2.

Structure Of Atom

Sub-atomic particles Discovery of Electron

Electron was discovered by J J Thomson by Cathode ray discharge tube experiment A
cathode ray tube is made of glass containing two thin pieces of metal (called electrodes) sealed in
it The electrical discharge through the gases could be observed only at very low pressures and at
very high voltages

When a very high voltage (about 10,000 volts) is applied between the two electrodes, no electric
discharge occurs at normal pressure When the pressure of the gas inside the tube is less than 1
mm of mercury, a dark space appears near the cathode When the pressure is reduced to 0 01 mm
Hg, it fills the whole tube When the pressure is further reduced (10-4 mm Hg), the electric discharge
passes between the electrodes and the tube begins to glow This is due to the striking of some
invisible rays from the cathode These rays which start from the cathode and move away from it,in
straight lines are called cathode rays or cathode ray particles
These rays can be further checked by making a hole in the anode and coating the tube behind
anode with phosphorescent material like zinc sulphide When these rays strike the zinc sulphide
coating, a bright spot on the coating is developed

Properties of Cathode Rays

i The cathode rays start from cathode and move towards the anode
ii They are invisible, but their behaviour can be observed with the help of fluorescent or
phosphorescent materials
iii In the absence of electrical or magnetic field, these rays travel in straight lines
iv In the presence of electric or magnetic field, the cathode rays behave similar to that of
negatively charged particles From this it is clear 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
vi These rays possess kinetic energy and hence can do mechanical work
vii They can produce x-rays when incident on metals with high atomic mass
 Charge to Mass Ratio of Electron
J J Thomson measured the ratio of electrical charge (e) to the mass of electron (m e ) by using
cathode ray tube and applying electrical and magnetic field perpendicular to each other as well as to
the path of electrons

In the absence of electric or magnetic field, the cathode rays hit the screen at point B When
only electric field is applied, the electrons deviate from their path and hit the cathode ray tube at
point A Similarly, when only magnetic field is applied, electron strikes the cathode ray tube at point
C By carefully balancing the electrical and magnetic field strength, it is possible to bring back the
electron beam to the point B From the strength of electric and magnetic field, Thomson was able to
calculate the value of e/me as:
e/me = 1.758 × 1011 C kg–1
Where me is the mass of the electron in kg and e is the magnitude of the charge on the electron in
coulomb (C)

Charge on the Electron (e)

R A Millikan determined the charge on the electrons by a method known as ‘oil drop
experiment’ He found that the charge on the electron to be 1.6022 × 10–19 C

Mass of electron (me)

The mass of the electron (me) was determined as follows:
Mass of electron (me)= e = 1 6022 x 10-19
e/me = 1 758 x 1011
= 9 1 x 10-31 kg

Discovery of Protons
E Goldstein modified the discharge tube experiment by perforated (with small holes)
cathode After evacuating the tube and on applying high voltage, he found that some rays were
emitting behind the cathode and moving in the opposite direction of cathode rays These rays
deflect to the negative plate of electric field So they carry positive charge and were called anode
rays or canal rays.
Properties of Canal rays
The characteristics of canal rays are:
i. They depend on the nature of gas present in the cathode ray tube These are positively charged
gaseous ions
ii. The charge to mass ratio of these particles depends on the nature of the gas
iii. Some of the positively charged particles carry a multiple of the fundamental unit of electric
charge
iv. The behaviour of these particles in the magnetic or electrical field is opposite to that observed for
cathode rays v They can produce heating effect and can do mechanical work
v. They are invisible and can be observed with the help of fluorescent or phosphorescent materials
vi. They also travel in straight lines
The smallest and lightest positive ion was obtained from hydrogen and was called proton.

