Question

The half life period of ${}^{234}U$ is $2.5\times {10}^5$ years. In how much time is the quantity of an isotope reduced to $25$% of the original amount?

Answer 1

Initial amount of this isotope $N_0=100$
Final amount of the isotope $N=25$
We know that, $N={\left(\frac{1}{2}\right)}^n{N}_0$
So, $25={\left(\frac{1}{2}\right)}^n\times 100$
or $\frac{25}{100}={\left(\frac{1}{2}\right)}^n$
or $\frac{1}{4}={\left(\frac{1}{2}\right)}^n$
or ${\left(\frac{1}{2}\right)}^2={\left(\frac{1}{2}\right)}^n$
or $n=2$
Time taken $T=n\times t_{1/2}$
$=2\times 2.5\times {10}^5=5\times {10}^5$ years