1101: Structure of the atom

Atom

• The existence of atoms has been proposed since the time of early Indian and Greek philosophers (400 B.C.) who were of the view that atoms are
   the fundamental building blocks of matter. 
• According to them, the continued subdivisions of matter would ultimately yield atoms which would not be further divisible. The word ‘atom’ has
   been derived from the Greek word ‘a-tomio’ which means ‘uncut able’ or ‘non-divisible’.
• These earlier ideas were mere speculations and there was no way to test them experimentally. These ideas remained dormant for a very long 
  time and were revived again by scientists in the nineteenth century.

Theories of Atomic Structure

• The atomic theory of matter was first proposed on a firm scientific basis by John Dalton, a British school teacher in 1803. His theory, called 
  Dalton’s atomic theory, regarded the atom as the ultimate particle of matter.
• The physical and chemical properties of elements and their compounds are essentially with the structure of atoms. A systemic knowledge of the 
  structure of atoms is, therefore, necessary for the study of modern inorganic chemistry.
• In this unit, we shall start with the experimental observations made by scientists towards the end of nineteenth and beginning of twentieth 
  century. These established that atoms can further be divided into subatomic particles i.e. electrons, protons and neutrons.

Fundamental Particles of Atom

• An atom is the smallest particles of an element having its own chemical identity and properties.
• Atom consists of still smaller particles, such as electron, proton, neutron etc. into which it can be subdivided. These particles are regarded as 
  fundamental particles, because these are the main constituents of atoms.

Classification of Fundamental Particles:

i) Permanent Fundamental Particles: The particles which play an important role for the formation of structure of an atom and are present in all atoms 
(except hydrogen where neutron is absent) are called permanent fundamental particles. There are three kinds of permanent fundamental particles such 
as electron, proton and neutron.

• Electron bears negative charge and revolves around the nucleus in a circular path. The mass of electron is very small.
• Proton bears the positive charge and stays at the nucleus with neutrons.
• Neutron has no charge, i.e. it is neutral. It is placed in the nucleus with proton.

ii) Temporary Fundamental Particles:

• The particles which exist in some atoms for short time are called temporary fundamental particles. There are about 100 numbers of temporary 
  fundamental particles in different atom like positron, neutrino, antineutrino, meson etc.

Discovery of Electron

Production of Cathode Rays:

In mid 1850s many scientists mainly Faraday began to study electrical discharge in partially evacuated tubes, known as cathode ray discharge tubes. 
It is depicted in the following Fig.

Fig. Production of cathode rays in a discharge tube.

The knowledge about the electron was derived as a result of the electric discharge in a discharge tube. The discharge tube consists of a glass tube with
 metal electrodes fused in the walls. The pressure inside the tube can be reduced with a pump.

When the electrodes are connected to a source of high voltage (10,000 volts), an electric discharge from high potential source is passed through a gas 
at pressure of 0.001 mm of Hg in the discharge tube and the glass wall opposite to the cathode begins to glow with a very faint greenish light (or bluish 
light depending on the composition of the glass of which the tube is made.)

This glow or fluorescence is formed due to the bombardment on the glass by certain rays which are emitted from the cathode surface and move 
towards the anode with tremendous velocity. These rays were called cathode rays by Goldstein since these were originated from the cathode.

The results of this experiment are summarized below.

(i) The cathode rays start from cathode and move towards the anode.

(ii) These rays themselves are not visible but their behavior can be observed with the help of certain kind of materials (fluorescent or phorosphescent) 
which glow when they are hit by the cathode rays.

(iii) In the absence of electrical or magnetic field, these rays travel in straight lines (Fig. 2.2)

Fig. Measurement of e/m for electrons.

(iv) In the presence of electrical or magnetic field, the behaviors of cathode rays are similar to that expected from negatively charged particles, 
suggesting that the cathode rays consist of negatively charged particles, called electrons.

(v) The characteristics of cathode rays (electrons) do not depend upon the material of electrodes and the nature of the gas present in the cathode ray 
tube.

Thus, we can conclude that electrons are basic constituent of all the atoms.

The rays which proceed from the cathode and move away from it right angles in straight lines are called cathode rays. They are deflected in 
electric and magnetic fields indicating that these rays carry electric charge.

