(Atoms... Atoms... Atoms)

          "Rome was not built in a single day" - So the structure of an ATOM.
Introduction
             From the earlier days, man has been wondering about the structure of "MATTER". It is now well known that all material bodies are made up of very very small particles called "ATOMS".
            The smallest particle of matter that can take part in a chemical reaction is called an atom. A substance that contains atom of only one kind is called an "element" those containing more than one kind of atoms are called "compounds". The smallest part of a compound is called a "Molecule". For example, Copper (element) consists of one kind of atoms i.e. Copper atoms.
             However, water (compound) composed of two kinds of atoms i.e. Hydrogen and Oxygen. The smallest part of water consists of two atoms of Hydrogen and one atom of Oxygen together (H₂O molecule).

            The word "ATOM" is derived from the Greek language. "A" means of "not" and "tomogeneous" means to cut. Together Atom - 'not to cut'. Atom is supposed to be spherical in shape (as nature prefers symmetry). According to John Dalton, father of Atomic theory atom is not only indivisible but also invisible further Dalton thought that the tiny atom spherical in shape is rigid like a marble. It is interesting to note that the assumptions of Dalton about "Atom" are proved to be wrong in later days. It is sad that Dalton never saw an atom in his life time!!
             Atoms of different elements differ in mass, size and chemical properties. For example an atom of Copper is heavier than that of Aluminium.
What is the structure of Atom?
i) Thomson's Model
            In 1897 J.J.Thomson discovered "Electron" a negatively charged particle from his experiments on the passage of electricity through gases at low pressure. Thomson argued that electron should be a part of matter i.e. an atom. Since matter is electrically neutral, Thomson concluded that atom also should have a positive charge equal and opposite to that of electrons. Then how the electrons and positive charge are arranged in an atom?

"Watermelon", an inspiration!
            On a fine summer evening, when J.J.Thomson was enjoying a "Watermelon" salad on a lake side, he observed that the black seeds in Watermelon are embedded in the red coloured juicy pulp. Then, he thought why not the electrons like seeds in the Watermelon are embedded in a sphere of uniform positive charge which is like the red juicy portion.

                                  
Note: Somewhere, it is told that Thomson's model of atom as "Thomson's Plum pudding model. Plums as "electrons" and pudding as "positive charge".

            Accordingly J.J.Thomson proposed his model of an atom in 1898.
            Thus an atom spherical in shape consists of positive and mass (since electrons are of negligible mass) distributed uniformly over the entire body of the atom with negative electrons embedded in this continuous positive charge.
            Although it is the first scientific study of "Structure of an atom" this model had following defects.
a) This is a "Static Model" i.e. in this model the positive and negative charges are stationary. So they will be attracted towards each other, thus destroying the individual charges.
b) It could not explain the large angle of scattering of "alpha particles". When they are passed close to the atom - A crucial experiment conducted by Ernst Rutherford, (interestingly J.J.Thomson's research student) in 1906.
c) It could not explain the presence of discrete spectral lines emitted by hydrogen and other atoms.

Rutherford's α - Particle Scattering Experiment
            In 1906, in order to detect the structure of the atom, Rutherford proposed a classic experiment and the apparatus to investigate the scattering of alpha (α) particles. A narrow beam of high energy α particles (Helium nuclei) from a radioactive source 'Radon' was incident on a thin sheet of gold (0.2 µm thick). An α particle is the nucleus of helium atom - the helium atom with its electrons removed. Therefore, it is a positively charged particle gold was used because, it can be cut into thin sheets easily and also the nucleus of gold is heavy and can produce large deflections of α particles.

         

            The range of α particles in air is limited to about 5 cm. Therefore, the whole apparatus was kept in vacuum. So that α particles would not be prevented from reaching the detector.
            The angle θ of the deviation of an α particle from its original direction is called scattering angle.
            From the experiment it was found that
i) a very few particles were scattered at an angle greater than 90º. Some even reflected back i.e. a deflection of 180º.
ii) Some of the α particles were deflected through small angles.
iii) Most of the α particles passed straight way through the gold foil with no change of direction.
Thomson model fails: The large angle scattering canot be explained on the basis of Thomson's model.

