Block 4 Nuclear Science Practice Exam

How is the decay constant λ related to a isotope's half-life T1/2?

λ = T1/2 / ln(2) λ = ln(2) / T1/2 λ = 2 / T1/2 λ = ln(1/2) / T1/2

Correct Answer:

λ = ln(2) / T1/2

Explanation:

Exponential decay is described by N(t) = N0 e^{-λ t}, where λ is the rate at which each nucleus decays. The half-life T1/2 is the time it takes for the quantity to drop to half its initial amount, so at t = T1/2 we have N = N0/2. Plugging in gives 1/2 = e^{-λ T1/2}. Taking natural logs of both sides yields ln(1/2) = -λ T1/2, and since ln(1/2) = -ln 2, we get λ = ln 2 / T1/2. This is the correct relation. Numerically, λ ≈ 0.693 / T1/2. Other forms would produce incorrect signs or values (for example, using ln(1/2)/T1/2 would be negative, which doesn’t fit a positive decay constant; T1/2 / ln 2 or 2 / T1/2 would not match the exponential relation).

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