Lecture 3
Multi-electron atoms & the periodic table
Suggested reading: Chapter 1
J
Journal
l article
ti l iis posted
t d online!
li ! 
Goal: Understanding chemical diversity
Diamond
Indium tin oxide
Graphite
Aluminum
Silicon
Cuprate ceramic
en.wikipedia.org (carbon, aluminum, superconductor); Edwards’ group, Oxford (ITO)
Recap from last class: Hydrogenic atoms
Schrodinger equation:
Potential of a
1-electron atom:
Electron wavefunctions:
2m
2m
   2 E  V   0

2
V (r ) 
 Z eff e 2
4 o r
 (r , ,  )  Rn ,l (r )Yl ,m ( ,  )
l
R: Radial
d w
wavefunction – d
depends
p d on two
w q
quantum numbers,
b , “n” and
d “l”
Y: Angular wavefunction – depends on another quantum number, “ml”
(A fourth quantum number, also in Y, arises from relativity: “ms”)
Quantum Numbers
 (r , ,  )  Rn ,l (r )Yl ,m ( ,  )
l
n
Principal quantum
number
n = 1, 2, 3,…
K, L, M,… shells
Quantizes the electron energy
of 1-electron
1 electron atoms
l
Orbital angular momentum
quantum number
l = 0, 1, 2,…(n-1)
s, p, d, … subshells
Quantizes the magnitude of
orbital angular momentum L
ml
Magnetic quantum
number
0 ±1,
±1 ±2,…
±2 ±l
ml = 0,
Q
Quantizes
the orbital
angular momentum along
a magnetic field B
ms
Spin magnetic quantum
number
ms = ±½
Quantizes the
h spin
angular momentum along
a magnetic field B
ms: Arises from relativistic quantum theory
Quantum Numbers: Shells & subshells
n Quantizes the electron energies
Knowing ψ, we can use the Schrodinger equation to find the electron
energies
i off 1-electron
1 l
atoms.
4
2
me Z
En   2 2 2
8o h n
(Z is atomic number, n is the quantum number, 1,2,3,…)
Ionization energy of hydrogen: energy required to remove the
electron from the ground state in the H-atom
4
me
18
E I  2 2  2.18 10 J  13.6 eV
8 o h
Electron energies of Hydrogen
Energies are more
closely spaced for
higher n
Energy transitions can occur via photons
Absorption of a photon
Emission of a photon
Example: Solar Spectrum
1829: JJosef von Fraunhofer
λdark1=656.3 nm
λdark2=486.1 nm
1
Z 2me4
En   2 2 2  E1( 2 )
8o h n
n
Convenient conversion: λ [eV]= 1241.341/λ [nm]
Example: Solar Spectrum
1829: JJosef von Fraunhofer
λdark1=656.3 nm
λdark2=486.1 nm
E3-E2 = -13.6[(1/32) -(1/22)]=1.89eV
= 656 nm
E4-E2 = 486 nm
1
Z 2me4
En   2 2 2  E1( 2 )
8o h n
n
l Quantizes the orbital motion of the electron
O bi l angular
Orbital
l momentum
L      1 
1/ 2
( = 0, 1, 2, ….n1)
Orbital angular momentum along an applied magnetic field Bz
Lz  m 
l Quantizes the orbital motion of the electron
O bi l angular
Orbital
l momentum
L      1 
1/ 2
( = 0, 1, 2, ….n1)
Orbital angular momentum along an applied magnetic field Bz
Lz  m 
l Quantizes the orbital motion of the electron
O bi l angular
Orbital
l momentum
L      1 
1/ 2
( = 0, 1, 2, ….n1)
Orbital angular momentum along an applied magnetic field Bz
Lz  m 
l Quantizes the orbital motion of the electron
O bi l angular
Orbital
l momentum
L      1 
1/ 2
( = 0, 1, 2, ….n1)
Orbital angular momentum along an applied magnetic field Bz
Lz  m 
=2
s Quantizes the spin momentum of the electron
Spin angular momentum
S  ss  1
1/ 2
1
s
2
Spin along a magnetic field
S z  ms
1
ms  
2
s Quantizes the spin momentum of the electron
Spin angular momentum
S  ss  1
1/ 2
1
s
2
Spin along a magnetic field
S z  ms
1
ms  
2
Magnetic behavior arises from L and S
Orbital magnetic moment
e
μ orbital  
L
2me
Spin magnetic moment
μ spin
e
 S
me
Orbiting/spinning electron is analogous to a current loop
(classical magnetic moment μ = current I*area A)
Towards multi-electron atoms: Helium (Z=2)
Potential energy
of one electron
in the He atom
2
2
2e
e
V (r1, r12 )  

4or1 4or12
r12 makes the Schrodinger equation non-separable: can only solve
with approximate techniques (not covered in this class)
The Orbital Approximation
•Assume each electron in a multi-electron atom occupies an
atomic orbital that resembles those found in hydrogenic atoms.
