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Quantum theory
and Atomic structure
CM1502
Chapter 1
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The Wave
nature
of light
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Amplitude is the height
of the crest or the depth
of the trough of each
wave.Amplitude is related to
the intensity of the
radiation which we
perceive as brightness inthe case of visible light.
The Wave nature of light
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Classical Distinction between
Wave and Particle
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Electromagnetic Spectrum
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Continuous Spectra
A warm solid, liquid and plasma will radiate atallwavelengths thus producing a continuousEM spectrum.
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Discrete Spectra 1
A warm gas emits EMR, but at certain specific
wavelengths thus producing a discrete EM
spectrum.
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Discrete Spectra 2
A gas can also absorb EMR and does so at
discrete wavelengths.
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Examples: Spectra of Stars
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Examples: Gas clouds in space 1
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Line spectra of Atoms
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Hydrogen Atom Spectrum 1
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Hydrogen Atom Spectrum 2
A high school maths teacher, Balmer, in 1885
noticed that the wavelengths of the visible lines of
Hs spectrum could be represented by the formula:
...5,4,3,1
4
112H
=
= nn
R
where RHis a constant.
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Hydrogen Atom Spectrum 3
Later Rydberg showed that all the H atoms absorption
and emission lines (not just those seen in the visible,
i.e., the Balmer series) could be represented by the
formula
where RHis, as before, a constant now known as theRydberg constant, 1/RH= 91.1763 nm.
...3,2,1...,3,2,1,111
111212
2
2
1
H +++==
= nnnnnnn
R
(1.2)
The Rydberg equation and the value of the constant are
based on data rather than theory
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The Unexplained
Why do the spectral lines of Hydrogen appear in apattern?
Is there any importance about this experimental
observation? Over a wide range of wavelengths, light is only
observed at certain discrete wavelengths
Light is quantized (?) It was taught before that light behaves as wave
only
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Explaining Rydbergs Formula
Quantization of Light
Max Planck first proposed that light could bequantized into little packets of energy.
The packets of energy, calledphotons, have an
energy value of,
hE=(1.3)
where his Plancks constant.
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Bohrs H atom 1 Bohr postulated that an atom could only exist in certain
allowed states of specific total energy Ewhich he called
stationary states. Atoms do not leak energy while in oneof its stationary states.
If an atom was not in its lowest energy state (groundstate), then it could make a downward transition, to astate of lower energy and in the process, emit a photon.
High energy atomic state = Eu
Low energy atomic state = El
.
Eu El= E=a photon energy = h.
Bohrs postulated fixed atomic states and energy levelsand this leads naturally to discrete spectra.
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The energy levels of hydrogenic
atoms/ions
En= -2.18 x 10-18 J Z2/n2
The negative sign for the energy appears
because we define the zero point of the atoms
energy when the electron is completely
removed from the attraction of the nucleus
A table top analogy for def in ing the
energy of a system.
If you define the zero point of your
textbooks potential energy when the
book is on the table, the energy is
negative when it is on the floor.
222
0
421
8 nh
eZE
n
= (1.4)
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The lowest energy state of hydrogen atom n=1 has
the energy of -13.6eV or -1312 KJ/mole.
The amount of energy needed to promote an atomfrom the ground state to a given excited state is
called excitation energy.
The amount of energy needed to remove an electronfrom an atom in ground state is called the ionization
energy.
The separation energy is energy needed to removean electron from an atom in any excited state.
H Atom Energy States
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n=1
n=
n=5
n=4
n=3
n=2 -13.6/22= -3.4 eV
-13.6/32= -1.51eV
-13.6/42=0.85eV
-13.6/52=-0.54eV
0
-13.6eV
1stexcited state
2nd excited state
Ground state
a
b
a: Ionization energy = +13.6eV
b: Excitation energy
c,d:Separation energies
Energy level diagram for hydrogen atom
c
d
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Applications of Bohrs equation for
energy levels of an atom
We can find the
-difference in energy between two levels.
-energy needed to ionize the H atom
-wavelength of the spectral line.
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Figure 7.11
The Bohr explanation of the three series of spectral lines.
E of emitted photon: UV series > VIS series > IR series
nin Rydberg equation23
A spectral line results because a photon of specific energy is emitted.
