RADIOACTIVITY -

The process of spontaneous ( i.e without external means by itself) disintegration of the nuclei of heavy elements with the emission of certain types of radiation is called RADIOACTIVITY.

A.H Becquerel discovered radioactivity in 1896 purely by accident. After illuminating some piece of uranium compound with visible light, he wrapped them in black paper and separated the package from a photographic plate by a piece of silver.

When after several hours of exposure, the photographic plate was developed, it showed blacking due to some things that must have been emitted by the compound and was able to penetrate both black paper and silver.

Experiments performed subsequently showed that radioactivity was a nuclear phenomenon in which an unstable nucleus undergoes a decay (referred to as radioactive decay.

There are three types of radioactive decay that occur in nature.

α decay

β decay and

γ decay

The total number of radioactive elements known at present is nearly about 40. Natural elements with atomic number greater than 82 are all radioactive.

NOTE

The lightest radioactive isotopes are tritium $\fn_cm \large \left (H_{1}^{3} \right )$

α – RAYS OR PARTICLE

It carries 2 unit +ve charge and 4 unit mass.
It is represented as helium nucleus or Helium ions. $\fn_cm \large _{2}^{}\textrm{He}^4\;or\;He^{++}$
Its velocity range between $\fn_cm \large 1.4\times10^7m/s\;to\;2.1\times10^7m/s\;\;(\frac{1}{10}^{th}\; of\;C)$
Its penetrating power is very small, they can easily stop through a few millimetres of Al.
Its ionising power is very high.
α particle affected photographic plate.
It deflected by an electric and magnetic field.

β – RAYS AND PARTICLE

It carries 1 unit -ve charge and mass is a $\fn_cm \large 9\times10^{-31}kg$ (known as an electron) represented by $\fn_cm \large e^-\;or\;(_{-1}^{}\textrm{e}^0)$
Also, it carries 1 unit +ve charge and mass same as $\fn_cm \large 9\times10^{-31}kg$ (known as a positron) represented by $\fn_cm \large e^+\;or\;(_{+1}^{}\textrm{e}^0)$
Its velocity range from 33% to 99% of speed of light
Its penetrating power is very large. They can easily pass through a few millimeters of Al.
Its ionising power is very low ( $\fn_cm \large \frac{1}{100}^{th}$ that of α particle)
They affected photographic plate
They are deflected by an electric and magnetic field.

γ – RAYS

The rest mass of γ ray photon is zero.
It carries no charge
It is an electromagnetic wave of a small wavelength.
its velocity is $\fn_cm \large 3\times10^8 m/s$
It has small ionising power.
They have a large penetrating power. They can pass through several centimetres of iron or lead.
They affected photographic plate more than β particle.
γ rays not deflected by an electric and magnetic field.
γ rays can produce a nuclear reaction.

LAWS OF RADIOACTIVE DECAY

The radioactive decay is spontaneous. It is not influenced by external conditions such as temperature, pressure etc
In any radioactive decay, either an α particle or β particle is emitted by the atom. Both the particles are not emitted simultaneously. Moreover, an atom doesn’t emit more than one α particle or more than one β particle at a time.
On emission of α particle or β particle, the new atom formed maty emit γ ray photon in case the nucleus is left in the excited state.
The emission of α particle from an atom will change it into a new atom whose charge number is reduced by 2 and mass number is reduced by 4.

for example

$\fn_cm \large _{Z}^{}\textrm{X}^A\rightarrow _{Z-2}^{}\textrm{Y}^{A-4}+_{2}^{}\textrm{He}^4$

parent daughter α particle

$\fn_cm \large \;\;\;_{92}^{}\textrm{U}^{238}\rightarrow _{90}^{}\textrm{Th}^{234}+_{2}^{}\textrm{He}^4$

$\fn_cm \large _{88}^{}\textrm{Ra}^{238}\rightarrow _{86}^{}\textrm{Rn}^{224}+_{2}^{}\textrm{He}^4$

5. the emission of β particle (electron, not orbital electron) from an atom will change it into a new atom whose charge number is raised by one, without any change in its mass number.

for example

$\fn_cm \large _{Z}^{}\textrm{X}^A\rightarrow _{Z+1}^{}\textrm{Y}^{A}+_{-1}^{}\textrm{e}^0$

parent daughter α particle

$\fn_cm \large _{6}^{}\textrm{C}^{14}\rightarrow _{7}^{}\textrm{N}^{14}+_{-1}^{}\textrm{e}^0$

