Radioactive Decay

Radioactive decay or radioactive disintegration is the process of emission of radioactive radiations from a nucleus. During decay, a radioactive element constantly breaks up into fresh radioactive atom till a stable element (generally an isotope of lead) is reached.

There are three types of decay. They are:

α-decay

α-decay is the process of emission of an α-particle from a nucleus. When a nucleus emits an α-particle, it loses two protons and two neutrons and a new nucleus is formed which has its mass number smaller by $4$ and its atomic number smaller by $2$. \[{}_zX^A \to {}_{z-2}Y^{A-4}+{}_2He^4\;\;\text{(α-particle)}\]
For example, \[{}_{92}U^{238} \to {}_{90}Th^{234}+{}_2He^4\] \[{}_{88}Ra^{226} \to {}_{86}Rn^{222}+{}_2He^4\]

β-decay

It is the process of emission of β-particle (an electron) from a nucleus. When β-particle is emitted, the mass number of resulting nucleus remains unchanged but its atomic number increases by $1$.
\[{}_zX^A \to {}_{z+1}Y^A + {}_{-1}e^0 \;\; \text{(β-particle)}+\bar{v}\]
The new nucleus remains unstable until γ-emission.
For example,
\[{}_{88}Ra^{228} \to {}_{89}Ac^{228}+{}_{-1}e^0+\bar{v}\] \[{}_{86}Rn^{222} \to {}_{87}Fr^{222}+{}_{-1}e^0+\bar{v}\]
Nuclei having excess neutrons (i.e. high value of $n/p$ ratio) are found to decay by β-emission.
The emission of β-particle from a nucleus does not mean that the nucleus contains electron. This is because a neutron inside the nucleus is converted into a proton, an electron and an antineutrino.
\[{}_1n^0 \to +{}_1p^1+{}_{-1}e^0+\bar{v}\]
The emission of β-particle from a nucleus is caused by the decay of a neutron into a proton.

γ-decay

The process of emission of γ-ray from nucleus is called γ-decay. In γ-decay, the mass number and the atomic number of the nucleus remain unchanged so no new nucleus is formed. Photons of suitable energy are emitted.
When an α-decay or β-decay occurs in a nucleus, the daughter nucleus is mostly in excited state, so to become stable it emits γ-rays (photons). Therefore, γ-decay occurs after α-decay or β-decay.

Laws of Radioactive Decay

In the year 1903, Rutherford and Soddy gave the following laws of radioactive decay;

  1. All the radioactive rays (α-rays, β-rays and γ-rays) are not emitted simultaneously but one after another.
  2. γ-rays are emitted after the emission of α-rays and β-rays.
  3. Radioactive decay is spontaneous i.e. it is not affected by external conditions such as temperature, pressure, electric and magnetic field.
  4. The rate of disintegration $(dN/dt)$ i.e. the number of atoms disintegrating per unit time is directly proportional to the number of atoms present at that time. \[\frac{dN}{dt} ∝ \text{Number of atoms}\] \[\frac{dN}{dt}=λ×\text{Number of atoms}\] \[\frac{dN}{dt}=-λN\] where, $λ$ is a constant called decay constant or disintegration constant. Its value depends upon the nature of the radioactive substance. The negative sign means that the number of atoms of the radioactive substance decreases with time.

Mathematical Treatment

Let $N_o$ be the number of atoms of a radioactive substance present at a time $t=0$ and $N$ be the number of atoms present at time $t$. If $dN$ atoms disintegrate in small time $dt$, then,
\[\text{Rate of decay}=\frac{dN}{dt}\]
From the laws of radioactive decay,
\[\frac{dN}{dt}=-λN\] \[\frac{dN}{N}=-λdt\]
Integrating, \[\int_{N_o}^N \frac{dN}{dt}=\int_0^t -λ\;dt\] \[[\log_eN]_{N_o}^N=-λ[t]_0^t\] \[\log_eN-\log_eN_o=-λ[t-0]\] \[\log_e\frac{N}{N_o}=-λt\] \[\frac{N}{N_o}=e^{-λt}\] \[N=N_oe^{-λt}\]

This relation shows that the number of atoms of radioactive substance decreases exponentially with time. It means that the number of atoms decreases rapidly in the beginning and then it becomes more slow and slow. And, $N$ becomes zero only when $t$ approaches infinity, so a radioactive substance will never disintegrate completely.

