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Dual Nature Of Matter And Radiation

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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

begin mathsize 12px style straight P equals straight h over straight lambda                  . .. left parenthesis 1 right parenthesis end style

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)



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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 begin mathsize 12px style fraction numerator straight h over denominator 2 straight pi end fraction end style
      • Thus, the angular momentum (L) of the orbiting electron id quantized, i.e.begin mathsize 12px style straight L equals fraction numerator straight n. straight h over denominator 2 straight pi end fraction end style, 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

        begin mathsize 12px style straight E equals hv equals straight E subscript straight i plus straight E subscript straight f end style
        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, begin mathsize 12px style 2 πr subscript straight n. end subscript end styleThus,

                   begin mathsize 12px style 2 πr subscript straight n equals nλ end style 

      Here begin mathsize 12px style straight lambda end style is the de Broglie wavelength of the electron moving in the nth orbit.

      • Thus,
        begin mathsize 12px style 2 πr subscript straight n equals nh over mv text   end text Or text  mv end text subscript straight n straight r subscript straight n equals fraction numerator nh over denominator 2 straight pi end fraction end style

      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
         table attributes columnalign left end attributes row cell straight K subscript straight n equals 1 half mv squared equals fraction numerator me to the power of 4 over denominator 8 element of subscript 0 squared straight n squared straight h squared end fraction end cell row cell straight U subscript straight n equals negative fraction numerator straight e squared over denominator 4 straight pi subscript straight e 0 end subscript straight r subscript straight n squared end fraction equals negative fraction numerator me to the power of 4 over denominator 4 subscript eo squared straight n squared straight h squared end fraction end cell end table
      • The total energy is the sum of the kinetic and potential energy:

        begin mathsize 12px style table attributes columnalign left end attributes row cell straight E subscript straight n equals straight K subscript straight n plus straight U subscript straight n end cell row cell straight E subscript straight n equals negative fraction numerator me to the power of 4 over denominator 8 element of subscript 0 squared straight n squared straight h squared end fraction end cell end table end style
      • 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,

        

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        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.

        

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      • Atomic Mass Unit: It is begin mathsize 12px style 1 over 12 end style th of the actual mass of a carbon atom of isotope begin mathsize 12px style blank subscript 6 straight C to the power of 12 end style
        It is denoted by amu or just by u.

        begin mathsize 12px style 1 text  amu=1.660565 end text cross times text 10 end text to the power of negative 27 end exponent Kg end style
        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 begin mathsize 12px style straight alpha end style -Particles, highly penetrating radiations are emitted, which consist of neutrons.

         begin mathsize 12px style He presubscript 2 presuperscript 4 plus Be presubscript 4 presuperscript 9 rightwards arrow straight n presubscript 0 presuperscript 1 plus straight C presubscript 1 presuperscript 12 end style

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

         begin mathsize 12px style straight n presubscript 0 presuperscript 1 rightwards arrow straight H presubscript 1 presuperscript 1 plus straight e presubscript negative 1 end presubscript presuperscript 0 plus straight v with bar on top end style 
      • Size of the Nucleus: It is found that a nucleus of mass number A has a radius

        

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        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 

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 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

        begin mathsize 12px style Δm equals left square bracket Zm subscript straight p plus left parenthesis straight A minus straight Z right parenthesis straight m subscript straight n right square bracket minus straight m end style 
      • 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 begin mathsize 12px style blank subscript straight z straight x to the power of straight A end style is given by

        begin mathsize 12px style BE equals left square bracket Zm subscript straight p plus left parenthesis straight A minus straight Z right parenthesis straight m subscript straight n minus straight m right square bracket straight C squared end style 
      • 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.

         begin mathsize 12px style straight B with bar on top equals fraction numerator straight B. straight E over denominator straight A end fraction equals fraction numerator left square bracket Zm subscript straight p plus left parenthesis straight A minus straight Z right parenthesis straight m subscript straight n minus straight m right square bracket straight C squared over denominator straight A end fraction end style
      • 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

                   

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      • Radioactivity: It is the phenomenon of spontaneous disintegration of the nucleus of an atom with the emission of one or more radiations such as begin mathsize 12px style straight alpha text -particles,  end text straight beta text -particles or  end text straight gamma text -rays end text end style .The substance which spontaneously emit penetrating radiation are called radioactive substances.

