 minutes
This approach is considerably faster and is highly recommended during competitive examinations.
Explanation of Every Option
Option A: 5 minutes
This option is incorrect because a half-life of 5 minutes would mean that in 30 minutes the reaction would complete six half-lives. The remaining reactant would be (1/2)6 = 1/64, which corresponds to approximately 98.4% completion. This is much greater than the given 87.5% completion.
Option B: 10 minutes
This is the correct answer. Three half-lives reduce the reactant concentration from 1 to 1/8. Since three half-lives occur in 30 minutes, each half-life must be 10 minutes. The same result is obtained using the integrated first-order rate equation.
Option C: 15 minutes
This option is incorrect because two half-lives would occur in 30 minutes. After two half-lives, the remaining reactant would be 1/4, corresponding to only 75% completion. This does not match the information provided in the question.
Option D: 20 minutes
This option is also incorrect. In 30 minutes only one and a half half-lives would have elapsed. The remaining reactant would be much greater than 1/8, so the reaction could not have reached 87.5% completion.
Final Answer
Half-life = 10 minutes
Correct Answer: 10 minutes
Key Concepts Covered
This problem reinforces several important concepts, including the integrated rate law of first-order reactions, the relationship between percentage completion and the fraction of reactant remaining, calculation of the rate constant, the half-life equation, and shortcut techniques based on successive halving. Mastering these ideas enables students to solve a wide range of kinetics problems quickly and confidently.
