Download Document

ATOMIC
STRUCTURE
Presented By:
Dr. Vatsala Soni
1
Heisenberg Uncertainty Principle
 In the world of very small particles, one cannot measure any
property of a particle without interacting with it in some way
 This introduces an unavoidable uncertainty into the result
 One can never measure all the
properties exactly
Werner Heisenberg (1901-1976)
2
Measuring the position and
momentum of an electron
 Shine light on electron and detect reflected
light using a microscope
Minimum uncertainty in position
is given by the wavelength of the
light
So to determine the position
accurately, it is necessary to use
light with a short wavelength
3
Measuring the position and momentum
of an electron (cont’d)
 By Planck’s law E = hc/λ, a photon with a short wavelength has a
large energy
 Thus, it would impart a large ‘kick’ to the electron
 But to determine its momentum accurately,
electron must only be given a small kick
 This means using light of long wavelength!
4
Fundamental Trade Off …
 Use light with short wavelength:
 accurate measurement of position but not
momentum
 Use light with long wavelength
 accurate measurement of momentum but not
position
5
Planck’s Distribution
•
Energies are limited to discrete value
– Quantization of energy
E  nh
•
, n  0,1,2,...
Max Planck
Planck’s distribution
dE  d

8hc
5 (e hc / kT  1)
• At high frequencies approaches the Rayleigh-Jeans
law
(e hc / kT  1)  (1 
hc
hc
 ....)  1 
kT
kT
• The Planck’s distribution also follows StefanBoltzmann’s Las
6
Wave-Particle Duality
-The particle character of wave
•
Particle character of electromagnetic radiation
– Observation :
• Energies of electromagnetic radiation of frequency v
can only have E = 0, h, v 2hv, …
(corresponds to particles n= 0, 1, 2, … with energy = hv)
– Particles of electromagnetic radiation : Photon
– Discrete spectra from atoms and molecules can be explained
as generating a photon of energy hn .
– ∆E = hv
7
•
Wave-Particle Duality
-The particle character of wave
Photoelectric effect
– Ejection of electrons from metals when
they are exposed to UV radiation
– Experimental characteristic
• No electrons are ejected, regardless
of the intensity of radiation, unless UV
its frequency exceeds a threshold
value characteristic of the metal.
• The kinetic energy of ejected
electrons increases linearly with the
frequency of the incident radiation
but is independent of the intensity of
the radiation .
• Even at low light intensities, electrons
are ejected immediately if the
frequency is above threshold.
electrons
Metal
8
Wave-Particle Duality
-The particle character of wave
•
Photoelectric effect
– Observations suggests ;
• Collision of particle – like projectile that carries energy
• Kinetic energy of electron = hν - Φ
Φ : work function (characteristic of the meltal)
energy required to remove a electron from the metal
to infinity
• For the electron ejection , hν > Φ required.
• In case hν < Φ , no ejection of electrons
9
Wave-Particle Duality
-The particle character of wave
• Photoelectric effect
10
Wave-Particle Duality
-The wave character of particles
• Diffraction of electron beam from metal
surface
– Davison and Germer (1925)
– Diffraction is characteristic property of
wave
– Particles (electrons) have wave like
properties !
– From interference pattern, we can get
structural information of a surface
LEED (Low Energy Electron Diffraction)
11
Wave Particle Duality
•
De Brogile Relation
(1924)
– Any particle traveling with a linear
Matter wave:
= mvlength
= h/l
momentum
p haspwave
– Macroscopic bodies have high
momenta (large p)
 small wave length
 wave like properties are not observed
12
Schrödinger equation
• 1926, Erwin Schrödinger (Austria)
– Describe a particle with wave function
– Wave function has full information about the
particle
Time independent Schrödinger equation
for a particle in one dimension
13
Schrodinger Wave Equation
In 1926 Schrodinger wrote an equation that
described both the particle and wave nature of the eWave function (Y) describes:
1. energy of e- with a given Y
2. probability of finding e- in a volume of space
Schrodinger’s equation can only be solved exactly
for the hydrogen atom. Must approximate its
solution for multi-electron systems.
14
Schrodinger Equation
General form
HY = E Y
H= T + V
: Hamiltonian
operator
15
The Schrodinger equation:
Kinetic
energy
+
Potential
energy
=
Total
energy
For a given U(x),
• what are the possible (x)?
• What are the corresponding E?
16
For a free particle, U(x) = 0, so
 (x)  Ae
ikx
Where k = 2
= anything real
2
2
k
E
2m
= any value from
0 to infinity

