Comparing mean lifetimes

Problem

Consider four radioactive isotopes: (A) carbon-14, $\tau_{1/2} \approx 5730\ \text{years}$; (B) iodine-131, $\tau_{1/2} \approx 8\ \text{days}$; (C) technetium-99m, $\tau_{1/2} \approx 6\ \text{hours}$; (D) uranium-238, $\tau_{1/2} \approx 4{,}5 \cdot 10^{9}\ \text{years}$. Rank the isotopes from fastest to slowest decay.

Solution

The decay rate is greater the smaller the half-life $\tau_{1/2}$ (equivalently, the larger the constant $\lambda = \ln 2 / \tau_{1/2}$).

Putting all the times on a common scale: $6\ \text{hours} < 8\ \text{days} < 5730\ \text{years} < 4{,}5 \cdot 10^{9}\ \text{years}$. So the fastest is technetium-99m and the slowest is uranium-238:

$C\ (6\ \text{h}) \ll B\ (8\ \text{d}) \ll A\ (5730\ \text{years}) \ll D\ (4{,}5 \cdot 10^{9}\ \text{years})$

The jumps are enormous: from technetium to uranium we go from hours to billions of years, spanning over twelve orders of magnitude.

$\ev{C \ll B \ll A \ll D}$

Connections

Topics: Nuclear physics Concepts: Radioactive decay Skills: Ranking reasoning