•Basically, reducing a many-electron problem to many “oneelectron” problems (and treating the electron-electron
interaction term as a small perturbation)
The charge experienced by each electron is the “effective
effective nuclear
•The
charge” Zeffe = (Z-σ)e: Shielding constant σ
•S l i for
•Solving
f the
th energies
i off the
th electrons
l t
in
i multielectron
lti l t
atoms
t
yields a dependence on n and 
Energy
y
Usually,
U
ll the
h order
d off
energy levels in a
shell is s<p<d<f
n
Atomic orbital energies versus atomic number Z
For Z‹21,
Z‹21 4s is lower in
energy than 3d
p
s
d
Effective nuclear charge Zeff
Fi t 3 groups off th
First
the periodic
i di table
t bl
Zeff decreases for “frontier” orbitals and also
increases across a period, down a group
How do electrons fill these energies?
P uli E
Pauli
Exclusion
clusion Principle
Principle: No ttwoo electrons in an
n atom
tom
can have the same four quantum numbers
If electrons are in the same orbital (with identical
n, , m), their spins will “pair.”
n=2
n=1
H
He
Li
Be
B
How do electrons fill these energies?
Hund’s Rule:
Rule Experimental
E periment l spectroscopic studies
indicate that electrons in the same n, orbitals prefer
their spins to be parallel (same ms)
Origin: If electrons enter the same m by pairing their
spin, they will occupy the same spatial distribution
(ψn,,m
, , ) and experience a strong repulsion
C
n=2
n=1
O
F
Important exceptions to these rules
1 Electron repulsion modifies the “atomic
1.
“ tomic orbit
orbital”
l” trends
for elements with an incomplete d-shell. Electrons in
such elements first occupy orbitals predicted to be higher
in energy (i.e., 4s instead of 3d)
General trend: [X]3dn4s2
However, all d-block cations and complexes have dn
configurations
2. Because electrons with the same ψn,,m experience a
strong repulsion,
l
h
half-filled
lf f ll d shells
h ll off electrons
l
with
h
parallel spins are particularly stable (spin correlation)
Ground state of Cr: [Ar]3d54s1 or [Ar]3d44s2
Ground State electron configuration of Ti
Click for answer
Ground State electron configuration of Ti
[Ar]3d
[A
]3d24s
42
Ground State electron configuration of Ti3+
Click for answer
Ground State electron configuration of Ti3+
[Ar]3d
[A
]3d1
Periodic Table Trends
In general…
1. Metals combine
with nonmetals to
give hard, nonvolatile solids
2. Nonmetals combine
with each other to
f
form
volatile
l til molecular
l l
compounds
33. Metals
M l combine
bi
with metals to give
alloys
y
Columns = “groups” Rows = “periods”
Rare earths: not as rare as you think!
Rare earths:
Ce is 26th most
abundant element
Lanthanoids
• Term “rare earth”
h refers
f to “hiding
h d behind”
b h d each
h other
h in minearls
l
• First discovered lanthanoid, Lanthanum, was found in a cerium
mineral
•All contain 4f-shell electrons, except Lanthanum (which is a d-block
element)
• All form trivalent cations: Ln33+
• All Lanthanoid ions are fluorescent, as a result of the forbidden nature
of f-f transitions
Applications of Lanthanoids
• Europium-doped Yttrium vanadate was the first red phosphor to
enable the development of color tv screens
• Lanthanoids deflect UV and IR radiation: used in production of
sunglass
l lenses
l
• Lasers, fiber amplifiers, transmission links for internet
Amplification
& upconversion
http://nanotechweb.org/cws/a
rticle/tech/41882
First color tv broadcast
in 1953
From WebMD: “Erbium
laser resurfacing is
designed to remove
superficial and
moderately deep lines
and wrinkles on the face
hands, neck, or chest.”
