Zumdahl’s Chapter 7
Atomic Structure
Atomic Periodicity
Chapter Contents
• EM Radiation
• EM Quantization
• H Spectrum
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Niels Bohr
L. deBroglie
W. Heisenberg
E. Schrödinger
• Quantum Nos.
• Orbital Shapes
– Degeneracies
• W. Pauli (spin)
• Multielectrons
• Periodic Table
– Aufbau
– Property Trends
– Groups
Electromagnetic
Radiation
• Oscillating E & M fields forever.
• Wavelength, , distance between
successive peaks.
• Period, τ, time between peaks.
• Speed, c =  / τ =  
–  = frequency (cycles per second)
–  = c /  and  = c / 
Electromagnetic
Quantization
• E = h = hc / 
• Equipartition Theorem demands kT
worth of thermal energy to all light
and matter modes. Leads to  energy.
• Vibration overtones in matter are
truncated by indivisible atoms.
•  in light energies overcome by Planck
with QUANTIZED energies.
• h = Planck’s constant = 6.6x10–34 Js
Hydrogen Atom
Spectrum
• White light is all colors (all );
diffraction in prisms or
raindrops gives continuous
spectra.
• Atomic excitation gives instead
discrete colors (few ).
• In H atom, E light = R (n2–2 – n1–2)
• Why so simple?
Niels Bohr
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F centripetal = m r ²
F attraction = – Z e² / r²
Balance: Z e² = m r³ ²
E = K+V = ½ m r² ² – Z e² / r
E = – ½ Z e² / r on substitution
E = – (½ Z e² / r) (Z e² / m r³ ²)
E = – Z² e4 m / (2 m² r4 ²)
Quantized Momenta
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E = – Z² e4 m / (2 m² r4 ²)
pθ = m r²  is angular momentum
E = – Z² e4 m / (2 pθ²)
pθ = n h / 2 Bohr’s postulate
En = – (2² Z² e4 m / h²) n–²
En = – RH Z² / n² YES!!
– But but but WHY quantize pθ ?!?
Louis deBroglie
• Just as light moves as a wave
but lives and dies as a discrete
energy packet (“photon”),
• Matter too has wave properties
only dominant for light masses:
•  = h / p = h / mv = h / (m r² )
• And n  = 2 or wave kills itself!
Werner Heisenberg
• Waves must S P A N .
• Attempts to narrow the wave,
reduce , increasing p = h / 
• So a minimum uncertainty x in
position MUST exist, and
• ( x) ( px)  ½ h / 2
– Heisenberg’s Uncertainty Principle
Edwin Schrödinger
• Bohr’s orbits are infinitesimally
thin trajectories. Being off them
must be infinitely uncertain!
• Need a full 3-D wave, , not 1-D.
• Schrödinger’s Wave Equation
solves H energy in 3-d and finds:
• En = – RH Z² / n² also and more!
Quantum Numbers
(n,
l ,ml ,ms)
• Principle Quantum Number, n
– n = 1, 2, 3, 4, 5, … , 
– Governs the number of nodes in
the electron’s matter wave, !
– # of nodes (where =0) is n – 1
– For “hydrogenic” ions (and H itself)
electron energy depends only on n.
– Nodes can be spherical or angular!
Angular Momentum
Quantum Number, l
•l
•
= 0, 1, 2, 3, 4, … , (n – 1)
s p d f g …
shorthand
–l
chemist
measures # of nodes that are
angular; so it must stop at n–1.
– Increasing angular nodes squeezes
waves, so E usually depends on l
– Z component of
l is also quantized!
Magnetic Quantum
Number, ml
• ml is the component of l along
(up or down) the Z axis in space.
– ml = – l, – l + 1, … , –1, 0, 1, … , l–1,
– Because the component can’t
exceed its vector.
– E only depends upon ml when a
magnetic field is applied.
l
3-D Shapes of Orbitals
• Governed by n, l, and ml
• l = 0 is spherical
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n = 1 means no nodes:
(n – 1) = # of nodes, all spherical.
n = 2 means one spherical node:
Wavefunction, , falls off in
intensity to zero at large distance
from +
Cross-section of 2s
Spherical
Node
Angular Orbitals
• l = 1 implies one angular node
– Cleave space with an x=0 plane
– But y=0 or z=0 work as well, so
there are three or 2l+1 suborbitals.