The free particle can be found anywhere, with
equal probability
17
Normalization
 When ψ is a solution, so is Nψ
 We can always find a normalization const. such that the
proportionality of Born becomes equality
N 2  * dx  1
*

  dx  1
*
*


dxdydz



  d  1
Normalization const. are
already contained in wave
function
18
Quantization
 Energy of a particle is
quantized
 Acceptable energy can be found
by solving Schrödinger equation
 There are certain limitation in
energies of particles
19
The information in a wavefunction
 Simple case
 One dimensional motion, V=0
 2 d 2

 E
2
2m dx
Solution
  Aeikx  Be ikx
k 2 2
E
2m
20
Probability Density
B=0
  Ae
ikx
  A
2
2
A=0
  Be
 ikx
 B
2
2
A=B
  2 Acos kx
  4 A cos 2 kx
2
nodes
21
Eigenvalues and eigenfucntions
 Eigenvalue equation
(Operator)(function) = (constant factor)*(same function)
̂  
Operator
Eigenfunction
Solution : Wave function
Eigenvalue
Allowed energy (quantization)
(operator correspond ing to observable )  (value of observable ) 
22
Quantum Mechanics and Atomic Orbitals
 The first orbital of all elements is spherical.
 Other orbitals have a characteristic shape and position as described
by 4 quantum numbers: n,l,ml,ms. All are integers except ms
 Principal Quantum Number (n): an integer from 1... Total # e in
a shell = n2.
 Angular quantum number (l). (permitted values l = 0 to n1):
the subshell shape.
 Common usage for l = 0, 1, 2, 3, 4, and use s, p, d, f, g,...
respectively.
 Subshell described as 1s, 2s, 2p, etc.
23

Magnetic quantum number,ml, (allowed l to +l )
directionality of an l subshell orbital.
 Total number of possible orbitals is 2l+1.
 E.g. s and p subshells have 1 & 3 orbitals,
respectively.
 Spin quantum number,ms (allowed values 1/2). Due to
induced magnetic fields from rotating electrons.
 Pauli exclusion principle: no two electrons in an atom
can have the same four quantum numbers.
24
Permissible Quantum States
25
Orbital energies of the hydrogen atom.
26
Shapes of orbitals (electron
probability clouds)
 s orbitals are spherical (1).
 p orbitals are dumbbell shaped (3).
 d orbitals have four lobes (5).
 f orbitals are very complex (7).
27
Orbital Energies of Multielectron Atoms
 All elements have the same number of orbitals (s,p,
d, and etc.).
 In hydrogen these orbitals all have the same energy.
 In other elements there are slight orbital energy
differences as a result of the presence of other
electrons in the atom.
 The presence of more than one electron changes
the energy of the electron orbitals
28
Shape of 1s Orbital
29
Shape of 2p Orbital
30
Shape of 3d Orbitals
31
Elements
and
Their
Electronic
Configurations
32
Electron Configuration
 The arrangement of electrons in an atom in the
ground state.
 Need to learn some simple rules or principles.
33
Rules are…
 Aufbau principle
 Pauli’s exclusion principle
 Hund’s Rule
34
Aufbau Principle
 German for building up.
 An electron occupies the lowest-energy orbital
that can receive it.
 In Hydrogen, the electron goes into the 1s orbital
because it’s the lowest energy orbital.
35
A general rule -they arrange
themselves to have
the lowest possible
energy.
Ground State
Configuration
36
Pauli Exclusion Principle
 No two electrons in the same atom can have the same
set of four quantum numbers.
 Each electron in the same atom has a unique set of
quantum numbers.
37
Hund’s Rule
Equivalent orbitals of equal
energy are each occupied by
one electron before any one
orbital is occupied by a
second electron.
38
Hund’s Rule (cont.)
All electrons in singly
occupied orbitals have
the same spin.
39
Writing Electron Configurations
40
Standard Notation
of Fluorine
Number of electrons
in the sub level 2,2,5
2
1s
2
2s
5
2p
Sublevels
41
Thank You
42