Question

A human body excretes (removes by waste discharge, sweating, etc.) certain materials by a law similar to radioactivity. If technetium is injected in some form in a human body, the body excretes half the amount in 24 hours. A patient is given an injection containing ⁹⁹Tc. This isotope is radioactive with a half-life of 6 hours. The activity from the body just after the injection is 6 μCi. How much time will elapse before the activity falls to 3 μCi?

Answer 1

Given:
Time taken by the body to excrete half the amount, t₁ = 24 hours
Half-life of radioactive isotope, t₂ = 6 hours
Initial activity, A₀ = 6 μCi
Let after time t, activity of the sample be A.
Half-life period $\left(T_{1/2}\right)$ is given by
$$\begin{gathered} T_{1/2}=\frac{t_1{t}_2}{t_1+t_2}=\frac{24\times 6}{24+6} \\ =\frac{24\times 6}{30}=4.8\mathrm{h} \end{gathered}$$
Activity $\left(A\right)$ at time t is given by
$$\begin{gathered} \therefore \mathrm{A}=\frac{{\mathrm{A}}_0}{2^{t/T_{1/2}}} \\ \Rightarrow 3\mathrm{\mu C}i=\frac{6\mathrm{\mu Ci}}{2^{t/4.8}} \\ \Rightarrow \frac{6\mathrm{\mu Ci}}{2^{t/4.8}}=3 \\ \Rightarrow t=4.8\mathrm{h} \end{gathered}$$