Unit-2 Structure of Atom

Developments Leading To The Bohr’s model of Atom

The interaction of matter with radiation gave information about the structure of atom.This
was utilized by Neil’s Bohr to propose his model, where two developments played a major
role in the formulation of his model of atom. These are:
1)Dual character of Electromagnetic radiation means that radiation possess wave like and
particle like properties
2)Experimental results of atomic spectra explained by assuming quantized energy levels
in atoms.
Wave nature of Electromagnetic radiation:
In early days according to Newton, light i.e form of radiation was considered to be made
of particles called corpuscules
But later James Maxwell explained about the electric and magnetic character of light. He
suggested that when electrically charged particles accelerates , it produces alternating
electrical and magnetic field which are emitted in the form of waves called
electromagnetic waves or electromagnetic radiations.
Properties of Electromagetic waves:
1)The oscillating electric and magnetic fields produced by oscillating charged particles are
perpendicular to each other and both are perpendicular to the direction of propagation of
the wave.
2)Unlike sound waves or water waves ,electromagnetic waves do not require medium and
can travel in vacuum.
3)There are many electromagnetic radiations , which differ from one another in
wavelength (or frequency).These constitute what is called electromagnetic spectrum.
Electromagnetic Spectrum:
The arrangement of different types of electromagnetic radiations in the order of
increasing wavelength (or decreasing frequencies) is called electromagnetic spectrum.

Different regions of spectrum

*Radio frequency region- Frequency around 106 Hz , used for broadcasting

*Microwave region- Frequency around 1010 Hz, used for radar

*Infrared region- Frequency around 1013 Hz, used for heating purpose
*Ultraviolet region- Frequency around 1016 Hz.It is a component of sun’s radiation.

*Visible light-The small portion having frequency around 1015Hz and wavelength 400nm
to 750nm is called visible light.This region can be seen by our eyes.
4)Different kinds of units are used to represent electromagnetic radiation.

Characteristics of Electromagnetic waves:

1) Wavelength(λ):It is the distance between any two successive peaks(crests) or
troughs of a wave.
S.I unit is meter.It is also expressed in terms of nm, Ao
1A0=10-10m
2) Frequency(ν):It is defined as the number of waves passing through a given point
in one second.it is also called cycles per second(s-1)
 S.I unit is Hertz(Hz,s-1)
3) Velocity(c):It is the distance travelled by the wave in one second.
S.I unit is m/s
In vacuum all types of electromagnetic radiations travel with the same velocity ie
3 X108m/s
4) Wave number(ν):It is defined as the number of wavelengths per unit length.
Unit is cm-1 or m-1

(ν)= 1/ λ
The frequency(v), wavelength(λ) and velocity (c) are related by the
equation:
C=v λ
Limitations of Electromagnetic wave theory
1)Inability to explain the nature of emission of radiation from hot bodies(Black body
radiation)
2)Inability to explain photoelectric effect
3) Inability to explain the variation of heat capacity of solids as a function of
temperature.
4)Inabilty to explain the line spectra of hydrogen atoms.

Particle Nature of Electromagnetic Radiation:

Planck’s Quantum Theory
According to Max Planck
1. Atoms or molecules emit or absorb energy in the form of discrete packets of
energyand not continuously
2. Each of such packet of energy is called quantum or photon (in case of light)
3. The energy of each quantum is directly proportional to the frequency of the
radiation, i.e. . E α υ or E= hv where h= proportionality constant called
Planck’s constant, whose value= 6.626x10-34Js

Planck was able to explain the distribution of intensity in the radiation from black
body as a function of frequency or wavelength at different temperatures.

Photoelectric effect
When certain metals( like potassium, rubidium ,caesium) are exposed to a beam
of light, they eject electrons from their surface.This phenomenon is called
photoelectric effect and the emitted electrons are called photoelectrons.
 Experimental results observed for the experiment of Photoelectric
effect-
*When beam of light falls on a metal surface electrons are ejected
immediately.
*Number of electrons ejected is proportional to intensity or brightness of light
*Threshold frequency (vo): For each metal there is a characteristic minimum
frequency below which photoelectric effect is not observed. Thisis called
threshold frequency.
 * If frequency of light is less than the threshold frequency there is no ejection
of electrons no matter how long it falls on surface or how high its intensity is.
*Photoelectric work function (Wo): The minimum energy required to eject
electrons is called photoelectric work function.Wo= hvo
*Since the striking photon has energy equal to hv and the minimum energy
required to eject electron is hvo then the difference in energy is transferred
as kinetic energy of the photoelectron.
Energy of the ejected electrons :

Where me = mass of electron, v= velocity of the ejected electron

Dual Behaviour of electromagnetic radiations:

Photoelectric effect shows particle nature of light .Diffraction and interference
explain wave nature of light. It shows that light possesses both particle as well as
wave like properties ie light has dual behavoiur.Depending on the experiment
,we find light behaves either as a particle(when it interacts with matter) or like
wave (interference or diffraction) it exhibits when it propagates.

