half-life

Meaning: Half-life (t½) is the time it takes for half of the undecayed nuclei in a sample to decay, which also halves the activity. Decay is random and first-order: each nucleus has the same chance of decaying per unit time, so the decay rate is proportional to the number of nuclei left. That gives exponential decay, and t½ stays the same whether you start with a large or small sample. Temperature and pressure do not change it at MCAT level. The same math describes first-order reactions in General Chemistry.

Equation: N = N₀(1/2)^(t/t½) and t½ = ln 2 / λ ≈ 0.693/λ, where N is the number of undecayed nuclei (unitless count), N₀ is the initial number, t is elapsed time (s), t½ is the half-life (s), and λ is the decay constant (s⁻¹).

On the MCAT: A passage gives a radioisotope used in imaging, therapy, or dating, with a table or decay curve. You find the fraction left after a given time (1/2ⁿ), the time needed to fall to 1/8, or t½ from the time at which the curve reaches half its starting height. A longer t½ means a smaller λ.

Common confusion: Two half-lives do not remove everything. Each half-life removes half of what is left, so 25% remains after two and 12.5% after three.