Conclusions of Rutherford's Experiment:
i) Rutherford argued that the large angle scattering of α- particles could happen only if the positively charged particles were repelled by a massive positive charge concentrated in very small region of space and this tiny positively charged core which Rutherford named as 'NUCLEUS' (Hence this model is also called Nucleus model) which contains 99.99% of the mass of the atom.
ii) The nucleus is surrounded by electrons some distance away. But if the electrons were at rest, they will fall into nucleus due to electrical attraction. 
         Necessity is the mother of invention. In order to avoid this eventuality, Rutherford proposed that the electrons are revolving in circular orbits around the nucleus just as planets are revolving around the Sun. If the planets are not moving around the Sun, they would have fallen into the Sun's core due to gravitational force of the Sun (Hence Ruther's model of the atom is also known as planetary model) (whenever we get a doubt about a problem, we scratch our head and look little upward. Probably, Rutherford when he got the doubt about the stability of atom, he would have done the same and raised his head and got the thought of the motion of the planets around the Sun as a flash!!!)

iii) As an atom is electrically neutral, the total positive charge on the nucleus is equal to the total negative charge on the electrons in the atom.
iv) Since very few alpha particles were scattered through large angles, the probability of head on approach is small which shows that nucleus occupies only a small portion of the available space. This shows that whole of the positive charge and entire mass of the atom is confined to an extremely small central core called 'NUCLEUS'.
v) The radius of nucleus is of the order of 10-15 m and that of the atom is 10-10 m. Thus atom consists mostly empty space dotted with electrons and a very tiny nucleus at the centre.
Defects in Rutherford's model:
i) In Rutherford's model, the centrifugal force acting on the electrons is balanced the electrostatic attraction between the nucleus and the electrons. But according to the classical electromagnetic theory, an electron going in a circle and subject to a normal acceleration, must radiate energy and thus lose some of its energy. Hence the electron should approach the nucleus in spiral path and finally fall into the nucleus. Thus, the atom cannot be stable. But it is well known that most of the atoms are stable.

ii) According to classical electromagnetic theory, the accelerating electron must radiate energy at a frequency equal to the mechanical frequency of the orbiting electron and hence proportional to the angular velocity of the electron. As the electron spirals towards the nucleus the angular velocity tends to infinity, there by the frequency of the emitted energy tends to infinity. This results in a continuous spectrum with all possible wavelengths. But what we observe experimentally is that atoms like 'hydrogen' emit line spectra of fixed wavelengths only. If classical electromagnetic theory were not to fail, Rutherford's model of the atom had to be given up.
Niels Bohr - The architect: It was at this juncture, that the Danish physicist Niels Bohr entered the field in 1913 and proposed his theory of the structure of atom (Bohr atom model) and the origin of spectra for which he was awarded Nobel Prize for Physics in 1922 (Rutherford was also awarded Noble in 1908 but surprisingly not for Physics but for Chemistry). Bohr extended Planck's Quantum theory to the Rutherford's nuclear atom. He suggested that the electrons revolve round the nucleus in 'fixed orbits'. It means that the orbits of definite radii only are present and not all orbits of all radii as suggested by Rutherford. He gave the name 'energy levels' to these orbits.

The electron cannot emit any energy when it moves on fixed orbit, known as 'stationary energy level'. The electron gives out energy in the form of electromagnetic radiations of definite frequency only when it jumps from higher energy level to a lower energy level. Thus, the energy levels are 'Quantized'.
Bohr's Theory of the Hydrogen Atom:
          In order to explain the spectrum emitted by Hydrogen, Bohr made certain basic postulates, to modify the model suggested by Rutherford. In the process, Bohr made a bold adventure in scientific field by bridging the gulf between Classical Physics and Planck's Quantum theory of radiation.
Postulate 1:
         An electron cannot revolve round the nucleus in all possible orbits, as suggested by classical theory. The electron can revolve round the nucleus only in certain selected or permissible (or allowed) orbits, satisfying the Quantum condition that the angular momentum of the electron in the orbit must be an integral multiple of  where h is Planck's constant  

