Question

Gallium citrate, which contains the radioactive nuclide ${}^{67}\mathrm{Ga}$ , is used in medicine as a tumor-seeking agent. Gallium-67 decays with a half-life of 77.9 hours. How much time is required for it to decay to 5.0% of its initial activity?

Short answer

Time required to decay to 5.0% of its initial activity is 336.6 hours.

Step-by-step solution

Decay constant of GA  

The decay constant is calculated from the half-life below.

$$\begin{gathered} k=\frac{0.693}{t_{1/2}} \\ =\frac{0.693}{77.9} \\ =0.0089 hr{s}^{-1} \end{gathered}$$

Fraction of sampling

Fraction of sample remaining =$\frac{5.0}{100}\times A_0=0.05A_0$

Since radioactive decay is first-order reaction, we can calculate the time required by the help of the following expression.

$$\begin{gathered} t=\frac{2.303}{k}\log \frac{A_0}{A_t} \\ =\frac{2.303}{0.0089}\log \frac{A_0}{0.05A_0} \\ =336.6hours \end{gathered}$$