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The Electronic Structure of Atoms
Electromagnetic Radiation
• A wave is a vibration by which energy
is transmitted
• The wavelength, , is the distance
between two identical points on the
wave
• The frequency, , is the number of
identical points on a wave that pass a
given point per second
• The amplitude is the height measured
from the median line to a crest or a
trough
Electromagnetic Radiation
• The velocity of the propagation of a wave, v, is given by
v  λν
• Electromagnetic radiation travels at 3.00 x 108 m s-1, the speed of light,
c,
c  λν
• Example: What is the length (in meters) of an electromagnetic wave
that has a frequency of 3.64 x 107 Hz (1 Hz = 1 s-1)
• Solution:
c  λν

λ
c
ν
3.00  10 8 m s 1
λ
 8.24 m
7 1
3.64  10 s
Electromagnetic Radiation
Planck’s Quantum Theory
• When a solid is heated, it emits radiation
• With classical physics, we cannot adequately describe this
phenomenon
• Planck’s hypothesis was that matter could emit (or absorb)
energy in discrete amounts , i.e., quanta
• A quantum is the smallest amount of energy that can be emitted
(or absorbed) in the form of electromagnetic radiation
Planck’s Quantum Theory
• The energy of a quantum, E, is given by
E  hν
• h is the Planck constant and its value is 6.63 x 10-34 J s
• According to Planck’s theory, emitted energy (or absorbed) is an
integer multiple of h (h, 2h, 3h, …) and never a fraction of
h
• Planck could not explain why energy is quantized, but his theory
was able to accurately describe the radiation emitted by a hot
object
• Nobel Prize in 1918
The Photoelectric Effect
• Classical physics cannot describe the
photoelectric effect, i.e., a certain minimum
frequency (frequency threshold) is required to
eject an electron from a metal and the energy
of the ejected electrons does not depend on the
intensity of light
– Not possible if the light is simply a wave
• Einstein proposed that light is a stream of
particles, called photons, and the energy of a
photon, E, is given by
E  hν
• N.B. Same formula as the Planck equation
The Photoelectric Effect
• To eject an electron from a metal, a photon
with enough energy (with a high enough
frequency) must hit the metal
• If the metal is irradiated with light of
too low of a frequency (even if the
light is intense), the photons do not
have enough energy to eject an
electron
• The energy of the ejected electron is equal to
the "surplus” energy that the photon had
• If the light is intense, more photons strike the
metal, and more electrons are ejected (if the
frequency is high enough), but their energy
remain the same
• Nobel Prize 1921
The Photoelectric Effect
• Einstein's work forced scientists of his time to accept the fact that light:
• Acts sometimes like a wave
• Acts sometimes like a beam of particles
• The wave/particle duality is not unique to light as matter also has it (we
will see this shortly).
• Example: The energy of a photon is 5.87 x 10-20 J. What is its
wavelength (in nanometers)?
• Solution:
E  hν
and
c  λν
ν
E
h

c E

λ h

λ
ch
E
(3.00  108 ms 1 )(6.63  10 34 Js)
6
λ

3.39

10
m  3390nm
 20
5.87  10 J
Bohr Model of the Atom
• The emission spectrum of a substance is a
spectrum, continuous or discontinuous, of the
radiation emitted by the substance
• We can observe the emission spectrum by
heating the substance at very high temperature
or by hitting it with a beam of energetic
electrons
• A condensed phase will typically have a
continuous emission spectrum
• However, atoms in a gaseous state emit light at
specific wavelengths (a line spectrum)
Bohr Model of the Atom
• In the Bohr model for the hydrogen atom, Bohr
assumed that the electron revolves around the
proton in a circular orbit like a planet around the
Sun.
• However, in the Bohr model, the
electron's circular orbit can just do well
defined radii
• The radiation emitted by a hydrogen atom is
attributed to the release of a quantum of energy
when the electron jumps from a higher orbit to a
lower orbit (and vice versa for the absorption of
light)
Bohr Model of the Atom
• Each orbit is associated with a principal quantum
number, n, which must be a positive integer (n = 1,
2, 3, …)
• The energy of an electron in the orbit with the
principle quantum number n is given by
R
En  2H
n
where RH is the Rydberg constant (2.18 x 10-18 J)
• The lowest energy level (n = 1) is the ground level
(or state)
• All of the other levels are excited levels (or
states)
• The energy of the electron increases as it is
farther from the nucleus and therefore more
weakly retained by the nucleus
The Dual Nature of the Electron
• In 1924, de Broglie proposed that if
light had a wave/particle duality, why
not matter as well
• According to de Broglie, the electron of
the hydrogen atom behaves as a
stationary wave, i.e., the positions of the
nodes are fixed
• Therefore:
2πr  nλ
where λ is the wavelength, r is the
radius of orbit n, and n is a positive
integer (n = 1, 2, 3, ….)
The Dual Nature of the Electron
• The work of Bohr found that
 h 

