Law of Radioactive Decay, Half-life and Mean Life (JEE Advanced)

“Law of radioactive decay; Decay constant; Half-life and mean life”

Source: JEE Advanced 2026 Information Bulletin, Annexure-I. Closest JEE Main topic: Composition and size of nucleus, atomic masses.

JEE Main only covers basic nuclear structure, but Advanced expects you to work with the exponential decay law, calculate activities, and solve problems involving half-life and mean life. You need to be comfortable with the mathematics of exponential functions and their applications to decay chains.

• Radioactive decay is statistical: you cannot predict when a single nucleus will decay, but you can predict how many will decay in a large sample
• The decay constant λ represents the probability of decay per unit time for a single nucleus
• Half-life (T₁/₂) is the time for half the nuclei to decay; mean life (τ) is the average lifetime of a nucleus before it decays
• Activity (A) is the number of decays per second, measured in Becquerel (1 Bq = 1 decay/s) or Curie
• After n half-lives, the fraction remaining is (1/2)ⁿ, which is useful for quick calculations
• The relationship T₁/₂ = τ × ln(2) connects half-life and mean life, so τ is always larger than T₁/₂

• dN/dt = −λN (fundamental decay law)
• N(t) = N₀ × e^(−λt) (number of undecayed nuclei at time t)
• A(t) = λN(t) = A₀ × e^(−λt) (activity at time t)
• T₁/₂ = ln(2)/λ = 0.693/λ (half-life)
• τ = 1/λ (mean life)
• T₁/₂ = τ × ln(2) ≈ 0.693τ
• N(t) = N₀ × (1/2)^(t/T₁/₂) (alternate form using half-life)
• Number decayed = N₀ − N(t) = N₀(1 − e^(−λt))

• Confusing half-life and mean life: mean life τ = 1/λ is about 44% larger than half-life T₁/₂ = 0.693/λ
• Forgetting that activity A = λN, not just N; activity also decays exponentially with the same time constant
• Using N = N₀/2 after one mean life instead of one half-life; after one mean life, N = N₀/e ≈ 0.37N₀
• In problems with two decaying species, forgetting that each has its own decay constant and they decay independently

Question. A radioactive sample has a half-life of 20 minutes. Find the ratio of the number of nuclei decayed in the first 20 minutes to the number decayed in the next 20 minutes.

Solved twice independently; both solutions agree.