Decay Constant

According to the laws of radioactive decay, we have,
\[\frac{dN}{dt}=-λN\] \[λ=-\frac{dN/dt}{dt}\]
Therefore, the rate of disintegration per unit atom present is called decay constant.
Also, \[N=N_oe^{-λt}\]
If $t=1/λ$, then, \[N=N_oe^{-λ\frac{1}{λ}}\] \[\frac{N}{N_o}=\frac{1}{e}\]
Thus, decay constant can also be defined as the reciprocal of time when $N/N_o$ falls to $1/e$.
Further, \[N=\frac{N_o}{2.718}\] \[N=0.37N_o\]
Hence, decay constant can also be defined as the reciprocal of time when the number of atoms decreases to about $37\%$ of its original value.

Half Life

Half life of a radioactive substance is the time interval in which its number of atoms is reduced to half of its original value.
When $t=T$ (half life), $N$ becomes $N_o/2$.
We know, \[N=N_oe^{-λt}\]
If $t=T$, then, \[\frac{N_o}{2}=N_oe^{-λT}\] \[\frac{1}{2}=e^{-λT}\] \[e^{λT}=2\] \[λT=\log_e2\] \[T=\frac{\log_e2}{λ}\]

This is the relation between half life and decay constant of a radioactive substance.
Radium has a life life of $1620$ years. It means that $1$ gram of Ra will be reduced to $0.5$ grams in $1620$ years and to $0.25$ grams in the further $1620$ years and so on.

In time $T$, the number of atoms left will be $\frac{N_o}{2}$. After a time $2T$, the number of atoms will be $\frac{1}{2}\frac{N_o}{2}=N_o\left(\frac{1}{2}\right)^2$. After time $3T$, the number of atoms will be $\frac{1}{2}N_o\left(\frac{1}{2}\right)^2=N_o\left(\frac{1}{2}\right)^3$. Therefore, after $n$ half lives, the number of atoms will be $N_o\left(\frac{1}{2}\right)^n$.
\[\therefore N=N_o\left(\frac{1}{2}\right)^n\] \[\frac{N}{N_o}=\left(\frac{1}{2}\right)^n\]

Average Life or Mean Life

Average life is the average time for which the nuclei of a radioactive substance exist. It is defined as the ratio of the sum of lives of all atoms to the original number of atoms.
\[\text{Average life} \;\; (\bar{T})=\frac{\text{Sum of lives of all atoms}}{\text{Original number of atoms}}\] Also, \[\bar{T}=\frac{1}{λ}\]
Average life of a radioactive substance is the reciprocal of its decay constant.

Relation between Half Life and Mean Life

\[\text{Mean Life}\;\;\bar{T}=\frac{1}{λ}\] \[\text{Half life}\;\;T=\frac{\log_e2}{λ}\] \[\therefore \bar{T}=\frac{T}{\log_e2}\] \[T=\log_e2\bar{T}\] \[T=0.693\bar{T}\]
Therefore, half life of a radioactive substance is equal to $69.3\%$ of its mean life.

Units of Radioactivity

A radioactive substance is specified by its activity. The rate of decay of a radioactive substance is its activity. \[\text{Rate of decay}=\frac{dN}{dt}=\text{Activity}\]
Following are the units of activity;

  1. Curie: It is defined as the amount of radioactive substance which gives $3.7×10^{10}$ disintegrations per second. \[1\:\text{Ci}=3.7×10^{10}\:\text{dis s}^{-1}\]
  2. Rutherford: It is defined as the amount of radioactive substance which gives $10^6$ disintegrations per second. \[1\:\text{Rd}=10^6\:\text{dis s}^{-1}\]
  3. Becquerel: It is defined as the amount of radioactive substance which gives one disintegration per second. This is the SI unit of radioactivity. \[1\:\text{Bq}=1\:\text{dis s}^{-1}\]

\[\therefore 1\:\text{Ci}=3.7×10^{10}\:\text{Bq}=3.7×10^4\:\text{Rd}\]

Also See: Radioactive Dating


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