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# Dual Nature Of Matter And Radiation

## Dual Nature of Matter and Radiation PDF Notes, Important Questions and Formulas

Modern Physics –1

1. NATURE OF LIGHT

It was a matter of great interest for scientists of know that what exactly from the light is made up of or how the light behaves. This is briefly described over here

1.1 Newton's Corpuscular Theory:
Newton was the first scientist who said that light is made 0 up tiny elastic particles called "Corpuscles" which travels with the velocity of light. So according to Newtons, light is a particle.

1.2 Huygen's Wave Theory:
Huygen was a scientist working parallel to Newton who come with a drastically different idea for nature of light & said that light is not a particle but a wave.

1.3 Maxwell's Electromagnetic Wave Theory:
During the time of Hymen, his views regarding nature of light were not accepted as Newton was a popular scientist of his time. But, when Maxwell asserted that light is a electromagnetic wave, scientists started believing that light is a wave.

1.4 Max Planck's Quantum Theory of Light:
Once again when scientists started believing that the light is a wave Max Planck came with different idea & asserted that light is not a wave but a photon (i.e. a particle) which he proved through black body radiation spectrum. At this time there was a great confusion about the nature of light which was solved by de-Broglie from where origin of theory of matter wave come into picture.

1.5 Debroglie Hypothesis
It supports dual nature of light (wave nature and particle nature). According to him the light consists of particles associated with definite amount of energy and momentum. These particles were later named as photons.

The photon posses momentum and is given by

Where c = speed of light

Debrogile relates particle property (momentum) with wave property (wavelength) i.e. he favours dual nature of light.

Electron volt: It is the energy gained by an electron when it is accelerated through a potential difference of one volt.

1 eV= 1.6 X 10-19Joule.

Now from eq. (2)

Where λ is in Å

Properties of Photon:

1. Photon travels with speed of light.
2. The rest mass of a photon is zero.
3. There is no concept of photon conservation.
4. All the photons of a particular frequency or wavelength posses the same energy irrespective of the intensity of the radiation.
5. The increase in the intensity of the radiation imply an increase in the number of photon's crossing a given area per second.
When light travels from one medium to another medium then
Frequency= const (because it is the property of source)
But v, λ   changes

1. PHOTOELECTRIC EFFECT

Electron Emission process:

When light is incident on a metal surface it was observed that electrons are ejected from a metal Surface sometimes even when incredicely dim light such as that from starts and distance galaxies incident on it and sometime electrons not comes out from the metal surface even high energetic or high intensity light falling on the metal surface.

This shows that the electron emission from a metal surface is not depends on the intensity of incident light but it is basically depends on the energy of the incident.

Photons no matters in number of photons are very less in a dim light, photo electric effect can be seen. During the phenomenon of photoelectric effect one incident photon on metal surface can eject at most only one electron.

A photon is an energy packet which is fully absorbed not partially. Thus one photon cannot be absorbed by more than one electron.

The minimum amount of energy of photon required to eject an electron out of a metal surface is called work function. It is denoted by ϕ.
The work function depends on the nature of the metal.

1. The electron emission from a metal is only depends on the work function or energy of one photons.
2. But how many electrons comes out from the metal is depends on intensity of the falling light on energy of the light.
3. Energy of photon incident on metal will not necessarily cause emission of an electron even if its energy is more than work function. The electron after absorption may be involved in many other process like collision etc in which it can lose energy hence the ratio of no. of electrons emitted to the no. of photons incident on metal surface is less than unity.

Atoms and Nuclei

Bohr’s Model of Atom

• The Rutherford nuclear model has two main difficulties in explaining the structure of the atom;
1. It predicts that atoms are unstable because the accelerated electrons revolving around the nucleus must spiral into the nucleus. This contradicts the stability of matter.
2. It cannot explain the characteristic line spectra of atoms of different elements.
• The classical electromagnetic theory states that the energy of an accelerating electron should continuously decrease, so the electron should move spirally inward and eventually fall into the nucleus. Thus, such an atom cannot be stable.

Bohr’s postulates of an atom

Bohr’s first postulate

• Bohr’s first postulate was that an electron in an atom could revolve in certain stable orbits without the emission of radiant energy.
• Each possible state has definite total energy. These are called the stationary states of the atom.

Bohr’s Second postulate

• This postulate states that the electron revolves around the nucleus in only those orbits for which the angular momentum is some integral multiple of
• Thus, the angular momentum (L) of the orbiting electron id quantized, i.e., which was later confirmed by de Broglie.

Bohr’s third postulate

• Bohr’s third postulate states that a photon is emitted when an electron makes a transition from one of its specified non-radiating orbits to another of lower energy having energy equal to the energy difference between the initial and final states.
• The energy of the emitted photon is given by

Where Eand Ef are the energies of the initial and final states, and Ei>Ef

Proof of Boh’s Second postulate by De Broglie

• Boh’s Second postulate states that the angular momentum of the electron orbiting around the nucleus is quantized.
• For an electron moving in the nth circular orbit of radius rn, the total distance is the circumference of the orbit, Thus,

Here  is the de Broglie wavelength of the electron moving in the nth orbit.