      • Radioactivity Displacement Law
      1. When a radioactive nucleus emits an begin mathsize 12px style straight alpha minus particle end style, the atomic number decreases by 2 and the mass number decreases by 4.
      2. When a radioactive nucleus emits a begin mathsize 12px style straight beta minus particle end style, the atomic number increases by 1 but the mass number remains the same.
      3. The emission of a begin mathsize 12px style straight gamma minus particle end style does not change the mass number or the atomic number of the radioactive nucleus. begin mathsize 12px style straight gamma minus particle end style emission by a radioactive nucleus lowers its energy state. 
      • Alpha Decay: It is the process of emission of an begin mathsize 12px style straight alpha minus particle end style from a radioactive nucleus. It may be represented as
         begin mathsize 12px style straight X presubscript straight Z presuperscript straight A equals straight Y presubscript straight Z minus 2 end presubscript presuperscript straight A minus 2 end presuperscript plus He presubscript 2 presuperscript 4 plus straight Q end style 
      • Beta Decay: It is the process of emission of an electron from a radioactive nucleus. It may be represented as

        begin mathsize 12px style straight X presubscript straight Z presuperscript straight A rightwards arrow straight Y presubscript straight Z plus 1 end presubscript presuperscript straight A plus straight e presubscript negative 1 end presubscript presuperscript 0 plus straight v with bar on top end style 
      • Gamma decay: It is the process of emission of a begin mathsize 10px style straight gamma minus ray end style photon during the radioactive disintegration of a nucleus. It can be represented as

                    begin mathsize 12px style table attributes columnalign left end attributes row cell straight X presubscript straight Z presuperscript straight A text                     end text rightwards arrow text                 end text straight X presubscript straight Z presuperscript straight A plus straight gamma end cell row cell left parenthesis Excited text  state end text right parenthesis text                    end text left parenthesis Ground text  state end text right parenthesis end cell end table end style

      • 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

        begin mathsize 12px style straight N left parenthesis straight t right parenthesis equals straight N left parenthesis 0 right parenthesis straight e to the power of negative λt end exponent end style
        Where N(0)is the number of nuclei at t=0 and begin mathsize 12px style straight lambda end style 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 

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 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.

        

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      • 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

        

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      • 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

        begin mathsize 12px style straight R left parenthesis straight t right parenthesis equals left square bracket dN over dt right square bracket equals λN left parenthesis straight t right parenthesis equals λN left parenthesis 0 right parenthesis straight e to the power of negative λt end exponent end style 
      • Curle: It is the SI unit of decay. One curie is the decay rate of  

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 disintegrations per second.

                 begin mathsize 12px style 1 text  Ci(curie)=3.70 end text cross times text 10 end text to the power of negative 10 end exponent end styledisintegrations/s 

      • Rutherford: One Rutherford is the decay rate of begin mathsize 12px style 10 to the power of 6 end style disintegrations per second. 
      • Natural Radioactivity: It is the phenomenon of the spontaneous emission of begin mathsize 12px style straight alpha minus comma text   end text straight beta text - or  end text straight gamma text -radiations end text end style 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: begin mathsize 12px style straight U presubscript 92 presuperscript 235 plus straight n presubscript 0 presuperscript 1 rightwards arrow Ba presubscript 56 presuperscript 141 plus Kr presubscript 36 presuperscript 92 plus 3 straight n presubscript 0 presuperscript 1 plus straight Q end style 
      • 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,  begin mathsize 12px style 2 πr subscript straight n equals nh over mv text  Or mv end text subscript straight n straight r subscript straight n text  = end text fraction numerator nh over denominator 2 straight pi end fraction end style             

      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.

       




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