2 π r  n 
 me v 
where me and v are the mass and
velocity of the electron and n is the
principle quantum number (n = 1, 2, 3,
….)
• Thus, de Broglie proposed that
λ
h
mev
for the electron, and, in general, for any
particle
h
λ
mv
The Dual Nature of the Electron
• de Broglie demonstrated that any moving particle has wave properties
• Nobel Prize 1929
• Example: Calculate the wavelength (a) of a tennis ball (60 g) that travels at
62 m/s and (b) of an electron that travels at 62 m/s.
• Solution:
(a)
h
6.63  10 34 J s
λ

 1.8  10 34 m
mv (0.060 kg)(62 m/s)
(b)
h
6.63  10 34 J s
5
λ


1.2

10
m
31
mv (9.11  10 kg)(62m/s)
Heisenberg’s Uncertainty Principle
• If a particle such as an electron, has an important wave behavior, how can we
describe its movement?
• Heisenberg proposed the Uncertainty Principle:
• It is impossible to know simultaneously, with certainty, the momentum
and the position of a particle
• Heisenberg’s Uncertainty Principle says that the more we know about the
position, the less we know about the momentum, and vice versa
• N.B. Heisenberg’s Uncertainty Principle is not due to experimental limitations
but rather a fundamental law of nature
• Nobel Prize 1932
The Schrödinger Equation
• In 1926, Schrödinger proposed a general method for describing the behavior
of microscopic particles
• The Schrödinger equation is H = E where  is the wave function, E is the
energy of the system, and H is the Hamiltonian of the system (this concept is
beyond this course)
• 2 gives us the probability of finding an electron at a point in space
• We can only speak of probability because of Heisenberg’s uncertainty
principle
• N.B. Classical Newtonian mechanics can describe the motion of a particle with
perfect precision, but Schrödinger’s quantum mechanics can only talk about
probabilities
• Nobel Prize 1933
Quantum Mechanics Applied to the Hydrogen Atom
• The Schrödinger equation for a hydrogen
atom gives all possible wavefunctions and
energies for the hydrogen atom
• The electronic density, 2, is the
probability density of the presence of an
electron per unit volume for a given state
described by the wavefunction
• Rather than talking about the orbit of an
electron, we’re talking about an atomic
orbital
• An atomic orbital is like the
wavefunction of an electron in an
atom
Polyelectronic Atoms
• No exact solution to the Schrödinger equation is known for
systems with two or more electrons
• We make the approximation that the electrons in a polyelectronic
atom are in atomic orbitals that resemble those found within the
hydrogen atom (for which the exact solutions are known)
• N.B. The 1s, 2s, 2p, 3s, 3p, 3d, 4s, …., orbitals exist only in the
hydrogen atom (or an atom with only one electron, like He+)
• It is only an approximation when discussing such orbitals
in a polyelectronic atom (even for something as simple as
He or H-)
Quantum Numbers
• In the solutions to the Schrödinger equation for the hydrogen
atom , there are integers which define the solution
• These are the quantum numbers
• If we know the value of the three quantum numbers of an
orbital, we can describe the structure/shape/orientation of
the orbital
• The quantum numbers obey very specific relationships (we
will see them soon)
• If we try to use a fractional number in the place of a whole
number or we do not obey a given specific relationship, we
obtain an "orbital" which is not a solution to the
Schrödinger equation and is therefore not truly an orbital
The Principal Quantum Number
• The principle quantum number (n) must be an integer value 1, 2,
3, ….
• For the hydrogen atom, the value of n determines the energy of
the orbital
– This is not strictly the case in a polyelectronic atom
• The principal quantum number determines the average distance
between an electron in a given orbital and the nucleus
– The greater the value of n, the greater the average distance of
an electron from the nucleus
The Secondary or Azimuthal Quantum
Number
• The secondary or azimuthal quantum number (l) defines the
shape of the orbital
• The possible values of l depend on the value of the principal
quantum number (n)
• l is any integer between 0 and n-1
•
•
•
•
If n = 1, l = 0
If n = 2, l = 0, 1
If n = 3, l = 0, 1, 2
etc.
The Secondary or Azimuthal Quantum
Number