Spectrum:
When a ray of white light is passed through a prism , the wave with shorter
wavelength bends more than the one with a longer wavelength.Since ordinary
white light consists of waves with all the wavelengths in the visible range, a ray
of white light is spread out into a series of coloured bands called spectrum. The
light of red colour which has the longest wavelength deviates the least , while the
violet light , which has the shortest wavelength is deviated the most.

Continuous spectrum
The spectrum of white light ranges from violet 7.50x1014 Hz to red 4x1014 Hz.This
spectrum is called continuous spectrum. In this spectrum ,there is continuity of
colours ie one colour merges into other without any gap or discontinuity example
violet merges into indigo, indigo into blue, blue into green etc
Emission Spectra
*When electromagnetic radiation interact with matter, atoms and molecules may
absorb energy and reach to a higher energy state called exited state. With higher
energy state ,these are in an unstable state. For returning to the normal energy
state (more stable ,lower energy state) , the atoms and molecules emit radition
in various regions of the electromagnetic spectrum.
*The spectrum of radiation emitted by a substance that has absorbed energy is
called emission spectrum.
*Emission spectrum consists of bright coloured lines separated by dark spaces.

Absorption spectra.
*When a continuous electromagnetic radiation (say white light) is allowed to pass
through a gas or a solution of some salt and the transmitted light is analysed , a
spectrum is observed in which dark spaces are observed in an otherwise
continuous spectrum. These dark spaces indicate the radiations of corresponding
wavelengths that have been absorbed by the substance from white light.Such a
spectrum containing dark spaces due to absorption of light is called absorption
spectrum.
*An absorption spectrum is a photographic negative of an emission spectrum. It
consists of dark spaces in a bright continuous spectrum.

Spectroscopy
The study of emission or absorption spectra is called spectroscopy.
Line spectra
The emission spectrum of atoms in gas phase do not show a continuous spread
of wavelength from red to violet , rather they emit light only at specific
wavelengths with dark spaces between them.Such spectra are called line spectra
or atomic spectra because the emitted radiation is identified by the appearance
of bright lines in the spectra.
 Each element has a unique line emission spectrum.
 The characteristic lines in atomic spectra can be used in chemical analysis to
identify unknown atoms as mush as finger prints are used to identify people.
 Elements like Rb, Cs,Tl, In, Ga and Sc were discovered by analyzing their
minerals by spectroscopic method.
 Helium was discovered in the sun by spectroscopic method.
 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 lines named after
the discoverers.
Balmer showed that if spectral lines are expressed in terms of wave number(v) ,
then the visible lines of the hydrogen spectrum obey the following formula

Where ‘n’is an integer equal to or greater than 3 (ie n=3,4,5…..).

The series of lines described by the above formula are called Balmer series .The
Balmer series of lines are the only lines of the hydrogen spectrum which appear in
the visible region of the electromagnetic spectrum.
Later Rydberg noted that all series of lines in the hydrogen spectrum could be
described by the following expression:

where n1=1,2,3 ……

n2=n1+1, n1+2, n1+3…….

The value 109677 cm-1 is called Rydberg constant for hydrogen.The first five series
of lines that correspond to n1=1,2,3,4,5 are called Lyman series, Balmer, Paschen,
Bracket and Pfund series respectively.
Series n1 n2 Spectral
Lyman 1 2,3….. Ultraviolet
Balmer 2 3,4…. Visible
Paschen 3 4,5…. Infrared
Brackett 4 5,6… Infrared
Pfund 5 6,7… Infrared
Bohr model for Hydrogen Atom:
Bohr model for hydrogen Atom is based on the following postulates.
1. The electrons in the hydrogen atom 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 concentrically arranged around the nucleus.
2. The energy of an electron in the orbit does not change with time.However electron
will move from a lower stationary state to a higher stationary when required amount
of energy is absorbed by the electron.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
energy states that differ in energy is given by