Properties of Cathode Rays

• They travel in straight lines away from the cathode. e.g. Shadow of aluminium crosses.
• They produce fluorescence (a glow) when they strike the glass wall of discharge tube.
• They produce X-rays when they strike a metallic target.
• Consist of moving particles with definite mass and velocity i.e. Momentum.
• They can ionize gases.
• They can pass through thin metal foil.
• The cathode rays are deflected towards the positive pole of the electric field which proves that the rays are associated with negative charges. 
  The particles making up the cathode rays were all identical and ultimately given the name electron by the Dutch Physicist H. A. Lorentz.

(a) In 1897 J.J. Thomson showed that when cathode rays are passed through an electric field (i.e. when the rays are passed between two electrically charged 
plates), they are deflected towards the positively charged plate [Fig. 1.2 (a)], showing thereby that the particles constituting the cathode rays are negatively 
charged. These negatively charged particles were called electrons by Thomson.

Fig. 1.2 (a) Cathode rays-are deflected in an electric field, showing- that the particles (i.e. electrons) constituting the cathode rays are negatively charged.

(b)In 1895 J. Perrin showed that if a magnetic field is applied on cathode rays from top to bottom, perpendicular to the plane of the paper as shown in Fig. 1.2 (b),
 cathode rays are deflected upwards in the plane of the paper. This experiment also shows that the particles constituting the cathode rays are negatively charged.

Fig. 1.2 (b) Cathode rays are deflected in a magnetic field, showing that the particles (i.e., electrons) constituting the cathode rays are negatively charged.

Fig. 1.2. Experiment to show that the particles (i.e. electrons) constituting the cathode rays are negatively charged.

Definition of an Electron

Having known the charge and mass of an electron, it can be defined as:

An electron is a subatomic particle which bears charge –1.60✕10–19 coulomb and has mass
9.1✕10–28 g.

Alternatively, an electron may be defined as:

A particle which bears one unit negative charge and mass 1/1835th of a hydrogen atom.

Since an electron has the smallest charge known, it was designated as unit charge by Thomson.

Discovery of Proton

• In 1886, German physicist , Eugen Goldstein used a discharge tube provided with a cathode perforated with extremely fine holes and observed 
  that when the electrodes are connected to a source of high voltage (10,000 volts), not only cathode rays of electrons were originated from the 
  cathode moving from cathode towards anode but also new type of colored rays were simultaneously originated from the anode moving from 
  anode towards cathode through the fine holes or canals in the cathode causing a glow on the wall opposite to anode.
• Initially these rays were called canal rays, since they pass through the canals or holes of the cathode. These are also called anode rays as they 
  seem to originate from the anode.
• Sir J. J. Thomson has shown that these rays consist of particles carrying a positive charge and hence he named them positive rays.

Fig. Production of positive (anode) rays

Explanation of the Production of Positive Rays

• When a high voltage is applied across the metallic electrodes of a discharge tube filled with a gas under reduced pressure, the molecules of the 
  gas are dissociated into atoms.
• Now when high speed electrons (i.e. cathode rays) originated from cathode strikes these atoms, the atoms are converted into cations which pass 
  through the holes of the cathode and constitute positive rays.
• The electrons obtained in the conversion of the gaseous atoms into cations move towards anode as negative rays.

Fig. Explanation of the production of positive rays (Illustration).

The characteristics of these positively charged rays are listed below.

1. Unlike cathode rays, the positively charged particles depend upon the nature of gas present in the cathode ray tube. These are simply the 
   positively charged gaseous ions.
2. Some of the positively charged particles carry a multiple of the fundamental unit of electrical charge.
3. The behavior of these particles in the magnetic or electrical field is opposite to that observed for electron or cathode rays.

The smallest and lightest positive ion was obtained from hydrogen and was called proton.
This positively charged particle was characterized in 1919.

Thomson Model of Atom

J. J. Thomson, in 1898, proposed that an atom possesses a spherical shape (radius approximately 10–10 m) in which the positive charge is uniformly 
distributed. The electrons are embedded into it in such a manner as to give the most stable electrostatic arrangement (Figure). Many different names 
were given to this model, for example, plum pudding, raisin pudding or watermelon.

