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Chapter 8: Electrons in Atoms Electromagnetic Radiation
• Electromagnetic (EM) radiation is a form of energy transmission
modeled as waves moving through space. (see below left)
Electromagnetic Radiation
• Any wave can be characterized by its:
o Speed: how fast the wave travels or propagates.
– EM radiation in vacuo propagates at a constant speed of c,
where c = λν = 3.00 × 108 m s−1, the ‘speed of light.’
o Wavelength (lambda or λ, in m): the distance between two
adjacent crests on a wave.
o Frequency (nu or ν, in s−1): the number of wavelengths
passing a point per second.
o Amplitude: maximum displacement of the wave.
• EM radiation spans a wide range of frequencies & wavelengths
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An FM radio station broadcasts on a frequency of 91.5 MHz. What is the
wavelength of these radio waves in meters? (Hz = Hertz = cycles per
second, or frequency)
• The amplitudes of EM waves are signed (+/−) and ‘additive.’
o When waves in phase are added, the amplitude sum
increases; this is constructive interference.
o When waves out of phase are added, the amplitude sum
decreases; this is destructive interference.
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• The colors in white light can be separated into a continuous
spectrum in two ways:
o Diffraction: reflecting the light off a grooved surface causes
the colored spectrum to appear due to interference.
o Refraction: passing the light through a prism causes colors to
travel at different speeds and separate into the spectrum.
Atomic Spectra
• The spectra produced by light from gas discharge tubes or salts
in a flame consists of a limited number of wavelengths or colors.
o This line spectrum is an 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
for 𝑛 = 3,4,5, ⋯
Quantum Theory
• At the end of the 19th century, some phenomena could not be
explained by classical physics.
• Attempting to explain these phenomena a new theory was
developed, quantum theory.
o Central to quantum theory is the proposal that energy is not
continuous, but rather is discrete in nature, that is quantized.
• Max Planck (1901) explained why the intensity of EM radiation
emitted by a ‘black body’ did not shoot off to infinity as
predicted by classical theory. He proposed that the energy is
emitted in definite (discrete) amounts called quanta.
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• 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 symbolized the new quantum physics
• Albert Einstein (1905) recognized that Planck's idea of light
appearing as quanta (bundles or photons) was the key to
understanding the mystery of the photoelectric effect.
o The photoelectric effect occurs when light strikes the surface
of a metal in vacuo causing an electron to be ejected.
o Researchers found the energy of the ejected electron did not
depend on the intensity of the light but rather on frequency.
The Bohr Atom
• Niels Bohr (1913) was the first to
apply quantum theory to atomic
structure. An impressive result of
Bohr’s 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 visible Balmer line
spectrum of atomic hydrogen.
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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 H atom, En = −2.69 x 10-20 J?
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• This model of an electron constrained to move in quantized
orbits around a central nucleus was called the Bohr Atom.
o The Ground state of the Bohr H atom occurs when the
electron occupies the orbit with the lowest energy (n = 1).
o An Excited state occurs when the electron occupies a higher
energy orbit with unfilled orbits closer to the nucleus.
• As an electron moves between orbits, it absorbs or releases
energy 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 = ∞).
1
1
∆𝐸 = 𝐸f − 𝐸i = 𝑅𝐻 � 2 − 2 � = 𝑅𝐻 = 2.179 × 10−18 J
1
∞
• The Bohr atom 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 example, n = 2 in Hg79+
𝐸2 =
−�802 �𝑅𝐻
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• The Bohr atom was a revolutionary application of the new
quantum theory, but it only worked for ‘atoms’ with one
electron! It did not work for multielectron atoms. These more
complex systems needed further refinements to quantum theory
Two Ideas Leading to a New Quantum Mechanics
• Louis de Broglie (1924) posited: If light waves can exhibit
particle-like properties (photons), then perhaps particles
(electrons) can exhibit wave-like behavior.
• The wavelength of these “matter waves” is
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𝜆=
ℎ
𝑝
=
ℎ
𝑚𝑚
Assuming Superman has a mass of 91 kg, what is his wavelength if he is
traveling at one-fifth the speed of light?
• Werner Heisenberg (1926) 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! It tells us that there is an
inherent limit to the precision of a
measurement.
ℎ
∆𝑥 ∆𝑝 = (∆position)(∆momentum) ≥
4𝜋
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 (1926) proposed that an electron’s matter
wave could be described by a wave function corresponding to a
standing wave (ψ or psi).
