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Chapter 8: Electrons in Atoms Electromagnetic Radiation
• Electromagnetic (EM) radiation is a form of
energy transmission propagated as
waves moving through space.
o We will use a simple model of the wave as
a disturbance in the propagating medium
An EM wave is generated by accelerating
a charged particle, such as an e−
Electromagnetic Radiation
• A wave is characterized by its:
o Wavelength (lambda, λ, in m) is the distance between two
adjacent crests on a wave.
o Frequency (nu, ν, in s−1) is the number of wavelengths
passing a point in one second.
o Amplitude is the maximum displacement of the wave.
o Speed is how fast the wave travels or propagates in the
medium.
–
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• EM radiation in vacuo propagates at a constant speed of c,
where c = λν = 3.00 × 108 m s−1, the ‘speed of light.’
• EM radiation spans a wide range of frequencies & wavelengths
Practice: An FM radio station broadcasts on a frequency of 91.5
MHz. What is the wavelength of these radio waves in meters?
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• A property of EM waves is that their amplitudes are “additive.”
o Treating amplitudes as signed quantities (+/−), when waves
are added the resulting amplitude can increase or decrease.
– When waves are in phase, the amplitude increases; this is
constructive interference
– When waves are out of phase, the amplitude decreases;
this is destructive interference
• White light contains all the colors of the rainbow; we can see
these colors in two ways
o Reflect the light off a grooved surface; interference in the
reflected light (diffraction) is observed as colorful patterns.
o Refract the light by passing it through a prism; the different
colored components travel at different speeds in glass and are
bent at different angles producing a rainbow spectrum.
Atomic Spectra
• Light generated by a gas discharge tube or heating salts in a
flame consists of a limited number of wavelengths or colors.
• Diffracting this light produces an atomic line spectrum.
o The line spectrum is the element’s ‘fingerprint.’
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• Balmer deduced a formula for the frequencies of the lines
observed in the atomic spectrum of hydrogen
𝜈line = 3.2881 × 1015 �
1
1 −1
−
�s
22 𝑛 2
4
for 𝑛 = 3,4,5, ⋯
Quantum Theory
• As the 19th century came to a close, physicists observed
phenomena that could not be explained by classical physics.
• Over the course of 30 years, a new theory was developed.
o At the heart of this was the proposal that energy is not
continuous, but rather is discrete in nature, that is quantized.
We will survey some of the people involved.
• Max Planck explained the puzzle of “black body radiation” by
proposing that radiant heat energy is emitted only in definite
amounts called quanta.
• We use Planck's result in an equation that shows the energy of
electromagnetic radiation is proportional to frequency
𝐸 = ℎ𝜈
where ℎ = 6.626 × 10−34 J s = Planck ' s constant
• h (Planck's constant) symbolizes the new quantum physics
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• When light strikes the surface of a metal in vacuo, an electron
may be ejected; this is called the photoelectric effect.
• Researchers noted that the energy of the ejected electron did
not depend on the intensity of the light but rather on the
frequency of the light.
• Albert Einstein realized Planck's idea of light appearing as
quanta (bundles or photons) was the key to understanding this
mystery. Einstein’s explanation is recognized as the first
scientific work utilizing quantum theory.
• Chemical reactions induced by photons (packets or “particles”)
of light are called photoreactions (photochemistry).
The Bohr Atom
• Niels Bohr (1913) was the first to apply the quantum theory to
atomic structure. The most impressive result of the so-called
Bohr theory was the way it predicted the spectral lines of atomic
hydrogen.
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• Bohr theory predicted the radii of the stable orbits allowed by
quantum theory
𝑟𝑛 = 𝑛2 𝑎0
where 𝑛 = 1,2,3, ⋯
and
𝑎0 = 53 pm
and also the energy of an electron in an orbit
−𝑅𝐻 −2.179 × 10−18 J
𝐸𝑛 = 2 =
𝑛2
𝑛
where 𝑅𝐻 = 2.179 × 10−18 J
• By constructing an energy-level diagram and calculating the
energies between orbits, Bohr explained the line spectrum of H
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Is it likely that one of the electron orbits in the Bohr atom has a
radius of 1.00 nm?
Is there an energy level for the hydrogen atom,
En = −2.69 x 10-20 J?
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• This model of electrons constrained to move in quantized orbits
around a central nucleus was called the Bohr Atom (viz H atom)
o Ground state occurs when the electron occupies the orbit with
the lowest energy (n = 1); closest to the nucleus.
o Excited state occurs when the electron occupies a higher
energy orbit with unfilled orbit(s) between itself and the
nucleus.
• Electron movement between orbits involves energy. This
energy can absorbed or released as EM radiation (E = hν).
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• In the Bohr atom, The energy difference between initial and
final energy levels ni and nf is given by
−𝑅𝐻
−𝑅𝐻
1
1
1
1
∆𝐸 = 𝐸f − 𝐸i = � 2 � − � 2 � = 𝑅𝐻 � 2 − 2 � = 2.179 × 10−18 � 2 − 2 � J
𝑛f
𝑛i
𝑛i 𝑛f
𝑛i 𝑛f
∆E > 0 if energy is absorbed (electron excitation)
∆E < 0 if energy is released (electron relaxation)
• The energy of the photon either absorbed or emitted is given by
(and given ∆E, this allows us to determine the EM frequency)
|∆𝐸 | = 𝐸 = ℎ𝜈
Determine the wavelength of light absorbed in an electron
transition from n=2 to n=4 in a hydrogen atom.