This atomic model was given up after some time, since it could not explain the large deflection suffered by α— particles in Rutherford’s experiment.

Discovery of Neutron

• Up to 1932, it was postulated that an atom is composed of only proton and electron.
• Mass of an atom is due to proton as mass of electron is negligible.
• Rutherford noticed that mass of different atoms cannot be explained if it is supposed that atom is composed of proton and electron. He predicted,
   in 1920, that some kind of neutral particles having mass equal to proton must be present in an atom.
• These particles were discovered by James Chadwick (1932) by bombarding a thin sheet of beryllium by α-particles when electrically neutral 
  particles having a mass slightly greater than that of the protons were emitted. He named these particles as neutrons.

Experiment

• A stream of alpha-particles was directed from a polonium source to a Beryllium target.
• A charge detector was placed behind the Beryllium target.
• A new radiation was produced which no charge on them had indicated by charge detector.
• The reaction in Chadwick’s experiment converted a ‘Be’ atom into a Carbon atom through the nuclear reaction.

Summary of Electron, Proton & Neutron

Rutherford’s α—particle scattering experiment

Rutherford and his students (Hans Geiger and Ernest Marsden) bombarded very thin gold foil with
α—particles. Rutherford’s famous α—particle scattering experiment is represented in the following Fig.  

A stream of high energy α—particles from a radioactive source was directed at a thin foil
(thickness ∼ 100 nm) of gold metal. The thin gold foil had a circular fluorescent zinc sulphide screen around it. Whenever α—particles struck the screen,
 a tiny flash of light was produced at that point.

Rutherford’s observation

Fig. Rutherford and Marsden's -particle scattering experiment.

It was observed that:

(i)  Most of the α-particles passed through the gold foil undeflected.

(ii)  A small fraction of the α-particles was deflected by small angles.

(iii)  A very few α-particles (∼1 in 20,000) bounced back, that is, were deflected by nearly 180°.

Fig. How nuclear atom causes scattering of -particles.

On the basis of the observations, Rutherford drew the following conclusions regarding the structure of atom:

1. Most of the space in the atom is empty as most of the α-particles passed through the foil undeflected.
2. A few positively charged α-particles were deflected. The deflection must be due to enormous repulsive force showing that the positive charge of 
   the atom is not spread throughout the atom as Thomson had presumed. The positive charge has to be concentrated in a very small volume that 
   repelled and deflected the positively charged α-particles.
3. Calculations by Rutherford showed that the volume occupied by the centre is negligibly small as compared to the total volume of the atom. 
   The radius of the atom is about 10-10 m, while that of centre is 10-15 m. One can appreciate this difference in size by realizing that if a cricket ball
    represents the centre, then the radius of atom would be about 5 km.

Rutherford’s Nuclear Model of Atom

On the basis of above observations and conclusions, Rutherford proposed the nuclear model of atom .

According to this model:

1. The positive charge and most of the mass of the atom was densely concentrated in extremely small region. This very small portion of the atom 
   was called nucleus by Rutherford.
2. As the atom is electrically neutral, so the equal number of electrons as that of protons surround the nucleus.
3. The nucleus is surrounded by electrons that move around the nucleus with a very high speed in circular paths called orbits. Thus, Rutherford’s 
   model of atom resembles the solar system in which the nucleus plays the role of sun and the electrons that of revolving planets.
4. Electron and the positive nucleus are held together by electrostatic forces of attraction. This attraction force is counter-balanced by the outward 
   force of electron.

Drawbacks of Rutherford Model

• Rutherford’s nuclear model of an atom is like a small scale solar system with the nucleus playing the role of the massive sun and the electrons
   being similar to the lighter planets. When classical mechanics is applied to the solar system, it shows that the planets describe well-defined orbits
   around the sun. The theory can also calculate precisely the planetary orbits and these are in agreement with the experimental measurements. 
  The similarity between the solar system and nuclear model suggests that electrons should move around the nucleus in well-defined orbits.
• According to the electromagnetic theory of Maxwell, charged particles when move around another charged particle, then they should emit 
  electromagnetic radiation. Therefore, an electron in an orbit will emit radiation; the energy carried out by radiation comes from electronic motion.
   The orbit will, thus, continue to shrink. But this does not happen. Hence, the Rutherford model cannot explain the stability of an atom.
• Another serious drawback of the Rutherford model is that it says nothing about the electronic structure of atoms i.e. how the electrons are 
  distributed around the nucleus and what are the energies of these electrons.
• It could not explain how the spectral lines are produced by the simplest atoms like hydrogen atoms when an electron jumps from one orbit to 
  another.