Discovery of Neutrons
Neutrons were discovered by Chadwick by bombarding a thin sheet of beryllium by α-particles
4 Be9 + 2He4 → 6C12 + 0n1
They are electrically neutral particles having mass slightly greater than that of the protons

Characteristics of sub-atomic particles

Sub atomic Symbol Discoverer Absolute Relative Mass/kg

particle Charge charge
(in Coulomb)
Electron e J J Thomson -1 6022 x 10-19 -1 9 01x10-31
Proton p E Goldstein +1 6022x10-19 +1 1 6726x10-27
Neutron n James 0 0 1 675x10-27
Chadwick

Some important terms relating to Atomic structure

Atomic Number: It is the number of protons present in the nucleus or number of electrons
present outside the nucleus It is denoted by the symbol ‘Z’
Atomic number (Z) = nuclear charge or number of protons (p)
= number of electrons (e)
Mass Number: It is the total number of protons and neutrons in atom Or, it is the total number
of nucleons in an atom It is denoted by ‘A’
i e Mass number (A) = no of protons (p) + no of neutrons (n)
or, A = p + n.
By knowing the atomic number and mass number, we can calculate the number of neutrons as: n
= A – Z If an element X has the atomic number Z and the mass number A, it is denoted as: zAX or
zX
A
Isotopes, Isobars and Isotones
Isotopes are atoms with same atomic number but different mass number That is, they
contain same number of protons but different number of neutrons Hydrogen has three isotopes:
Protium (1H1), Deuterium (1H2 or 1D2) and Tritium (1H3 or 1T3) Among these, Protium is the ordinary
hydrogen and Tritium is the radioactive isotope of Hydrogen
The number of protons, neutrons and electrons present in the 3 types of hydrogen are:
Isotope Number of protons Number of Number of neutrons
electrons
Protium 1 1 0
Deuterium 1 1 1
Tritium 1 1 2
Almost all the elements have isotopes All the isotopes of a given element have same
chemical properties, but they differ in their physical properties
Isobars are atoms of different elements having same mass number but different atomic
number i e they have different number of protons but have equal sum of the protons and neutrons
e g 6C14 and 7N14 18Ar and 20Ca
40 40

Isotones are atoms having same number of neutrons but have different atomic numbers
Some examples are:

Isotones p e n
6 C 14
6 6 8
7 N15 7 7 8
8 O16 8 8 8

ATOM MODELS

Thomson’s Model of Atom

J J Thomson proposed the first atom model, which is known as the plum pudding or raisin
pudding or watermelon model According to this model, an atom has a spherical shape in which the
positive charge is uniformly distributed The electrons are distributed in it, just like the seeds are
distributed in a water melon or plums are distributed in a pudding An important feature of this
model is that the mass of the atom is assumed to be uniformly distributed over the atom Also the
total positive charge in an atom is equal to the total negative charge and hence the atom is
electrically neutral

Rutherford’s Nuclear Model of Atom

Rutherford proposed an atom model based on his α–particle scattering experiment He bombarded
a very thin gold foil (approximately 10-7m thickness) with α–particles
 The Experiment: A stream of high energy α–particles from a radioactive source was directed at a
thin gold foil 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

Observations: The important observations made by Rutherford are:

1. Most of the α– particles passed through the gold foil without any deviation
2. A small fraction of the α–particles was deflected by small angles
3. A very few α– particles ( 1 in 20,000) bounced back, that is, were deflected by nearly
180°

Conclusions: From the above observations, Rutherford made the following conclusions:
1. Since most of the α–particles passed through the foil without any deviation, most space in
the atom is empty
2. A few positively charged α– particles were deflected This is because the positive charge of
the atom is concentrated in a very small volume at the centre called nucleus
3. The volume occupied by the nucleus 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 nucleus is 10–15 m
On the basis of above observations and conclusions, Rutherford proposed the nuclear model
(Planetary model) of atom According to this model:
1. All the positive charge and most of the mass of the atom are concentrated in an extremely
small region called nucleus
2. Electrons are revolving round the nucleus with a very high speed in circular paths called
orbits
3. Electrons and the nucleus are held together by electrostatic forces of attraction

