A Geiger counter is used to measure the radioactive disintegrations from a sample of a certain radioisotope. The count rate recorded is shown in the graph belowWhat is the half-life of this sample?

Question

BlackTom AI · Accepted Answer

Radioactive decay follows A(t) = A0 (1/2)^(t/T1/2), so the half-life is the time required for the activity to fall to half its initial value.

1. Relevant knowledge points and concepts
- Exponential decay: A(t) = A0 e^(-λt) = A0 (1/2)^(t/T1/2).
- Definition of half-life T1/2: time when A(t) = A0/2.
- Graph reading: vertical axis is Activity in units of 10^12 Bq; horizontal axis is time in hours.

2. Step-by-step reasoning
- Read the initial activity at t = 0 from the graph: A0 = 5 (×10^12 Bq).
- Compute half of the initial activity: A0/2 = 5/2 = 2.5 (×10^12 Bq).
- Find the time at which the curve equals 2.5 on the plot. The plotted curve crosses 2.5 at t = 10 hours.

3. Calculations and intermediate steps
- Using the exponential form to confirm: if A(10) = 2.5 and A0 = 5,
  A(10)/A0 = 2.5/5 = 0.5 = (1/2) = (1/2)^(10/T1/2).
  Thus (1/2) = (1/2)^(10/T1/2) implies 10/T1/2 = 1, so T1/2 = 10 hours.
- Alternatively with decay constant λ: λ = (1/t) ln(A0/A) = (1/10) ln(2) → T1/2 = ln 2 / λ = 10 h.

4. Why the correct answer is correct
- The graph explicitly shows activity halving from 5 to 2.5 in 10 hours, which is the operational definition of half-life. The exponential formula and decay-constant check both yield T1/2 = 10 h.

5. Why the other choices are incorrect
- 20 hours: would correspond to two half-lives (activity ≈1.25), not a single half reduction to 2.5.
- 15 hours: no matching point on the graph; activity at 15 h is ≈1.6–1.8, not 2.5.
- 5 hours: would be too short; at 5 h the activity is ≈3.5, not half of 5.

Conclusion: The half-life is 10 hours, so the correct answer is (a) 10 hours.

A Geiger counter is used to measure the radioactive disintegrations from a sample of a certain radioisotope. The count rate recorded is shown in the graph belowWhat is the half-life of this sample?单项选择题

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