Lecture 3Lecture 3

Multi-electron atoms & the periodic table

Suggested reading: Chapter 1

J l ti l i t d li ! Journal article is posted online!

Goal: Understanding chemical diversity

Diamond Graphite Silicon

Cuprate ceramicAluminumIndium tin oxide

en.wikipedia.org (carbon, aluminum, superconductor); Edwards’ group, Oxford (ITO)

2m

Recap from last class: Hydrogenic atoms

022

2 VEm

Schrodinger equation:

eZV eff)(

2Potential of a r

rVo

eff

4)( Potential of a

1-electron atom:

)()()( YrRr Electron wavefunctions:

R: Radial wavefunction – depends on two quantum numbers, “n” and “l”

),()(),,( ,, lmlln YrRr Electron wavefunctions:

d w d p d w q b , dY: Angular wavefunction – depends on another quantum number, “ml”

(A fourth quantum number, also in Y, arises from relativity: “ms”)

Quantum Numbers

)()()( ll YrRr ),()(),,( ,, lmlln YrRr

n Principal quantum number

n = 1, 2, 3,… Quantizes the electron energy of 1 electron atoms

l

number

Orbital angular momentum l = 0, 1, 2,…(n-1)

K, L, M,… shells of 1-electron atoms

Quantizes the magnitude of orbital angular momentum L

m

quantum number

Magnetic quantum 0 ±1 ±2 ±l

s, p, d, … subshells orbital angular momentum L

Quantizes the orbital ml Magnetic quantum number

ml = 0, ±1, ±2,… ±lQangular momentum along a magnetic field B

hms Spin magnetic quantum number

ms = ±½ Quantizes the 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 i f 1 l energies of 1-electron atoms.

24Zme2228 nh

ZmeEo

n

Ionization energy of hydrogen: energy required to remove the

(Z is atomic number, n is the quantum number, 1,2,3,…)

184me

Ionization energy of hydrogen: energy required to remove the electron from the ground state in the H-atom

eV13.6J1018.28

1822

hmeE

oI

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: Josef von FraunhoferJλdark1=656.3 nmλdark2=486.1 nm

)1(8 21222

42

nE

nhmeZE

on

Convenient conversion: λ [eV]= 1241.341/λ [nm]

Example: Solar Spectrum

1829: Josef von FraunhoferJλdark1=656.3 nmλdark2=486.1 nm

E3-E2 = -13.6[(1/32) -(1/22)]=1.89eV= 656 nm

E4-E2 = 486 nm

)1(8 21222

42

nE

nhmeZE

on

lQuantizes the orbital motion of the electron

O bi l l

( = 0, 1, 2, ….n1) 2/11 L

Orbital angular momentum

Orbital angular momentum along an applied magnetic field Bz

mLz

O bi l l

lQuantizes the orbital motion of the electron

( = 0, 1, 2, ….n1) 2/11 L

Orbital angular momentum

Orbital angular momentum along an applied magnetic field Bz

mLz

O bi l l

lQuantizes the orbital motion of the electron

( = 0, 1, 2, ….n1) 2/11 L

Orbital angular momentum

Orbital angular momentum along an applied magnetic field Bz

mLz

O bi l l

lQuantizes the orbital motion of the electron

( = 0, 1, 2, ….n1) 2/11 L

Orbital angular momentum

Orbital angular momentum along an applied magnetic field Bz

mLz

2 = 2

s Quantizes the spin momentum of the electron

2/1

Spin angular momentum

2/11 ssS

12

s

sz mS Spin along a magnetic field

sz

21

sm2

s Quantizes the spin momentum of the electron

2/1

Spin angular momentum

2/11 ssS

12

s

sz mS Spin along a magnetic field

sz

21

sm2

Magnetic behavior arises from L and S

LeOrbital magnetic moment

Lμem2orbital

Spin magnetic moment

Sμ espin

emspin

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 222)( eerrV

in the He atom121

121 44),(

rrrrV

oo

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 “one-electron” problems (and treating the electron-electron

interaction term as a small perturbation)

•The charge experienced by each electron is the “effective nuclear The charge experienced by each electron is the effective nuclear charge” Zeffe = (Z-σ)e: Shielding constant σ

•S l i f th i f th l t i lti l t t •Solving for the energies of the electrons in multielectron atoms yields a dependence on n and

yE

nerg

y

U ll h d f Usually, the order of energy levels in a

shell is s<p<d<f

n

shell is s<p<d<f

Atomic orbital energies versus atomic number Z

For Z‹21 4s is lower in For Z‹21, 4s is lower in energy than 3d

p d

s

Effective nuclear charge Zeff

Fi t 3 f th i di t blFirst 3 groups of the periodic table

Zeff decreases for “frontier” orbitals and also increases across a period, down a group

How do electrons fill these energies?

