The half-life of a radioactive substance is 2 seconds. Calculate the decay constant.

Analyze the question: We're given the half-life (T½ = 2 s) and asked for the decay constant (λ).

Concept: The half-life and decay constant are related by:

λ = ln 2 / T½ = 0.693 / T½

This comes from the decay law N = N₀e^(−λt). At t = T½, N = N₀/2, so ½ = e^(−λT½), which gives λT½ = ln 2.

Solution:

λ = 0.693 / 2 λ = 0.3465 s⁻¹ ≈ 0.347 s⁻¹

Check against the options:

• A (2.890) – too large
• B (1.445) – this is 1/(ln 2 × 1), not what we need
• C (0.576) – incorrect
• D (0.347) – matches our result
• E (0.152) – incorrect

Tip: Memorize λ = 0.693 / T½. Quick estimate: since 0.693 is about 0.7, divide 0.7 by the half-life. Here 0.7 ÷ 2 = 0.35, which is closest to 0.347. Also, don't mix up decay constant with mean life (τ = 1/λ = T½/0.693), which would be 2.885 s here. That's the trap behind option A (2.890).