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𝑛𝑛𝑛
2
𝜓𝑛 (𝑥 ) = � sin �
�
𝐿
𝐿
where n = 1,2,3,…
• The simplest model that shows how ψ relates to the energy
levels of an electron is the one-dimensional ‘particle in a box.’
This model gives quantized energies for ψ.
o Note how the wave function changes signs at the nodes
• 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 =
8𝑚 𝐿2
where 𝑛 = 1,2,3, ⋯
• Note that the kinetic energy is never zero since n≥1.
o Lowest possible energy when n=1 is the zero-point energy.
o This is consistent with the Heisenberg Uncertainty Principle
because zero KE would violate ∆x ∆p ≥ h/4π.
• The most compact form of the Schrödinger equation is Eψ = Hψ.
• Schrödinger did not provide physical interpretation for ψ, but
Max Born (1926) did: ψ2 is the probability of finding an electron
at some point in space.
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• 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 (shell or energy level)
Allowed values: n = 1, 2, 3, 4, …
Meaning: The distance of the orbital from the nucleus
Relationship to Periodic Table: Period number = n
o l = orbital angular momentum quantum number (subshell)
– Allowed values: l = 0, 1, 2, 3, … , n−1
(if n=3, l =0,1,2)
– Meaning: The shape of the orbital
– Relationship to Periodic Table: Sets of group numbers = l
– Letters often used: l =0 = s, l = 1 = p, l = 2 =d, l = 3 = f
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
– Relationship to Periodic Table: No permanent association.
but we can say that an orbital spans two groups
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The Fourth and Last Quantum Number!
• Wolfgang Pauli (1924) proposed a fourth quantum number
(ms=magnetic spin) to explain how two electrons could occupy
the same space (i.e. orbital) at the same time.
• 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).
Summary of the Four 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.
o In a hydrogen atom (only one electron) all orbitals in a shell
(principal energy level) are degenerate.
• In a multielectron atom orbitals, within a subshell are
degenerate, but subshells within a shell are not degenerate.
Interactions between electrons remove degeneracy.
o As degeneracy is removed in multielectron atoms, subshells
separate in energy and the lowest energy subshells fill first.
The filling order is observed in the Periodic Table!!
subshells in a shell
degenerate
subshells
not degenerate
greater
separation
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penetration
Electron Configurations
• We use the energy ordering of shells and subshells, and the
number of orbitals per subshell, to write electron configurations.
o The electron configuration of an atom or ion lists subshells
that are occupied (i.e. have electrons in them). See the PT!
• 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
The
The
The
The
s subshell holds
p subshell holds
d subshell holds
f subshell holds
• Let’s look at the guiding principles for filling orbitals:
o Aufbau principle: ‘building up’ principle; electrons fill lowest
energy orbitals first; 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.
• The Ground State electron configuration can be written a few
different ways. Here are examples for oxygen (O has 8 e−):
spdf notation (condensed)
spdf notation (expanded)
orbital diagram
Noble gas shorthand
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You can mix and match these!
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? _______
(Condensed is default!)
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
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
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4d
5p
6s
4f
5d
6p
• Valence electrons are those electrons in the highest numbered
shell and any partially filled d or f subshells. (Only valence
electrons are involved in chemical bonding and reactions.)
• 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.)
• Electron configurations of Cr and Cu and the stability of halffilled subshells.
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• Penetration occurs when the radial probability of an outer
subshell brings e−‘s closer to the nucleus than an inner subshell.
o The 4s subshell fills before the 3d because the 4s subshell
penetrates inside the 3d subshell.
• Penetration also explains the non-degeneracy of subshells
within a shell. This accounts for the energy ordering of the
sublevels: d>p>s.
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• Assessing electron configurations:
o Is it a ground state configuration?
o Is it an excited state configuration?
o Is it a ‘legal’ electron configuration?
– If not, what filling rules have been violated?
Assess the following configurations for nitrogen (7 electrons).
1s2 2s2 2p3
1s2 2s2 2p2 3s1
1s2 2s2 2p2 2d1
1s2 2s2 ↑↑ ↑_ __
2p
1s2 2s2 ↑↓ ↑_ __
2p
1s2 2s2 ↓_ ↓_ ↓_
2p
1s2 2s2 ↑_ ↓_ ↑_
2p
1s1 2s2 2p3 7g1
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