Radioactive decay and half-lifeAQA A-Level Physics: Revision notes

Radioactive decay is random: it is impossible to predict when a particular nucleus will decay, and decay is not affected by temperature, pressure or chemical state. Every nucleus of a given isotope has the same constant probability of decay per unit time, called the decay constant $\lambda$ (unit s⁻¹).

The number of decays in a short time $\Delta t$ is proportional to the number of undecayed nuclei $N$:

$\frac{\Delta N}{\Delta t} = -\lambda N$

The activity $A$ is the number of decays per second (unit becquerel, Bq): $A = \lambda N$.

Solving $\frac{\Delta N}{\Delta t} = -\lambda N$ gives an exponential decrease:

$N = N_0e^{-\lambda t}$

Because $A = \lambda N$, activity (and the corrected count rate) follows the same law:

$A = A_0e^{-\lambda t}$

If a question gives mass, use the molar mass and the Avogadro constant: $N = \frac{m}{M} \times N_A$.

The half-life $T_{1/2}$ is the time taken for the number of undecayed nuclei (and the activity) to fall to half its value. Putting $N = N_0/2$ into $N = N_0e^{-\lambda t}$ gives

$T_{1/2} = \frac{\ln 2}{\lambda}$

A large $\lambda$ means a short half-life. After $n$ half-lives the fraction left is $(1/2)^n$.

A graph of $N$ or $A$ against $t$ is an exponential curve: the half-life can be read off several times and is constant.

Taking logs of $A = A_0e^{-\lambda t}$:

$\ln A = \ln A_0 - \lambda t$

A graph of $\ln A$ against $t$ is a straight line with gradient $-\lambda$ and intercept $\ln A_0$. Then $T_{1/2} = \ln 2/\lambda$. This is more accurate than reading half-lives from the curve.

Decay can be modelled with dice or a spreadsheet using a constant probability of decay for each nucleus in each time step. If each of $N$ nuclei has probability $p = \lambda\Delta t$ of decaying in a step, then about $pN$ decay. With small numbers the results fluctuate randomly around the exponential curve, which shows decay is a random process; with large numbers the curve is smooth.

Radioactive dating: living things take in carbon-14 at a constant proportion, and when they die the activity falls with a half-life of 5730 years. Comparing the activity with that of living material gives the age: $t = \frac{1}{\lambda}\ln\frac{A_0}{A}$. It assumes the atmospheric proportion has been constant.

Waste storage: short half-life isotopes are very active but fall quickly, so can be stored temporarily; long half-life isotopes stay active for thousands of years and need secure long-term storage deep underground.

Worked example: a sample has $4.0\times10^{12}$ nuclei and $\lambda = 5.0\times10^{-5}$ s⁻¹. $A = \lambda N = 2.0\times10^{8}$ Bq and $T_{1/2} = 0.693/5.0\times10^{-5} = 1.4\times10^{4}$ s.