Drawbacks or Limitations of Rutherford’s atom model

1. Rutherford’s model cannot explain the stability of the atom
2. He cannot explain the electronic structure of atom

Wave nature of Electromagnetic Radiation

James Maxwell suggested that when electrically charged particle moves under acceleration,
alternating electrical and magnetic fields are produced and transmitted These fields are
transmitted in the forms of waves called electromagnetic waves or electromagnetic radiation (emr)
These are the radiations associated with electric and magnetic fields E g light
The important characteristics of these radiations are:
 1. The oscillating electric and magnetic fields are perpendicular to each other and both are
perpendicular to the direction of propagation of the wave
2. The electromagnetic waves do not require a medium for propagation and can move in
vacuum
3. There are many types of electromagnetic radiations, which differ from one another in
wavelength (or frequency) These constitute electromagnetic spectrum The important
electromagnetic radiations in the increasing order of wavelength are:
Cosmic rays, Gamma rays, X-rays, Ultra-violet rays, Visible light, Infra red rays, Microwaves,
radio waves
4. All electromagnetic radiation travel through vacuum with a constant speed of 3x10 8 m/s

Some important terms relating to electromagnetic radiations

a) Wavelength (λ): It is the distance between two neighbouring crests or troughs of the waves It is
denoted by Greek letter lambda (λ)

b) Frequency (ν): It is defined as the number of waves which pass through a given point in one
second It is represented by the Greek letter ν (nu) Its units are cycles per second (cps) or Hertz
(Hz)

c) Velocity (c): The distance travelled by a wave in one second is called velocity of the wave It is
denoted by letter c Velocity is related to frequency and wavelength of the wave by the following
relation:
c= νλ or ν=c/λ
Velocity of all electromagnetic radiations in space or in vacuum is same and is equal to 3 x 108
ms-1 or 3 x 1010 cms-1

d) Wave number (ῡ): It is defined as the number of wavelengths per unit length It is equal to the
inverse of wavelength Wave number is denoted by ῡ (nu bar)
Wave number ῡ = 1
Wavelength (λ)
ῡ=1
λ
It is generally expressed in cm-1 or m-1

e) Amplitude (α): It is the height of the crest or depth of the trough of a wave It is expressed by the
letter ‘α’ It determines the intensity or brightness of radiation
Particle Nature of Electromagnetic Radiation: Planck’s Quantum Theory
Some of the experimental phenomenon like diffraction and interference can be explained by
the wave nature of the electromagnetic radiation But some phenomena like black body
radiation, photoelectric effect, variation of heat capacity of solids with temperature, line spectra
of atoms etc could not be explained by wave nature of emr

Black body radiation

An ideal body which emits and absorbs all frequencies of radiations is called a black body and
the radiation emitted by such a body is called black body radiation The frequency distribution of
radiation emitted from a black body depends only on its temperature At a given temperature,
intensity of radiation emitted increases with decrease of wavelength, reaches a maximum value at
a given wavelength and then starts decreasing
The phenomenon of black body radiation was first explained by Max Planck by his Quantum theory
According to this theory:
1. Atoms and molecules could emit (or absorb) energy not in a continuous manner,
but discontinuously in small packets of energy called quanta or photons
2. The energy (E ) of a quantum of radiation is proportional to its frequency (ν ) It is
expressed by the equation, E = hν
Where ‘h’ is known as Planck’s constant and its value is 6 626×10–34 J s

Photoelectric effect
It is the phenomenon of ejection of electrons by certain metals (like potassium, rubidium,
caesium etc ) when light of suitable frequency incident on them The electrons ejected are called
photoelectrons This phenomenon was first observed by H Hertz The important characteristics of
photoelectric effect are:
1. The electrons are ejected from the metal surface as soon as the beam of light strikes the
surface i e , there is no time lag between the striking of light beam and the ejection of
electrons from the metal surface
2. The number of electrons ejected is proportional to the intensity or brightness of light
3. For each metal, there is a minimum frequency (known as threshold frequency [ν0]) below
which photoelectric effect is not observed
4. The kinetic energy of the ejected electrons is directly proportional to the frequency of the
incident light

Explanation of photoelectric effect

A satisfactory explanation to photoelectric effect was first given by Albert Einstein using
Planck’s Quantum theory According to him, when a photon of sufficient energy strikes the metal
surface, it transfers its energy to the electron of the atom of the metal instantaneously and the
electron is ejected without any time lag A part of the energy is used to eject the electron from the
metal surface (i e to overcome the attractive force of the nucleus [work function, hν0]) and the
other part is given to the ejected electron in the form of kinetic energy Greater the energy
possessed by the photon, greater will be transfer of energy to the electron and greater the kinetic
energy of the ejected electron
 Since the striking photon has energy equal to hν and the minimum energy required to eject the
electron is hν0 (also called work function, W0) then the difference in energy (hν – hν0) is transferred
as the kinetic energy of the photoelectron
Following the law of conservation of energy principle, the kinetic energy of the ejected electron is
given by
K E = hν - hν0
Or, hν = hν0 + ½ mev2
Where me is the mass of the electron and v is the velocity of the ejected electron
A more intense beam of light contains larger number of photons, so the number of electrons
ejected is also larger

