Question
The radio isotope, tritium $$\left( {_1{H^3}} \right)$$ has a half-life of $$12.3$$ $$yr.$$ If the initial amount of tritium is $$32$$ $$mg,$$ how many milligrams of it would remain after $$49.2$$ $$yr?$$
A.
4$$\,mg$$
B.
8$$\,mg$$
C.
1$$\,mg$$
D.
2$$\,mg$$
Answer :
2$$\,mg$$
Solution :
$$\eqalign{
& {t_{\frac{1}{2}}} = 12.3\,yr \cr
& {\text{Initial amount}}\left( {{N_0}} \right) = 32\,mg \cr
& {\text{Amount left}}\left( N \right) = ? \cr
& {\text{Total time}}\left( T \right) = 49.2\,yr \cr
& \frac{N}{{{N_0}}} = {\left( {\frac{1}{2}} \right)^n} \cr
& {\text{where, }}n = {\text{total number of half - lives}} \cr
& n = \frac{{{\text{Total time}}}}{{{\text{Half - life}}}} \cr
& \,\,\,\,\, = \frac{{49.2}}{{12.3}} \cr
& \,\,\,\,\, = 4 \cr
& {\text{So,}}\,\,\,\frac{N}{{{N_0}}} = {\left( {\frac{1}{2}} \right)^n} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\frac{N}{{32}} = {\left( {\frac{1}{2}} \right)^4} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\frac{N}{{32}} = \frac{1}{{16}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,N = \frac{{32}}{{16}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 2\,mg \cr} $$