6. The number of atom disintegration per second (i.e rate of disintegration) at any instant is directly proportional to the number of radioactive atoms actually present in the sample at that instant. This is called radioactive law

Let $\fn_cm \large N_0\rightarrow$ Total no of atoms present in a sample at a time $\fn_cm \large t=0$

$\fn_cm \large N\rightarrow$ Total no of atoms left in the sample at a time

$\fn_cm \large \therefore \;\;\;\;-\frac{\mathrm{d} N}{\mathrm{d} t}\propto N$

$\fn_cm \large \therefore \;\;\;\;-\frac{\mathrm{d} N}{\mathrm{d} t}\propto \lambda N$ Where $\fn_cm \large \lambda$ is decay constant

$\fn_cm \large \int_{N_0}^{N}\frac{\mathrm{d} N}{\mathrm{N}}=-\lambda \int_{0}^{t}dt$

$\fn_cm \large \left [ log_e N \right ] _{N_0}^{N}=-\lambda \left [ t \right ]_{0}^{t}$

$\fn_cm \large log_e\frac{N}{N_0}=-\lambda t$

$\fn_cm \large \frac{N}{N_0}=e^{-\lambda t}$

$\fn_cm \large \left [ N=N_0 e^{-\lambda t} \right ]$

NOTE

Since the decay of a radioactive substance obeys exponential law, the rate of disintegration is rapid in the beginning and become slower and slower with the passes of time.

DISINTEGRATION CONSTANT

Let $\fn_cm \large t=\frac{1}{\lambda}$ , we get

$\fn_cm \large N=N_0 e^{-\lambda t}$

$\fn_cm \large N=N_0 e^{-1}$

$\fn_cm \large \left [ N=\frac{N_0}{e} \right ]$

Thus the disintegration constant of a radioactive element is the reciprocal of time at the end of which the number of atoms left under decayed in a radioactive sample reduces to $\fn_cm \large \left ( \frac{1}{e} \right )$ times the original number of atoms $\fn_cm \large \left ( N_0 \right )$ in the sample.

HALF-LIFE PERIOD

It is defined as the time during which half the number of atoms presents initially in the sample of the element decay. It is represented by $\fn_cm \large T_{1/2}$

i.e when $\fn_cm \large t=T_{1/2}, \;\;\;\;\; N=\frac{N_0}{2}$ \

$\fn_cm \large \frac{N_0}{2}=N_0 e^{-\lambda T_{1/2}}$

$\fn_cm \large e^{-\lambda T_{1/2}}=\frac{1}{2}$

$\fn_cm \large e^{\lambda T_{1/2}}=2$

$\fn_cm \large \lambda T=log_e 2$

$\fn_cm \large \lambda T=2.3026 \times log_{10}2$

$\fn_cm \large \lambda T=2.3026 \times 0.3010$

$\fn_cm \large \lambda T=0.6931$

$\fn_cm \large \left [ T_{1/2}=\frac{0.6931}{\lambda} \right ]$

Half-lives of radioactive elements vary over a very wide range.

for example

for Polonium $\fn_cm \large T_{1/2}=3 \times 10^{-7}sec$

for lead $\fn_cm \large T_{1/2}=1.4 \times 10^{7}sec$

NOTE

let $\fn_cm \large t=0\;\;\;\;\;\;\;\;\;\;N=N_0$

After one half-life $\fn_cm \large \left ( t=T \right ), N=\frac{N_0}{2}=N_0\left ( \frac{1}{2} \right )^1$

After two half-life $\fn_cm \large \left ( t=2T \right ), N=\frac{N_0}{4}=N_0\left ( \frac{1}{2} \right )^2$

After three half-life $\fn_cm \large \left ( t=3T \right ), N=\frac{N_0}{8}=N_0\left ( \frac{1}{2} \right )^3$

………………………………………….

