Download QUANTUM MECHANICAL MODEL OF THE ATOM

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DUAL NATURE OF MATTER
• The French physicist, de Broglie in 1924 proposed
that matter, like radiation, should also exhibit
dual behaviour i.e., both particle and wave like
properties.
• de Broglie gave the following relation between
wavelength (λ) and momentum (p) of a material
particle 𝒉
•
h
λ=
•
orλ=
𝒎𝒗
p
Where, m = mass of electron, v= velocity of
electron,
λ = wavelength of matter wave
associated with electron
DUAL NATURE OF MATTER
• The de Broglei concept is significant only for
microscopic particles not for large object
because the wavelengths associated with
ordinary objects are so short that their wave
properties cannot be detected.
• The wavelengths associated with electrons
and other subatomic particles (with very
small mass) can however be detected
experimentally.
Nature
DUAL Dual
NATURE
OF MATTER
de Broglie
Matter has both wave
According to Planck’s
and particle nature.
quantum theory
E  h
(1)
According to Einstein’s equation
E  mc2
Equating (1) and (2)
mc2  h
c

For any particle =
(2)
mc2  h
h

mc
h
mv
de Broglie
wavelength
DUAL NATURE OF MATTER
• The wave nature of electron was
experimentally proved by Davison & Germer
in his X-Ray diffraction experiment
Proof:- Quantisation of Angular Momentum
Bohr had just invented his “Quantum Hypothesis”, because it
explained the Hydrogen-spectrum. He gave no explanation, as to
why angular momentum should be “quantized”.
2r  n
According to de Broglie equation
h

mv
2r 
On rearranging we get;
nh
mvr 
2
nh
mv
Heisenberg’s uncertainty principle
it is impossible to determine
simultaneously, the exact position and exact
momentum (or velocity) of an electron.
It
states
that
The product of their uncertainties is always equal to or greater
than h/4π.
Mathematically,
𝒉
Δx.Δp ≥
𝟒π
Where , Δx = uncertainty in measurement of position
Δp = uncertainty in measurement of momentum
Heisenberg’s uncertainty principle
Since Δp = Δ(mv) = m.Δv
𝒉
HenceΔx.mΔv ≥
𝟒π
Or
Δx.Δv ≥
𝒉
𝟒πm
Heisenberg’s uncertainty principle rules out the existence of
definite paths or trajectories of electrons and other similar
particles
QUANTUM MECHANICAL MODEL OF THE ATOM
• 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 is based on a fundamental equation
which is called Schrodinger equation.
• 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:
QUANTUM MECHANICAL MODEL OF THE ATOM
• ψ gives us the amplitude of electron wave. The value of ψ
has no physical significance.
• Ψ2 gives us the region in which the probability of finding an
electron is maximum. It is called probability density.
• 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.
Quantum mechanical model of atom
• The energy of electrons in atoms is quantised.
•
The number of possible energy levels for
electrons in atoms of different elements is a
direct consequence of wave-like properties of
electrons.
•
The position and momentum of an electron
cannot be determined simultaneously.
•
Electrons of different energies are likely to be
found in different regions.
• The region of space around the nucleus where the
probability of finding an electron is maximum is called
an atomic orbital.
Orbit and Orbital
Orbit is a fixed circular path around the nucleus
in which electron moves(proposed by Bohr)
whereas orbital is the quantum mechanical
concept and refers to the wave function.
Nodal Plane
The plane where the probability of finding the
electron is almost zero.
Total nodes in a shell = (n -1)
Angular nodes = l
Spherical nodes= (n –l -1)
Quantum Numbers
They specify the address (energy & position) of
each electron in an atom. These are four types.
Quantum Numbers
Azimuthal
Principal
Magnetic
Spin
Principal Quantum Number (n)
• Average distance of the electron from
the nucleus
• Energy Level of electron
• Possible values (n=1,2,3…..)
• Maximum number of electrons in any
shell is 2n2.
Azimuthal or Angular Quantum
number:(l)
• It identifies sub-shell (sub energy level), the shape
of orbitals, and orbital angular momentum.
• For a given value of n, l = 0 to n-1
• Total number of subshells in a particular shell is
equal to the value of n.
s
p
d
f
Values of l 0
1
2
3
Subshell
Azimuthal or Angular Momentum
Quantum number:(l)
p
s
d
f
Azimuthal or Angular Momentum
Quantum
Numbers
Quantum number:(l)
Orbital angular momentum of an
electron is given by
h

