l15 chapt9-2 web

Chemistry 5 Chapter-9 Electrons in Atoms Part-2 28 October 2002

Wave-Particle Duality ƒ If light energy has particle-like properties, does matter have wave-like properties? Rem: photoelectric effect where individual

photons must have energy >threshold to observe electron ejected from metal surface.

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

photon:

E(photon) = hν and c = νλ

•

matter:

E = mc2 hν = mc2 hν/c = mc But c = νλ, and mc is momentum, p h/λ = p Now for particle of mass, m, and velocity, u:

h h λ= = mu p

de Broglie wavelength

Does electron exhibit wavelike properties? ƒ G.P. Thomson passed an electron beam through a sheet of gold foil and observed intensity vs. deflection angle: θ electron gun gold foil

ƒ 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 called diffraction and is due to the wavelike properties of electrons.

Viewing Atoms and Electrons: Tunneling Microscope

ion pump

sample preparation

sample transfer

sample storage

I

ion pump

Vb

optical table air legs

UHV

vacuum can

Au

dewar

Ground

LHe

STM microscope

9-11 T Solenoid

Electrons Waves & Interference! -0.447 eV -0.414 eV

(dI/dV)/(I/V) (a.u.)

-0.384 eV

El Ek Ej Ei

Reikx x

eikx

-0.293 eV 0.171 eV 0.231 eV

Teikx

0.311 eV

0 0.368 eV 0.415 eV

Au

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

0.0

1.0

2.0 Distance (nm)

3.0

Ouyang, Huang, and Lieber, Science

Wave-Particle Duality: Summary ƒ Mass-Wavelength: • MASS INCREASES Æ Wavelength gets shorter • MASS DECREASES Æ Wavelength gets longer

ƒ How important is this? • What are the “de Broglie wavelengths” of a 0.10 kg baseball moving at 35 m/s and an electron moving at 1.0 x 107 m/s?

h λ= mv Baseball

h = 6.626 x 10-34 J s 1J = kg m2 s-2 Electron

− 34

6.626 × 10 Js λ= (0.10kg )( 35m / s ) = 1.9 x 10-34 m

6.626 × 10− 34 Js λ= (9.11 × 10− 31 kg )(1 × 107 m / s )

= 7.3 x 10-11 m

More massive particle– baseball– has immeasurably small wavelength!

The Uncertainty Principle ƒ Laws of classical physics enable precise predictions of position and velocity………however, you cannot pin an electron down! ƒ Heisenberg postulated the following: • •

then:

∆x is the uncertainty in the particle’s position ∆p is the uncertainty in the particle’s momentum

h ∆x∆p ≥ 4π

Effect of mass included in momentum:

∆p = m ∆v

ƒ 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 any meaningful precision at the same time!

The Uncertainty Principle: Examples ƒ What is the uncertainty in velocity for baseball and hydrogen atom, if we assume that position is known to 1%? ƒ Baseball:

Size ~ 0.5 m Î ∆x =

1 × 0.05m = 5 × 10−4 m 100

h 1 1 ∆v ≥ × × 4π m ∆x

= 1 x 10-30 m/s

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

ƒ Hydrogen Atom: Assume we estimate position to 1% of radius of H-atom, then

1 ∆x = × 0.05nm = 5 × 10− 4 nm = 5 × 10−13 m 100

h 1 1 ∆v ≥ × × ∆v = 1.15 × 108 m / s 4π m ∆x The uncertainty in the velocity is enormous!!

Quantum Mechanics ƒ We cannot know precisely where electrons are! ƒ This means we cannot describe the electron as following a known path such as a circular orbit. ƒ What about Bohr Model? Bohr’s model is therefore fundamentally incorrect in its description of how the electron behaves. ƒ Wave Description– Look at demonstration first: • 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

Was Bohr upset with Heisenberg? BOHR

HEISENBERG

AND THIS IS?? Niels Bohr and Werner Heisenberg dining (1934)

Wave Functions ƒ Schrodinger equation: • 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

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

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.

Quantum Numbers (beyond Bohr model) ƒ Quantum Numbers (are part of orbital/wave function description) •

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

• orbital angular momentum, l -- zero or positive integer with values l = 1, 2, 3, …, n-1 •

magnetic, ml -- negative or positive integer, including zero with values ml = 0, ±1, ±2, ±3, …±l

ƒ Principle Shells and Subshells • 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

s orbitals l=0 ml = 0 one s orbital in s subshell

p orbitals l=1 ml = 0, ±1 three p orbitals in p subshell

d orbitals l=2 ml = 0, ±1, ±2 five d orbitals in d subshell

Wave functions, orbitals & probabilities ƒ Wave functions and probabilities: • Wave function itself does not have physical significance, although sign (+/-) tells

us 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!

ƒ s-orbitals

n-1 radial nodes