• Thus,

Bohr’s model of Hydrogen atom

The reasons which make Bohr’s model still useful are

1. The model is based on just three postulate but accounts for almost all the general features of the hydrogen spectrum.
2. The model incorporate many of the concepts we have learnt in classical physics.
3. The models demonstrate how a theoretical physicist occasionally must quite literally ignore certain problems of approach on the hope of being able to make some predictions.

Energy levels

• Kinetic and potential energies Kn and Un in the nth orbit are

• The total energy is the sum of the kinetic and potential energy:

• Substituting the value of m, e, and h with n=1, we get the least energy of the atom in the first orbit, -13.6 eV. Hence,

Atomic Numbers: The number of proton in the nucleus is called the atomic number. It is denoted by Z.

• Mass Number: The total number of proton and neutrons present in a nucleus is called the mass number of the element. It is denoted by A. Number of proton in an atom=Z Number of nucleons in an atom= A Number of neutrons in an atom=N=A-Z

• Nuclear Mass: the total mass of the protons and neutron present in a nucleus is called the nuclear mass.
• Nuclide: A nuclide is specific nucleus of an atom characterized by its atomic  number Z and mass number A. It is represented as zxA,
Where X=chemical symbol of the element
Z=atomic number
A=mass number

Isotopes: The atoms of an element which have the same atomic number but different mass number are called isotopes. Isotopes have similar chemical properties but different physical properties.
• Isobars: The atoms with the same mass number but different atomic number are   called isobars.
• Isotones: The nuclides with the same number of neutron are called isotones.
• Isomers: These are nuclei with the same atomic number and same mass number but are in different energy states.

• Electron Volt: It is defined as the energy acquired by an element when it is accelerated through a potential difference of 1 volt and is denoted by eV.

• Atomic Mass Unit: It is  th of the actual mass of a carbon atom of isotope
It is denoted by amu or just by u.

The energy equivalence of 1 amu is
1 amu=931MeV

• Discovery of Neutron: Neutrons were discovered by Chadwick in 1932. When beryllium nuclei are bombarded by  -Particles, highly penetrating radiations are emitted, which consist of neutrons.

A free neutron decays spontaneously, with a half-life of about 900 s into a proton, Electron and antineutrino.

• Size of the Nucleus: It is found that a nucleus of mass number A has a radius

This implies that the volume of the nucleus, which is proportional to R3, is proportional to A.
The density a nucleus is consist and independent of A for all nuclei, and the density of nuclear matter is approximately  which is very large as compared to ordinary matter, say water, which is 103Kg m-3
• Mass-Energy Equivalence: Einstein proved that it is necessary to treat mass as another form of energy. He gave mass-energy equivalence relation as E=mc2
Where m is the mass and c is the velocity of light in vacuum.
• Mass Defect: The difference between the rest mass of a nucleus and the sum of the rest masses of its constituent nucleons is called its mass defect. It is given by

• Binding Energy: It may be defined as the energy required to break up a nucleus into its constituent protons and neutrons and to separate them to such a large distance that they may not interact with each other.

It may also be defined as the surplus energy which the nucleus gives up by virtue of their attractions which they become bound together to form a nucleus.
The binding energy of a nucleus  is given by

• Binding Energy per Nucleon: It is the average energy required to extract one nucleon from the nucleus. It is obtained by dividing the binding energy of a nucleus by its mass number.

• Nuclear Forces: These are the strong attractive forces which hold protons and neutrons together in a tiny nucleus. These are short range forces which operate over a very short distance of about 2-3 fm of separation between any two nucleons. The nuclear force does not depend on the charge of the nucleon.

• Nuclear Density: the density of a nucleus is independent of the size of the nucleus and is given by

• Radioactivity: It is the phenomenon of spontaneous disintegration of the nucleus of an atom with the emission of one or more radiations such as  .The substance which spontaneously emit penetrating radiation are called radioactive substances.

1. When a radioactive nucleus emits an , the atomic number decreases by 2 and the mass number decreases by 4.
2. When a radioactive nucleus emits a , the atomic number increases by 1 but the mass number remains the same.
3. The emission of a  does not change the mass number or the atomic number of the radioactive nucleus.  emission by a radioactive nucleus lowers its energy state.
• Alpha Decay: It is the process of emission of an  from a radioactive nucleus. It may be represented as

• Beta Decay: It is the process of emission of an electron from a radioactive nucleus. It may be represented as

• Gamma decay: It is the process of emission of a  photon during the radioactive disintegration of a nucleus. It can be represented as

• Radioactive Decay Law: It is states that the number of nuclei undergoing decay per unit time is proportional to the number of undecayed radioactive nuclei present at that instant. It may be written as

Where N(0)is the number of nuclei at t=0 and  is the disintegration constant.
• Decay or Disintegration Constant: It may be defined as the reciprocal or the time interval in which the number of active nuclei in a given radioactive sample reduces to  of its initial value.
• Half-life: The mean life of a radioactive sample is defined as the ratio of the combined age of all the disintegrations. It is inversely proportional to the decay constant of the radioactive substance.

