Half-life of a radioactive substance is $$12.5\,h$$  and its mass is $$256\,g.$$  After what time, the amount of remaining substance is $$1g$$ ?

Solution :
The mass of radioactive substance remained is,
$$M = {M_0}{\left( {\frac{1}{2}} \right)^n}$$
Here, final mass, $$M = 1\,g,$$   initial mass, $${M_0} = 256\,g,$$   half life period, $${T_{\frac{1}{2}}} = 12.5\,h$$
So, $$1 = 256{\left( {\frac{1}{2}} \right)^n}\,\,{\text{or}}\,\,\frac{1}{{256}} = {\left( {\frac{1}{2}} \right)^n}$$
or $${\left( {\frac{1}{2}} \right)^8} = {\left( {\frac{1}{2}} \right)^n}$$
Comparing the powers on both the sides, we get
$$n = 8 = \frac{t}{{{T_{\frac{1}{2}}}}}$$
$$\eqalign{ & \therefore t = 8{T_{\frac{1}{2}}} \cr & = 8 \times 12.5 \cr & = 100\,h \cr} $$