After n half-life $\fn_cm \large \left ( t=nT \right ),\;\;\;\;\;\;\;\;\;\;\;\;\;\; N=N_0\left ( \frac{1}{2} \right )^1$

$\fn_cm \large \left [ N=N_0\left ( \frac{1}{2} \right )^n \;\;\;\;\;\;\;where \;\;n=\frac{t}{T} \right ]$

AVERAGE ( OR MEAN ) LIFE

All the atoms in a radioactive substance do not disintegrate at the same time. Some atoms disintegrate earlier whereas other disintegrate after a long time. So we calculate the average life.

i.e The average life of a radioactive substance is the sum of the lives of all the atoms of the substance divided by the total number of atoms presents initially in the substance.

i.e $\fn_cm \large T_{avg}=\frac{sum\;of\;lives\;of\;all\;atoms}{total\;number\;of\; original\; atoms\;(N_0)}$

Let $\fn_cm \large N_0\rightarrow$ Number of atoms present initially (t=0)

$\fn_cm \large N\rightarrow$ Number of atoms present after time t

Let $\fn_cm \large dN$ atoms further disintegrate between t and (t+dt)

i.e each of the dN atoms has lived for a time t.

$\fn_cm \large \therefore$ The total life of dN atoms = Number of atoms × life of each atom

= dN.t

$\fn_cm \large \therefore$ The total life of all atom= $\fn_cm \large \int_{N=0}^{N=N_0}t.dN$

$\fn_cm \large \because when\;N=N_0, t=0\;and\;\;when\;N=0, t=\infty$

$\fn_cm \large =\int_{\infty}^{0}t(- \lambda N dt)\; \; \; \; \; \; \; [\because -\frac{dN}{dt}=\lambda N]$

$\fn_cm \large =\int_{0}^{\infty}t \lambda[N_0 e^{- \lambda t}]dt$

$\fn_cm \large = \lambda N_0 \int_{0}^{\infty}[t.e^{- \lambda t}]dt$

$\fn_cm \large =\lambda N_0\left [ \frac{e^{-\lambda t}}{- \lambda} \;t\;\;-\; \int \frac{e^{-\lambda t}}{-\lambda } dt \right ]_0 ^{\infty}$ (integration by parts)

$\fn_cm \large =\lambda N_0\left [ \frac{e^{-\lambda t}}{- \lambda} \;t\;\;-\; \frac{e^{-\lambda t}}{\lambda^2 } \right ]_0 ^{\infty}$

$\fn_cm \large =-\frac{N_0}{\lambda}\left [ \lambda t e^{-\lambda t}+e^{-\lambda t} \right ]_0 ^{\infty}=\frac{N_0}{\lambda}$

$\fn_cm \large \therefore T_{av}=\frac{total\;life\;of\;all\;atoms}{N_0}$

$\fn_cm \LARGE {\color{DarkRed} \left [ T_{av}=\frac{1}{\lambda} \right ]}$

Thus the average life of a radioactive substance is equal to the reciprocal of its decay constant.

NOTE

$\fn_cm \large \left [ T_{1/2}=0.693 T_{av} \right ]$

ACTIVITY

The rate of disintegration in a radioactive substance is known as its activity.

i.e $\fn_cm \large A=-\frac{\mathrm{d} N}{\mathrm{d} t}$

According to decay law

$\fn_cm \large - \frac{\mathrm{d} N}{\mathrm{d} t} \propto N$

$\fn_cm \large - \frac{\mathrm{d} N}{\mathrm{d} t} =\lambda N$

$\fn_cm \large \therefore A=\lambda N$

$\fn_cm \large at \;t=0,\; A_0=\lambda N_0\;and \; at\; t=t, A=\lambda N$

$\fn_cm \large \therefore \frac{A}{A_0}=\frac{N}{N_0}=\frac{N_0 e^{-\lambda t}}{N_0}$

$\fn_cm \large \left [ A=A_0 e^{-\lambda t} \right ]$

Note that activity also decreases exponentially at the same rate as does

Now $\fn_cm \large A=\lambda N$

$\fn_cm \large \left [ A=\frac{0.693 N}{T_{1/2}} \right ]$

It is clear that the activity of a radioactive substance is inversely proportional to its half-life.