• The value of l is often designated by a letter,
• l = 0 is an s orbital
• l = 1 is a p orbital
• l = 2 is a d orbital
• l = 3 is an f orbital
• l = 4 is a g orbital
• etc.
• A set of orbitals with the same value of n is a shell
• A set of orbitals having the same values ​of n and l is a subshell
• The fact that l < n explains why there aren’t 1p, 1d, 2d, 3f, etc.
orbitals
The Magnetic Quantum Number
• The magnetic quantum number (m) describes the orientation of an orbital in
space (eg.; m distinguishes between the px, py, and pz orbitals)
• The possible values of m depend on the value of the azimuthal quantum
number (l)
• m is any integer value between -l and +l
• If l = 0, m = 0
• If l = 1, m = -1, 0, +1
• If l = 2, m = -2, -1, 0, +1, +2
• If l = 3, m = -3, -2, -1, 0, +1, +2, +3
• etc.
• This explains why subshells have only 1 s orbital, or 3 p orbitals, or 5 d
orbitals, or 7 f orbitals, etc.
The Spin Quantum Number
• The first three quantum numbers define the
orbital occupied by the electron
• Experiments indicate that a fourth quantum
number exists; the spin quantum number (s)
• Electrons act like microscopic magnets
• The spin quantum number describes the
direction that the electron spins
• The possible values are +1/2 and -1/2
The s Orbitals
• An s orbital is a spherical structure
• Since we are dealing with
probabilities, it is difficult to
describe the size of an orbital and
give it a specific form
• In principle each orbital
extends from the nucleus to
infinity
• The probability of finding an
electron increases when
approaching the nucleus
The s Orbitals
• Often, an orbital is shown
with a contour surface
(surface of constant
probability of finding the
electron) which defines a
border encompassing 90% (or
95%, or any percentage) of
the electron density for the
orbital in question
• All of the s orbitals are
spherical
• Their sizes increase as the
principal quantum number
increases
The p Orbitals
• The p orbitals exist only if the principal
quantum number is greater than or equal
to 2
• Each p subshell has three orbitals: px, py,
and pz
• The subscript indicates the axis
along which each orbital is
oriented
• Apart from their orientation, the
three orbitals are identical
• Each p orbital consists of two lobes, with
the nucleus being positioned where the
two lobes join
The d and f Orbitals
• The d orbitals only exist if the principal
quantum number is equal to or greater
than 3
• Each d subshell has five orbitals:
d x 2  y 2 , d z 2 , d xy , d xz , d yz
• Each f subshell has seven orbitals
• Their structure is difficult to
visually represent
• The electrons in f orbitals only
play an important role in the
elements with atomic numbers
greater than 57
Atomic Orbitals
• Example: Give the values of the quantum numbers associated with the orbitals in
the 3p subshell.
• Solution:
pour
uneorbital:
orbitale : n  3, l  1, m  1
For
one
pour
uneorbital:
orbitale : n  3, l  1, m  0
For
one
pour
une
orbitale : n  3, l  1, m  1
For
one
orbital:
• Example: What is the total number of orbitals associated with the principal
quantum number n = 4?
• Solution: There is one 4s orbital, three 4p orbitals, five 4d orbitals, and seven 4f
orbitals. It is impossible to have 4g, 4h, …. orbitals since the value of l would be
equal or greater than n, and this is not allowed. The total number of orbitals
associated with the principle quantum number n = 4 is therefore 1 + 3 + 5 + 7 = 16.
Orbital Energies
• In a hydrogen atom, the energy
of an orbital is entirely
determined by the principal
quantum number, n, even if the
forms of the orbitals are different
1s < 2s = 2p <
3s = 3p = 3d <
4s = ...
Orbital Energies
• In a polyelectronic atom, the
energies of orbitals with the same
value of n but different values of l
are not identical
s < p < d < f < ….
• This order is observed because the
other electrons hide the nucleus and
this “screening effect” becomes
more important when going from s
to p to d to f…
Orbital Energies
• The screening effect becomes so
important that the energetic order of the
orbitals depends primarily on the value
of (n+1) rather than just n
• Klechkowski’s Rule:
• The filling of subshells in a
polyelectronic atom is always
in the order of increasing sum
of n and l
• If two subshells have the same