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Limitations of Bohrs Model
Works only for hydrogenic species(one electronspecies) such as H, He+, Li2+etc
Fails for atoms with more than one electron
because the e-e repulsions and additional nucleuselectron attractions create more complexinteractions.
He assumed that an atom has only certain energylevels in order to explain line spectra. However hehas no theoretical basefor the assumption.
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Louis de Broglieconsidered other
systems that displayonly certain allowedmotions such as thevibrations of a pluckedguitar string.
He proposed that ifenergy is particle-like,perhaps matter is wavelike.
If electrons havewavelike motion inorbits of fixed radii, theywould have only certainallowable frequenciesand energies.
de Broglie and Standing Waves
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Fundamental or 1
st
Harmonic
2ndHarmonic or 1stOvertone 3rdHarmonic or 2ndOvertone
Animations of standing waves
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Perhaps the electrons, fixed to occupy specific orbits,behaved like standing waves?
Took the formulae E= mc2and set it equal to E= hc/to obtain for a photon = h/mc, mis equivalentmass (not actual) of a photon.
For a particle, he substituted v(velocity of the particle)for c.
de Broglies Matter Waves
mv
h=particle (1.5)
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Light and matter can interact.
Light can be absorbed and emitted by matter. The absorption and emission can be continuous ordiscrete.
Light is quantized, E= h.
The energy states of atoms (and molecules) arequantized.
Discrete spectra can be explained by atomsundergoing a transition and emitting or absorbing aphoton of energy.
Matter has wave properties.
The energy stationary-states of hydrogenic atoms(only) can be explained by electrons existing instanding-waves that surround the nucleus.
Summary
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The Schrdinger Wave Equation
Schrdinger eventually came up with the famous equation:
is the total energy operatoror Hamiltonianoperator. Itrepresents a set of mathematical operations that when
carried out with a particular wave function, yields one of the
allowed energy states of the atom. Thus each solution of
the equation gives an energy state associated with a givenatomic orbital.
is the wave function also a mathematical function.
Eis the total energy of the systemunder consideration it
is simply a number with units of J.
nnn EH =
(1.6)
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The Wave Function,
is just a function, like x2, or sinx, or ex. This function depends on the positions of all the
particles in the system under consideration.
For the hydrogen atom, it is a mathematical function of
the position of the electron and proton. If we were looking at H2O instead, then is a function
of the positions of the two H nuclei,and the O nucleusand the positions of each and every one of the 10electrons.
For each value of nwe have, we will have a different n. That is, 1, 2, 3etc.
Later we will see that these are the 1s, 2s, 3s,2pand 3detc. atomicorbitals.
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Solving the Schrdinger Equation
Solving the Schrdinger, therefore, involves finding the
correct mathematical function such that when we operate onthat function with we get a constant times the original
function back again.
If we can do this then the constant is the energy of thesystem.
Upon solving the Schrdinger equation it was found
thatthree integers, denoted as n, l, mlfully characterized
the functions that solved the Schrdinger equation.(According to Bohr, only n was needed)
( ) ( ) ( ) ,,,,,,,
ll mllnmln
YrRr =
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The Total Energy, En
Interestingly, the formula for the energies was the same as derivedfrom the Bohr and Bohr-de Broglie models, and did not depend on l,nor ml.
Recall that Enrepresented the energy of the H atom when its electronwas in orbit number n.
Thus the nhere reminds us that our system, here the H atom, couldexist in any number of energy states.
Each energy state being labeled by n, with the lowest being n= 1.
222
0
421
8 nh
eZE
n
=
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one electron
system
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Quantum Numbers n, l, and ml n the principal quantum number (QN). The energy of
hydrogenic atoms depends only on this quantumnumber. Can take the values 1, 2, 3,
l azimuthal QN, or orbital angular momentum QN. Is
associated with the allowed angular momentum of anelectron in an orbital. Can take the values, 0, 1, 2, , (n - 1)
ml magnetic QN, or orbital angular momentumprojection QN. Is associated with the orientation of theorbital angular momentum Can take the values, -l, -(l-1),-(l-2), , 0, 1, 2, , l, i.e., mlgoes
fromlto l in steps of one.
