Chapter 7

Quantum Theory and Atomic Structure

Announcements

--Exam 3 Oct 3

--Chapter 7/8/9/10

Chapter 7: Skip Spectral Analysis p. 226-227. Skip calculations for de Broglie and Heisenburg, conceptual understanding only.

The good thing is the material is much more descriptive. Chapter 10 will be done as a part of the lab with me.

--Quiz 6 Next Week Tuesday Chapter 6

Quantum Theory and Atomic Structure

7.1 The Nature of Light

7.2 Atomic Spectra

7.3 The Wave-Particle Duality of Matter and Energy

7.4 The Quantum-Mechanical Model of the Atom

is

having

amplitude energy frequency wavelength

related by

units calleda wave and a particle

related by

Quanta

EM radiation

are

emittedabsorbed

electrons

involve energy changes inatomsatoms

moleculesdescribed by

Wave functions e- configuration comprising

determined by

Aufbau Rules

E = hv c = !v

e- fillinggives

having quantum uunmbers

Wave Function (Orbital)

described by

described by

which are

spdf electronic configuration

Aufbau Rules

determined by

which involve

comprising

Core Electrons

Valence Electrons

Periodic Table

basis for

which summarizes

Periodic Properties

Hund’s Rule

OribitalEnergy

PauliExclusion

e- filling

Quantum Numbers

Quantum Numbers

Principaln = 1,2,3,..

Angular momentum, l

Magneticml

Spin, ms

defines

defines

defines

defines

defines

Orbital size & energy

Electron spin

Orbitalorientation

Orbitalshape

The wavelength, λ, is the crest-to-crest-distance in space.

The frequency ν, is the number of times per second that a crest passes a given point on the x-axis.

!In 1873, Maxwell found that light was an electromagnetic wave that travels at the at the speed of light, c.

! = wavelength

Electric field is perpendicular to an oscillating magnetic field both perpendicular direction to both direction of propagation

speed of light = c = ! " = 3.00 x 108 m/sec

Text

!

The frequency ν, is the number of times per second that a crest passes any given point on the x-axis. It is related to the period (sec) of the wave by f = 1/T

The wavelength, λ, is the crest-to-crest-distance in space of the waveform taken at an instant in time.

The amplitude, A of the electric field is the maximum disturbance of the waveform.

The speed of light which is a constant relates the frequency ν, and wavelength:

Amplitude

x distance

Amplitude

time

c = ! x "

The wavelength (!) of electromagnetic radiation is expressed in many different units of length...know the conversions between them.

10 Angstroms = 1 nm

!Electromagnetic radiation exists over a broad range of frequencies.

Radiation

! (nm)

" (Hz)

103 105 107 109 1011 10130.10 1010-3

X-rays UV IR Microwave Radio WavesGamma

104106108101010121014101610181020

400nm 500nm 600nm 700nm

RedOrange

YellowGreen

BlueIndigo

Violet

700 nm 430 nm525 nm 370 nm

The Visible Spectrum of Light (wavelength)Interconverting Wavelength and Frequency

A dental hygienist uses x-rays (! = 1.00A˚) to take a series of dental radiographs while the patient listens to a radio station (! = 325 cm) and looks out the window at the blue sky (!= 473 nm). What is the frequency (in s-1) of the electromagnetic radiation from each source? (Assume that the radiation travels at the speed of light, 3.00 x 108 m/s.)

Sample Problem 7.1 Interconverting Wavelength and Frequency

c = !"

A dental hygienist uses x-rays (! = 1.00A˚) to take a series of dental radiographs while the patient listens to a radio station (! = 325 cm) and looks out the window at the blue sky (!= 473 nm). What is the frequency (in s-1) of the electromagnetic radiation from each source? (Assume that the radiation travels at the speed of light, 3.00 x 108 m/s.)

10-2 m1 cm

= 325 x 10-2 m325 cm X

10-9 m1 nm = 473 x 10-9 m473 nm X

= 1.00x10-10 m1.00 A X 10-10 m1 A " =

3x108 m/s1.00x10-10 m

= 3x1018 s-1

" =

" =

3x108 m/s325 x 10-2 m

= 9.23x107 s-1

3x108 m/s473 x 10-9 m

= 6.34x1014 s-1

!Experiments over 300 years show that electromagnetic radiation (light) exhibits both wave-like and particle-like properties.

WAVE PARTICLE

Waves “bend”

Particles “fly straight”

!Experiments over 300 years show that electromagnetic radiation (light) exhibits both wave-like and particle-like properties.

