Question

Mantles for gas lanterns contain thorium, because it forms an oxide that can survive being heated to incandescence for long periods of time. Natural thorium is almost \(100\% \)\({}^{232}Th\) , with a half-life of \(1.405 \times {10^{10}}\,{\rm{y}}\). If an average lantern mantle contains \(300\,{\rm{mg}}\) of thorium, what is its activity?

Short answer

The activity is \(1.2 \times {10^3}\,{\rm{Bq}}\).

Step-by-step solution

Define radioactivity

Radioactivity is a phenomenon in which a few substances spontaneously release energy and subatomic particles. The nuclear instability of an atom causes radioactivity.

Explanation

The molar mass is \(M = 232\,{\rm{g}}\) of \({}^{{\rm{232}}}{\rm{Th}}\). As a result, the amount of \({\rm{Th}}\)atoms in \(m = 300\,{\rm{mg}}\) is,

\(\begin{align}{}N & = \frac{m}{M}{N_A}\\ & = \frac{{300 \times {{10}^{ - 3}}\,{\rm{g}}}}{{232\,{\rm{g}}}}(6.02 \times {10^{23}}\,{\rm{atoms}})\\ & = 7.78 \times {10^{20}}\,{\rm{atoms}}\end{align}\)

\({}^{{\rm{232}}}{\rm{Th}}\) Has a half-life of\({t_{1/2}} = 1.405 \times {10^{10}}\,{\rm{y}} = 4.43 \times {10^{17}}\,{\rm{s}}\). As a result, the activity is,

\(\begin{align}{}R & = \frac{{0.693N}}{{{t_{1/2}}}}\\ & = \frac{{0.963(7.78 \times {{10}^{20}}\,{\rm{atoms}})}}{{4.47 \times {{10}^{17}}\,{\rm{s}}}}\\ & = 1.2 \times {10^3}\,{\rm{atoms/s}}\\ & = 1.2 \times {10^3}\,{\rm{Bq}}\end{align}\)

Therefore, the activity is \(1.2 \times {10^3}\,{\rm{Bq}}\).