Radon-222 has a half-life of 3.82 days. Find its decay constant in h⁻¹…

Step 1. The decay constant. $T = 3.82 \times 24\ \text{h} = 91.7$ h. So $\lambda = \dfrac{0.693}{91.7\ \text{h}} = 7.56 \times 10^{-3}\ \text{h}^{-1}$.

Step 2. The exact value. After $1$ h the fraction decayed is $1 - e^{-0.00756} = 0.00753$, which is $0.753\ \%$.

Step 3. The approximation. $\lambda\,\Delta t = 7.56 \times 10^{-3}\ \text{h}^{-1} \times 1\ \text{h} = 0.00756$, which is $0.756\ \%$.

Why the approximation fails for a long time. For $\Delta t = 91.7$ h, $\lambda\,\Delta t = 0.693$, as if $69\ \%$ had decayed. In fact exactly $50\ \%$ has decayed. The approximation works only when $\lambda\,\Delta t$ is much smaller than $1$, as in the first hour here, where the two answers differ by $0.4\ \%$.