Postulate 2:
          The privileged orbits are called 'stationary orbits' and the electrons revolving in these oribts do not radiate energy.
Explanation: For an electron of mass 'm', moving with a speed 'v', in an orbit of radius 'r', the angular momentum  where n is called 'Principal Quantum number'. It takes integral values except zero; n = 1, 2, 3, 4, .... v = rω, where ω is angular velocity.
Postulate 3:
           'Only when electron jumps'
          An atom radiates energy only when an electron 'jumps' from a stationary orbit of higher energy to one at lower energy. If the electron jumps from an initial orbit of energy Ei to a final orbit of energy Ef where Ei > Ef, a photon of frequency

  This is known as "Bohr's frequency condition". The state of least energy is the one defined for n = 1 and is called ground or normal state as it is the lowest energy state. The states where 
n = 2, 3, 4, ... are called excited states because the atom then has more energy than it has in normal state.
 In Bohr's model, there is no explanation why the atom does not radiate energy when it is in stationary state. This is simply taken as a Postulate!
Radii of orbits
           Let us apply Bohr's Postulates to an atom having a nucleus with a positive charge Ze and mass M.
          for Hydrogen Z = 1
         Let an electron of charge (- e) and mass 'm' move round the nucleus in an orbit of radius 'r'. Since M > > m, the nucleus is stationary. Hence the mass of the nucleus does not appear in the calculations.

        The electrostatic force of attraction between the nucleus and the electron

           From equation (iii) it is clear that r  n₂ i.e. the radii of the privileged orbits are directly proportional to the square of the natural numbers 1, 2, 3, etc ...... These numbers 1, 2, 3, etc which decide the angular momentum of the electron in its orbit are called 'Principal Quantum Numbers'.  
Total Energy in Different Orbits:
           The total energy of the electron in any orbit is the sum of its kinetic and potential energies.

           

             The potential energy (P.E.) of the electron is considered to be zero, when it is at an infinite distance from the nucleus. So the potential energy of an electron in an orbit is the work done in bringing the electron from infinity to that orbit. This amount of work is obtained by integrating the electrostatic force of attraction between the nucleus and the electron with in the limits infinity () to r.

   

      
      Negative values of energy shows that the electron is bound to the nucleus.

      
      i.e., the outer orbits will have greater energies than the inner orbits.
Interpretation of Hydrogen Spectrum
        If an electron jumps from an outer initial orbit n₂ of higher energy to an inner orbit n₁ of lower energy. The frequency of the radiation emitted is given by

         
The wave number  of a radiation is defined as the reciprocal of its wavelength 'λ' in vacuum and gives the number of waves contained in unit length in vacuum.

            
Rydberg Constant

      

Spectral Series of Hydrogen Atom
1. Lyman Series: When an electron jumps from second, third etc. orbits to the first orbit; the spectral lines are in the ultraviolet region. Here n₁ = 1 and n₂ = 2, 3, 4, 5, ......

                
These are called Lyman series.

2. Balmer Series
       When the electron jumps from outer orbits to the second orbit.
       Here n₁ = 2 and n₂ = 3, 4, 5, ....... etc.

               
          This series is called 'Balmer Series' and lies in the visible region of the spectrum. The first line in the series n₂ = 3 is called the Hα line, the second n₂ = 4, the Hβ line and so on.

3. Paschen Series: When the electron jumps from the outer orbits to the third orbit
     i.e. n₁ = 3 and n₂ = 4, 5, 6, ..... etc. and these series are in the infrared region.

             

4. Brackett Series
      If n₁ = 4 and n₂ = 5, 6, 7 etc., we get Brackett series 

          
5. Pfund Series
     If n₁ = 5, n₂ = 6, 7, 8 etc., we get Pfund series

         
           Brackett and Pfund series lie in the very far infrared region of hydrogen spectrum.
           By putting n₂ = ∞, in each one of the series, we get the last line in the series which is the wave number of the series limit. Different spectral series of Hydrogen atom are shown in figure.

