Question

If \[{N_0}\] is the original mass of the substance of half life period $T_{\frac{1}{2}} = 5 years$, then the amount of substance left after 15 years is
A. \[\dfrac{{{N_0}}}{8}\]
B. \[\dfrac{{{N_0}}}{{16}}\]
C. \[\dfrac{{{N_0}}}{2}\]
D. \[\dfrac{{{N_0}}}{4}\]

Answer 1

Hint: Half life of the substance is the time in which the amount (weight/moles/fraction) of substance becomes half of its initial value. So if the initial amount of the substance with half life of 5 years is \[{N_0}\]then after 5 years the remaining amount will be \[\dfrac{{{N_0}}}{2}\].


Formula used:
\[\dfrac{N}{{{N_0}}} = {\left( {\dfrac{1}{2}} \right)^n}\]
Here, N = Remaining amount of substance \[{N_0} = \]Initial amount of substance, ${n = \dfrac{t}{T_{\frac{1}{2}}}}$



Complete answer:
Given here is a substance of initial amount with half life $T_{\frac{1}{2}} = 5 years$, we have to calculate the amount of the substance remaining after time t = 15 years.
As we know that, \[\dfrac{N}{{{N_0}}} = {\left( {\dfrac{1}{2}} \right)^n}\,.....(1)\]
And ${n = \dfrac{t}{T_{\frac{1}{2}}}}$
=> $n = \dfrac{15}{5} = 3$
Substituting n=3 in equation (1) we get,
\[\begin{array}{l}\dfrac{N}{{{N_0}}} = {\left( {\dfrac{1}{2}} \right)^3}\\\dfrac{N}{{{N_0}}} = \dfrac{1}{8} \Rightarrow N = \dfrac{{{N_0}}}{8}\end{array}\]
Hence, the remaining amount of substance after 15 years is\[\dfrac{{{N_0}}}{8}\].
Therefore, option A is the correct option.


Note: Half life of radioactive substances is useful in many ways, in case of nuclear waste it can be used to determine the time period after which it can be disposed safely and it also gives time up-to which the material will be active for use in medical or experimental facilities.