l(l  1)
2
Illustrative example
The orbital angular momentum of an electron is 4s orbital is
1 h
(a) +
4 2p
(c)
h
2p
Solution
(b) zero
(d)
h
4
2p
Orbital angular momentum =
AIEEE 2003
l  l  1 .
h
2
For s electrons, l = 0
For 4s electrons, orbital angular momentum is zero.
Hence, answer is (b)
Magnetic quantum number(m)
Magnetic
Quantum Number (ml)

Orientations of an orbital in space.

Explains the Stark and Zeeman effect.

Takes (2l+1) values,m=- l to + l.
Specifies
Orbitals
the exact orbital within each sublevel
combine to form a spherical shape:
2s
2px
2py
2pz
Spin Quantum Number(s)
Describes the direction of spin of an electron
Clockwise or anticlockwise (+1/2,-1/2)
ms = +½
ms = -½
Spin angular momentum=
Magnetic moment =
h
s(s  1)
2
n(n  2)
n= no. of unpaired electrons

1
2
Relation between quantum numbers
•For every value of n, l = 0 to (n-1)
•For every value of l, m = -l to +l
•For every value of m, s =

1
2
Quantum Numbers
lucknow
n
Area
l
Sector
House number
m
s
lucknow
aliganj
J
(shell)
(sub shell)
(orbital)
42
(spin)
Pauli’s exclusion principle
It is impossible for two electrons in a given
atom to have same set of four quantum
numbers.
For example:
In case of 1s2,there are two electrons in the 1s orbital.
The quantum numbers of the two electrons are:
n=1 , l=0 , m=0 , s=+1/2
n=1 , l=0 , m=0 , s=-1/2
Atomic Orbital Shapes
Atomic Orbital Shapes
Atomic Orbital Shapes
Atomic Orbital Shapes
Atomic Orbital Shapes
Atomic Orbital Shapes
The five d orbitals in ψ(x, y, z)2 form, with a combination diagram showing how
they fit together to fill space around an atomic nucleus.
Atomic Orbital Shapes
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.
Aufbau’s principle
In the ground state of the atoms, the orbitals are filled in order of
their increasing energies .
n+l rule
Aufbau’s principle
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s
Aufbau’s principle
For example:
Consider 3d and 4s orbitals, the electron will
first enter the orbital having minimum value
of (n+l).
Electron will therefore enter 4s orbital
(4+0=4) before entering 3d orbital (3+2=5).
Aufbau’s principle
Incase (n+l) values are same!!!
Incase of 3d orbital (3+2=5) and 4p orbital (4+1=5), the
(n+l) values are same.
In such a case,electron enters the orbital for
which n is minimum.
The electron will thus enter 3d orbital before
entering 4p orbital.
Hund’s rule
•The most stable arrangement of electrons in sub
shells is the one with the greatest number of parallel
spins.
•Electron pairing starts only after all the degenerate
orbitals are filled with electrons having same direction
of spin.
For example:
Nitrogen (Atomic number=7)
Electronic configuration
1s2
2s2
2p3
Degenerate refers to orbitals having same energy.
Electronic configuration some elements
Electronic configuration some elements
Electronic configuration some elements
Exceptional electronic configuration
Orbitals in the same sub shell tend to become completely
filled or half filled since such orbitals are more stable.
Such as electronic configuration of Cr(24): [Ar]3d44s2
But actually it is [Ar]3d54s1
electronic configuration of Cr(24): [Ar]3d44s2
But actually it is [Ar]3d54s1
Thank you