        

Wavelength: The wavelength is defined as the distance between two successive crests or troughs of a wave.

Wavelength is denoted by the Greek letter λ(lambda).

Frequency: The frequency is the number of waves which pass a given point in one second.

Frequency is denoted by the letter ν(nu) and is expressed in hertz (hz).

Spectrum

A spectrum is an array of waves or particles which is spread out according to the increasing or decreasing of some properties such as wavelength or 
frequency.

Types of Spectra

Depending on the nature of the source emitting the radiation there are two types of spectra.

(1) Emission spectra

-Emission spectra are further of two types:

(a) Continuous spectrum

(b) Discontinuous spectrum which may be -

- Band spectrum

or, - Line spectrum  (atomic spectrum)

(2) Absorption spectra

1. Emission Spectra

Emission spectra can be obtained from the substances which emit light on excitation. Their excitation can be done as follows:

1. By heating the liquid or solid substances in a flame at high temperature. These substances become incandescent at high temperature i.e. when 
   these substances are heated at high temperature they emit light. Electric light, lime light, arc light are the examples of such solid substances.
2. By passing an electric discharge through a gas at low pressure.
3. By passing electric current through thin filament of a high melting point metal like tungsten.

Emission spectra are of the following two types:

1. Continuous Spectrum

When a narrow beam of sunlight or any white light is allowed to pass through a prism and then allowed to fall on a screen, it gets resolved into seven 
colours i.e. violet, indigo, blue, green, yellow, orange and red (VIBGYOR). This phenomenon is called dispersion and the band of seven colours 
observed on the screen is called visible spectrum.

In this spectrum, since one colour merges into another without any break or discontinuity (i.e. one colour changes gradually into another), the 
spectrum thus obtained is called continuous spectrum. This series of bands that form a continuous rainbow of colours, is called a Continuous 
Spectrum.

The Wave Nature of Light

• Refraction is the bending of light when it passes from one medium to another of different density.
• Speed of light changes.
• Light bends at an angle depending on its wavelength.
• Light separates into its component colors.

2. Discontinuous spectrum

Discontinuous spectrum may be band or line spectrum.

Band spectrum:

The band spectrum consists of a number of bright bands separated by dark spaces. Each band is sharp at one end and fades gradually in the other 
end. Band spectrum is the property of molecules and generally given by the compounds or gases like nitrogen and oxygen molecules at low 
temperature and pressure. Since this spectrum is characteristics of molecules it is also called molecular spectrum.

                  

Line spectrum:

When an atom of an element absorbs energy, it gets excited. The excited atom emits light radiations of a characteristic colour. When this emitted 
radiation is allowed to pass through a prism, it is resolved into several individual lines. The pattern of different lines of the radiation is called line 
spectrum. It is also called atomic spectrum because this spectrum is given by the atoms  of an element.

- - Helium was first discovered in the sun in 1868 with the help of its spectrum.

2. Absorption spectra

If, in the arrangement to get a continuous spectrum of white light, a substance is placed between the white light source and the prism, we get an 
absorption spectrum. This spectrum appears in the form of dark lines in a particular region of the continuous spectrum of white light. For example, 
if a solution of NaCl is placed between the white light source and prism, the continuous spectrum of white light is found to be crossed by two dark lines
 in the yellow region of the spectrum. If the white light source is removed, the whole of the continuous spectrum disappears leaving behind only two 
bright yellow light. These two dark lines produced by NaCl solution in the continuous spectrum of white light are known as absorption line spectrum.  

Hydrogen Spectrum (Origin of spectral line)

• When an electric discharge is passed through gaseous hydrogen in a discharge tube, the H2 molecules dissociate and the energetically excited 
  hydrogen atoms produced emit electromagnetic radiation of discrete frequencies.
• The hydrogen spectrum consists of several series of lines named after their discoverers. Balmer showed in 1885 on the basis of experimental 
  observations that if spectral lines are expressed in terms of wavenumber ( ν ), then the visible lines of the hydrogen spectrum obey the following 
  formula :

Where, n is an integer equal to or greater than 3 (i.e., n = 3,4,5,….)