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Allowed Values of the Quantum
Numbers
Will denote here as (n, l, ml)
Some allowed values:
(1,0,0), (5,4,-1), (2,1,0), (2,0,0),
(4,3,3) Some impossible values:
(1,1,0), (5,4,-5), (2,1,-2), (2,0,1),
(0,3,3), (0,0,0)
By convention lower-caseletters have been used to
designate the lQN.
l Letter
0 s
1 p2 d
3 f
4 g5 h
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Another Way
At around the same time Schrdinger developed thematter wave equation, another scientist wasformulating quantum mechanics in an entirely differentway.
Heisenbergused linear algebra, or matrix algebra, anddeveloped quantum matrix mechanics.
A very important, finding from this approach is
Heisenberg Uncertainty Principle.
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The Heisenberg Uncertainty Principle
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The Heisenberg Uncertainty Principle
Because energy is directly related to p, what theuncertainty principle means for us is that becausethe energy of an atom is known with considerableaccuracy, the location of the electron within the atomis not known at all, accurately.
This means that nice circular orbits of electronsaround nuclei can not be correct.
Worse is that electron positions can only ever beknown in terms of probabilities rather than assigningto them a special spot around the atom.
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Th P b bl L ti f th l t
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The Probable Location of the electron While we cannot know the exact
position of the electron, we can know
where it probably is. i.e. where itspends most of its time.
2is called the probability density, ameasure of probability of finding the
electron in some tiny volume of theatom.
Electron probability density in theground state H-atom is shown in the
figure.
Theprobability density decreaseswith r but does not reach zero.
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Radial probability distribution
The total probability of finding the electron atsome distance rfrom the nucleus is called
radial probability distribution.
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Analyzing the Dart BoardZone N holes Area holes Prob %
50 169 3.14 53.79 11
40 358 9.42 37.98 24
30 401 15.71 25.53 27
20 268 21.99 12.19 18
10 154 28.27 5.45 10
Probability per unit area is
highest in the 50 zone 2
Most likely place to find
a dart, however, is in
the 30 zone RDF
Not as many holes per unit area as,but a lot more places for the holes to go.
Even more places for the holes to go, but far
too few holes per unit area
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Shape of s orbitals
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Shape of s orbitals
An orbital with l=0
has
a spherical shape
with the nucleus at
its centre is called s
orbital. Because asphere has only one
orientation, an s
orbital has only one
mlvalue.
Node is the regionwhere the probability
of finding the
electron drops to
zero.
( ) 0/2/300,0,11
,, ar
ear =
( ) 02/
0
2/3
00,0,2 2
24
1,,
are
a
rar
=
3,0,0(r,,) =
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Shape of 2p orbital
An orbital with l=1 is called p orbital and
has two regions of high probability, oneon the either side of the nucleus.
there are three possible ml values
Ml=-1,0,+1. Hence three possible
orientations in mutually perpendicular
directions.
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Shape of 3d orbital
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Shape of 3d orbital
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Shape of orbitals with higher l values
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Shape of orbitals with higher l values
Orbitals with l=3 are f orbitals.
They have 7 orientations.Given figure shows one of the seven
orientations.
What does an horbital look like?
Check out these site: http://www.orbitals.com/orb/orbtable.htm
http://winter.group.shef.ac.uk/orbitron/
Just for interest
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Radial Distribution
http://www.orbitals.com/orb/orbtable.htmhttp://winter.group.shef.ac.uk/orbitron/http://winter.group.shef.ac.uk/orbitron/http://www.orbitals.com/orb/orbtable.htm8/12/2019 CM1502 chapter 1 2013-14
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Radial Distribution
Functions/Bohrs radius
Note the probability maxima
occurs at the same orbit
radius fixed by Bohr
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Radial Distribution Functions
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Summary
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Summary
Atomic orbitals are solutions to the Schrdinger equationfor hydrogenic atoms.
Atomic orbitals are characterized by three quantumnumbers (QN). nthe principal QN and ranges from 1 up to infinity. The larger
the n, the more extended the orbital.
lthe orbital angular momentum QN, and ranges from 0 up to n-1.
lgives the shape of the orbital. l= 0 is an sorbital, l= 1 is a porbital, l= 2 is a dorbital, etc. mlthe orbital angular momentum projection QN, and ranges from
lup to lin steps of 1. mlgives the orientation of the orbital.
2is the probability density of finding the electron atposition (r,,) or (x,y,z).
The radial distribution function gives the probabilitydensity of finding the electron at a distance rfrom thenucleus, regardless of direction.
We finally arrive at the shapes of different atomicorbitals. CM1502 Sem2-2013-14