Dual Behaviour of Electromagnetic Radiation

Electromagnetic radiations possess both particle and wave nature This is known as
dual nature of Electromagnetic Radiation

Atomic spectrum
When a ray of white light is passed through a prism, we get a series of coloured bands called
spectrum This spectrum is called continuous spectrum, because here violet merges into blue,
blue into green and so on
Similarly, when electromagnetic radiation interacts with matter, atoms and molecules may
absorb energy and reach to a higher energy unstable state To attain stability, they emit radiations in
the form of spectrum Such a spectrum is called atomic spectrum

Evidence for The Quantized Electronic Energy Levels: Atomic

Spectra:
When a white light is passed through a prism, it splits into a series of coloured bands known as
spectrum Spectrum is of two types: continuous and line spectrum
(a) The spectrum which consists of all the wavelengths is called continuous spectrum
(b) A spectrum in which only specific wavelengths are present is known as a line spectrum It
has bright lines with dark spaces between them
Electromagnetic spectrum is a continuous spectrum It consists of a range of electromagnetic
radiations arranged in the order of increasing wavelengths or decreasing frequencies It extends
from radio waves to gamma rays
Emission and Absorption Spectra
The spectrum of radiation emitted by a substance that has absorbed energy is called an
emission spectrum. Atoms, molecules or ions that have absorbed radiation are said to be
“excited” To produce an emission spectrum, energy is supplied to a sample by heating it or
irradiating it and the wavelength (or frequency) of the radiation emitted is recorded
An absorption spectrum is like the photographic negative of an emission spectrum Here
a continuum of radiation (like white light) is passed through a sample which absorbs radiation of
certain wavelengths The missing wavelengths leave dark spaces in the bright continuous
spectrum The study of emission or absorption spectra is referred to as spectroscopy
The emission spectra of atoms in the gas phase do not form a continuous spectrum The
excited atoms emit light only at specific wavelengths with dark spaces between them Such spectra
are called line spectra or atomic spectra.
Line emission spectra are very useful in the study of electronic structure of atoms Each
element has a unique line emission spectrum The characteristic lines in atomic spectra can be
used in chemical analysis to identify unknown atoms in the same way as finger prints are used
to identify people So line emission spectra are also called finger print of atoms.

Line Spectrum of Hydrogen

When an electric discharge is passed through gaseous hydrogen, 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 The first five series of lines are Lyman, Balmer, Paschen, Brackett
and Pfund series Among these lines, the Balmer series is the only series that we can be visible
(since it lies in the visible region of emr)
 Johannes Rydberg proposed an equation for finding the wave number of the different lines in
Hydrogen spectrum
The expression is:
ῡ = 1/ λ = 109677 (1/n12 -1/n22) cm-1
Where n1 = 1, 2, 3,… and n2 = n1 + 1, n1 + 2, ……
The different spectral lines, their n1 and n2 values and their spectral region are:

Series Spectral n1 n2
region
Lyman Ultra violet 1 2,3,4…
Balmer Visible 2 3,4,5…
Paschen Infra red 3 4,5,6…
Brackett Infra red 4 5,6,7…
Pfund Infra red 5 6,7,8…