                                   
Energy Level Diagram

             
called the 'energy level diagram'.

                
            The lowest energy level E₁ is called normal or ground state of the atom and the higher energy levels E₂, E₃, E₄ ...... etc. are called excited states. As n increases, En increases. As n increases, the energy levels crowd. In the energy level diagram, the discrete energy states are represented by horizontal lines and the electron jumps between these states by vertical lines.

                       
         We have seen that the energy associated with an electron in the n^th orbit of hydrogen atom is  eV. Thus the energies of the first, second, third, ...... ∞ orbits are respectively −13.6, −3.4, −1.51, ...... 0 eV. The energy required to raise the atom from the ground state (n = 1) to the first excited state is 13.6 − 3.4 = 10.2 eV. The energy required to raise it to the second excited state is 13.6 − 1.51 = 12.09 eV and so on. 13.6 eV is called the ionisation potential. 10.2, 12.09 eV are called excitation potentials.

            In case of hydrogen, there is only one ionisation potential, where as there are several excitation potentials.
Ionisation potential (or eneregy): It is the energy, in electron volts required to remove an electron from a given orbit to an infinite distance from the nucleus.
Excitation potential (or energy): It is the energy, in electron volts required to raise an atom from its normal state into an excited state.
Shells: The electrons in the energy levels with n = 1, 2, 3, etc are said to be in K, L, M etc shells respectively. An atom with atomic number Z contains Z electrons. The electrons are distributed among different shells, starting with the K shell and filling up shells of higher energy. Each shell can accommodate only a definite maximum number of electrons.
              The maximum number of electrons which can be accommodated in a shell is 2n². Thus the K, L, M etc shells can contain a maximum of 2, 8, 18 etc electrons.

Why should the angular momentum have values that are integral multiples of   
            (De Broglie's explanation of second postulate of Quantisation)
According to second postulate of Bohr's model of the atom, the angular momentum (L) of the electron orbiting around the nucleus is quantized i.e. L =  where n = 1, 2, 3, etc which means the angular momentum have values that are integral multiples of  . 
           After 10 years of Bohr proposed his model the french physicist Louis De Broglie explained this aspect in the year 1923.
          According to De Broglie, matter and its particles like electrons will also have 'wave nature'. Latter, Davisson and Germer verified experimentally the wave nature of electrons. De Broglie argued that electron (particle) orbiting around the nucleus of an atom should be seen as a wave.
What Type of Wave? When a string fixed at both ends under tension is plucked (in any stringed musical instrument) a large number of wavelengths are excited. But only those wavelengths which have nodes (no displacement) at ends will exist and form a stationary on standing wave in the string. It means in a string only, stationary waves are formed in one wavelength, two wavelengths or any integral number of wavelengths.

          Similarly for an electron moving in nth circular orbit of radius rn, the total distance in the circumference of the orbit 2Πrn.
          Thus 2Πrn = nλ, where n = 1, 2, 3 etc.
          For example, let us consider a standing particle wave on a circular orbit, for n = 4 i.e., 2Πrn = 4λ where λ is the De Broglie wavelength of electron moving in the nth orbit.
          We know from the De Broglie's relation λ =  where P is the electron's momentum. If the speed of the electron is much less than the speed of light, the momentum is mvn

            

            This is the Quantum condition proposed by Niels Bohr for the angular momentum of the electron. This equation is the basic equation for explaining the discrete orbits and energy levels of the hydrogen atom.
            Thus De Broglie's concept of matter waves provided an explanation of Bohr's second postulate of Quantisation of angular momentum of the orbiting electron.
Limitations of Bohr's model
i) Bohr's model is applicable only for hydrogen atom with single electron. It cannot be extended even for two electron atoms like helium. The reason is that Bohr's model involves the electrical force between negatively charged electron and positively charged nucleus. It does not involve the electrical forces between electrons.
ii) Though the Bohr's model predicts exactly the frequencies of the light emitted by hydrogenic atom, but it is not able to explain the relative intensities of the frequencies in the spectrum and unable to account for the intensity variations. Bohr's model gives a brilliant picture of an atom and cannot be extended to complex atoms. For complex atoms, we have to apply a rigorous theory based on quantum mechanics. But the Bohr's theory of atom is the basis for explaining the structure of any atom. Thus Neils Bohr who synthesised brilliantly the traditional classical mechanics and revolutionary quantum theory is the true architect of the structure of the atom.