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• The Ionization Energy of a hydrogen atom is the energy needed
to remove an electron from the atom (ni = 1, nf = ∞).
∆𝐸 = 𝐸f − 𝐸i = 𝑅𝐻 �
1
1
−
� = 𝑅𝐻 = 2.179 × 10−18 J
2
2
1
∞
o This works for any ‘atom’ with one electron, such as He+ or
Li2+ by introducing nuclear charge into the equation.
The energy of an orbit in such an atom is
−𝑧 2 𝑅𝐻
𝐸𝑛 =
𝑛2
where z = atomic number = nuclear charge = # of protons
For n = 2 in Hg79+
𝐸2 =
−�802 �𝑅𝐻
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o The Bohr atom was a revolutionary application of the new
quantum theory, but the model only worked for ‘atoms’ with
one electron! It did not work for multielectron atoms.
o More complex systems needed further refinements to
quantum theory
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Two Ideas Leading to a New Quantum Mechanics
• Louis de Broglie posited: If light waves can exhibit particle-like
properties (photons), then perhaps particles (electrons) can
exhibit wave-like behavior.
• The wavelength associated with these “matter waves” is
𝜆=
ℎ
ℎ
=
𝑝 𝑚𝑚
Practice: Assuming Superman has a mass of 91 kg, what is the
wavelength associated with him if he is traveling at one-fifth the
speed of light?
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• Werner Heisenberg established that there is a limit to how
precisely we can measure the position of a particle (x) and the
momentum of that particle (p) simultaneously. The Heisenberg
Uncertainty Principle is deep! There is an inherent limit to the
precision of a measurement.
ℎ
∆𝑥 ∆𝑝 = (∆position)(∆momentum) ≥
4𝜋
Heisenberg is out for a drive
when he's stopped by a traffic cop.
The cop says,
"Do you know how fast you were going?“
Heisenberg says,
Wave Mechanics
• All of these new concepts culminated in the formulation of the
‘Wave Mechanics,’ the laws for the motion of electrons in atoms.
• A standing wave is a wave motion that reflects back on itself in
such a way that the wave contains a certain number of points
(nodes) that undergo no motion.
• Erwin Schrödinger proposed that an electron’s matter wave
could be described by a wave function corresponding to a
standing wave (ψ or psi).
• The simplest model that shows how ψ relates to the energy
levels of an electron is the one-dimensional ‘particle in a box.’
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o This model gives quantized energies for ψ.
o Note how the wave function changes signs at the nodes
𝑛𝑛𝑥
2
𝜓𝑛 (𝑥 ) = � sin �
�
𝐿
𝐿
where n = 1,2,3,…
• By applying de Broglie’s relationship (λ=h/p) to the kinetic
energy of the particle in the box, we obtain this expression for
the kinetic energy of an electron in ψn:
𝑛2 ℎ 2
𝐾𝐾𝑛 =
where 𝑛 = 1,2,3, ⋯
8𝑚 𝐿2
o Note that the kinetic energy is never 0 since n≥1
– The lowest possible energy when n=1 is the zero-point
energy.
– This is consistent with the Heisenberg Uncertainty Principle
because zero KE would violate ∆x ∆p ≥ h/4π.
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• The most compact form of the Schrödinger equation is Eψ = Hψ.
• Schrödinger did not provide physical interpretation for ψ, but
Max Born did: ψ2 is the probability of finding an electron at
some point in space.
• Wave function solutions to the Schrödinger equation are called
orbitals (‘like an orbit’) to distinguish them from Bohr’s ‘orbits.’
• We will not go into further mathematics of the Schrödinger
Equation except to point out that the solutions yield three
quantum numbers that specify a particular orbital; we will look
at these quantum numbers, allowed values, and their meaning.
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Quantum Numbers and Electron Orbitals
• Here are the 3 quantum numbers from Schrödinger Equation
o n = the principal quantum number
– Allowed values: n = 1, 2, 3, 4, …
– Meaning: The distance of the orbital from the nucleus
– Alternate names: All orbitals with the same value of n are
in the same shell or principal electronic shell or principal
energy level
– Relationship to PT: The period number = n
– In electron configuration: leading number (2p)
o l = orbital angular momentum quantum number
– Allowed values: l = 0, 1, 2, 3, … , n−1
(if n=3, l =0,1,2)
– Meaning: The shape of the orbital
– Alternate names: All orbitals with the same values of n and
l are in the same subshell or sublevel
– Relationship to PT: Sets of group numbers = l
(for example, groups 1-2  l = 0)
– In electron configuration: the letter; to wit
l =0 = s, l = 1 = p,
l = 2 =d,
l=3=f
(after this, it’s alphabetical!)
o ml = magnetic quantum number
– Allowed values: ml = −l, −l+1,…, 0,…, l−1, +l
(example: for l = 2 or 5d, ml = −2, −1, 0, +1, +2)
– Meaning: The orientation of the orbital in 3 dimensions
– Alternate names: This is the true “orbital” occupied by
electrons; examples are px, py, pz
– Relationship to PT: No permanent association, but we can
say that an orbital spans two groups
– In electron configuration: no designation, but they do show
up as the short line segments in an energy level diagram
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17
The Fourth and Last Quantum Number!