Where E2 and E1 are the energies of the lower and higher allowed energy states
respectively. This expression is called Bohr’s frequency rule.
4. The angular momentum of an electron in a given stationary state can be expressed
as

h=Planck’s constant
Thus an electron can move only in those orbits fow which the angular momentum is
an integral multiple of h/2∏
According to Bohr’s theory for Hydrogen atom
 The stationary states for electron are numbered n=1,2,3….
These integral numbers are called principal quantum numbers.
 The radii of the stationay states are expressed as
rn= n2 a0
where ao=52.9pm
The radius of the first stationary state called Bohr orbit is 52.9pm (n=1)
As n increases , r will increase ie electron will be present fr away from the
nucleus.
 The energy of the stationary state is given by the expression:

where RH iscalled Rydberg constant and its value is 2.18x10-18 J

 Bohr’s theory could be applied to the ions (hydrogen like ions) ie containing
only one electron example Li2+,He+, Be3+etc
The energies of the stationary states are given by the expression

and radii is given by the expression

rn=52.9(n2)
Z
Where Z= atomic number and has value for He=2, Li=3, Be=4
 It is possible to calculate the velocities of electrons moving in these orbits.The
magnitude of velocity of electron increases with increase of positive charge on
the nucleus and decreases with increase of principal quantum number.
Explanation of Line Spectrum of hydrogen
Let ni be the initial orbit , nf be the final orbit.
The energy gap between the two orbits is given by the equation:
  In case of absorption spectrum, nf> ni, ∆E = positive and energy is absorbed.

 In case of emission spectrum, ni> nf, ∆E = negative and energy is released.

Limitations of Bohr’s Model

1 It fails to explain the finer details of hydrogen spectrum(doublet ie
two closely spaced lines)

2. Bohr’s model was also unable to explain spectrum of atoms containing

more than one electron.

3. It was unable to explain the splitting of spectral lines in presence of
magnetic field(Zeeman effect) and electric field( Stark effect)
4. It could not explain the ability of atoms to form molecules by chemical bonds.
Towards Quantum Mechanical Model of the Atom
Two important developments which contributed significantly in the formulation of such a
model were
1. Dual behaviuor of matter
2. Heisenberg uncertainty principle.

Dual behavior of matter: de Broglie proposed that as radiation (or

photon) matter (or material particles such as electron,proton, atom,
 molecules)also possesss dual behavior i.e.it behaves both as particle and
wave
de Broglie gave the following relationship between wavelength(λ),
momentum(p) of a material particle

Where m is mass of particle, v=velocity, p is its momentum.

 This was confirmed experimentally when it was found that an electron beam
undergoes diffraction, a phenomenon characteristic of waves.
 This fact was used in the making of electron microscope which is based on the wave
like behavior of electrons just as an ordinary microscope utilizes the wave nature of
light.
 According to de Broglie , every object in motion has wave character. Wave character
associated with macroscopic objects are short and negligible because of their large
size where as in case of microscopic objects (like sub atomic particles) the
wavelengths are long and can be detected.

Heisenberg uncertainty principle.

It states that it is impossible to determine simultaneously, the exact position and exact
momentum (or velocity) of an electron.The product of their uncertainties is always equal to or
greater than h/4π.

Significance of Heisenberg uncertainty principle:

 Heisenberg’s uncertainty principle rules out the existence of definite paths

or trajectories of electrons and other similar particles
 The effect of Heisenberg Uncertainty principle is significant ony for the
motion of microscopic objects and is negligible for that of macroscopic
objects.

Reasons for the failure of Bohr’s model:

a) It ignores the dual behaviour of matter.
b) It contradicts Heisenberg’s uncertainty principle

Quantum Mechanical model of Atom

Classical mechanics is based on Newton’s laws of motion. It successfully describes the

motion of macroscopic particles but fails in the case of microscopic particles.
Reason: Classical mechanics ignores the concept of dual behaviour of matter
especially for sub-atomic particles and the Heisenberg’s uncertainty principle.
Quantum mechanics is a theoretical science that deals with the study of the
motions of the microscopic objects that have both observable wave like and particle
like properties.
 Quantum mechanics was developed independently by Werner
 Heisenberg and Erwin Schrodinger
 The fundamental equation for quantum mechanics was
developed by Schrodinger.
 Schrodinger’s equation: For a system (such as an atom or a molecule
whose energy does not change with time) the Schrödinger equation is
written as:

 When Schrödinger equation is solved for hydrogen atom, the solution

gives the possible energy levels the electron can occupy and the
corresponding wave function(s) of the electron associated with each
energy level.Out of the possible values, only certain solutions are
permitted. Each permitted solution is highly significant as it corresponds
to a definite energy state. Thus, we can say that energy is quantized.

