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Nucleus — Physics STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 SciencePhysicsNucleus5 Marks
Question
Explain principle of Rutherford and Soddy in relation with redioactive decay.
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Answer

When a radioactive element emits $\alpha-$ or $\beta-$ particle the original atom of the element called the parent atom changes into a new atom of another element called the daughter atom and the parent atom is said to be disintegrated. Now, if the daughter atom is also radioactive, the process of disintegration continues till a stable element usually lead is obtained. This phenomenon is known as radioactive disintegration and this gives rise to a series of radioactive elements. There are three main series namely uranium, thorium and actinium series.
In 1903 Rutherford and Soddy together made an exhaustive study of radioactive disintegration and summed up their conclusions in the form of two laws as follows:
(1) Soddy's group displacement law : It may be stated in the following two parts :
(i) When an element is disintegrated by the emission of an $\alpha$ - particle $\left({ }_2^4 He \right)$, the daughter element obtained has an atomic mass number less by 4 and atomic number less by 2 and it falls in a group of periodic table two coloumns to the left of the parent element.
In this way if a parent element X of mass number A and atomic number Z emits an $\alpha$ - particle to form a daughter element Y then the disintegration process can be expressed as follows :
${}_{Z}^{A}X \rightarrow {}_{Z-2}^{A-4}Y + {}_{2}^{4}He$ ($\alpha$ - particle)
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(ii) When an element is disintegrated by the emission of a $\beta$-particle $\left({ }_{-1}^0 \beta\right)$ , the daughter element obtained has an atomic mass number same as that of parent element and atomic number increased by 1 and it falls in a group of periodic table one coloumn to the right.
In this way if a parent element X with mass number A and atomic number Z emits a $\beta$-particle to form a daughter element Y then this disintegration process is expressed as follows :
${}_{Z}^{A}X \rightarrow {}_{Z+1}^{A}Y + {}_{-1}^{0}\beta + \bar{v}$
Image
Although $\gamma$-rays are emitted with the emission of a-particle and $\beta$-particle both, but if only $\gamma$-rays are emitted from some nucleus there is no change in mass number and atomic number of the parent nucleus but it falls in its excited state.
(2) Law of Decay : When a radioactive element disintegrates by emitting $\alpha-$ or β-particles, the number of atoms of the parent element goes on decreasing with passage of time because disintegration occurs spontaneously i.e. it is a continuous process. The continuous decrease in number of atoms of parent element is called radioactive decay. All the atoms of a radioactive element do not disintegrate simultaneously. Which atom will disintegrate first is simply a matter of chance. Thus, the radioactive disintegration process is statistical in nature.
The law of radioactive decay established by Rutherford and Soddy experimentally is stated as follows :
"The rate of disintegration of a radioactive element (i.e., number of atoms disintegrating per second) at any instant is directly proportional to the number of atoms present in the element at that instant."
This law is known as Rutherford-Soddy's law of radioactive decay.
Since radioactive disintegration is a continuous process hence the number of atoms present decreases with time. Therefore, the rate of disintegration decreases with time. According to this law
$\frac{ d N }{ d t }=-\lambda . N$ ...(1)
If N be the number of atoms of a radioactive element present at any given instant $t$ and $d N$, be the number of atoms that disintegrate during the time interval $d t$ (from $t$ to $t+d t)$, and $\lambda$ is called decay constant of radioactive element. The above relation (1) can also be expressed as follows :
$N = N _0 \cdot e^{-\lambda t}$ ...(2)
Or $N = N _0(1 / 2)^n$ ...(3)
where $n$ = number of half-life in time '$t$'.
A graph between undecayed atoms and time is shown in the figure given here. The radioactive substance decreases exponentially with time i.e. more rapidly first and slowly afterwards.
Image

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