• Tags: decay-law, half-life, mean-life
• Difficulty: Moderate

Radioactive decay is a statistical process. For a large number N of identical nuclei, the decay rate dN/dt = -lambdaN, where lambda is the decay constant (probability of decay per unit time). Solution: N(t) = $N_0$e^(-lambdat). Activity A(t) = |dN/dt| = lambdaN = $A_0$e^(-lambdat). Half-life t_$\frac{1}{2}$ = ln$\frac{2}{lambda}$ = 0.693/lambda is the time for half the nuclei to decay. Mean life (average lifetime) tau = 1/lambda = t_$\frac{1/2}{ln}$(2) = 1.44*t_$\frac{1}{2}$. Physical meaning: if no decay occurred during the mean life, all nuclei would have decayed ($N_0$ = $A_0$*tau, so total decays = $N_0$). After n half-lives: N = $N_0$/2^n. The fraction decayed = 1 - 1/2^n. For non-integer half-lives, use the exponential form. Activity units: 1 Becquerel (Bq) = 1 decay/s; 1 Curie (Ci) = 3.7 x 10^10 Bq (activity of 1 g of Ra-226).