Q31.5-48PE - College Physics (Urone)

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?

Step-by-Step Solution

Answer

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

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,

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

\({}^{{\rm{232}}}{\rm{Th}}\) 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,

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

 

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