• The fourth quantum number was provided by Wolfgang Pauli to
explain how two electrons could occupy the same space (i.e.
orbital) at the same time.
ms=magnetic spin QN
• The Pauli Exclusion Principle: two electrons can occupy the
same orbital only if they have opposite spins (↑ or ↓).
Corollary: No two electrons in an atom can have the same four
quantum numbers (Quantum Numbers = n, l, ml, ms).
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Summary of the 4 Quantum Numbers
• n = the principal quantum number (shell)
o n = 1, 2, 3, 4, …
o Distance of the orbital from the nucleus
• l = orbital angular momentum quantum number (subshell)
o l = 0, 1, 2, 3, … , n−1
o Shape of the orbital; l=0=s, l=1=p, l=2=d, l=3=f
• ml = magnetic quantum number
o ml = −l, −l+1, …, 0, …, l−1, +l
o Orientation of the orbital in 3D
• ms = electron spin quantum number
o ms = +½, −½ (also denoted by ↑ and ↓)
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Degenerate and Non-Degenerate Orbitals
• Degenerate orbitals all have the same energy.
• In a hydrogen atom all orbitals in a shell (principal energy level)
are degenerate.
• In a multi-electron atom orbitals within a subshell are
degenerate, but subshells with a shell are not degenerate.
Interactions between electrons remove degeneracy.
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Electron Configurations
• We use the energy ordering of shells and subshells, and the
number of orbitals per subshell to write electron configurations
o The electronic configuration of an atom or ion lists orbitals
that are occupied (i.e. have electrons in them)
• Subshells are occupied in the following order
1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p
• How do we remember this? The Periodic Table!
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• Subshells have these maximum occupancies:
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10 6p6 7s2 5f14 6d10 7p6
• Let’s look at some guiding principles for filling orbitals:
o Aufbau principle: ‘building up’ principle; electrons occupy the
lowest energy orbitals first; this yields the ground state
configuration.
o Hund’s Rule: when degenerate orbitals (within a subshell) are
being filled, the electrons occupy the orbitals singly first, all
with the same spin, before pairing up.
o Pauli exclusion principle: an orbital can hold at most two
electrons; electrons in same orbital must have opposite spins.
• GS electron configurations can be written a few different ways:
spdf notation (condensed)
spdf notation (expanded)
orbital diagram
Noble gas shorthand
You can mix and match these!
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Identify the element having the electron configuration
1s2 2s2 2p6 3s2 3p6 4s2 3d2
Use spdf condensed notation to show the electron configuration of
iodine. How many electrons does the iodine atom have in its 3d
subshell? How many unpaired electrons are in an iodine atom?
1s
2s
2p
3s
3p
4s
3d
4p
5s
4d
5p
6s
4f
Represent electron configuration of iron with an orbital diagram
__
1s
__
2s
__ __ __
2p
__
3s
__ __ __
3p
__
4s
__ __ __ __ __
3d
Represent the electron configuration of bismuth with an orbital
diagram; use the Noble gas shorthand
__
6s
__ __ __ __ __ __ __
4f
__ __ __ __ __
5d
__ __ __
6p
Practice: For an atom of Sn, indicate the number of (a) electron
shells that are either filled or partially filled; (b) 3p electrons;
(c) 5d electrons; and (d) unpaired e−
1s
2s
2p
3s
3p
4s
3d
4p
5s
4d
5p
6s
4f
5d
6p
For home practice, look at the ‘Concept Assessment’ on page 323.
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• We have a few concepts left to consider
o Valence electrons are those electrons in the highest
numbered shell and any unfilled d or f subshells. (Only
valence electrons are involved in chemical bonding and
reactions.)
o Core electrons are those electrons that are not valence
electrons! Electrons in filled shells that may have empty d or
f subshells. (Core electrons are too tightly held to get
involved in chemical bonding or reactions.)
o Penetration refers to the fact that the radial probabilities of
subshells interpenetrate; this explains the following. (See
also pictures below.)
o The 4s subshell fills before the 3d because the 4s subshell
penetrates inside the 3d subshell. An electron fills the 4s
subshell first because it will be closer to the nucleus and have
lower energy (Aufbau principle).
o Shielding (screening) means that core electrons ‘shield’
valence electrons from the attraction of the full nuclear
charge (more in chapter 9).
o Electron configurations of Cr and Cu and the stability of halffilled subshells.
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In multielectron atoms, the 3s electrons penetrate most deeply into the inner orbitals,
are least shielded, and experience the greatest effective nuclear charge. The 3d
electrons penetrate least. This accounts for the energy ordering of the sublevels: d>p>s.
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Assessing electron configurations
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