The most common isotope of uranium, \({ }_{92}^{238} \mathrm{U},\) produces
radon \({ }_{86}^{222} \mathrm{Rn}\) through the following sequence of decays:
$$\begin{array}{c}{ }^{238} \mathrm{U} \rightarrow{ }^{234}
\mathrm{Th}+\alpha,{ }^{234} \mathrm{Th} \rightarrow{ }^{234}
\mathrm{~Pa}+\beta^{-}+\bar{\nu}_{e}, \\\\{ }_{91}^{234} \mathrm{~Pa}
\rightarrow{ }_{92}^{234} \mathrm{U}+\beta+\bar{\nu}_{e},{ }^{234} \mathrm{U}
\rightarrow{ }^{230} \mathrm{Th}+\alpha ,\\\\{ }_{91}^{230} \mathrm{Th}
\rightarrow{ }_{90}^{226} \mathrm{Ra}+\alpha,{ }_{88}^{226} \mathrm{Ra}
\rightarrow{ }_{86}^{222} \mathrm{Rn}+\alpha,\end{array}$$. 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?
