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Quantum or Wave MechanicsQuantum or Wave Mechanics

de Broglie (1924) proposedthat all moving objectshave wave properties.

For light: E = mc2

E = hν = hc / λ

Therefore, mc = h / λ

and for particles

(mass)(velocity) = h / λ

dede Broglie Broglie (1924) proposed (1924) proposedthat all moving objectsthat all moving objectshave wave properties.have wave properties.

For light: E = mcFor light: E = mc22

E = h E = hνν = = hc hc / / λλ

Therefore,Therefore, mc mc = h / = h / λλ

and for particlesand for particles

(mass)(velocity) = h / (mass)(velocity) = h / λλ

L. deL. de Broglie Broglie(1892-1987)(1892-1987)

Baseball (115 g) atBaseball (115 g) at100 mph100 mph

λλ = 1.3 x 10 = 1.3 x 10-32-32 cm cm

e- with velocity =e- with velocity =1.9 x 101.9 x 1088 cm/sec cm/sec

λλ = 0.388 = 0.388 nm nmExperimental proof of waveExperimental proof of waveproperties of electronsproperties of electrons

Quantum or Wave MechanicsQuantum or Wave MechanicsSchrodingerSchrodinger applied idea of e- applied idea of e-

behaving as a wave to thebehaving as a wave to theproblem of electrons in atoms.problem of electrons in atoms.

He developed the He developed the WAVEWAVEEQUATIONEQUATION

Solution gives set of mathSolution gives set of mathexpressions called expressions called WAVEWAVEFUNCTIONS, FUNCTIONS, ΨΨ

Each describes an allowed energyEach describes an allowed energystate of an e-state of an e-

QuantizationQuantization introduced naturally. introduced naturally.

E.E. Schrodinger Schrodinger1887-19611887-1961

Quantum or Wave MechanicsQuantum or Wave Mechanics

WAVE FUNCTIONS, WAVE FUNCTIONS, ΨΨ

• ΨΨ is a function of distance and twois a function of distance and two

angles.angles.

• Each • Each ΨΨ corresponds to an corresponds to an ORBITALORBITAL— the region of space within which an— the region of space within which anelectron is found.electron is found.

• • ΨΨ does NOT describe the exact does NOT describe the exact

location of the electron.location of the electron.

• • ΨΨ22 is proportional to the probability of is proportional to the probability of

finding an e- at a given point.finding an e- at a given point.

Uncertainty PrincipleUncertainty PrincipleProblem of defining nature

of electrons in atomssolved by W. Heisenberg.

Cannot simultaneouslydefine the position andmomentum (= m•v) of anelectron.

We define e- energy exactlybut accept limitation thatwe do not know exactposition.

Problem of defining natureProblem of defining natureof electrons in atomsof electrons in atomssolved by W.solved by W. Heisenberg Heisenberg..

Cannot simultaneouslyCannot simultaneouslydefine the position anddefine the position andmomentum (= m•v) of anmomentum (= m•v) of anelectron.electron.

We define e- energy exactlyWe define e- energy exactlybut accept limitation thatbut accept limitation thatwe do not know exactwe do not know exactposition.position.

W.W. Heisenberg Heisenberg1901-19761901-1976

Types ofTypes ofOrbitalsOrbitals

s orbitals orbital

p orbitalp orbital

d orbitald orbital

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OrbitalsOrbitals•• No more than 2 e- assigned to anNo more than 2 e- assigned to an

orbitalorbital

•• OrbitalsOrbitals grouped in s, p, d (and f) grouped in s, p, d (and f)subshellssubshells

ss orbitals orbitals

dd orbitals orbitals

pp orbitals orbitals

ss orbitals orbitals

dd orbitals orbitals

pp orbitals orbitals

ss orbitals orbitals pp orbitals orbitals dd orbitals orbitals

No.No.orbs.orbs.

No.No.e-e-

11 33 55

22 66 1010

SubshellsSubshells & Shells & Shells

•• SubshellsSubshells grouped in shells. grouped in shells.

•• Each shell has a number calledEach shell has a number calledthethe PRINCIPAL QUANTUMPRINCIPAL QUANTUMNUMBER, nNUMBER, n

•• The principal quantum numberThe principal quantum numberof the shell is the number of theof the shell is the number of theperiod or row of the periodicperiod or row of the periodictable where that shell begins.table where that shell begins.

SubshellsSubshells & Shells & Shells

n = 1n = 1

n = 2n = 2

n = 3n = 3

n = 4n = 4

QUANTUM NUMBERSQUANTUM NUMBERS

Each orbital is a function of 3 quantumEach orbital is a function of 3 quantumnumbers:numbers:

n n (major)(major) ---> shell---> shell

ll (angular) (angular) ---> ---> subshellsubshell

mmll (magnetic)(magnetic) ---> designates an orbital ---> designates an orbital within awithin a subshell subshell

SymbolSymbol ValuesValues DescriptionDescription

n (major)n (major) 1, 2, 3, ..1, 2, 3, .. Orbital sizeOrbital sizeand energyand energy

where E = -R(1/nwhere E = -R(1/n22))

l (angular)l (angular) 0, 1, 2, .. n-10, 1, 2, .. n-1 Orbital shapeOrbital shapeor type or type ((subshellsubshell))

mmll (magnetic) (magnetic) -l..0..+l-l..0..+l Orbital Orbital orientationorientation

