Half-life

Key ideas

Radioactive decay is random for any individual nucleus, but for a large sample it follows a predictable pattern. The half-life ($t_{1/2}$) is the time taken for half of the radioactive nuclei in a sample to decay.

• Half-life is a constant for a given isotope — it does not change as the sample ages or as conditions (temperature, pressure) change.
• After one half-life, half the original nuclei remain. After two half-lives, a quarter remain. After three, an eighth remain — the amount halves every half-life, it does not fall by equal amounts.

The half-life equation

$N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$

• $N$ — number of radioactive nuclei (or mass of the sample) remaining at time $t$
• $N_0$ — initial number of radioactive nuclei (or initial mass), at $t = 0$
• $t$ — elapsed time
• $t_{1/2}$ — the half-life of the isotope

The same relationship applies to activity (decay rate, in becquerels) or count rate (readings on a detector), since both are proportional to the number of undecayed nuclei present:

$A = A_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$

• $A$ — activity remaining at time $t$
• $A_0$ — initial activity

Reading a decay curve

• A decay graph plots $N$ (or activity/count rate) on the vertical axis against $t$ on the horizontal axis. It starts high and curves down toward zero, never quite reaching it.
• To find the half-life from a graph: read the initial value $N_0$, find half of it on the vertical axis, trace across to the curve, then trace down to read the corresponding time — that time is $t_{1/2}$.
• To check for a constant half-life: pick any starting point on the curve, find the time taken for the value to halve from there — it should be the same $t_{1/2}$ no matter where on the curve you start.

Practice question

A sample starts with a mass of $160$ g of a radioactive isotope with a half-life of $3.0$ hours. Find the mass remaining after $12$ hours.

Worked solution: $n = 12/3.0 = 4$ half-lives. $N = 160 \times (\tfrac{1}{2})^4 = 160/16 = 10$ g.