Question
The half-life of radium is about $$1600\,yr.$$ Of $$100\,g$$ of radium existing now, $$25\,g$$ will remain unchanged after
A.
$$4800\,yr$$
B.
$$6400\,yr$$
C.
$$2400\,yr$$
D.
$$3200\,yr$$
Answer :
$$3200\,yr$$
Solution :
Amount of substance remained is
$$\eqalign{
& M = {M_0}{\left( {\frac{1}{2}} \right)^n}\,\,\left[ {_{{M_0} = \,{\text{initial amount}}}^{M = {\text{ substance remained}}}} \right] \cr
& {\text{Given,}}\,\,{M_0} = 100\,g,M = 25\,g, \cr} $$
Half-life of radioactive substance $${T_{\frac{1}{2}}} = 1600\,yr$$
$$\eqalign{
& {\text{So,}}\,\,25 = 100{\left( {\frac{1}{2}} \right)^n} \cr
& {\text{or}}\,\,\frac{{25}}{{100}} = {\left( {\frac{1}{2}} \right)^n}{\text{or }}{\left( {\frac{1}{2}} \right)^2} = {\left( {\frac{1}{2}} \right)^n} \cr} $$
Comparing the power, we have
$$\eqalign{
& n = 2 \cr
& {\text{or}}\,\,\frac{t}{{{T_{\frac{1}{2}}}}} = 2 \cr
& {\text{or}}\,\,t = 2{T_{\frac{1}{2}}} = 2 \times 1600 \cr
& = 3200\,yr \cr} $$