Chapter 40

The most common isotope of uranium, ${}_{92}^{238}\mathrm{U},$ produces radon ${}_{86}^{222}\mathrm{Rn}$ through the following sequence of decays: $$\begin{array}{c}{}^{238}\mathrm{U}\to {}^{234}\mathrm{Th}+\alpha ,{}^{234}\mathrm{Th}\to {}^{234}\mathrm{Pa}+{\beta }^{-}+{\overline{\nu }}_e,\\ \\ {}_{91}^{234}\mathrm{Pa}\to {}_{92}^{234}\mathrm{U}+\beta +{\overline{\nu }}_e,{}^{234}\mathrm{U}\to {}^{230}\mathrm{Th}+\alpha ,\\ \\ {}_{91}^{230}\mathrm{Th}\to {}_{90}^{226}\mathrm{Ra}+\alpha ,{}_{88}^{226}\mathrm{Ra}\to {}_{86}^{222}\mathrm{Rn}+\alpha ,\end{array}$$. A sample of ${}_{92}^{238}\mathrm{U}$ will build up equilibrium concentrations of its daughter nuclei down to ${}_{88}^{226}\mathrm{Ra};$ the concentrations of each are such that each daughter is produced as fast as it decays. The ${}_{88}^{226}\mathrm{Ra}$ decays to ${}_{86}^{222}\mathrm{Rn},$ which escapes as a gas. (The $\alpha$ particles also escape, as helium; this is a source of much of the helium found on Earth.) In high concentrations, the radon is a health hazard in buildings built on soil or foundations containing uranium ores, as it can be inhaled. a) Look up the necessary data, and calculate the rate at which $1.00\mathrm{kg}$ of an equilibrium mixture of ${}_{92}^{238}\mathrm{U}$ and its first five daughters produces ${}_{86}^{222}\mathrm{Rn}$ (mass per unit time). b) What activity (in curies per unit time) of radon does this represent?