The series of lines described by this formula are called the Balmer series. The Balmer series of lines are the only lines in the hydrogen spectrum 
which appear in the visible region of the electromagnetic spectrum.

Lyman found a series of lines in the ultraviolet region of hydrogen spectrum. Later, three more scientists Paschen, Bracket and Pfund found three more
 series of lines in the infrared region of hydrogen spectrum which were named after them.

The Swedish spectroscopist, Johannes Rydberg, noted that all series of lines in the hydrogen spectrum could be described by the following expression
 :

Where, n1 = 1,2……..

  n2 = n1 + 1

The value 109,677 cm-1 is called the Rydberg constant for hydrogen and the value of n1 is fixed for a particular series of lines.

The five series of lines that correspond to n1 = 1, 2, 3, 4, 5 are known as Lyman, Balmer, Paschen, Bracket and Pfund series, respectively. 
Table 2.3 shows these series of transitions in the hydrogen spectrum.

The five series of lines that correspond to n1 = 1, 2, 3, 4, 5 are known as Lyman, Balmer, Paschen, Bracket and Pfund series, respectively. 
Table 2.3 shows these series of transitions in the hydrogen spectrum.

Fig : shows the Lyman, Balmer and Paschen series of transitions for hydrogen atom.

Fig. Hydrogen spectral series on a Bohr atom energy diagram.

Of all the elements, hydrogen atom has the simplest line spectrum. Line spectrum becomes more and more complex for heavier atom. There are, 
however, certain features which are common to all line spectra

1. Line spectrum of an element is unique.
2. There is a regularity in the line spectrum of each element i.e. the radiations emitted from an atom have a fixed frequency and wavelength in the 
   spectrum.

Problem

3. What are the frequency and wavelength of a radiation emitted during a transition from n = 5 state to the n = 2 state in the hydrogen atom?

Bohr’s Atomic Model

Neils Bohr (1913) was the first to explain quantitatively the general features of hydrogen atom structure and its spectrum. Bohr’s model for hydrogen 
atom is based on the following postulates:

1. The electron in the hydrogen atom can move around the nucleus in a circular path of fixed radius and energy. These paths are called orbits, 
   stationary states or allowed energy states. These orbits are arranged concentrically around the nucleus.
2. The energy of an electron in the orbit does not change with time. However, the electron will move from a lower stationary state to a higher 
   stationary state when required amount of energy is absorbed by the electron or energy is emitted when electron moves from higher stationary 
   state to lower stationary state.
3. The frequency of radiation absorbed or emitted when transition occurs between two stationary states that differ in energy by ΔE, is given by :

ΔE = (E2 – E1) = hν   —————————————(1)

Where, E1 and E2 are the energies of the lower and higher allowed energy states respectively. This expression is commonly known as 
Bohr’s frequency rule.

4. The angular momentum of an electron in a given stationary state can be expressed as in equation-

mevr = n.h/2π ——————————————(2)

5. Thus, an electron can move only in those orbits for which its angular momentum is integral multiple of h/2π. That is why, only certain fixed 
   orbits are allowed.

Reasons for the Failure of the Bohr Model

1. Bohr model can explain only hydrogen atom spectrum but it can’t explain the spectrum of atoms having several electrons.

2. An electron is regarded as a charged particle moving in well-defined circular orbits around the nucleus. But modern research reveals that electrons
 may revolve in elliptical orbits also.

3. Further, an orbit is a clearly defined path and this path can be completely defined only if both the position and the velocity of the electron are known
 exactly at the same time. This is not possible according to the Heisenberg’s uncertainty principle. (Its position and momentum cannot be precisely 
pinned down)

4. For the transformation of electrons from one energy level to another only one line should be obtained in the spectrum. But after analyzing the 
spectrum, several fine lines are obtained. Bohr’s atom model can’t explain these features.

Orbit & Orbital

• Orbit and orbital are not synonymous.

• Orbit: An orbit, as proposed by Bohr, is a circular path around the nucleus in which an electron moves.
• Orbital: It is a region of space around the nucleus where the probability of finding an electron is maximum.