Actinoids
• All are man
man-made
made, except for thorium and uranium
• All are radioactive
• First synthesized as part of the Manhattan project in 1944
• Some
S
h
have electrons
l t
iin 6d orbitals,
bit l b
butt in
i compounds
d th
the 6s
6
electrons and any d electrons are lost, leaving the ions with an
electronic configuration [Rn]5fn
• Need particle colliders, nuclear reactors, or supernova for their
synthesis
A pellet of 238PuO2 to be used in a
radioisotope thermoelectric generator for
either the Cassini
C ssini or Galileo
G lileo mission.
mission The pellet
produces 62 watts of heat and glows because of
the heat generated by the radioactive decay
(primarilyy α).
(p
) Photo is taken after insulatingg
the pellet under a graphite blanket for minutes
and removing the blanket. (from Wikipedia)
Blocks of the Periodic Table
S-Block
• Except for H and He, electrons are easily lost for
form positive ions
• He
H iis exceedingly
di l stable
bl and
dh
has no known
k
stable
bl
compounds
• All other s-block elements are very powerful
reducing agents  never occur naturally in the free
state
• The metallic forms of these elements can only be
extracted by electrolysis of a molten salt (Sir
Humphry Davy)
• All are fi
fire h
hazards
d and
d show
h b
be stored
t d iin Ar
A
• React vigorously with H2O to liberate hydrogen
(Mg, Li, and Be react relatively slowly)
Halogens: part of the p-Block
• Highly
Hi hl reactive:
i found
f
d in
i the
h environment
i
only
l as
compounds or ions
• Only periodic table group that contains elements in
all
ll 3 states off matter: F and
d Cl:l gases, Br: liquid;
l
d I and
d
Astatine, solids
• F is one of the most reactive elements, attacking
otherwise inter materials like glass and forming
compounds with the heavier noble gases. Once is does
react, the resulting molecule is very inert. Teflon: F+C
• Hydrogen halides form a series of very strong acids
Noble Gases: part of the p-Block
•Odorless
Odorless, colorless
colorless, monatomic gases
• Non-flammable,
• Low chemical reactivity:
N < He
Ne
H < Ar
A < Kr
K < Xe
X < Rn
R
•First noble gas compounds: XeF4 and XeF2
(used to etch Si)
d-Block
Co
Cr
Ni
Cu
Mn
Partly filled d-shell results in unique qualities:
1 Formation of compounds and complexes whose color is due to
1.
d-d transitions
2. Formation of compounds in many oxidation states, due to low
reactivity of unpaired d-electrons
3. Formation of many paramagnetic compounds
Trend 1: Effective Nuclear Charge
Efffective nucclear charg
ge
The net positive charge experienced by an electron in a multimulti
electron atom (shielding prevents outermost electrons from
feeling full nuclear charge)
Effective nuclear charge
Trend 2: Atomic Radius
The distance from the nucleus to the outermost stable electron
orbital (here in pm).
Increases down a group due to addition of a new energy shell.
D
Decreases
across a period
i d because
b
effective
ff i nuclear
l charge
h
increases, attracting electrons
Trend 2: Atomic Radius
The distance from the nucleus to the outermost stable electron
orbital (here in pm).
Increases down a group due to addition of a new energy shell.
D
Decreases
across a period
i d because
b
effective
ff i nuclear
l charge
h
increases, attracting electrons
Trend 3: Ionization Energy
The minimum energy required to remove one electron from each
atom in a mole of atoms in the gaseous state.
Trend 4: Electron affinity and electronegativity
Electron affinity: the energy change when a gas
gas-phase
phase atom gains an
electron
Electronegativity:
g
y the abilityy of an atom to attract electrons when it is
part of a compound
Polarizability α
Ability of an atom to be distorted by an electric field
Polarizability is high if the separation of frontier orbitals is small
L
Large,
highly
hi hl charged
h
d anions
i
are easily
il polarized
l i d
Cations that do not have noble-gas configurations are easily polarized
Small, highly charged cations easily distort the electron distribution of
neighboring ions: strong polarizing ability
Trend 5: Metallic character of the elements
Trend Summary
M t lli character
Metallic
h
t
Effective n
nuclear chaarge
Electron affinity &
electronegativity
Effective nuclear charge
Atomic radius
Ioniization eneergy
Atoomic radiuss
Ionization energy