Law of Radioactive Decay - JEE

The term radioactivity was introduced by A.H. Becquerel in 1896 purely by accident. Radioactivity defines a phenomenon in which an unstable nucleus undergoes decay and this was observed accidentally by him during an experiment. The story started when while studying fluorescence and phosphorescence of compounds irradiated with visible light, Becquerel observed something interesting. 

He wrapped some pieces of uranium-potassium sulphate in black paper after illuminating them with visible light and separated the package from a photographic plate by a piece of silver. After several hours of exposure, when the photographic plate was developed, it showed blackening due to something that must have been emitted by the compound and was able to penetrate both black paper and the silver.

What is Radioactive Decay?

The phenomenon in which an unstable radioactive nucleus undergoes a decay is referred to as radioactive decay. Radioactive decays are classified into three categories which are explained in the next heading.

Types of Radioactive Decay

There are basically three types of radioactive decay:-

1. Alpha (\[\alpha \])- decay 

In alpha decay, a helium nucleus ${}_{2}^{4}He$ is emitted.

2. Beta ($\beta $)- decay

In beta decay, electrons and positrons (particles having the same mass as electrons but opposite charge) are emitted.

3. Gamma ($\gamma $)- decay

In gamma decay, high-energy photons are emitted.

Laws of Radioactive Decay

According to the law of radioactive decay, when any radioactive sample undergoes $\alpha ,\beta \text{ or }\gamma $- decay, the number of nuclei undergoing the decay per unit of time in that radioactive sample is proportional to the total number of nuclei in the sample. The law of radioactive decay is given by the formula,

$\frac{\Delta N}{\Delta t}\propto N$

Or, $\frac{\Delta N}{\Delta t}=\lambda N$,

where $\lambda $is the radioactive decay constant, $N$ is the number of nuclei in the sample, and $\Delta N$undergoes decay in the time $\Delta t$.

Let’s assume the change in the number of nuclei in the sample is $dN=-\Delta N$ in time \[\Delta t\]. Thus, the rate of change of $N$ is (in the limit $\Delta t\to 0$)

$\frac{dN}{dt}=-\lambda N$

Or, $\frac{dN}{N}=-\lambda dt$

On, integrating both sides of the above equation, we get,

\[\int\limits_{{{N}_{o}}}^{N}{\frac{dN}{N}=-\lambda \int\limits_{{{t}_{o}}}^{t}{dt}}\]

Or, $\ln N-\ln {{N}_{o}}=-\lambda (t-{{t}_{o}})$

(${{N}_{o}}$ is the number of radioactive nuclei in the sample at some arbitrary time ${{t}_{o}}$ and $N$ is the number of radioactive nuclei at any subsequent time $t$) 

Setting ${{t}_{o}}=0$and rearranging the equation gives us

$\ln \frac{N}{{{N}_{o}}}=-\lambda t$

Which gives

$N(t)={{N}_{o}}{{e}^{-\lambda t}}$

Hence, the law of radioactive decay is represented by $N(t)={{N}_{o}}{{e}^{-\lambda t}}$.

What is the Decay Rate?

The term decay rate itself defines that it is the rate of the number of nuclei disintegrating per unit of time. 

To derive the formula of decay rate R of a sample, let’s assume that in the time interval $dt$, the decay count measured is $\Delta N$. Thus, $dN=-\Delta N$.

So,  

$R=-\frac{dN}{dt}$

On substituting the value of N

$R=-\frac{d({{N}_{o}}{{e}^{-\lambda t}})}{dt}$

And after differentiating, we get,

$R=\lambda {{N}_{o}}{{e}^{-\lambda t}}$

Or, $R={{R}_{o}}{{e}^{-\lambda t}}$

Thus, it’s clear that${{R}_{o}}=\lambda {{N}_{o}}$is the decay rate at $t=0$.

The decay rate (R) at a certain time (t) and the number of undecayed nuclei (N) at the same time are related by

$R=\lambda N$

The decay rate is also called activity and its SI unit is Becquerel, named after the discoverer of radioactivity, Henri Becquerel.

Half-Life and Mean Life Time 

The half-life time of any element is defined as the time interval in which the number of atoms (nuclei) of the given radioactive element reduces to half of its original value (number). 

Thus, when $t=T,\text{ N}=\frac{{{N}_{o}}}{2}$

Using the law of radioactive decay,

$N={{N}_{o}}{{e}^{-\lambda t}}$

$N=\frac{{{N}_{o}}}{2}={{N}_{o}}{{e}^{-\lambda T}}$ [Substituting the value of $t$ and $N$]

Now, on solving

$\frac{1}{2}={{e}^{-\lambda T}}$

$2={{e}^{\lambda T}}$

${{\log }_{e}}2=\lambda T{{\log }_{e}}e$

$\lambda T={{\log }_{e}}2$

$T=\frac{{{\log }_{e}}2}{\lambda }$

$T=\frac{2.303\times {{\log }_{10}}2}{\lambda }$

$T=\frac{2.303\times 0.3010}{\lambda }$

$T=\frac{0.6931}{\lambda }$

Hence, the half-life time (T) of a radioactive element is $\frac{0.6931}{\lambda }$.

Also, the half-life time of a radioactive element is inversely proportional to its decay constant.

Mean life time 

The mean life time or average life time of a radioactive element is defined as the ratio of total life time (S) of all atoms to the total number of atoms (N0) present in the element at zero time (t=0).

Thus, mean lifetime is

$(\tau )=\frac{\text{total life time of all atoms}}{\text{total number of atoms}}$

$\tau =\frac{S}{{{N}_{o}}}$

Summary

Radioactivity is a process in which an unstable particle undergoes decay. It is also known as radioactive decay. The radioactive decay is of three types – alpha (\[\alpha \]) decay, beta ($\beta $) decay, and gamma ($\gamma $) decay. According to the law of radioactive, the number of nuclei undergoing the decay per unit of time in a radioactive sample is proportional to the total number of nuclei in the sample. It can be represented by the equation $N(t)={{N}_{o}}{{e}^{-\lambda t}}$. 

The half-life time of any element is defined as the time interval in which the number of atoms (nuclei) of the given radioactive element reduces to half of its original value (number) and it is calculated by the formula $T=\frac{0.6931}{\lambda }$. The mean life time or average life time of a radioactive element is defined as the ratio of total life time $(S)$of all atoms to the total number of atoms $({{N}_{o}})$present in the element at zero time $(t=0)$and it is calculated by the formula $\tau =\frac{S}{{{N}_{o}}}$.