Question

If $8g$ of a radioactive substance decays into $0.5g$ in $1h$, then the half-life of the substance is:

• A. $45min$
• B. $15min$
• C. $10min$
• D. $30min$

Answer 1

The correct option is B $15min$

Half-life of a radioactive element is defined as the time during which half the number of atoms present initially in the sample of the element decay or it is time during which number of atoms left undecayed in the sample is half the total number of atoms present initially in the sample. It is denoted by $T$.
In $t=T$
$N=\frac{N_0}{2}$
In another half-life
$N=\frac{1}{2}\frac{N_0}{2}=\frac{N_0}{4}$
After yet another half-life
$N=\frac{1}{2}\left(\frac{N_0}{4}\right)=\frac{N_0}{4}$
$=N_0{\left(\frac{1}{2}\right)}^3$ and so on
Hence, after $n$ half-lives
$N=N_0{\left(\frac{1}{2}\right)}^n=N_0{\left(\frac{1}{2}\right)}^{t/T}$
$\therefore 0.5=8{\left(\frac{1}{2}\right)}^n$
or $\frac{0.5}{8}={\left(\frac{1}{2}\right)}^n$
or ${\left(\frac{1}{2}\right)}^4={\left(\frac{1}{2}\right)}^n$
$\therefore n=4$
As, $n=\frac{t}{T}$
$\therefore T=\frac{t}{n}=\frac{1\times 60}{4}$
$=15min$