Radioactivity and Nuclear Reaction Note | Class 12 Physics | NEB | NEPALI EDUCATE

Radioactivity and Nuclear Reaction

Radioactivity:

The process by which an unstable nucleus becomes stable by emitting radiations like α, β, and ϒ radiations is called radioactivity. The substance of such nucleus is called a radioactive substance.

Example: Uranium, Polonium, Thorium, etc. are radioactive elements.

Usually nuclei having atomic no. (Z) > 83 are unstable.

Types of Radioactivity:

1. Natural radioactivity:
   

   This includes radioactivity shown by natural radioactive sources. In natural radioactivity, an element spontaneously transforms into another element by emitting radiations like α-particles, β –particles and ϒ radiations and they have very high penetrating power.
   The example of natural radioactive elements is Uranium, Polonium, and Thorium, etc.

2. Artificial radioactivity
   

   This includes radioactivity shown by artificial sources. In artificial radioactivity an element is made radioactive by striking by a fast moving particle like neutron. They emit electrons, positrons and other particles and as well as ϒ-radiations but they have less penetrating power. The examples of artificial radioactive isotopes are C¹⁴, Na²³, P³⁰, Co⁶⁰ etc. Such elements are lighter than natural radioactive elements.

Cause of Radioactivity:

In a nucleus of an atom, two types of forces are acting there; one is an attractive strong nuclear force between the nucleons and the other is a repulsive electrostatic force between the protons. If the attractive force dominates the repulsive force, the nucleus becomes stable. Still, if the repulsive force is too much greater than the attractive force, the nucleus becomes unstable. Such an unstable nucleus emits radioactive radiations in the form of α-particle, β-particle, and ϒ-ray. Hence, the unstable nucleus is the main cause of radioactivity.

Properties of radioactive radiation:

Property	α-particles	β-particles	γ-rays
Origin	Nuclei of helium atom (He++)	Electrons of nuclear origin	Electromagnetic waves of very short wavelength
Charge	+2e (3.2x10-19 C)	-1.6x10-19 C	Chargeless
Rest Mass	4mp (6.6x10-27 kg)	9.1x10-31 kg	Massless
Speed	Much less than the speed of light	Less than the speed of light	Equal to the speed of light
Deflection by Fields	Deflected by electric and magnetic fields	Deflected by electric and magnetic fields	Not deflected by electric and magnetic fields
Ionizing Power	High	About 1/100th of α-particle	About 1/100th of α-particle and β-particle
Penetrating Power	Very low	About 100 times less than α-particle	About 100 times more than α-particle and β-particle
Photographic Effect	Yes	Yes	Yes
Fluorescence on Substances	Can cause fluorescence on certain substances	Can cause less fluorescence	Can cause fluorescence on certain substances

Laws of radioactive disintegration:

1. The radioactive decay is a spontaneous phenomenon and is not affected by external conditions such as temperature, pressure, electric field etc.

2. In any radioactive decay, either one α-particle or one β-particle is emitted by the atom at a time. When a nuclide emits α-particle, its mass number is reduced by four and atomic number by two.
   _ZX^A→_Z-2X^A-4 + ₂He⁴(α-particle)
   When a nuclide emits β-particle, its mass number remains unchanged and atomic number increases by one.
   _ZX^A→_Z-1X^A-2 + ₁e⁰(β-particle)
   When a nuclide emits a ϒ-ray, neither the mass number nor the atomic no. changes.
   _ZX^A* → _ZX^A + ϒ-ray
   Where,
   _ZX^A* = excited nucleus These laws are called displacement law.

3. Decay Law:
   

   The rate of disintegration of radioactive substance is directly proportional to the number of atoms present at that instant .This is called decay law.
   Let us consider at time t=0, a radioactive sample contains N_o number of atoms and after time t, N number of atoms are left due to disintegration of the atoms. Further let us consider that dN number of atoms disintegrate in time between t and t+dt. Then rate of disintegration at time t will be
   According to decay law,
   $\frac{dN}{dt}\ α N $
   $\frac{dN}{dt}\ = -λN$………………..(1)
   Where λ is proportionality constant called as disintegration constant or decay constant. The –ve sign indicates that the no. of atoms N decreases as time increases .The rate of disintegration \(\frac{dN}{dt}\) is called the activity of the radioactive sample.
   Equation (1) can be written as $\frac{dN}{dt}\ = -λN$
   Integrating the equation on both sides, we get
   $\begin{array}{l}\int_{{N_0}}^N {\frac{{dN}}{{dt}}} = \int_0^t {{\rm{ - }}\lambda } t\\or,{\rm{ }}\;{\rm{[ln N]}}_{{N_0}}^N = {\rm{ - }}\lambda [t]_0^t\\or,{\rm{ ln N - ln }}{N_0}{\rm{ = - }}\lambda (t - 0)\\or,{\rm{ ln }}\frac{N}{{{N_0}}}{\rm{ =- }}\lambda t\\or,{\rm{ }}\frac{N}{{{N_0}}}\,{\rm{ = }}{{\rm{e}}^{{\rm{ - }}\lambda t}}\\or,\;{\rm{ }}N{\rm{ = }}{N_0}{{\rm{e}}^{{\rm{ - }}\lambda t}}\end{array}$
   This equation is known as decay equation .It shows that the decay of radioactive atoms obeys exponential law.

