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Chemistry 5

Chapter-9

Electrons in Atoms

Part-2

28 October 2002

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Wave-Particle Duality

If light energy has particle-like properties, does matterhave wave-like properties?

de Broglie: Small particles of matter may display wavelike properties!

• photon:

• matter:

Rem: photoelectric effect where individual photons must have energy >threshold to observeelectron ejected from metal surface.

E(photon) = hν and c = νλ

E = mc2

hν = mc2

hν/c = mc

But c = νλ, and mc is momentum, ph/λ = p

Now for particle of mass, m, and

velocity, u:

h hmu p

λ = = de Broglie wavelength

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Does electron exhibit wavelike properties?

θ

gold foil

electron gun

G.P. Thomson passed an electron beam through a sheet of gold foil and observed intensity vs. deflection angle:

Key Observations & Implications:

•

•

“Diffraction Pattern”

As detector is moved the intensity of transmitted electron varies(similar behavior is observed when X-ray source is used)

These results imply that the electrons (X-rays) interfere when they pass through the gold film. This interference phenomena is calleddiffraction and is due to the wavelike properties of electrons.

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Viewing Atoms and Electrons: Tunneling Microscope

optical table

dewar

9-11 T

Solenoid

Ground

air legs

LHe

v a c u u m

c a n

UHV

STM

microscope

ion

pump

ion

pump

sample

storage

s a m

p l e

p r e p a r a t i o n

sample

transfer

I

Au

V b

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-0.447 eV

-0.414 eV

-0.384 eV

-0.293 eV

0.171 eV

0.231 eV

0.311 eV

0.368 eV

0.415 eV

0.0 1.0 2.0 3.0Distance (nm)

( d I / d V ) / ( I / V ) ( a . u . )

Electrons Waves & Interference!

Ouyang, Huang, and Lieber, Science

Au

eik x

Reik x

x

Teik x

Ei

E j

Ek

El

0

ψ(k ,x)=e-ik x+|R|e-i(k x+δ)ρ(k ,x)= |ψ(k ,x)|2=1+|R|2+2|R|cos(2k x+δ)

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Wave-Particle Duality: Summary

Mass-Wavelength:• MASS INCREASES Wavelength gets shorter

• MASS DECREASES Wavelength gets longer

How important is this?

• What are the “de Broglie wavelengths” of a 0.10 kg baseballmoving at 35 m/s and an electron moving at 1.0 x 107 m/s?

vm

h=

1J = kg m2 s-2

h = 6.626 x 10-34 J s

)/35)(10.0(10626.6 34

smkg Js×=

= 1.9 x 10-34 m

Baseball Electron

)/101)(1011.9(10626.6

731

34

smkg Js×

×=−

−

λ

= 7.3 x 10-11 mMore massive particle– baseball– has immeasurably small wavelength!

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The Uncertainty Principle

Laws of classical physics enable precise predictions of

position and velocity………however, you cannot pin an

electron down!

Heisenberg postulated the following:

• ∆ x is the uncertainty in the particle’s position

• ∆ p is the uncertainty in the particle’s momentum

then:∆ ∆ x p

h≥

4π

Implications?

• there is a fuzziness intrinsic to all small things

• or put another way– for particle like an electron

we can never know both position and velocity to anymeaningful precision at the same time!

vm p

Effect of mass included in momentum:

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The Uncertainty Principle: Examples

What is the uncertainty in velocity for baseball andhydrogen atom, if we assume that position is known to 1%?

Baseball:

Hydrogen Atom:

x m

h

∆×

11

4v

π= 1 x 10-30 m/s

mm x

4

10505.0100

1 −×

Size ~ 0.5 m

Can play baseball without worrying about Heisenberg’s Uncertainty Principle!

Assume we estimate position to 1% of radius of H-atom, then

mnmnm x 134

10510505.0100

1 −×

smv /1015.18

× x m

h

∆×

11

4v π

The uncertainty in the velocity is enormous!!

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Quantum Mechanics

We cannot know precisely where electrons are!

This means we cannot describe the electron asfollowing a known path such as a circular orbit.

What about Bohr Model?

Wave Description– Look at demonstration first:

•

•

•

•

Bohr’s model is therefore fundamentally incorrect

in its description of how the electron behaves.

Waves have amplitude that depends on position

Wavefunctions can have different shapes– defined by nodes– between boundaries

Can identify well-defined/repeatable nodal structures

The number of nodes depends on energy added to system

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Niels Bohr and Werner Heisenberg dining (1934)

BOHR

HEISENBERG

AND THIS IS??

Was Bohr upset with Heisenberg?

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Wave Functions

Schrodinger equation:

Description of Orbital (vs. Bohr’s orbits):

•

•

•

• Schrodinger showed that functions– now called wave functions– describing a quantum system can be obtained by solving a wave equation.

• These wave functions are called orbitals

ψ(r, θ, φ) = R(r)Y(θ, φ)

radial wave function – R(r) – depends only on distance from nucleus

angular wave function – Y(θ, φ)– depends on

angular part of polar coordinates

Each wave function has three numbers,

called quantum numbers that define the

general functional form of R(r) and

Y( θ , φ )– and hence an orbital shape.

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Quantum Numbers (beyond Bohr model)

Quantum Numbers (are part of orbital/wave function description)

• principle, n

• orbital angular momentum, l

• magnetic, ml

Principle Shells and Subshells

•

•

s orbitals p orbitals d orbitals

-- positive, nonzero integer values n = 1, 2, 3, …

-- zero or positive integer with values l =1, 2, 3, …, n-1

-- negative or positive integer, including zero with values

ml = 0, ±1, ±2, ±3, …±l

l = 0

ml = 0one s orbital ins subshell

l = 1

ml = 0, ±1three p orbitalsin p subshell

Orbitals with the same value of n, are in same principle electronic shell

Orbitals with the same values of n and l , are in same subshell

l = 2

ml = 0, ±1, ±2five d orbitals ind subshell

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Wave functions, orbitals & probabilities

Wave function itself does not have physical significance, although sign (+/-) tellsus about phase and is important for describing chemical bonds.

The square of the wavefunction, ψ2, has physical meaning as electron probability

density (or charge density)and is used to describe shape…….lets examine this point!

Wave functions and probabilities:•

•

s-orbitals

n-1 radial nodes

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