Differences between Orbit and Orbital (home work)

Quantum Numbers

A large number of orbitals are possible in an atom. Qualitatively these orbitals can be distinguished by their size, shape and orientation. An orbital of 
smaller size means there is more chance of finding the electron near the nucleus. Atomic orbitals are precisely distinguished by what are known as 
 quantum numbers. Each orbital is designated by three quantum numbers labeled as n, l and m.

Thus, the numbers which express the size, shapes, direction of the orbital from the nucleus and the spins of the electron on their own axis are called  
quantum numbers.

There are four types of quantum numbers.

1. Principal quantum number

2. Subsidiary quantum number or Azimuthal quantum number

3. Magnetic quantum number

4. Spin quantum number

1. Principle Quantum Number:

• This number expresses the orbits or principal energy levels in which the electron revolves round the nucleus. This is also referred to as major 
  energy level. It is denoted by n ( n =1, 2, 3 etc). The value of ‘n’ represents the shells or energy levels.
• For the first shell, n =1, for the second shell, n = 2 and so on. Thus, with the increase in the value of ‘n’, the number of allowed orbit increases. 
  All the orbitals of a given value of ‘n’ constitute a single shell of atom and are represented by the following letters
• n = 1 2 3 4 …………
• Shell = K L M N …………
• Principal quantum number represents any particular orbit. The value of n gives roughly the binding force and the distance between the nucleus 
  and the electron and also the energy it possesses. The first energy level is the one nearest the nucleus.

Consequently, it is strongly bound with nucleus and possesses the lowest energy.

• Size of a shell increases with increase of principal quantum number ‘n’.

In other words, the electron will be located away from the nucleus.

• Maximum possible number of electrons in a shell is 2n2.

2. Azimuthal or Subsidiary Quantum Number:

The main energy levels (or shells) of electron may be considered as being made up of one or more
sub-levels (sub-shells). The quantum number which represents the existence of an electron in a sub-shell is called the subsidiary or Azimuthal quantum 
number. It is designated by the letter ‘l’.

• It defines the three dimensional shape of the sub-shell. For a given value of n, l can have values ranging from 0 to (n – 1),

that is, for a given value of n, the possible value of l are: l = 0, 1, 2, ………. (n-1)

• For example, when n = 1, value of l is only 0.
• For n = 2, the possible value of l are 0 and 1.
• For n = 3, the possible values of l are 0, 1 and 2.
• The number of sub-shells in a principal shell is equal to the value of n.
• For example in the first shell (n = 1), there is only one sub-shell which corresponds to l = 0.
  For (n = 2), there are two sub-shells (l = 0, 1) in the second shell; for (n = 3) three (l = 0, 1, 2) in third shell and so on.
• Sub-shells corresponding to different values of l are represented by the following symbols.
• Value for l: 0 1 2 3 …………
• Notation for sub-shell: s p d f …………
• Maximum number of electrons in a sub-shell is 2(2l+1). Where, l = 0,1,2,3…..etc.
  

Example: If n = 7, what are the possible values of l?

The following table shows the permissible values of ‘l’ for a given principal quantum number and the corresponding sub-shell notation.

n	l	Sub-shell notation
1	0	1s
2	0	2s
2	1	2p
3	0	3s
3	1	3p
3	2	3d
4	0	4s
4	1	4p
4	2	4d
4	3	4f

3. Magnetic Quantum Number:

• The magnetic quantum number was introduced to explain the orientation of orbitals in space particularly under the influence of an applied 
  magnetic field. This is designated by ‘m’.
• The magnetic quantum number  ‘m’ determines the number of orbitals and their orientation within a subshell. The permitted values of m are 
  dependent upon l.

For any sub-shell (defined by ‘l’ value), 2l+1 values of m are possible and they are from –l to +l including 0.

• Thus, for l = 0, the only permitted value of m = 0, [2(0)+1 = 1, one s orbital].
• For l = 1, m can be -1, 0 and +1 [2(1)+1 = 3, i.e. -2, -1, 0, 1, or 2. three p orbitals; the value of 2l+1 will be three and there will be three 
  different orbitals. ].
• For l = 2, m = -2, -1, 0, +1 and +2, [2(2)+1 = 5, five d orbitals].
• Each orbital in an atom, therefore, is defined by a set of values for n, l and m. An orbital described by the quantum numbers n = 2, l = 1, m = 0 
  is an orbital in the p sub-shell of the second shell.