What Einstein told ...
            The famous scientist Albert Einstein told "without Niels Bohr we cannot imagine how our knowledge about atom will be. He expresses his opinion as if he was involved in rigorous research with great commitment throught out his life and not as a great person who has found 'great truths' long ago."
            According to Niels Bohr, in subatomic systems both the particle and wave nature of matter should be considered. This concept will help to explain the free human will and the basic life styles and also to analyse the theory of 'multiuniverses.' This concept proposed by Bohr is well known as 'Copenhagen interpretation.'
Bohr - a Sports Person: Niels Bohr is not only a great scientist but a brilliant sports person! An expert in sports like scheming and boat races. In student days he participated in the game of football on behalf of Denmark. (His brother played football in Olympics)
            In 1960, Niels Bohr came to India to participate in Indian Science Congress Conferences. During that period he visited all important places in our country.

Important writings of Niels Bohr
             1) Theory of Spectra and Atomic Constitution
             2) Atomic Theory and Description of the Nature
             3) Atomic Physics and Human Knowledge

        Chemistry deals with structure, composition, properties of matter. Matter can be classified into mixtures, compounds and elements. All of them consists molecules and atoms. According to Greek philosophers atoms are fundamental blocks of the matter. The word atom was derived from Greek word "atomio", which means "non-divisible". According to Daltons atomic theory, atoms can be further divided into sub atomic particles like protons, electrons and neutrons. This theory successfully explained the law of Conservation of mass, Law of Constant composition and law of multiple proportion.
       Electron was discovered by J.J. Thomson in cathode rays experiment in a discharge tube. Proton was discovered by Goldstein in canal ray's experiment. Neutron was discovered by James Chadwick by bombarding a thin sheet of Be by α -particles. Experimental observations suggested few atomic models to explain the structure of atom i.e., distribution of these charged particles in an atom.

      According to Thomson watermelon model of atom, positive charge is uniformly distributed in an atom just like red mass present in watermelon and the electrons are embedded in it (like seeds). Rutherford's nuclear model of atom resembles the solar system. In which nucleus is compared with Sun and electrons with the revolving planets.  This model could not explain the stability of atom and electronic structure of atom. 

 Maxwell suggested that when electrically charged particle moves under acceleration, electro-magnetic waves are produced. They are propagating in perpendicular directions in one other, can move in vacuum. The spectrum of electromagnetic radiation consists γ-rays, X-rays, u.v. rays visible I.R., microwave, radio and long radio waves.

      The distance between two successive crests or troughs in a wave is called wavelength (λ). The number of waves passing through a point in one second is called frequency (ν). The number of waves present in Unit length are called wave number (). The height of the crest or depth of a trough is called amplitude.
                 Velocity of light (c) = ν . λ          

        Light is electromagnetic radiation which possesses both the particle nature (black body radiation & photo electric effect) and wave nature (diffraction and interference). Particle nature of electromagnetic radiation was explained by Plank's quantum theory.
Plank's quantum theory:
    Black body radiation was successfully explained by Max Plank in 1900. A hallow metallic sphere coated inside with platinum black with a fine hole acts as black body. Which is a perfect absorber and perfect radiator of energy. Radiation is associated with energy. The energy is emitted or absorbed by a body discontinuously in the form of a small packet called "quantum". Energy is directly proportional to the frequency of radiation.   E ∞  ν
                       E = h ν
       Propagation of radiant energy in the form of quanta is called "quantization of energy".
          E = n . h ν (n = integer)
      Curves are obtained by plotting 'E' against 'λ'. As the temperature increases, the peak of the curve shifts to lower λ.