BOHR’S MODEL FOR HYDROGEN ATOM

The general features of the structure of hydrogen atom and its spectrum was first explained
by Niels Bohr The important postulates of his theory are:
1. The electron in the hydrogen atom can move around the nucleus in circular paths of fixed
radius and energy These paths are called orbits or stationary states or allowed energy
states These energy levels are numbered as 1,2,3 etc or as K, L, M, N, etc These numbers
are known as Principal quantum numbers
 2. The energy of an electron in an orbit does not change with time However, when an electron
absorbs energy, it will move away from the nucleus (i e to a higher energy level) and when it
loses energy, it will move towards the nucleus (i e to a lower energy level)
3. The radius of orbits can be given by the equation: rn = a0 n2 where a0 = 52 9 pm
Thus the radius of the first stationary state is 52 9 pm (called the Bohr radius) As n
increases, the value of r will increase
4. The energy of electron in an orbit is given by the expression: En = -RH (1/n2), where n =
1,2,3…… and RH is a constant called Rydberg constant Its value is 2 18x10 -18 J The energy of
the lowest state (the ground state) is given by E1 = –2 18×10–18J As the value of n increases,
the energy of the electron also increases
5. The frequency of radiation absorbed or emitted when transition occurs between two
stationary states that differ in energy by ΔE, is given by:
V=ΔE = E2-E1
h h
Where E1 and E2 are the energies of lower and higher energy levels respectively This expression
is commonly known as Bohr’s frequency rule
6. The angular momentum of an electron is an integral multiple of h/2π i e mevr = nh
2π
Where me is the mass of electron, v is the velocity of electron and r is the radius of Bohr
orbit n = 1,2,3 Thus, an electron can move only in those orbits whose angular momentum
is an integral multiple of h/2π So only certain fixed orbits are allowed

Significance of negative energy of electron

When the electron is free from the influence of nucleus, its energy is taken as zero In this
situation, the electron is at the orbit with n = ∞ When the electron is attracted by the nucleus and is
present in orbit n, the energy is emitted and its energy is lowered That is the reason for the
presence of negative sign in equation

Explanation of Line Spectrum of Hydrogen

According to Bohr atom model, radiation is absorbed if the electron moves from lower
energy to higher energy level and radiation is emitted if the electron moves from higher orbit to
lower orbit The energy gap between the two orbits is given by equation:
The wave number (ῡ)=109677 1 - 1 cm-1
n 1 2 n2 2

In case of absorption spectrum, n2 > n1 and the term in the bracket is positive and energy is
absorbed On the other hand, in case of emission spectrum n1 > n2, ΔE is negative and energy is
released

Expression for calculating radius of an orbit

In C G S units r = n2 h2
4πme2ZK
The radius of the nth orbit of hydrogen atom

In S I units rn = n2 x 0 53 Å

In C G S units r = n2 x 0 53 x 10-10 m

The radius of the nth orbit of any other atom

In S I units rn = n2 x 0 53 Å
Z

rn = n2 x 0 53 x 10-10 m
Z

Expression for calculating Energy of an electron

En = -2π2me4Z2K2
n2 h2
where K = Coulomb’s law constant; Z = atomic number of the element; e = charge on electron; m =
mass of an electron; n = principal quantum number; h = Planck’s constant and En = energy of the
electron in the nth principal shell

Expression for calculating Velocity of an electron in H-atom

v = 2πe2
nh
= 2 18 x 108 cm/s
n

Limitations of Bohr Atom Model

Bohr atom model could explain the stability and line spectra of hydrogen atom and hydrogen like
ions (e g He+, Li2+, Be3+ etc) But it has the following limitations:
1. It could not explain the fine spectrum of hydrogen atom
2. It could not explain the spectrum of atoms other than hydrogen
3. It was unable to explain the splitting of spectral lines in the presence of electric field (Stark
effect) and in magnetic field (Zeeman effect).
4. It could not explain the ability of atoms to form molecules by chemical bonds
5. It did not consider the wave character of matter and Heisenberg’s uncertainty principle
Dual Behaviour of Matter – de Broglie’s equation
de Broglie proposed that like radiation, matter also exhibit dual behaviour i e , both particle and
wave like properties This means that electrons should also have momentum as well as
wavelength He gave the following relation between wavelength (λ) and momentum (p) of a material
particle
λ= h = h
mv p Where m is the mass of the particle, v is the velocity and p is the
momentum The above equation is known as de Broglie’s equation.
Just like electromagnetic radiations, an electron beam also undergoes diffraction This is an
evidence for the wave nature of electrons An electron microscope works on the principle of wave
nature of electron
According to de Broglie, every moving object has a wave character The wavelengths
associated with ordinary objects are so short (because of their large masses) that their wave
properties cannot be detected The wavelengths associated with electrons and other subatomic
particles (with very small mass) can be detected experimentally

