Question

What do you mean by half life? Half life of radium is $1600$ years. After how much time will its $\frac{3}{4}$ part disintegrate?

Answer 1

Half life of a radioisotope is defined as the time by which half of the number of initially presented nuclei decays.
Given : $T_{1/2}=1600years$
Let the initial activity of nucleus be $R_o$.
Fraction of nucleus decayed is $\frac{3}{4}$.
So, fraction of nucleus remained $R=\frac{R_o}{4}$
Using $R=\frac{R_o}{2^n}$
where $n=\frac{t}{T_{1/2}}$
Or $\frac{R_o}{4}=\frac{R_o}{2^n}$
$⟹n=2$
Or $\frac{t}{T_{1/2}}=2$
$⟹t=2T_{1/2}=2\times 1600=3200years$