#### 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 ^{99}Tc. 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?

#### Solution

Given :

Time taken by the body to excrete half the amount, t_{1} = 24 hours

Half-life of radioactive isotope, t_{2} = 6 hours

Initial activity, A_{0} = 6 μCi

Let after time t, activity of the sample be A.

Half-life period (`T_"1/2"`) is given by

`T_"1/2" = (t_1t_2)/(t_1 + t_2) = (24 xx 6)/(24 + 6)`

= `(24 xx 6)/30` = 4.8 h

Activity (A) at time t is given by

`therefore A = A_0/2^(t/T_"1/2"`

⇒ `3"μCi" = (6"μCi")/2^(t/4.8)`

⇒ `(6"μCi")/2^(t/4.8) = 3`

⇒ t = 4.8 h