E.3.22 (HL)—Half-life and decay constant

Use the half-life relation

Half-life and decay constant are related by $T_{1/2}=\frac{\ln2}{\lambda}$. A larger decay constant means a shorter half-life.

Worked example — convert time first

For $T_{1/2}=6.0\,\mathrm{h}=2.16\times10^4\,\mathrm{s}$, $\lambda=0.693/(2.16\times10^4)=3.21\times10^{-5}\,\mathrm{s^{-1}}$. The inverse-second unit matches a probability rate.

Convert units first

If half-life is given in days, hours or years but lambda is required in $\mathrm{s^{-1}}$, convert the time to seconds before dividing $\ln2$ by it.

Check the scale

The product $\lambda T_{1/2}$ should equal approximately 0.693. Use this as a quick unit and order-of-magnitude check.

Common trap

Do not use $1/\lambda$ as the half-life; it is the characteristic time and differs by the factor $\ln2$.