Question

The half-life of a radioactive substance is 3 second. Initially it has 8000atoms. Find (1) its decay constant and (2) time after which 1000 atoms will remain undecayed.

Answer 1

Recall the expression for the decay constant of substance in terms of the half life of the material and then you could simply substitute the given values to get the answer for the first part. Now find the expression for the ratio of number of remaining atoms to the initial number of atoms in terms of number of half lives. The number of half lives multiplied by the half life of the material gives you the required time.

Formula used:
Expression for half-life,
${{T}_{\dfrac{1}{2}}}=\dfrac{0.693}{\lambda }$

Complete step-by-step answer:
We are given the half-life of a radioactive substance as 3seconds. That is,
${{T}_{\dfrac{1}{2}}}=3s$ ……………………………….. (1)
The initial number atoms is also given,
${{N}_{0}}=8000atoms$ ………………………………… (2)
In the first part we are asked to find the decay constant of this substance. Let us recall what exactly decay constant is. For that let us recall the law of radioactive decay which states that the number of nuclei undergoing decay per unit time is proportional to the total number of nuclei in the sample. That is,
$\dfrac{\Delta N}{\Delta t}\propto N$
$\Rightarrow \dfrac{\Delta N}{\Delta t}=\lambda N$
Where, ‘$\lambda$’ is the radioactive disintegration constant or decay constant.
But we have a relation connecting half life of a substance with its decay constant that is given by,
${{T}_{\dfrac{1}{2}}}=\dfrac{\ln 2}{\lambda }$
$\Rightarrow {{T}_{\dfrac{1}{2}}}=\dfrac{0.693}{\lambda }$
Now substituting (1) gives,
$\lambda =\dfrac{0.693}{3}$
$\Rightarrow \lambda =0.231{{s}^{-1}}$
Therefore, the decay constant is found to be $0.231{{s}^{-1}}$
For the second part we are asked to find the time taken (t) after which the remaining number of atoms to be decayed will be 1000.
We have the relation,
$\dfrac{N}{{{N}_{0}}}={{\left( \dfrac{1}{2} \right)}^{n}}$ ……………………………………. (3)
Where, N is the number of atoms that remains after time t, ${{N}_{0}}$is the total number of atoms in the sample initially and n is the number of half lives. Also, t is given by,
$t=n\times {{T}_{\dfrac{1}{2}}}$ …………………………… (4)
Substituting (2) in (3),
$\dfrac{1000}{8000}={{\left( \dfrac{1}{2} \right)}^{n}}$
$\Rightarrow n=3$
Now (4) becomes,
$t=3\times 3=9s$
Therefore, the time after which 1000 atoms will remain undecayed is 9s.

Note: The process by which an unstable atomic nucleus loses energy by radiation is radioactive decay. Radioactive material is that material that contains unstable nuclei. Alpha decay, beta decay and gamma decay are the most common types of decay. When the proton number changes during a decay, an atom of different chemical elements is created.