Heisenberg’s Uncertainty Principle

Werner Heisenberg proposed the uncertainty principle which is the consequence of dual
behaviour of matter and radiation It states that “it is impossible to determine simultaneously,
the exact position and exact momentum (or velocity) of a moving microscopic particle like
electron” Mathematically, it can be given as in equation:
Δx Δp ≥ h
4π
Or, Δx mΔv ≥ h
4π

Or, Δx Δv ≥ h
4πm
Where Δx is the uncertainty in position and Δp (or, Δv) is the uncertainty in momentum (or velocity)
of the particle
If the position of the electron is known with high degree of accuracy (Δx is small), then the
velocity of the electron will be uncertain [Δv is large] and vice versa

Significance of Uncertainty Principle

Heisenberg Uncertainty Principle is significant only for motion of microscopic objects and is
not applicable to macroscopic objects According to this Principle, we cannot determine the exact
position and momentum of an electron Thus, it rules out the existence of definite paths or orbits of
electrons We can only say the probability of finding an electron at a given point
Reasons for the Failure of the Bohr Model
In Bohr model, electrons are moving in well defined circular orbits about the nucleus The
wave character of the electron is not considered in Bohr model Further, an orbit is a clearly defined
path and this path can completely be 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
uncertainty principle Therefore, Bohr model of the hydrogen atom not only ignores dual behaviour
of matter but also contradicts Heisenberg uncertainty principle

QUANTUM MECHANICAL MODEL OF ATOM

On the basis of dual nature of matter and the uncertainty principle, Erwin Schrodinger and
Werner Heisenberg proposed a new model of atom called Quantum mechanics The fundamental
equation of quantum mechanics was developed by Schrödinger and is known as Schrödinger
equation It is written as:
Ĥ ψ = Eψ
where Ĥ is a mathematical operator called Hamiltonian operator, E is the total energy of the system
(K E + P E) and ψ is called the wave function On solving the above equation, we get different values
for E and ψ
When Schrödinger equation is solved for hydrogen atom, the solution gives the possible
energy levels the electron can occupy and the corresponding wave function (ψ) These quantized
energy states and corresponding wave functions are characterized by a set of three quantum
numbers

Significance of ψ
The wave function (ψ) is a mathematical function and it has no physical meaning Wave
functions of hydrogen or hydrogen like species with one electron are called atomic orbitals All the
information about the electron in an atom is stored in its orbital wave function ψ It may be positive
or negative
But ψ2 has some physical significance It gives the probability of finding an electron at a
point within an atom So ψ2 is known as probability density and is always positive From the value
of ψ2, it is possible to predict the probability of finding the electron around the nucleus

Orbitals
(i) An atomic orbital is the wave function for an electron in an atom
(ii) An orbital is defined as a three dimensional region in space around the nucleus where
there is higher probability (95%) of finding an electron of specified energy

Quantum Numbers
These are certain numbers used to explain the size, shape and orientation of orbitals Or,
Quantum numbers are the address of an electron There are four quantum numbers which describe
the electron in an atom They are Principal Quantum number (n), Azimuthal Quantum number (Ɩ),
Magnetic Quantum number (m or mƖ) and Spin Quantum number (s)
1. Principal Quantum Number (n)
The following informations are obtained from n
1 It gives the size the orbit
2 It gives the energy of electron in an orbit
3 It gives the shell in which the electron is found
4 It also gives the average distance between the electron and the nucleus As the value of n
increases, the distance between the electron and the nucleus also increases

The possible values of n are 1,2,3,4,5 etc

If n=1 the electron is in K shell
n=2 the electron is in L shell
n=3 the electron is in M shell and so on
n 1 2 3 4
Shell No. K L M N
Total number of orbitals in a shell = 1 4 9 16
n2
Maximum number of electrons = 2n2 2 8 18 32

