E.3.19 (HL)—Radioactive decay law

Use the exponential law

The number of undecayed nuclei after time $t$ is $N=N_0e^{-\lambda t}$. The same factor applies to the remaining mass when each daughter product is stable and the sample starts pure.

Worked example — arbitrary time

For $N_0=1.0\times10^{10}$, $\lambda=0.0126\,\mathrm{s^{-1}}$ and $t=60\,\mathrm{s}$, $N=(1.0\times10^{10})e^{-(0.0126)(60)}=4.70\times10^9$ nuclei. About $5.30\times10^9$ parent nuclei have decayed.

Find daughter amount

If every parent decay produces one daughter nucleus, the number formed is $N_{daughter}=N_0-N$. Define whether the question asks for remaining parent or accumulated daughter before substituting.

Control units

Use seconds when $\lambda$ is in $\mathrm{s^{-1}}$. Convert minutes, days or years before evaluating the exponential.

Common trap

Do not use $N_0e^{-\lambda t}$ for daughter amount directly; it gives the parent nuclei remaining.