Question

The half-life of tritium is 12.5 years. What mass of tritium of initial mass $64mg$ will remain undecayed after 50years?
A. $32mg$
B. $8mg$
C. $16mg$
D. $4mg$

Answer 1

Hint: The time required for half the atoms of a given amount of a radioactive substance to disintegrate is known as half-life of a substance.


Formula Used: The formula used for calculating the remaining mass of a radioactive substance is given by, $N\left( t \right) = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{{t_{\dfrac{1}{2}}}}}}}$, $N\left( t \right)$ is remaining mass,${N_0}$ is initial mass, $t$ is the time passed and ${t_{\dfrac{1}{2}}}$ is half-life of the substance.


Complete step by step answer:
Given half-life of tritium is 12.5 years and initial mass is $64mg$,
Therefore,
$N\left( t \right) = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{{t_{\dfrac{1}{2}}}}}}}$
Replace ${t_{\dfrac{1}{2}}} = 12.5$, ${N_0} = 64$ and $t = 50$ in the above equation
$
  N\left( t \right) = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{{t_{\dfrac{1}{2}}}}}}} \\
  \Rightarrow N\left( t \right) = 64{\left( {\dfrac{1}{2}} \right)^{\dfrac{{50}}{{12.5}}}} \\
  \Rightarrow N\left( t \right) = 4 \\
 $
So, the mass which is undecayed after 50 years is $N\left( t \right) = 4mg$.

Additional Information:
A material consisting unstable nuclei is considered as radioactive. Radioactive substances have unstable nuclei and they radiate energy by radiation. Three types of decay are alpha, beta and gamma radiation.
In gamma decay, there is emission of beta or alpha particles and after that the nucleus emits gamma photons.
In beta decay, there are two types of emission: beta-minus decay and beta-plus decay. In beta-minus decay the nucleus is converted from neutron to proton and in beta-plus decay the nucleus is converted from proton into neutron.
Alpha decay, there is emission of alpha particles i.e. helium nucleus.
As the time taken by substance to decay half of its substance is known as half-life therefore, Archeologists use half-life to know the age of organic objects known as carbon dating.

Note: It is important to note that half-life tells us about the time in years after which half of the substance is decayed of the radioactive material. The time taken is always in same unit in the formula $N\left( t \right) = {N_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{{{t_{\dfrac{1}{2}}}}}}}$.