2. Azimuthal Quantum Number [Subsidiary or orbital angular

momentum Quantum number] (Ɩ)
The following informations are obtained from Ɩ
1. It gives the shape of the orbital
2. It gives the sub shell or sub level in which the electron is located
3. It also gives the orbital angular momentum of the electron
For a given value of n, Ɩ can have n values ranging from 0 to n – 1 That is, for a given value of n, the
possible value of Ɩ are : Ɩ = 0, 1, 2, (n-1)
For example, when n = 1, value of Ɩ is only 0 For n = 2, the possible value of Ɩ can be 0 and 1
For n = 3, the possible Ɩ values are 0,1 and 2
Ɩ= 0 represents s orbital, Ɩ = 1 represents p orbital, Ɩ = 2 represents d orbital and Ɩ = 3
represents f orbital
The number of sub shells in a principal shell is equal to the value of n For
example,
When n = 1, Ɩ= 0 i e K shell contains only one sub shell - s subshell
when n = 2, Ɩ = 0 and1 i e L shell contains two sub shells - s and p sub
shells when n = 3, Ɩ = 0, 1 and 2 i e M shell contains three sub shells – s, p
and d subshells
when n = 4, Ɩ = 0, 1, 2 and 3 i e N shell contains four sub shells – s, p,d and
f subshells
l Subshell Shape of Subshell
0 s Spherical
1 p Dumbell
2 d Double dumbbell
3 f Complex shape
 Subshell notation s p d f G
Value of ‘l’ 0 1 2 3 4
Number of 1 3 5 7 9
orbitals

3.Magnetic Quantum Number (m or mƖ)

It gives information about the orientation of orbitals in space For a given ‘Ɩ’ value, there are
2Ɩ+1 possible values for m and these values are given by :
m = – Ɩ to 0 to + Ɩ
Thus for Ɩ = 0, mƖ = 0 [2(0)+1 = 1] i e s sub shell contains only one orbital called s orbital
For Ɩ = 1, mƖ = –1, 0 and +1 [2(1)+1 = 3] i e p subshell contains three orbitals called p orbitals (p x, py
and pz)
For Ɩ = 2, mƖ = –2, –1, 0, +1 and +2, [2(2)+1 = 5] i e d subshell contains five orbitals called d orbitals
(dxy, dxz, dyz, dx2- y2 and dz2)

4. Spin Quantum Number (s or ms)

It is the only experimental Quantum number and it gives the spin orientation of electrons This
spin may be either clockwise or anticlockwise So the values for s may be +½ or -½ +½ represents
clock-wise spin and-½ represents anticlock-wise spin

Shapes of orbitals
The region where the probability density function reduces to zero is called nodal surfaces or
simply nodes
Radial nodes: Radial nodes occur when the probability density of wave function for the electron is
zero on a spherical surface of a particular radius
Number of radial nodes = n – l – 1
Angular nodes: Angular nodes occur when the probability density wave function for the electron is
zero along the directions specified by a particular angle
Number of angular nodes = l
Total number of nodes = n – 1
Degenerate orbitals: Orbitals having the same energy are called degenerate orbitals

1. s-orbital:
For s-orbitals, Ɩ = 0 and hence mƖ = 0 So there is only one possible orientation for s orbitals
They are spherically symmetrical The plots of probability density (ψ2) against distance from
the nucleus (r) for 1s and 2s atomic orbitals are as follows:
 For 1s orbital the probability density is maximum at the nucleus and it decreases with increase
in r But for 2s orbital the probability density first decreases sharply to zero and again starts
increasing After reaching a small maximum it decreases again and approaches zero as the value of r
increases The region where the probability density (ψ2) reduces to zero is called nodal surface or
node.
For 1s orbital, there is no node, for 2s orbital there is only one node, for 3s orbital there are 2
nodes and so on In general, for an ns-orbital there are (n – 1) nodes
All the s-orbitals are spherically symmetrical and their size increases with increase in n The
boundary surface diagrams for 1s, 2sand 3s orbitals are as follows:

