What Is Half-Life and How Do You Calculate Activity/Count Rate Over Time?

Half-life is the time taken for the number of undecayed nuclei, the activity, or the corrected count rate of a radioactive sample to fall to half its previous value. At SL, calculate repeated halving over whole half-lives; at HL, you may also use exponential decay equations.

The Reasoning and Calculation

Radioactive decay is random and spontaneous, but a large sample follows a predictable statistical pattern. Activity is the number of nuclear decays per unit time:

$$A=\frac{\Delta N_{\text{decayed}}}{\Delta t}$$

Its SI unit is the becquerel, where $1\text{ Bq}=1\text{ decay s}^{-1}$. Count rate is the number of events detected per unit time. It is usually lower than the activity because a detector does not record every emitted particle or photon.

For $n$ elapsed half-lives:

$$A=A_0\left(\frac{1}{2}\right)^n,\qquad n=\frac{t}{T_{1/2}}$$

The same relationship applies to corrected count rate. For example, a detector records $860\text{ counts min}^{-1}$, including background radiation of $60\text{ counts min}^{-1}$. The initial net count rate is $800\text{ counts min}^{-1}$. After three half-lives:

$$800\left(\frac{1}{2}\right)^3=100\text{ counts min}^{-1}$$

The expected measured count rate is therefore $100+60=160\text{ counts min}^{-1}$.

At HL, use the decay constant $\lambda$:

$$A=A_0e^{-\lambda t},\qquad T_{1/2}=\frac{\ln 2}{\lambda}$$

Since activity is proportional to the number of undecayed nuclei, both quantities have the same half-life. For example, if $T_{1/2}=4.0\text{ days}$, $A_0=640\text{ Bq}$, and , then and . This method works when the elapsed time is not a whole number of half-lives.

Exam Technique

For E.3 Radioactive decay, state that decay is random but follows an exponential trend. A common misconception is to halve the total measured count rate: always subtract background first, apply the decay calculation, and then add the background back if the question asks for the detector reading.