Question

Question: Compare the relative health risks of contact with a given amount of ${}^{\mathbf{226}}\mathbf{Ra}$, which has a half-life of $\mathbf{1622}$years and emits $\mathbf{4}\mathbf{.}\mathbf{78}\mathbf{}\mathbf{MeV}$alpha particles, with the same amount of ${}^{\mathbf{14}}\mathbf{C}$which has a half-life of $\mathbf{5730}$years and emits beta particles with energies upto $\mathbf{0}\mathbf{.}\mathbf{155}\mathbf{}\mathbf{MeV}$.

Short answer

${}^{226}\mathrm{Ra}$poses higher health risks than ${}^{14}\mathrm{C}$.

Step-by-step solution

Decay constant of C14

The decay constant can be calculated from the half-life of with the help of the following expression:

$\begin{array}{l}{\mathrm{t}}_{1/2}=\frac{0.693}{\mathrm{k}}\\ \mathrm{k}=\frac{0.693}{5730}{\mathrm{yr}}^{\mathrm{\_}1}\\ \mathrm{k}=1.2\times {10}^{-4}{\mathrm{yr}}^{-1}\end{array}$

Decay constant of Ra226

he decay constant can be calculated from the half-life of with the help of the following expression:

$\begin{array}{l}{\mathrm{t}}_{1/2}=\frac{0.693}{\mathrm{k}}\\ \mathrm{k}=\frac{0.693}{1622}{\mathrm{yr}}^{\mathrm{\_}1}\\ \mathrm{k}=4.2\times {10}^{-4}{\mathrm{yr}}^{-1}\end{array}$

Comparison of the relative health risks

Decay constant of ${}^{14}\mathrm{C}$localid="1660979550957" $=1.2\times {10}^{-4}{\mathrm{yr}}^{-1}$

Energy of beta particles emitted by${}^{14}\mathrm{C}$localid="1660979573400" $=0.155\mathrm{MeV}$

Decay constant of ${}^{226}Ra$localid="1660979590416" $=4.2\times {10}^{-4}{\mathrm{yr}}^{-1}$

Energy of alpha particles emitted by${}^{226}Ra$localid="1660979606599" $=4.78\mathrm{MeV}$

From the above data it is evident that decays faster and emits alpha particles of higher energy as compared to the beta particles emitted by. Thus poses higher health risks.