 These quantized energy states and the corresponding wavefunctions are

characterized by a set of three quantum numbers (principal quantum
number n, azimuthal quantum number l, and magnetic quantum number
ml)

 ψ gives us the amplitude of wave. The value of ψ has no physical

significance. Ψ2gives us the region in which the probability of finding an
electron ismaximum. It is called probability density.
 Orbital: The region of space around the nucleus where the probability of
finding an electron is maximum is called an orbital.
An orbital cannot have more than two electrons.
Quantum numbers
A large number of orbitals are possible in an atom. These orbitals can be
distinguished by their size, shape and orientation .Atomic orbitals are
precisely distinguished by a set of numbers –
1.Principal quantum number(n)
2. Azimuthal Quantum number(l)
3.Magnetic Quantum number(ml)
Further the spin of electron on its own axis is represented by the 4th quantum
number i.e
4. Spin Quantum number(ms)
Quantum numbers are defined as set of four numbers which gives all the
information about all the electrons in an atom.
The principal Quantum Number(n)
It identifies shell, determines size and energy of orbitals.As the value of n increases , the
size and energy of the orbital increases.

energy of orbitals

Azimuthal Quantum Number(‘l’)

Also known as orbital angular momentum or subsidiary quantum number
It represents the subshell to which an electron belong and determines the three
dimensional shape of orbital.
For a given value of n, l can have values ranging from 0 to n-1
The number of orbitals in a subshell = 2l + 1.
Total number of subshells in a particular shell is equal to the value of n.

Subshell S P d f g
notation
Value of ‘l’ 0 1 2 3 4
 Number of 1 3 5 7 9
orbitals

Shell(n) l Subshell Number of

notation subshells
1 0 1s one
2 0 2s Two
1 2p
3 0 3s Three
1 3p
2 3d
4 0 4s Four
1 4p
2 4d
3 4f

Magnetic quantum number (ml)

It gives information about the spatial orientation of the orbital with respect to
standard set of co-ordinate axis
For any sub-shell (defined by ‘l ‘value) 2l +1 values of ml are possible.
For each value of l, ml can have values from -l ……to zero to …...+l

Shell(n) l Subshell ml Number Total Total

notation of number number of
orbitals of electrons=
in a orbitals (2n2)
given in a
subshell given
(2l+1) energy
level
1 0 1s 0 one 1 2
2 0 2s 0 one 4 8
1 2p -1,0,+1 three
3 0 3s 0 One 9 18
1 3p -1,0,+1 Three
2 3d -2,-1,0,+1,+2 Five
4 0 4s 0 One 16 32
1 4p -1,0,+1 Three
2 4d -2,-1,0,+1,+2 Five
3 4f -3,-2,- seven
1,0,+1,+2,+3
 Electron spin quantum number (ms): It refers to orientation of the spin of
the electron. An electron spins around its own axis It has besides charge and
mass, intrinsic spin angular momentum.Spin angular momentum of the
electron -a vector quantity can have two orientations .These two orientations
are distinguished by the spin quantum number which can take the two values
+1/2 and -1/2 called the two spin states of the electron,+1/2 identifies the
clockwisespin ( ) and -1/2 identifies the anti- clockwise spin ( )
Note :An orbital cannot hold more than two electrons and these electrons
must have opposite spin.

Shapes of orbitals
We know ψ is a mathematical function which represents coordinate axis of an
electron.The plot of wave function as a function of r(ie the distance from the
nucleus) for different orbitals are different.
For s orbital
The plot of 1s (n=0 , l =0) and 2s (n=2, l =0) are given.
1s 2s

For 1s the probability density ψ2 is maximum at the nucleus and decreases sharply
as we move away from the nucleus.
For 2s the probability density ψ2 first decreases sharply to zero and again starts
increasing reaches a small maxima and then decreases to zero as the value of r
increases (ie as we move away from the nucleus)
Nodal surface /Node The region where this probability density function
 reduces to zero is called nodal surfaces or simply nodes.
Probability density can be represented using :
1. Charge cloud diagram- Density of dots represents the density of electrons
in that region.

2. Boundary surface diagram:It is obtained by drawing boundary surface or

counter surface for the orbitals where the probability density ψ2 is
constant.Here ψ2 is more ie upto 90%

Shape of s-orbital
l=0 , ml=0
ψ2 is equal in all direction, Shape is spherically symmetrical, non directional.

p-orbital
l=1, ml= -1,0,+1. Total of three p orbitals designated as px, py,pz
Shape is dumb bell
 It has two lobes on either side of the plane passing through the nucleus.