Decay Constant

The decay constant is defined as the ratio of its instantaneous rate of disintegration to the number of atoms present at that instant.
i.e λ ${\rm{=}}\frac{{-\frac{{dN}}{{dt}}}}{N}$
Again, the decay equation is
N=N_o𝑒^−λt
If λ = $\frac{1}{t}\$, then
N=N_o𝑒⁻¹

$\lambda {\rm{ = }}\frac{{{N_0}}}{e}$ $= \frac{{{N_0}}}{{2.718}}$
or,N=0.37N_o
i.e. N =37% of N_o
Hence decay constant may also be defined as the reciprocal of time during which the no. of radioactive atoms of radioactive substance falls to 37% of its original value.

Half-Life:

The half-life of a radioactive substance is defined as the time during which half of the atoms of a radioactive substance will disintegrate. It is denoted by T_1/2 or T. Its value is different for different substance. The decay equation is N=N_oe^−λt--------(1)
Where, N₌₀ initial no. of atoms of radioactive substance
N=no. of atoms after time (t)
λ=decay constant
Then, from defination of half life, i.e, when t=T then $\frac{{{N_0}}}{{2}}$
So, Equation (1) becomes
$\begin{array}{l}\frac{{{N_0}}}{2} = {N_0}{{\rm{e}}^{{\rm{ - }}\lambda T}}\\or,\;{{\rm{e}}^{{\rm{ - }}\lambda T}}{\rm{ = }}\frac{1}{2}\\or,{\rm{ }}{{\rm{e}}^{\lambda T}} = 2\\{\rm{or, ln }}{{\rm{e}}^{\lambda T}}{\rm{ = ln2}}\\{\rm{or, }}\lambda T{\rm{ = 0}}{\rm{.693}}\\{\rm{or, T = }}\frac{{0.693}}{\lambda }\end{array}$
Thus, half-life of a radioactive substance is inversely proportional to its decay constant and is a characteristic property of its nucleus and cannot be altered by any known method.

Average life or mean life

The mean life of a radioactive substance is equal to the total life of the atoms divided by the total number of atoms in the element. $\frac{\text{sum of life of all the atoms}}{\text{total number of atoms}}$
It can be shown that the mean life of a radioactive substance is equal to the reciprocal of the decay constant.
T_mean=$\frac{1}{λ}$
or, T_mean=$\frac{1}{0.693}=1.443T$
[Since, λ=$\frac{1}{0.693}$, where, T is the half life of the substance]
Thus, the mean life of a radioactive substance is longer than its half-life.

Activity of Radioactive Substance:

The rate of decay of a radioactive substance is called the activity \(A\) of the substance.

\[ A = \frac{dN}{dt} = -\lambda N \]

Or, \( |A| = \lambda N \)

If \(A_0\) is the activity of a substance at time \(t=0\), then

\[ A_0 = \lambda N_0 \]

Now,

\[ \frac{A}{A_0} = \frac{N}{N_0} = \frac{N_0e^{-\lambda t}}{N_0} \]

Or, \(A = A_0e^{-\lambda t}\)

Thus, \(\frac{A}{A_0} = \frac{N}{N_0} = \frac{N_0e^{-\lambda t}}{N_0}\)

Or, \(A = A_0e^{-\lambda t}\)

Number of Atoms Left Behind After \(n\) Half-Lives:

Let \(N_0\) be the total number of atoms of a radioactive substance present at time \(t=0\). Then,

• After one half-life, the number of atoms present \(= \frac{N_0}{2}\)
• After two half-lives, the number of atoms present \(= \frac{N_0}{2} \times \frac{1}{2} = \frac{N_0}{2^2}\)
• After \(n\) half-lives, the number of atoms present (\(N\)) \(= \frac{N_0}{2^n}\)

If \(A\) be the activity after \(n\) half-life times and \(A_0\) be the initial activity, then

\[ A = \frac{1}{2^n}A_0 \]

Units of Radioactivity:

The activity of a radioactive substance is measured in terms of disintegration per second.