       At a given temperature, the intensity of the radiant energy increases with the λ, reaches maximum and then decreases.
        In 1905, Einstein replaced the word quantum by "photon". He explained photo electric effect. Emission of electrons from a clean metal surface when light (photons) with suitable λ falls on it is called "photo electric effect". It is readily exhibited by alkali metals.
                  h ν = W + K.E.
                  h ν = energy of photon
                  W = work function
                  K.E. = Kinetic energy of emitted electron.
     The series of coloured bands obtained by splitting of electromagnetic radiation when it is passed through prism is called "spectrum". The spectrum which is produced due to excitation and de-excitation of electrons of atoms is called "line spectrum". This spectrum has sharp, well defined and distinct lines. If the spectrum is produced by molecules, it is called "Band spectrum", which has closely spaced lines (bands). The study of emission or absorption spectra is known as "spectroscopy" and is useful in chemical analysis.

             "Absorption spectrum" is produced due to excitation of atoms or molecules or ions, when they absorb energy. This spectrum consists dark lines on a bright background. "Emission spectrum" is produced due to emission of light by excited ions or atoms or molecules. This spectrum consists of bright lines on a dark background.

Bohr's Model of Atom
        Bohr's model of atom is a modification of Rutherford's model. It is based on Plank's quantum theory and hydrogen spectrum.
Postulates:      

* Electrons revolve round the nucleus in fixed, circular paths called 'orbits'.
* Orbits are denoted by 1, 2, 3, 4... or K, L, M, N ...
* Each orbit is associated with definite amount of energy called Energy levels.
* Electrons neither emit nor absorb energy when they revolve in a orbit called 'stationary orbit'.
* As 'n' value increases the size, energy of orbit increases. The angular momentum of an electron is integral multiple of   i.e. mvr = 
* Energy absorbed or emitted by electron is given by ∆E = E₂ - E₁ = hν.

Explanation of emission spectrum of H-atom:
       Of all the atomic spectra, the hydrogen spectrum is the simplest spectrum. H atom has only one electron, but it gives 5 series of spectral lines. When H₂ gas is heated or exposed to light or subjected to electric discharge, electrons of different H atoms get excited to different higher orbits and deexcites to different lower orbits in different manner with the emission of energy and give 5 series of spectral lines. λ of a spectral line in H atom can be calculated by using Rydberg's equation

    
         R = 2π² me₄/ch₃

  

    if n₂ - n₁ = 1 (Hα)
       n₂ - n₁ = 2 (Hβ)
       n₂ - n₁ = 3 (Hγ)
From Lyman series to Pfund series λ increases, E, ν decreases. RH value is 109677 Z² cm^-1. The spectral lines get closer and closer as we move from n₂ = 2 to 3 to 4 to 5 to 6 etc., Number of spectral lines formed when electron is coming from higher orbit n₂ to lower orbit n₁ is   where n = n₂ - n₁.

Limitations of Bohr's model
     * This theory explains only spectra of H and H like species.
     * This theory could not explain fine structure of H- spectrum.
     * This model failed to explain Zeeman effect and Stark effect.
     * This theory could not explain the ability of atoms to form molecules by chemical bonds.
     * This model failed to explain why angular momentum should always be an integral multiple of 
Derivation of equation for radius of nth orbit (rn):
       H atom has 1 proton in its nucleus with +e charge. An electron with -e charge revolves round the nucleus in a circular orbit of radius 'r'. As per Coulomb's law, electrostatic force of attraction between the nucleus and the electron is given by centripetal (attractive) force 

       To make atom stable, an equal and opposite centrifugal force must act away from the nucleus

 

    by substituting (4) in (3)

       
                                                  = 0.529 × 10^-8 n₂ cm
Derivation of equation for Energy of electron in nth orbit    

Quantum Mechanical Model of Atom
     Modern theory of atomic structure was proposed on the basis of quantum mechanics. The branch of science that takes into account this dual behaviour of matter is called quantum mechanics. It was developed by Heisenberg and Schrodinger. Quantum mechanical model of atom is the picture of the structure of the atom. Main features of this model are:
   * The energy of electrons in atoms is quantized.
   * The existence of quantized electronic energy levels is a result of the wave nature of electrons and also allowed solutions of Schrodinger wave equation.
   * It is impossible to know exact position and momentum of an electron in an atom can not be determined simultaneously (Heisenberg uncertainty principle)
                                             ∆x . ∆p ≥  
   * Atomic orbital is the wave function Ψ for an electron in an atom. Ψ Must be finite, continuous and single value.
   * The probability of finding electron at a given point in an atom is proportional to Ψ₂ (orbital).

