Question
A radioactive element has half-life period $$800\,yr.$$ After $$6400\,yr,$$ what amount will remain ?
A.
$$\frac{1}{2}$$
B.
$$\frac{1}{{16}}$$
C.
$$\frac{1}{8}$$
D.
$$\frac{1}{{256}}$$
Answer :
$$\frac{1}{{256}}$$
Solution :
Number of atoms left after $$n$$ half-lives is given by
$$N = {N_0}{\left( {\frac{1}{2}} \right)^n}\,\,{\text{or}}\,\,\frac{N}{{{N_0}}} = {\left( {\frac{1}{2}} \right)^n}$$
Number of half-lives,
$$\eqalign{
& n = \frac{t}{T} = \frac{{6400}}{{800}} = 8 \cr
& \therefore \frac{N}{{{N_0}}} = {\left( {\frac{1}{2}} \right)^8} = \frac{1}{{256}} \cr} $$
Alternative
Let the initial part be unity
So, after 800 years, it will remain $$ = \frac{1}{2}$$
after 1600 years, it will remain $$ = \frac{1}{4}$$
after 2400 years, it will remain $$ = \frac{1}{8}$$
after 3200 years, it will remain $$ = \frac{1}{16}$$
after 4000 years, it will remain $$ = \frac{1}{32}$$
after 4800 years, it will remain $$ = \frac{1}{64}$$
after 5600 years, it will remain $$ = \frac{1}{128}$$
after 6400 years, it will remain $$ = \frac{1}{256}$$