2. p-orbital
For p-orbitals, Ɩ = 1 and mƖ = -1, 0, +1 i e , there are three possible orientations for p orbitals
So there are 3 types of p-orbitals – px, py and pz Each p orbital consists of two lobes The probability
density function is zero on the plane where the two lobes touch each other
The size, shape and energy of the three orbitals are identical They differ only in the orientation
of the lobes For px orbital, the lobes are along the x-axis, for py, they are along the y-axis and for pz,
they are along the z-axis
All the p-orbitals have dumb-bell shape
The number of radial nodes for p-orbitals are given by (n –2), that is number of radial node is 1 for
3p orbital, two for 4p orbital and so on Besides the radial nodes, the probability density functions
for the np orbitals are zero at the plane, passing through the nucleus (origin) For example, in the
case of pz orbital, xy-plane is a nodal plane These are called angular nodes and number of angular
nodes is given by ‘Ɩ’
Number of radial nodes = n-l-1
Number of angular nodes = l
Total number of nodes = n-1
The boundary surface diagrams for three 2p orbitals are as follows:
 3. d-orbitals
For d-orbitals, Ɩ = 2 and mƖ = -2, -1, 0, +1 and +2 i e , there are five possible orientations for d -
orbitals So there are 5 types of d-orbitals They are dxy, dxz, dyz, dx2- y2 and dz2 The shapes of the first
four d-orbitals are double dumb-bell and that of the fifth one, dz2, is dumb-bell having a circular
collar in the xy-plane The five d-orbitals have equivalent energies For d-orbitals the number of
radial nodes is 2 and the total number of nodes is n-2 Boundary surface diagrams for d-orbitals are
as follows:

4. f-orbitals
For f-orbitals, Ɩ = 3 and mƖ = -3, -2, -1, 0, +1, +2 and +3 i e , there are seven possible
orientations for f orbitals So there are 7 types of f-orbitals They are fx3, fy3, fz3, fx(y2-z2), fy(z2-
x2), fz(x2-y2) and fxyz They have diffused shapes

Shielding Effect or Screening Effect

Due to the presence of electrons in the inner shells, the electron in the outer shell will not
experience the full positive charge on the nucleus So, due to the screening effect, the net positive
charge experienced by the electron from the nucleus is lowered and is known as effective nuclear
charge Effective nuclear charge experienced by the orbital decreases with increase of azimuthal
quantum number (l)

Rules for Filling of electrons in various orbitals

The filling of electrons into the orbitals of different atoms takes place according to the 3
rules – Aufbau principle, Pauli’s exclusion principle and the Hund’s rule of maximum
multiplicity

1. Aufbau principle:
The German word aufbau means ‘build up’ The building up of orbitals means the filling up of
orbitals with electrons It states that the orbitals are filled in order of their increasing energies In
other words, electrons first occupy the lowest energy orbital and then to higher energy orbitals
This rule has two sub rules:
a) The various orbitals are filled in the increasing order of their (n+Ɩ) value
b) If two orbitals have the same (n+Ɩ) values, the orbital with the lower n value is filled first
 The increasing order of orbitals is as follows:
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s

2. Pauli’s Exclusion Principle

It states that no two electrons in an atom can have the same set of four quantum numbers i e
an orbital can accommodate a maximum of only 2 electrons with opposite spin
If 2 electrons have same values for n, Ɩ and m, they should have different values for s i e if s =
+½ for the first electron, it should be -½ for the second electron

3. Hund’s rule of maximum multiplicity

It states that electron pairing takes place only after partially filling all the degenerate orbitals
Orbitals having same energies are called degenerate orbitals For example the electronic
configuration of N is 1s2 2s2 2px1py1pz1 and not 1s2 2s2 2px2py1
Electronic Configuration of Atoms
The distribution of electrons into various orbitals of an atom is called its electronic
configuration The electronic configuration of different atoms can be represented in two ways
(i) sa pb dc notation
(ii) Orbital diagram
The electrons in the completely filled shells are known as core electrons and the electrons in
the outer most shell are called valence electrons
For Example: Potassium (Z=19)

Stability of Completely filled and Half filled Subshells

(i) Symmetrical distribution of electrons: The completely filled or half-filled sub-shells have
symmetrical distribution of electrons in them and are more stable
(ii) Exchange energy: The two or more electrons with the same spin present in the degenerate
orbitals of a sub-shell can exchange their position and the energy released due to this
exchange is called exchange energy The number of exchanges is maximum when the
subshell is either half filled or completely filled As a result the exchange energy is maximum
and so is the stability

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