Radioactive decay
Nuclear & Particle Physics - OCR A-Level Physics
Worked Example
A radioactive sample has a half-lifeThe time taken for half the number of radioactive nuclei in a sample to decay, or for the activityThe number of nuclear decays per unit time. Measured in becquerels (Bq), where 1 Bq = 1 decay per second. to halve. of 8.0 days. Initially there are 6.0 × 10²⁰ atoms. Calculate (a) the decay constantThe probability of decay of a nucleus per unit time. Measured in s⁻¹., (b) the initial activityThe number of nuclear decays per unit time. Measured in becquerels (Bq), where 1 Bq = 1 decay per second., and (c) the number of atoms remaining after 24 days.
Show Solution
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(a) Convert half-lifeThe time taken for half the number of radioactive nuclei in a sample to decay, or for the activityThe number of nuclear decays per unit time. Measured in becquerels (Bq), where 1 Bq = 1 decay per second. to halve. to seconds
t₁/₂ = 8.0 × 24 × 3600 = 6.91 × 10⁵ s.
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$\lambda = ln 2 / t_{1}/_{2} = 0.693 / 6.91 \times 10⁵ = 1.00 \times 10⁻⁶ s⁻¹.$
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(b) A₀ = λN₀ = 1.00 × 10⁻⁶ × 6.0 × 10²⁰ = 6.0 × 10¹⁴ Bq.
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(c) 24 days = 3 half-lives. After 3 half-lives: $N = N_{0} \times (1/2)^{3} = 6.0 \times 10^{2}⁰ / 8 = 7.5 \times 10¹⁹$.
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Alternatively
N = N₀ e^(-λt) = 6.0 × 10²⁰ × e^(-(1.00 × 10⁻⁶)(24 × 86400)) = 6.0 × 10²⁰ × e^(-2.074) = 7.5 × 10¹⁹.
Answer
(a) λ = 1.0 × 10⁻⁶ s⁻¹. (b) $A_{0} = 6.0 \times 10¹⁴ Bq. (c) N = 7.5 \times 10¹⁹$ atoms.