   * Schrodinger wave equation gives the probability of finding electron around nucleus

                 
            Ψ = wave function, m = mass of electron
             E = total energy of electron, U = potential energy of e-
De Broglie Theory
      All micro particles moving with high velocities are associated with wave characteristics
      E = hν (plank's quantum theory) .............. (1)
      E = mc² (Einstein's theory) ........... (2)
      from (1) & (2)
      mc² = hν =  

         
    If circumference of the electron orbit 2 πr = nλ, electron wave is in phase. If 2πr ≠ nλ, electron wave is not in phase.

Quantum Numbers
A set of numbers used to provide complete description of an electron (energy and its complete address) in an atom are called quantum numbers. Four quantum numbers, i.e. Principal, Azimuthal, Magnetic and Spin quantum numbers are required for this purpose.
1. Principal Quantum number (n):
     * It was proposed by Neils - Bohr.
     * Is is denoted by 'n'.
     * n values can be denoted by K, L, M, N... or 1, 2, 3, 4...
     * The size and energy of the orbit increases with the increase of n.
     * It also represents the distance between the electron and nucleus.
     * The number of electrons present in an orbit = 2 n₂.
     * Angular momentum of an electron in an orbit =  
     * This quantum number indicates the size and energy of the orbit.

2. Azimuthal Quantum number (l):
     * It was proposed by sommerfeld.
     * It is denoted by 'l'.
     * The values of l = 0, 1, 2, 3... (n-1)
     * It represents sub - shells (s, p, d, f) in a shell.
     * The number of subshells in an orbit are equal to 'n'.
     * When l = 0, 1, 2, 3... electron belongs to the subshell s, p, d, f respectively.
     * The shapes of s, p, d, f orbitals are spherical, dumb-bell, double dumb-bell and four fold dumb-bell.
     * Energy of these sub-shells: s < p < d < f.
     * The number of electrons present in a sub-shell are equal to 2(2l + 1), i.e. 2 electrons in s, 6 electrons in p, 10 electrons in d and 14 electrons are present in f sub-shells.
     * This quantum number indicates the shape of the orbital.

3. Magnetic quantum number (m):
     * It was proposed by Lande to explain Zeeman effect.
     * It is denoted by m.
     * m values are ranging from -l to 0 to +l.
     * m has (2l + 1) values.
     * m values are 1, 3, 5, 7 if l values are 0, 1, 2, 3 respectively.
     * The energy of all the orbitals present in a sub-shell is same.
     * m value indicates the total number of orbitals in a sub-shell.

     * The no. of orbitals in s, p, d, f sub-shells are 1, 3, 5, 7 respectively.
     * No. of orbitals in any orbit = n₂

  

     * This quantum number indicates orientation of an orbital.
4. Spin quantum number (s):
     * It was proposed by George Uhlenbeck and Samuel Goudsmith.
     * It is denoted by s.
     * Electron can spin on its own axis either in clock wise direction (is denoted by +  or ↑) &  in anti clock wise direction (is denoted by -  or ↓). 