1. Curie (Ci):
   It is defined as the activity of a radioactive substance which gives \(3.7 \times 10^{10}\) disintegrations per second.
   i.e. 1 Ci = \(3.7 \times 10^{10}\) disintegrations/second
2. Rutherford (Rd): 1 Rd = \(10^6\) disintegrations/second
3. Becquerel (Bq): 1 Bq = 1 disintegration/second

Radio Carbon Dating:

The process of estimating the age of archaeological and geological objects by measuring the proportion of \(^{14}C\) to \(^{12}C\) in a specimen is known as carbon dating.

Atmosphere contains \(^{12}C\) and its radioisotope \(^{14}C\). Neutrons produced by cosmic rays react with nitrogen in the air to form \(^{14}C\) as below:

\[ ^{7}_{14}N + ^{1}_{0}n \rightarrow ^{14}_{6}C + ^{1}_{1}H \]

Both \(^{14}C\) and \(^{12}C\) are absorbed by living organisms (plants and animals). When the living organism dies, the intake of \(^{14}C\) stops, and the decay of \(^{14}C\) starts with a half-life of 5730 years. By determining the number of atoms of \(^{14}C\) and \(^{12}C\) in a sample of a dead organism at a given time, its age can be estimated.

Suppose the activity of some dead material was \(A_0\) at the time of death and it has been reduced to \(A\) after \(t\) years. Then according to the law of radioactive decay, we have:

\[ A = A_0 \cdot e^{-\lambda t} \]

Or, rearranging for \(e^{-\lambda t}\):

\[ e^{\lambda t} = \frac{A_0}{A} \]

Taking the natural logarithm (ln) of both sides:

\[ \ln(e^{\lambda t}) = \ln\left(\frac{A_0}{A}\right) \]

Which simplifies to:

\[ \lambda t = \ln\left(\frac{A_0}{A}\right) \]

Now, solving for \(t\):

\[ t = \frac{1}{\lambda} \ln\left(\frac{A_0}{A}\right) \]

Alternatively, expressing in terms of the half-life (\(T\)):

\[ t = \frac{T}{0.693} \ln\left(\frac{A_0}{A}\right) \]

Where \(\lambda = \frac{0.693}{T}\) and \(T\) is the half-life of the radioisotope \(^{14}C\). Hence, by measuring the activity \(A_0\) of a living organism and activity \(A\) of its dead material, the age \(t\) of that object can be determined.

Radioisotopes

The isotopes of an element which are also radioactive are called radioactive isotopes. For example: \(^{14}C\), \(^{11}C\) are the radioisotopes of carbon, and \(^{23}Na\), \(^{24}Na\) are the radioisotopes of sodium.

Use of Nuclear Radiation in Medical:

1. Diagnosis:
   • Radio iodine \(I^{31}\) is used to determine the condition of the human thyroid gland.
   • Radio mercury \(Hg^{203}\) is used to study the disorder of the kidney and liver.
   • Radioiodine and some other isotopes are used in the diagnosis of brain tumors.
   • Radio iron (\(Fe^{59}\)) is used to diagnose many diseases caused by the deficiency of red blood cells in the human body.
2. Therapy:
   • Gamma radiations emitted by radio cobalt (\(Co^{60}\)) are used to destroy cancer tumors in the body and for sterilization of materials.
   • Radio phosphorous (\(P^{32}\)) is used for treating skin diseases and leukemia.
   • Radio gold (\(Au^{198}\)) and radio cobalt are used in the treatment of some forms of cancer.
   • Radio iodine is used to treat an overactive thyroid gland.

Radiation (Health) Hazard and Safety Precautions:

The adverse effect of radiation on living organisms is called radiation hazard. Exposure to radiation may damage cells and tissues, leading to the death of living organisms. The following types of hazards can be caused by radiation:

1. The UV radiation can damage the retina of our eye.
2. The UV radiation can cause skin burn leading to skin cancer.
3. Strong exposure to β-particles can lead to the death of living organisms.
4. Strong exposure to α-particles can cause lung cancer.
5. Strong exposure to radiation can damage the fetus of pregnant ladies.
6. Exposure to α-particles, β-particles, and neutrons can damage red blood cells (RBC).

Safety precautions for radiation hazards are as follows:

1. The radioactive sources should be kept in containers with thick walls of lead.
2. Researchers and workers should wear lead aprons.
3. Radioisotopes or elements should be handled with remote control devices.
4. Nuclear reactors and research laboratories should be installed away from the city.
5. Normal access to radioactive substances should be avoided.