• Mean Life: The mean life of a radioactive sample is defined as the combined age of all the atoms and the total number of atoms in the given sample. It is given by

• Rate of decay or Activity of a Radioactive Sample: It is defined as the number of radioactive disintegrations taking place per second in a given sample. It is expressed as

• Curle: It is the SI unit of decay. One curie is the decay rate of   disintegrations per second.

disintegrations/s

• Rutherford: One Rutherford is the decay rate of  disintegrations per second.
• Natural Radioactivity: It is the phenomenon of the spontaneous emission of  from the nuclei of naturally occurring isotopes.
• Artificial or Induced Radioactivity: It is a phenomenon of inducing radioactivity in certain stable nuclei by bombarding them by suitable high-energy sub-atomic particles.
• Nuclear Reaction: It is a reaction which involves the change of stable nuclei of one element into the nucleus of another element.
• Nuclear Fission: It is the process in which a heavy nucleus when excited gets split into two smaller nuclei of nearly comparable masses.

Example:
• Nuclear Reactor: It is a device in which a nuclear chain reaction is initiated, maintained and controlled.
• Nuclear Fusion: It is the process of fusion of two smaller nuclei into a heavier nucleus with the liberation of a large amount of energy.
• Critical Size and Critical Mass: The size of the fissionable material for which the reproduction factor is unity is called critical size and its mass is called critical mass of the material. The chain reaction in this case remains steady or sustained.
• Moderator: Any substance which is used to slow down fast-moving neutrons to thermal energies is called a moderator. The commonly used moderators are water, heavy water (D20) and graphite.

Atoms

Results of alpha particle gold foil experiment.

• The α-particles pass through the foil.
• Only about 0.14% of the incident α-particles scatter by more than 1° and about 1 in 8000 deflects by more than 90°.
• This force could be provided if the greater part of the mass of the atom and its positive charge were concentrated tightly at its centre.
• Hence, the positive charge concentrated at the centre of the atom is called the nucleus.
• The size of the nucleus to be about 10−15 m to 10−14 m, by kinetic theory, the size of an atom was known to be 10−10 m; hence, most of an atom is empty space.

Trajectory of an alpha particle

• The charge of the gold nucleus is Ze, where Z = 79 is the atomic number of the atom.
• Nucleus of gold is about 50 times heavier than an α-particle, it remains stationary throughout the scattering process.
• The trajectory traced by an α-particle depends on the impact parameter (b) of collision.

Impact parameter (b)

• The impact parameter is the perpendicular distance of the initial velocity vector of the α-particle from the centre of the nucleus.
• α-Particles having a small parameter experience large scattering.

Bohr’s Model of Atom

• The Rutherford nuclear model has two main difficulties in explaining the structure of the atom:
1. It predicts that atoms are unstable because the accelerated electrons revolving around the nucleus must spiral into the nucleus. This contradicts the stability of matter.
2. It cannot explain the characteristic line spectra of atoms of different elements.
• The classical electromagnetic theory states that the energy of an accelerating electron should continuously decrease, so the electron should move spirally inward and eventually fall into the nucleus. Thus, such an atom cannot be stable.

Bohr’s postulates of an atom

Bohr’s first postulate

• Bohr’s first postulate was that an electron in an atom could revolve in certain stable orbits without the emission of radiant energy.
• Each possible state has definite total energy. These are called the stationary states of the atom. Bohr’s second postulate
• This postulate states that the electron revolves around the nucleus in only those orbits for which the angular momentum is some integral multiple of h 2π
• Thus, the angular momentum (L) of the orbiting electron is quantised, i.e. L = n.h 2π, which was later confirmed by de Broglie. Bohr’s third postulate
• Bohr’s third postulate states that a photon is emitted when an electron makes a transition from one of its specified non-radiating orbits to another of lower energy having energy equal to the energy difference between the initial and final states.
• The energy of the emitted photon is then given by E = hv = Ei − Ef Where Ei and Ef are the energies of the initial and final states, and Ei > Ef.

Proof of Bohr’s second postulate by De Broglie

• Bohr’s second postulate states that the angular momentum of the electron orbiting around the nucleus is quantised.
• For an electron moving in the nth circular orbit of radius rn, the total distance is the circumference of the orbit, 2πrn. Thus,
• 2πrn=nλ Here λ is the De Broglie wavelength of the electron moving in the nth orbit.
• Thus,

Bohr’s model of Hydrogen atom

• The reasons which make Bohr’s model still useful are
1. The model is based on just three postulates but accounts for almost all the general features of the hydrogen spectrum.
2. The model incorporates many of the concepts we have learnt in classical physics.
3. The model demonstrates how a theoretical physicist occasionally must quite literally ignore certain problems of approach in the hope of being able to make some predictions.