(n+1) sum, the subshell with
the smallest value of n is filled
first
Electronic Configurations
• Each atomic orbital has three quantum numbers: n, l, m
• Each electron has four quantum numbers: n, l, m, s
• A compact notation for giving the values of the quantum numbers of
an electron is the following: (n, l, m, s)
• Example: Give the possible quantum numbers for an electron in a 5p
orbital.
• Solution:
 5, 1,  1,  12 
 5, 1, 0,  12 
 5, 1,  1,  12 
 5, 1,  1,  12 
 5, 1, 0,  12 
 5, 1,  1,  12 
Electronic Configurations
• The electronic configuration of an atom indicates how the electrons
are distributed in different atomic orbitals
• The ground state is the electronic configuration that provides the
lowest possible energy for that atom
• N.B. For an atom, the number of electrons it contains is equal to its
atomic number
• N.B. To indicate the spin of an electron, we use  and  rather that
+1/2 and -1/2
• N.B. If we have a single unpaired electron,  and  are energetically
equivalent
• N.B. If we have an electron and three empty p orbitals, or 5 empty d
orbitals, or 7 empty f orbitals, …., we can choose to fill any orbital in
the subshell first, i.e., each orbital in the subshell has the same energy
The Pauli Exclusion Principle
• The Pauli Exclusion principle states that two electrons in the same
atom can not be represented by the same set of quantum numbers
• A consequence of the Pauli Exclusion principle is that an atomic
orbital can contain only two electrons and one electron must has a
spin of +1/2 and the other a spin of -1/2
• Because an atomic orbital can only have two electrons and the
two electrons have opposite spins, the electronic configurations of
the first five elements are
•
•
•
•
•
H (1s1)
He (1s2)
Li (1s22s1)
Be (1s22s2)
B (1s22s22p1)
1s 
1s 
1s 
1s 
1s 
2s 
2s 
2s 
2px 
Hund’s Rule
• When we get to carbon, we have three options
• C (1s22s22p2) 1s 
2s 
2px 
• C (1s22s22p2) 1s 
2s 
2px 
2py 
• C (1s22s22p2) 1s 
2s 
2px 
2py 
• Hund’s Rule states that the most stable electronic arrangement of a subshell
is that which contains the greatest number of parallel spins
• The third option respects Hund’s rule
• The configurations of the other elements in the second row are, therefore:
• C (1s22s22p2)
1s  2s 
2px 
2py 
• N (1s22s22p3) 1s 
2s 
2px 
2py 
2pz 
• O (1s22s22p4) 1s 
2s 
2px 
2py 
2pz 
• F (1s22s22p5)
1s 
2s 
2px 
2py 
2pz 
• Ne (1s22s22p6) 1s 
2s 
2px 
2py 
2pz 
Diamagnetism and Paramagnetism
• An atom or molecule is paramagnetic if it
contains one or more unpaired electrons
• A paramagnetic species is attracted by the
magnetic field of a magnet
• An atom or a molecule is diamagnetic if it has
no unpaired electrons
• A diamagnetic species is repelled by the
magnetic field of a magnet
• H, Li, B, C, N, O, F are paramagnetic
• He, Be, Ne are diamagnetic
Aufbau Principle
• The Aufbau principle (German for “building up”) means that the electronic
configuration of an element is achieved by filling the subshells one after the
other
• To represent the electronic configuration in a compact fashion, an inert gas
configuration is represented by bracketing the atomic symbol of the noble gas
• For example, the configuration of Al is changed from 1s22s22p63s23p1 to
[Ne] 3s23p1 as [Ne] replaces 1s22s22p6
• On the periodic table, each element in a group (a column) shares the same
configuration for its valence subshells
• For example, for the first column (alkali metals)
• Li: [He]2s1
• Na: [Ne]3s1
• K: [Ar]4s1
• Rb: [Kr]5s1
• Cs: [Xe]6s1
Transition Metals
• Transition metals have incomplete d subshells or they easily form
cations with incomplete d subshells
• In the first row (Sc to Cu), the Aufbau principle and Hund’s rule are
respected except in two cases:
• Cr should be [Ar]4s23d4 but instead it is [Ar]4s13d5
• Cu should be [Ar]4s23d9 but instead it is [Ar]4s13d10
• These deviations for Cr and Cu are attributed to the particular stability
of a half-filled d subshell (the electrons are all the same spin) or a
filled d subshells
• The other rows of transition metals also show such particularities
• For the following atoms/ions, give the number of electrons that
would have the following quantum numbers:
a) P: m = +1
b) Ar: l = 1, m = 0
c) Pt: l = 0, m = +1
d) As: m = -1
e) Br = m = 0, s = -½