UNIT OF ACTIVITY OR RADIOACTIVITY

a. Curie (Ci)

$\fn_cm \large 1\;Curie=3.7\times 10^{10}\;disintegration/sec$

b. Becquerel (Bq) [S.I unit]

$\fn_cm \large 1\;Bq=1\;disintegration/sec$

$\fn_cm \large 1\;Curie=3.7\times10^{10}\;Bq$

c. Rutherford (Rd)

$\fn_cm \large 1\;Rd=10^6 disintegration/sec$

DECAY PROCESS

The radioactive substance decay via three process α-decay, β-decay and γ-decay. the emission of any of the α,β changes the original nucleus called the parent nucleus into a new nucleus called the daughter nucleus. The daughter nucleus may be radioactive and disintegrate further to form a still new nucleus. The process continues till a stable nucleus ( $\fn_cm \large _{82}^{}Pb^{208}$ ) is achieved.

ALPHA DECAY (α-Decay)

The phenomenon of emission of α-particle from a radioactive nucleus is called α-decay.

$\fn_cm \large _{Z}^{}\textrm{X}^A\;\overset{-\alpha }{\rightarrow}\;_{Z-2}^{}\textrm{Y}^{A-4}\;+\;_{2}^{}\textrm{He}^4\;+\;Q$

Where Q is the energy released in the decay due to mass defect.

i.e $\fn_cm \large Q=\left \{ m_x - m_y - m_{He} \right \}c^2$

The energy released (Q) is shared by daughter nucleus Y and α particles. The bulk of the K.E is carried by the α particle.

ex- $\fn_cm \large _{92}^{}\textrm{U}^{232}\;\overset{-\alpha }{\rightarrow}\;_{90}^{}\textrm{Th}^{228}\;+\;_{2}^{}\textrm{He}^4\;+\;5.4\;MeV$

In this decay, the α particle has K.E of 5.3 MeV and the daughter nucleus has K.E of 0.1 MeV

$\fn_cm \large _{92}^{}\textrm{U}^{227}\;\overset{-\alpha }{\rightarrow}\;_{88}^{}\textrm{Ra}^{223}\;+\;_{2}^{}\textrm{He}^4\;+\;6.15\;MeV$

In this decay, the α particle has K.E of 6.04 MeV and the daughter nucleus has K.E of 0.11 MeV

BETA DECAY (β-DECAY)

the phenomenon of emission of an electron (not orbital electron) from a radioactive nucleus is called β- decay.

$\fn_cm \large _{Z}^{}\textrm{X}^{A}\;\overset{-\beta }{\rightarrow}\;_{Z+1}^{}\textrm{Y}^{A}\;+\;_{-1}^{}\textrm{e}^0\;+Q$

Experiments show that the energy of emitted electrons varies continuously from zero to maximum.

To account for the variable β-particle energy, Pauli in 1930, postulated the existence of an uncharged particle called antineutrino ( $\fn_cm \large \bar{\nu }$ ), which is emitted along with the β particles. The energy Q is shared by the β particle and antineutrino.

When antineutrino carries maximum energy, the energy of β particle is minimum and vice versa.

i.e $\fn_cm \large _{Z}^{}\textrm{X}^{A}\rightarrow \;_{Z+1}^{}\textrm{Y}^{A}\;+\;_{-1}^{}\textrm{e}^0\;+\bar{\nu }+Q\;\;\;\;\;\;\;\;(\bar{\nu }\rightarrow antineutrino)$

Note that in some β decay, a positron ( $\fn_cm \large \nu$ ) instead of an electron is emitted.

$\fn_cm \large _{Z}^{}\textrm{X}^{A}\rightarrow \;_{Z+1}^{}\textrm{Y}^{A}\;+\;_{+1}^{}\textrm{e}^0\;+\nu +Q\;\;\;\;\;\;\;\;(\nu \rightarrow neutrino)$

A positron is smaller than an electron except that it has a charge +e instead of -e.

NOTE

Antineutrino is emitted with an electron and neutrino is emitted with a positron in β- decay.

GAMMA DECAY (γ-DECAY)

The phenomenon of emission of gamma-ray photons from a radioactive nucleus is called gamma (γ) decay.

This occurs when an excited nucleus makes a transition to a state of lower energy, and emit γ ray in the form of electro magnetic wave.

$\fn_cm \large _{Z}^{}\textrm{X}^A\; (excited)\rightarrow _{Z}^{}\textrm{X}^A+\gamma$

After an α or β decay, the daughter nucleus is usually in an excited state and it achieves stability by the emission of one or more γ rays photons.