Question

A radioactive material has an initial mass of $16g$. After $120$ days it reduces to $0.5g$. Half-life of the radioactive elements is:

• A. $30$ days
• B. $24$ days
• C. $20$ days
• D. $18$ days

Answer 1

The correct option is B $24$ days

The amount of material after $t$ day $Ni=16g$ (initial amount)
$N=N_i{e}^{-kt}$ where $k=$ decay constant $=\frac{ln\left(2\right)}{\frac{t_1}{2}}$
$\frac{N}{N_i}=e^{-kt}⟹ln\left(e^{-kt}\right)=ln\left(\frac{N}{N_i}\right)$
or $-kt=ln\left(\frac{N}{N_i}\right)$
or $k=\frac{-1}{t}ln\left(\frac{N}{N_i}\right)⟹\frac{ln\left(2\right)}{\frac{t_2}{2}}=\frac{-1}{t}ln\left(\frac{N}{N_i}\right)$
or $\frac{t_1}{2}=\frac{-tln\left(2\right)}{ln\left(\frac{N}{N_i}\right)}$
when $t=120,N=0.5g$
So, $\frac{t_1}{2}=\frac{120\times ln\left(2\right)}{ln\left(\frac{0.5}{16}\right)}=24$ days.