Question

The decay constant of a radioisotope is $\lambda$ . If ${A_1}$​ and ${A_2}$​ are its activities at times ${t_1}$ and ${t_2}$ respectively, the number of nuclei which have decayed during the time $\left( {{t_1} - {t_2}} \right)$

$$A.{\text{ }}{A_1}{t_1} - {A_2}{t_2} \\\ B.{\text{ }}{A_1} - {A_2} \\\ C.{\text{ }}\left( {{A_1} - {A_2}} \right)/\lambda \\\ D.{\text{ }}\lambda \left( {{A_1} - {A_2}} \right) \\\$$

Answer 1

In order to solve the given problem and to find the number of nuclei that decays between the given time period; First we will understand the meaning of decay constant of a radioisotope and further we will see the relation between the activity, time period and the number of nuclei remaining of the isotope. We will consider the number of nuclei left at the two given time periods in terms of some unknown variable and further by the help of the relation we will find the difference in terms of known variables and will select the correct option amongst the given options.

Complete step by step answer:
Decay constant: The continuous ratio of radionuclide’s number of atoms of atoms decaying over a given period of time relative to the cumulative number of the same form present at the beginning of that period. The law on nuclear decay states that the probability that a nucleus will decay per unit time is constant, regardless of time. This constant is called the constant of decay and is referred to as $\lambda$ “lambda”.
Given that $\lambda$ is the decay constant. And ${A_1}$​ and ${A_2}$​ are its activities at times ${t_1}$ and ${t_2}$ respectively.
To find the number of nuclei which have decayed during the time $\left( {{t_1} - {t_2}} \right)$ .
Let the number of nuclei at time ${t_1}$ and at time ${t_2}$ is ${N_1}$ and ${N_2}$ respectively.
As we know the relation between the activity A and number of nuclei “N” at that time is given as:
$A = \lambda N$
Let us use the above formula to find the number of nuclei at time ${t_1}$ and at time ${t_2}$ respectively.
$\Rightarrow {A_1} = \lambda {N_1} - - - - (1) \\\ \Rightarrow {A_2} = \lambda {N_2} - - - - (2) \\\$
Now we have two relations. And we have to find the nuclei decayed between the time ${t_1}$ and at time ${t_2}$ respectively.
So we have to find the difference of ${N_1}$ and ${N_2}$ respectively.
Let us subtract equation (2) from equation (1) to find the nuclei decayed in the given time period as a function of known variables.
$\Rightarrow \lambda {N_1} - \lambda {N_2} = {A_1} - {A_2} \\\ \Rightarrow \lambda \left( {{N_1} - {N_2}} \right) = {A_1} - {A_2} \\\ \Rightarrow \left( {{N_1} - {N_2}} \right) = \left( {\dfrac{{{A_1} - {A_2}}}{\lambda }} \right) \\\$
Hence, the number of nuclei which have decayed during the time $\left( {{t_1} - {t_2}} \right)$ is $\left( {\dfrac{{{A_1} - {A_2}}}{\lambda }} \right)$ .
So, the correct answer is “Option C”.

Note: In order to solve such types of problems students must consider the unknown terms as variables and should proceed mathematically. Students must remember the basic definition and these relations for solving such problems. This constant of decay will vary for different types of nuclei and so different radioactive isotopes will have different decline periods. These decline constants depend upon various environmental factors such as temperature, pressure etc.