The number of radioactive nuclei in a particular sample decreases to one-eighth of its original number in 9 days. What is the half-life of these nuclei?

Question

BlackTom AI · Accepted Answer

Radioactive decay follows an exponential law: N = N0 (1/2)^{t/T1/2}.

1. Relevant knowledge points and concepts
- Radioactive decay is exponential and characterized by a half-life T1/2: after time T1/2 the number of nuclei falls to half.
- After n half-lives the remaining fraction is (1/2)^n.
- Equivalently, using decay constant λ: N = N0 e^{-λt} and T1/2 = ln(2)/λ.

2. Complete reasoning process step by step
Given: after t = 9 days the remaining fraction is 1/8 of the original: N/N0 = 1/8.
Use the half-life form: N/N0 = (1/2)^{t/T1/2}. Substitute known values:
1/8 = (1/2)^{9/T1/2}.

3. Calculations with intermediate steps
Recognize 1/8 = 1/2^3, so:
(1/2)^3 = (1/2)^{9/T1/2}.
Therefore exponents are equal: 3 = 9/T1/2.
Solve for T1/2: T1/2 = 9/3 = 3 days.

(Alternative method using natural logs:
1/8 = e^{-λ·9} → ln(1/8) = -λ·9 → -ln(8) = -λ·9 → λ = ln(8)/9 = (3 ln2)/9 = (ln2)/3.
Then T1/2 = ln2/λ = ln2 / ((ln2)/3) = 3 days.)

4. Why the correct answer is correct
The sample has decreased to one-eighth = (1/2)^3, so exactly three half-lives have elapsed in 9 days. Thus one half-life equals 9/3 = 3 days.

5. Why other options are incorrect
- 2 days: would make 9 days correspond to 4.5 half-lives, giving remaining fraction (1/2)^{4.5}, not 1/8.
- 9/8 days (~1.125 days): would give many more than three half-lives in 9 days, wrong.
- 8 days: would give (1/2)^{9/8} remaining, not 1/8.
- None of these: incorrect because a valid option (3 days) matches the condition.

6. Conclusion
The half-life is 3 days, which matches the provided correct answer.

The number of radioactive nuclei in a particular sample decreases to one-eighth of its original number in 9 days. What is the half-life of these nuclei?单项选择题

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