     * For every value of m, there can be two 's' values.
     * Maximum number of electrons in an orbital = 2 (one electron makes clock wise spin, other electron makes anti clock wise spin).
     * The difference between the 2 spin quantum numbers is 1.
     * Spin quantum number indicates the spin of the electron in an orbital.
Shapes of Atomic Orbitals
       Atomic orbital is the space around the nucleus where the probability of finding an electron (Ψ₂) is maximum (95%). The probability of finding electron within the radial space around the nucleus is called radial probability distribution. It can be calculated by using the formula
4πr² dr Ψ₂. The radial space at which the radial probability of finding electron is Zero is called "node" or "radial node" or "nodal region". It is the space between two similar orbitals. The number of nodes are equal to n-l-1. The probability of finding electron at the nucleus is Zero and is called "nodal point". The plane passing through this point is called "nodal plane" (or angular node). The number of nodal planes of orbital are equal to "l". The total number of radial nodes and angular nodes for any orbital are equal to (n -1). The nodal planes of px orbital is yz, py is zx, pz is xy, dxy are yz & zx, dyz are zx & xy, dzx are yz & xy and dx²-y² are yz & zx.

The shape of the orbital depends on n & l values.
s - orbital: The shape of s orbital is spherical. Its
l = 0. It is non directional orbital.
p - orbital: The shape of p orbital is dumb-bell and having 2 lobes. Its l = 1. These orbitals are degenerate orbitals and are directional. If they orient along x, y, z axes they are called px, py, and pz orbitals.    
Orbital:     px      py      pz
m value:   ±1      ±1      0
p orbital has one nodal plane.

d-orbital: The shape of d orbital is double dumb-bell and having 4 lobes. Its l = 2.
dxy, dyz, dzx orbitals are oriented in between the axes. Where as dx²-y², dz² orient along the axes.
Orbital:     dxy     dyz     dzx    dx²-y²    dz²
m value:    ±2       ±1      ±1       ±2         0

f-orbital: The shape of f orbital is 4 fold dumb-bell and having 8 lobes. Its l = 3. Seven f orbitals are f_x³, f_y³, f_z³, f_xyz, f_x (y² - z²), fy (z² - x²) and fz (x² - y²).
              The energy order of the orbital is s < p < d < f.

Electronic Configuration
     The distribution of electrons in different orbitals present in an atom of an element is called "electronic configuration". It is governed by Aufbau principle, Paulis exclusion principle and Hund's rule.
Aufbau Principle
Electrons enter into various atomic orbitals in an atom in increasing order of the energy of atomic orbitals. Energy sequence of atomic orbitals can be remembered either by Moeller's diagram or by (n + l) rules. Electron first enters into atomic orbital whose n + l value is lowest. If n + l values are same for different atomic orbitals, orbital with lowest n value will be filled first. The order of filling electrons is
1s < 2s < 2px = 2py = 2pz < 3s < 3px = 3py = 3pz < 4s < 3dxy = 3dyz = 3dzx = 3dx²-y² = 3dz²< 4px = 4py = 4pz < 5s <...........

Pauli's exclusion principle:
      No two electrons in an atom will have the same set of values for 4 quantum numbers. Mean while an orbital can accommodate maximum of 2 electrons with opposite spins.
                                  n    l    m      s
eg: He 1st electron: 1    0    0    +   
          2nd electron: 1   0     0    -  
 

Hund's rule of maximum multiplicity:
      Pairing of electrons in atomic orbitals of the same sub-shell takes place only after filling each orbital with one electron. The pairing of electrons in p, d, f orbitals start with the entry of 4th, 6th and 8th electron respectively.
      By keeping all these 3 rules in mind one can write electronic configuration by nlx method.
Stability of completely filled and half filled sub shells:
       Sub-shells either half filled or completely filled are more stable due to lowering of energy (due to the exchange energy) and symmetrical distribution of electrons. More repulsions are observed when two electrons are with parallel spins.

          Lowering energy = total possible exchangeable sets × K
          Where K = Average exchange energy for a set of 2 parallel spins
           possible sets = nC² =               eg. For Cr - 24
          If valency shell configuration = 4s² 3d4

                                                                                    
          No. of sets =  
        Lowering of energy = 10 K
       If valency shell configuration = 4s¹3d⁵

                                                                                  
        no. of sets =  
         Lowering of energy = 15 K
         For Chromium 3d⁵ 4s¹ configuration is more stable than that of 3d⁴ 4s².