– The ml sequence always gives 2l+1
– ml differentiates directions in
space for chemical bonding!
Degeneracies
• In the absence of applied
magnetic field, all suborbitals of
a given l have the same energy.
• This identity of energies is
called “degenerate.”
• Even nearly degenerate orbitals
may be mixed to give new ones.
P.A.M. Dirac
Wolfgang Pauli
• Dirac applied Einstein’s fixed c
to Schrödinger’s Equation and
found new quantum number, ms.
– ms is electron spin number and
takes on only two values, ½.
• Pauli Principle says only two
electrons can occupy any
orbital, and their ms must differ,
– without which NO CHEMISTRY.
Multielectronic Atoms
• Beyond H, repulsions BETWEEN
electrons compete with nuclear
attraction & complicate spectra.
– Hund’s Rule: if electrons have the
choice between degenerate
orbitals, they choose NOT to
double occupy them.
• It minimizes electronic repulsion.
Repulsive
Consequences
• Energies are now a function of l,
the angular quantum number.
• The “filling sequence” shows
the new energy order:
– 1s<2s<2p<3s<3p<4s<3d<4p<5s<4d
– <5p<6s<4f<5d<6p<7s<5f<6d etc.
– Periodic Table exemplifies it, but a
simple pattern emerges:
Filling Sequence
Mnemonic
1s
2s
3s
4s
5s
6s
7s
8s
2p
3p
4p
5p
6p
7p
8p
3d
4d
5d
6d
7d
8d
4f
5f
6f
7f
8f
And that’s as far
as the known
elements go.
5g
6g
7g
8g
6h
7h
8h
7i
8i
8k
Periodic Table
• Aufbau (filling sequence) follows
that table:
– 1s² 2s² 2p6 3s² 3p6 4s² 3d10 4p6 5s²
– 4d10 5p6 6s² 4f14 5d10 6p6 7s² 6d10
– and the latest elements among 7p6
• Irregularities occur where ½
filled suborbitals acquire greater
stability than a predecessor:
Aufbau Hiccups
• Vanadium: [Ar] 4s² 3d3 suggests
• Chromium: [Ar] 4s² 3d4 is next,
BUT IT ISN’T SO!
• Chromium: [Ar] 4s1 3d5 lowers
its energy by borrowing a 4s to
complete a ½–filled d suborbital.
• Manganese: [Ar] 4s² 3d5 follows.
Periodic Properties
• Rows are called “periods” on the
Periodic Table.
– Columns are called “groups.”
• Progression along rows implies
adding new electrons & protons.
– They get added 1:1 for neutrality.
– But new repulsions keep pace with
new attractions. Which wins?
Effective Potential
• Protons exert attraction only
toward the atom’s center.
• Electrons exert repulsion from
all over their wavefunctions.
– “Core” electrons are located very
close to the nucleus where they
repel outer electrons as effectively
as their number of protons attract.
Effective Charge =
–
Protons – Effective e
• So nucleus’s effective charge is
Z – (# of all core electrons) less
the effect of “outer” electrons.
• While core are 100% effective,
“valence” electrons are LESS by
virtue of spanning a greater
fraction of the atom. Only their
inner portion is 100% effective.
Inefficiency Wins
• Across a row, added electrons
are valence, not core. So they
repel one another less than the
added protons attract them.
• Effective potential INCREASES
across the row, binding the
subsequent electrons ever more
tightly!
Trends on Periods
• Increased electron binding
along rows (to the right),
generally results in:
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Increasing ionization potentials
Decreasing atomic sizes
Growing electron affinities
Increasing electronegativies
Groups
• Elements in the same column
have the same number and type
of valence electrons, differing
only by n.
• Because increasing n by 1 puts
one more node in , dimensions
of  increase, vaulting electrons
outside their predecessors.
Group Trends
• Dropping down a group (column)
increases efficiency of core and
distance of valence from center.
Both conspire to weaken the
nucleus’s grasp.
– Atomic size increases.
– Ionization potential decreases.
– Electronegativity decreases.