# of # of orbitals orbitals in in subshell subshell = 2 l + 1 = 2 l + 1

QUANTUM NUMBERSQUANTUM NUMBERS

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Shells and SubshellsShells andShells and Subshells SubshellsWhen n = 1, then l = 0 and mWhen n = 1, then l = 0 and m ll = 0 = 0Therefore, in n = 1, there is 1 type ofTherefore, in n = 1, there is 1 type of

subshellsubshelland thatand that subshell subshell has a single orbital has a single orbital(m(mll has a single value ---> 1 orbital) has a single value ---> 1 orbital)

ThisThis subshell subshell is labeled is labeled ss (“ (“essess”)”)

Each shell has 1 orbital labeled s,Each shell has 1 orbital labeled s,and it is and it is SPHERICALSPHERICAL in shape.in shape.

s Orbitalsss Orbitals Orbitals

See Figure 7.14 on page 319 andScreens 7.10 and 7.11.

See Figure 7.14 on page 319 andSee Figure 7.14 on page 319 andScreens 7.10 and 7.11.Screens 7.10 and 7.11.

All sAll s orbitals orbitals are spherical in shape are spherical in shape ..

1s Orbital1s Orbital

2s Orbital2s Orbital 3s Orbital3s Orbital pp Orbitals OrbitalsWhen n = 2, then l = 0 and 1Therefore, in n = 2 shell

there are 2 types oforbitals — 2 subshells

For l = 0 ml = 0 this is a s subshellFor l = 1 ml = -1, 0, +1

this is a p subshell with 3 orbitals

When n = 2, then l = 0 and 1When n = 2, then l = 0 and 1Therefore, in n = 2 shellTherefore, in n = 2 shell

there are 2 types ofthere are 2 types oforbitalsorbitals — 2 — 2 subshells subshells

For l = 0For l = 0 mmll = 0 = 0 this is a s this is a s subshell subshellFor l = 1 mFor l = 1 mll = -1, 0, +1 = -1, 0, +1

this is a this is a pp subshell subshell with with 33 orbitals orbitals

planar node

Typical p orbital

planar node

Typical p orbital

See Screens 7.11 and 7.13See Screens 7.11 and 7.13

When l = 1, there isWhen l = 1, there isaaPLANAR NODEPLANAR NODEthruthruthe nucleus.the nucleus.

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p Orbitalspp Orbitals Orbitals

A p orbitalA p orbital

pz

py

px90o

The three pThe three porbitalsorbitals lie 90 lie 90oo

apart in spaceapart in space

2p2pxx Orbital Orbital 2p2pyy Orbital Orbital

2p2pzz Orbital Orbital 3p3pxx Orbital Orbital 3p3pyy Orbital Orbital

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3p3pzz Orbital Orbital d Orbitalsdd Orbitals Orbitals

When n = 3, what are the values of l?When n = 3, what are the values of l?

l = 0, 1, 2l = 0, 1, 2and so there are 3and so there are 3 subshells subshells in the shell. in the shell.For l = 0, mFor l = 0, mll = 0 = 0 ---> s ---> s subshell subshell with single orbital with single orbitalFor l = 1, mFor l = 1, mll = -1, 0, +1 = -1, 0, +1 ---> p---> p subshell subshell with 3 with 3 orbitals orbitalsFor l = 2, mFor l = 2, mll = -2, -1, 0, +1, +2 = -2, -1, 0, +1, +2

---> ---> dd subshell subshell with 5 with 5 orbitals orbitals

d Orbitalsdd Orbitals Orbitals

ss orbitals orbitals have no planar have no planarnode (l = 0) and so arenode (l = 0) and so arespherical.spherical.

pp orbitals orbitals have l = 1, and have l = 1, andhave 1 planar node,have 1 planar node,

and so are “dumbbell”and so are “dumbbell”shaped.shaped.

This means dThis means d orbitals orbitals (with (withl = 2) havel = 2) have

2 planar nodes2 planar nodes

typical d orbital

planar node

planar node

See Figure 7.16See Figure 7.16See Figure 7.16

3d3dxyxy Orbital Orbital 3d3dxzxz Orbital Orbital 3d3dyzyz Orbital Orbital

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3d3dzz22 Orbital Orbital 3d3dxx

22- y- y

22 Orbital Orbital f Orbitalsff Orbitals OrbitalsWhen n = 4, l = 0, 1, 2, 3 so there are 4When n = 4, l = 0, 1, 2, 3 so there are 4

subshellssubshells in the shell. in the shell.For l = 0, mFor l = 0, mll = 0 = 0 ---> s ---> s subshell subshell with single orbital with single orbitalFor l = 1, mFor l = 1, mll = -1, 0, +1 = -1, 0, +1 ---> p---> p subshell subshell with 3 with 3 orbitals orbitalsFor l = 2, mFor l = 2, mll = -2, -1, 0, +1, +2 = -2, -1, 0, +1, +2 ---> d---> d subshell subshell with 5 with 5 orbitals orbitals

For l = 3, mFor l = 3, m ll = -3, -2, -1, 0, +1, +2, +3 = -3, -2, -1, 0, +1, +2, +3 ---> f---> f subshell subshell with 7 with 7 orbitals orbitals

11 22 33

No.No.Sub-Sub-shellsshells

No.No.OrbitalsOrbitals

11 22 33

11 44 99

No. e-No. e-

Shell Principal Quantum Number, nShell Principal Quantum Number, n

22 88 1818

= n= n

= n= n22

= 2 n= 2 n22

Relate to nRelate to n