ψ2 is equal and maximum on both sides of the lobes and zero where the two lobes
meet each other

Size, shape and energy of the orbitals are identical but they differ in
orientation of orbitals ie 2px along x-axis, 2py along y axis and 2pz along z-
axis.These orbitals are called degenerate orbitals.
Size of p orbitals increases with increase in ‘n’
Ie 4p>3p>2p

Shape of d-orbital
l=2, ml=-2,-1,0,+1,+2

Total of five d orbitals designated as dxy,dyz,dxz, dx2-y2, dz2

These orbitals have same energy, but different orientations.They are called
degenerate orbitals.Shape is double dumb bell shape.

Radial nodes: Radial nodes occur when the probability density of wave
functionfor the electron is zero on a spherical surface of a particular radius.
Numberof radial nodes = n – l – 1
Angular nodes: Angular nodes occur when the probability density wavefunction
for the electron is zero along the directions specified by a particular angle.
Angular nodes are the number of planes that bisect the orbital .The value
 of l gives the angular nodes.
Number of angular nodes = l
Numberof radial nodes = n – l – 1
Total number of nodes = Radial nodes + Anguar nodes
= n – l -1 + l
= n-1
Orbital Angular node Radial node Total nodes(n-1)
1s 0 0 0
2s 0 1 1
2p 1 0 1
3p 1 1 2
3d 2 0 2
4d 2 1 3
Energies of orbitals:
In case of hydrogen atom , the energies of an electron depend on the principal quantum
number(n)
The energies of orbitals increases in the order 1s<2s=2p<3s=3p=3d<4s=4p=4d=4f……
In case of atoms containing multielectron, the energy of the electron depend on ‘n’ and ‘l ’
For a given principal quantum number, the orbitals have different energies.This is because of
the mutual repulsion between electrons besides the presence of attraction between the electrons
and nucleus.
The attractive force increases with increase in positive charge (Ze) on the nucleus.

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 of
the nucleus because the inner electrons screens the effect of the nucleus on the
outer shell electrons. This is known as shielding or sceening effect.
So, due to the screening effect, the net positive charge experienced by the outer
electron from the nucleus is lowered and is known as effective nuclear
charge.(Zeff)
Effective nuclear charge experienced by the orbital decreases with increase of
azimuthal quantum number (l).
Order of shielding in orbitals
s>p>d>f
Since s orbital is spherical in shape, it shields the electrons from the nucleus more
effectively than p, d,f
Order of energy of orbitals
s<p<d<f
The orbitals which are close to the nucleus are attracted more towards the nucleus a
a result the energy of that orbital becomes less.

Since s orbital is close to the nucleus, it is attracted more and hence its energy is
less compared to p which has less energy than d
Energy of electron depend on n+l rule.
n+l rule-According to n+l rule orbitals with lower value of (n+l) have lower energy. If
two orbitals have the same value of (n+l) then orbital with lower value of n will have
lower energy.
The order in which the orbitals are filled is as follows: 1s, 2s,
2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 4f, 5d, 6p, 7s...

Filling of orbitals in Atom:

The filling of electrons into the orbitals of different atoms takes place
according to Aufbau principle which is based on Pauli’s exclusion
principle, the Hund’ s rule of maximum multiplicity and the relative
energies of the orbitals.
Aufbau Principle: In the ground state of the atoms, the orbitals are filled
inorder of their increasing energies.
The order in which the orbitals are filled is as follows: 1s2s,
2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 4f, 5d, 6p, 7s...

Pauli Exclusion Principle: No two electrons in an atom can have the same
setof four quantum numbers. Only two electrons may exist in the same
orbital and these electrons must have opposite spin
Hund’s Rule of maximum multiplicity:
Pairing of electrons into the orbitals belonging to the same subshell (p, d or
f) does not take place until each orbital belonging to that subshell has got
one electron each i.e., it is singlyoccupied.
Electronic configuration of atoms:Arrangement of electrons in different
orbitals of an atom. The electronic configuration of different atoms can
be represented in two ways.
a. sapbdc...... notation.
b. Orbital diagram:, each orbital of the subshell is represented by a box and
the electron is represented by an arrow (↑) a positive spin or an arrow (↓) a
negative spin.
Stability of completely filled and half filled subshells:
a. Symmetrical distribution of electrons- the completely filled or half filled
sub-shells have symmetrical distribution of electrons in them and are
more stable.
b. Exchange energy-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.