P uli E clusion Principle No t o electrons in n tom Pauli Exclusion Principle: No two electrons in an atom 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 E periment l spectroscopic studies Hund’s Rule: Experimental spectroscopic studies indicate that electrons in the same n, orbitals prefer

their spins to be parallel (same m )their spins to be parallel (same ms)

Origin: If electrons enter the same m by pairing their 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 O F

n=2

n=1

Important exceptions to these rules

1 Electron repulsion modifies the “ tomic orbit l” trends 1. Electron repulsion modifies the “atomic orbital” trends for elements with an incomplete d-shell. Electrons in

such elements first occupy orbitals predicted to be higher such elements first occupy orbitals predicted to be higher in energy (i.e., 4s instead of 3d)

General trend: [X]3dn4s2 General trend: [X]3d 4s

However, all d-block cations and complexes have dn

configurations

2. Because electrons with the same ψn,,m experience a l h lf f ll d h ll f l h

configurations

strong repulsion, half-filled shells of electrons with 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

[A ]3d24 2[Ar]3d24s2

Ground State electron configuration of Ti3+

Click for answer

Ground State electron configuration of Ti3+

[A ]3d1[Ar]3d1

Periodic Table Trends

1. Metals combine with nonmetals to

In general…

with nonmetals to give hard, non-volatile solids

2. Nonmetals combine with each other to f l til l l

3 M l bi

form volatile molecular compounds

3. Metals combine with metals to give alloysy

Columns = “groups” Rows = “periods”

Rare earths: not as rare as you think!

Rare earths: Ce is 26th most Ce is 26th most

abundant element

Lanthanoids

h f h d b h d h h l• Term “rare earth” refers to “hiding behind” each other in minearls• 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: Ln3+• All form trivalent cations: Ln3

• All Lanthanoid ions are fluorescent, as a result of the forbidden nature of f-f transitions

• Europium-doped Yttrium vanadate was the first red phosphor to

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 l lsunglass lenses

• Lasers, fiber amplifiers, transmission links for internet

Amplification & upconversion& upconversionhttp://nanotechweb.org/cws/article/tech/41882

First color tv broadcast in 1953

From WebMD: “Erbium laser resurfacing is designed to remove superficial and in 1953 superficial and moderately deep lines and wrinkles on the face hands, neck, or chest.”

• All are man-made except for thorium and uranium

Actinoids

All are man made, except for thorium and uranium• All are radioactive

• First synthesized as part of the Manhattan project in 1944S h l t i 6d bit l b t i d th 6 • Some have electrons in 6d orbitals, but in compounds the 6s

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 C ssini or G lileo mission The pellet either the Cassini or Galileo mission. The pellet produces 62 watts of heat and glows because of the heat generated by the radioactive decay (primarily α). Photo is taken after insulating (p y ) gthe 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

H i di l bl d h k bl • He is exceedingly stable and has no known stable 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 The metallic forms of these elements can only be extracted by electrolysis of a molten salt (Sir Humphry Davy)

All fi h d d h b t d i A• All are fire hazards and show be stored in Ar• React vigorously with H2O to liberate hydrogen (Mg, Li, and Be react relatively slowly)

Halogens: part of the p-Block

Hi hl i f d i h i l • Highly reactive: found in the environment only as compounds or ions• Only periodic table group that contains elements in

ll f d l l d dall 3 states of matter: F and Cl: gases, Br: liquid; I and 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 colorless monatomic gasesOdorless, colorless, monatomic gases• Non-flammable,• Low chemical reactivity:

N H A K X RNe < He < Ar < Kr < Xe < Rn•First noble gas compounds: XeF4 and XeF2

(used to etch Si)

d-Block

CrCo

CrNi Cu Mn

Partly filled d-shell results in unique qualities:

1 Formation of compounds and complexes whose color is due to 1. Formation of compounds and complexes whose color is due to 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

The net positive charge experienced by an electron in a multi

Trend 1: Effective Nuclear Charge

The net positive charge experienced by an electron in a multi-electron atom (shielding prevents outermost electrons from

feeling full nuclear charge)

gecl

ear

char

gfe

ctiv

e nu

cE

ff

Effective nuclear charge

Trend 2: Atomic RadiusThe 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 i d b ff i l h Decreases across a period because effective nuclear charge

increases, attracting electrons

Trend 2: Atomic RadiusThe 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 i d b ff i l h Decreases across a period because effective nuclear charge

increases, attracting electrons

Trend 3: Ionization EnergyThe minimum energy required to remove one electron from each 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-phase atom gains an Electron affinity: the energy change when a gas phase atom gains an electron

Electronegativity: the ability of an atom to attract electrons when it is g y ypart of a compound

Polarizability αAbility of an atom to be distorted by an electric fieldAbility of an atom to be distorted by an electric field

Polarizability is high if the separation of frontier orbitals is small

L hi hl h d i il l i dLarge, highly charged anions are easily polarized

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 h t

Ionization energy

Metallic character

s

Electron affinity & electronegativity

Ioniarge

omic

rad

ius ization enenu

clea

r ch

a

Ato

ergyE

ffec

tive

n

Effective nuclear chargeEffective nuclear charge

Atomic radius