The following chart gives the relation between the sub-shell and the number of orbitals associated with it.

Values of   l	0	1	2	3	4	5
Sub-Shell Notation	s	p	d	f	g	h
Number of orbitals	1	3	5	7	9	11

4. Spin Quantum Number:

• The three quantum numbers are used to define its size/energy, shape and orientation. But all these quantum numbers are not enough to explain 
  the line spectra observed in the case of multi-electron atoms, that is, some of the lines actually occur in doublets (two lines closely spaced), 
   triplets (three lines, closely spaced) etc. This suggests the presence of a few more energy levels than predicted by the three quantum numbers.
• In 1925, George Uhlenbeck and Samuel Goudsmit proposed the presence of the fourth quantum number known as the spin quantum number
   (ms).

• In other words, an electron has, besides charge and mass, intrinsic spin angular momentum. Spin angular momentum of the electron, a vector 
  quantity, can have two orientations relative to the chosen axis. These two orientations are distinguished by the spin quantum numbers ms which 
  can take the values of + or -. These are called the two spin states of the electron and are normally represented by two arrows, ↑ (spin up) and ↓ 
  (spin down).

Two electrons that have different ms values (one +1/2 and the other -1/2) are said to have opposite spins.
An orbital cannot hold more than two electrons and these two electrons should have opposite spins.

Summary of Quantum Numbers

To sum up, the four quantum numbers provide the following information :

1. ‘n’ defines the shell, determines the size of the orbital and also to a large extent the energy of the orbital. There are n subshells in the nth shell.
2. ‘l’ identifies the subshell and determines the shape of the orbital. There are (2l+1) orbitals of each type in a subshell, that is, one s orbital (l = 0), 
   three p orbitals (l = 1) and five d orbitals
   (l = 2) per subshell. To some extent l also determines the energy of the orbital in a multi-electron atom.
3. ‘m’ designates the orientation of the orbital. For a given value of l, m has (2l+1) values, the same as the number of orbitals per subshell. 
   It means that the number of orbitals is equal to the number of ways in which they are oriented.
4. ms refers to orientation of the spin of the electron.

Principal Quantum number (n)	Subsidiary Quantum number (l) L=0 to n-1	Sub-levels	Magnetic Quantum number  (m) M=+l to –l including 0	Orbital number
1(K)	0	1s	0	1
2(L)	0 1	2s 2p	0 +1, 0, -1	1 3
3(M)	0 1 2	3s 3p 3d	0 +1, 0, -1 +2, +1,0,-1,-2	1 3 5
4(N)	0 1 2 3	4s 4p 4d 4f	0 +1, 0, -1 +2, +1,0,-1,-2 +3, +2, +1,0,-1,-2,-3	1 3 5 7

Distribution of Electrons in Atoms:

• The electrons are arranged among the known elements in seven main energy levels designated by the principal quantum numbers n=1, 2, 3, 4, 5,
   6 and 7. These principal levels are divided into sub-levels indicated by s, p, d, and f.
• The first energy level (n=1) which has only one sub-level is designated as 1s. The second energy level (n=2) has two sub-levels designated as 2s 
  and 2p, the third (n=3) energy level has three sub-levels designated as 3s, 3p and 3d and the fourth energy level n=4 has four sub-levels 4s, 4p, 
  4d and 4f.
• The sub-levels are further divisible into orbitals. An s sub-level is made up of one orbital; a p sub-level - three orbitals; a d sub-level-five orbitals 
  and f sub-level - seven orbitals.
• Each electron orbital can accommodate a maximum of two electrons of opposed spins.
  Thus, one s orbital can hold a maximum of two electrons. The three p orbitals can hold a maximum of 6 electrons etc.

Electronic Configuration of Atom

According to quantum mechanics, the arrangement of electrons of an atom in different orbitals is called the electronic configuration. This electronic
 configuration follows some rules i.e.

