Q.5.89 - An Introduction to General, Organic, and Biological Chemistry

Question

A nuclear technician was accidentally exposed to potassium-42 while doing brain scans for possible tumours. The error was not discovered until 36 h later when the activity of the potassium-42 sample was 2.0 $\mu$Ci. If potassium- 42 has a half-life of 12 h, what was the activity of the sample at the time the technician was exposed? 

Step-by-Step Solution

Answer

The activity of the sample at the time the technician was exposed ${\mathrm{N}}_0=16 \mu \mathrm{Ci}$

Step 1: Given Information

The activity of the potassium-42 sample =$2.0 \mu Ci$

The half-life of potassium- 42 = $12 h$

Step 2: Explanation

Given,

The activity of the potassium-$42$ sample is $\mathrm{N}=2.0 \mu \mathrm{Ci}$

The half-life is

$$\begin{gathered} {\mathrm{T}}_{1/2}=12 \mathrm{h} \\ \mathrm{t}=36 \mathrm{h} \end{gathered}$$

Using,

$$\begin{gathered} \frac{N\left(t\right)}{N_0}={\left(\frac{1}{2}\right)}^{\frac{t}{T_1}} \\ \frac{{\displaystyle 2}}{{\displaystyle {\mathrm{N}}_0}}={\left(\frac{{\displaystyle 1}}{{\displaystyle 2}}\right)}^{\frac{{\displaystyle 36}}{{\displaystyle 12}}} \\ \Rightarrow {\mathrm{N}}_0=16 \mu \mathrm{Ci} \end{gathered}$$

The activity of the sample at the time the technician was exposed is $16 \mu Ci$.