1. Pauli’s Exclusion Principle
2. Aufbau’s law
3. Hund’s rule of maximum multiplicity

Pauli’s Exclusion Principle

• The number of electrons to be filled in various orbitals is restricted by the exclusion principle given by the Austrian scientist Wolfgang Pauli.
• Pauli’s exclusion principle states that, ‘no two electrons in an atom can have the same set of values for all four quantum numbers.’
• This principle can also be stated as-“only two electrons may exist in the same orbital and they must have opposite spin.”
• It means that the two electrons of the same atom can have the same values for three of their quantum numbers, but the fourth quantum number 
  must be different for the two electrons. Thus, two electrons may have orbits of the same size, shape and orientation in space provided they have
   opposed spins.

For example, helium has two electrons. According to Pauli’s exclusion principle, the values of the quantum numbers for these two electrons will be as 
follows-

For electron 1, e1: n = 1; l = 0; m = 0; s = +1/2

For electron 2, e2: n =1; l = 0; m = 0; s = -1/2

Aufbau’s Law

Aufbau is a German word which means build up or construction. It is for this reason that the principle is also often called building up principle or 
construction principle. This principle states as follows

The orbitals are filled up with electrons in the increasing order of their energy.

The principle can also be restated as, ‘During electronic configuration, electrons first go to the orbital of low energy and then systematically to the 
orbital of higher energy.’ This happens because the orbital with lower energy is more stable.

The energy of various orbitals of an atom a multi-electron atom increases as

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s ......

The energy of an orbital is determined by the sum of principal quantum number (n) and the azimuthal quantum number (l).

This rule is called (n + l) rule. There are two parts of this rule:

(a) The orbitals with the lower value of (n + l) have lower energy than the orbitals of higher (n + l) value. And hence is filled up first with electrons.

(b) When two or more orbitals have the same value of (n + 1), the orbital with lower value of n is lower in energy and hence is filled up first with 
electrons.

Example: Considering 4s and 3d orbital:

For the 4s orbital, the value of, n=4 and the value of l=0 (s). The total value of (n+l)= 4+0= 4

Again, for the 3d orbital, the value of, n=3 and the value of l=2 (d). The total value of  (n+l)= 3+2= 5

So, electrons will go to 4s orbital first between 4s and 3d orbital.

On the other hand, If the two orbital have the same value of (n+l) then electrons will go to that orbital first which has lower principal quantum number. 

Considering 3d and 4p orbital:

For the 3d orbital, the value of n=3 and the value of l=2 (d). The total value of (n+l) = 3+2=5

For the 4p orbital, the value of n=4 and the value of l=1(p). The total value of (n+l) = 4+1=5

Hence, electrons will go to the 3d orbital first.

Hund’s Rule of Maximum Multiplicity

This rule is applied to the electronic configuration in the orbitals of same energy. The distribution of electron in a set of degenerate orbitals like px, py, 
pz set; dxy, dyz, dzx, dz2, dx2, dy2 set etc. of a given sub-shell takes place in such a way to give the maximum number of unpaired electrons and these 
electrons must have the same direction of spins.

This rule can also be stated as- ‘The orbital of a given sub-shell is first filled singly and then the pairing of electrons in each orbital begins.’

Illustration of the principle:

In order to illustrate the principle, let us consider the electronic configuration of oxygen atom whose atomic number is eight. According to this principle 
the degenerated 2px, 2py and 2pz orbitals will be first filled singly with electrons numbered as 5, 6 and 7. These 3 electrons will have the same spin. 
Electron number 8 which is the last electron will pair with the electron number 5 in 2px orbital.

Fig. Electronic configuration of oxygen atom (atomic number = 8). The boxes represent orbitals, The arrows indicate the electrons while the numbers shown below the 
arrows represent the electron number.

Stable configuration:

It’s been observed that half-filled or fully filled orbitals with same energy have more stable configuration. That’s why np3, np6, nd5, nd10, nf7 or nf14 
orbitals are more stable.

For this reason, d5s1 configuration occurs rather than d4s2 and d10s1 occurs rather than d9s2.

Example,

Cr(24)=1s2 2s2 2p6 3s2 3p6 4s2 3d4 (Aufbau’s law)

=1s2 2s2 2p6 3s2 3p6 4s1 3d5 (Stable configuration)

Here, the latter one occurs in case of Chromium to acquire more stable configuration.

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