WAVE PARTICLE

Experiments over 300 years show that electromagnetic radiation (light) exhibits both wave-like and particle-like properties.

Light Intensity

SlitScreen

Two Slits

Light Intensity

Screen

When two slits spaced closely together a pattern of bright and dark appears showing that light is an wave that interfers.

When light is passed through a tiny single slit one bright line appears.

It has been known since the 1600’s that light acts like a wave; it can be diffracted by passing light through small slits producing an interference pattern.

Known in 1600’s by Huygens

Interference Pattern

Wall of Screen Projection of Wall

Dark spot

Bright spot

LightWaves

FilmWith Slit

In the early 1900‘s there were mysteries involving light and matter that the physicists could not explain including:

I. Black-body Radiation Problem - the frequencies and intensities of light emitted by heated solids at a given temperature.

II. The Photo-Electric Effect - the ejection of electrons from certain metals upon exposure of light depended on the frequency of light, not its intensity (how bright).

III. Line Emission and Absorption Spectra - compounds, gases emitted only emitted or absorb very specific frequencies (colors) of light.

electric heating element

smoldering coal

light bulb filament

BLACKBODY PROBLEM: a black body is an idealized object that absorbs all electromagnetic radiation falling on it. Blackbodies absorb and incandescently re-emit radiation in a characteristic, continuous spectrum.

1000K

3000K

5000K

7000K

9000K

Classical physics predictsObserved

Wavelength (nm)0 500 1000 1500 2000 2500 3000

electric heating element

smoldering coal

light bulb filament

BLACKBODY PROBLEM: All solids heated above T > 0 Kelvin emit electromagnetic radiation. Higher T = higher frequencies (lower wavelength) and higher intensities of light. Classical physics could not model or fit these experimental observations.

1000K

3000K

5000K

7000K

9000K

Classical physics predictsObserved

Wavelength (nm)0 500 1000 1500 2000 2500 3000

Inte

nsity

E = n h "

Energy (joules) n=

wholenumber

h= Plank’s constant = 6.63 x 10-34 J s

"= frequency of light (Hz = sec-1)

Max Planck

Planck postulated that energy and matter exist in fixed quantities or “quanta” of energy and not continuous energy states.

MYSTERY #2: When a photoelectric tube is bombarded with electromagnetic radiation (light), only specific frequencies eject electrons from the surface of the light sensitive plate to generate current.

evacuated tube

light sensitive metal plate

Current meter

Light

Freed e-

+ pole

battery

Al Einstein

Einstein finds that light acts like a particle. He calls the light particles “photons” and each photon has an energy = h".

E = h "

energy (joules)

h= Plank’s constant = 6.63 x 10-34 J s

"= frequency of light (sec-1)

Albert Einstein explained the photoelectric effect in 1905.

• Light acts like a particle and are “quantized energy packets” called photons.

• 1 photon has energy = E = h".

• Photon has to have a certain minimum energy to eject an electron from the metal.

• If a photon does not have enough energy no electrons are ejected no matter how intense the light is (the number of photons per unit time)

• Greater intensity of the wrong frequency (more photons) does not overcome the electron binding energy. It is the energy (frequency) and not the intensity that is key.

0

50

incoming red light

emitter

no e- ejected

no current

ammeter

battery

current flowsbattery

incoming blue lightcollector

e- ejected

collector

+

+

Learning Check: Photon-Energy Calculations

Calculate the energy, in joules per photon, of radiation that has a frequency of 4.77 x 1014 s-1.

1.

2. If the energy of an infrared krypton laser is 2.484 x 10-19 joules per photon, determine the wavelength of the radiation.

Note: The value of Planck’s constant is 6.63 x 10-34 J/s per photon.

A cook uses a microwave oven to heat a meal. The wavelength of the radiation is 1.20 cm. What is the energy of one photon of this microwave radiation?

3.

MYSTERY #3: Narrow bands of colors of light are emitted when gases are excited by high voltage. The colors are characteristic and reproducible for different elements.

The light generated by the H2 gas is dispersed into its component wavelengths by a prism and projected on a screen

High voltagetube containinghydrogen gas

Prism

Screen

All elements display a characteristic emission spectra that we can use to fingerprint all elements.

LiNaK

CaSrBaZnCdHgH

HeNeAr

gases

Metals

alkali metals

alkalineearth

Hydrogen Helium Lithium Sodium Potassium

Emission spectra of elements

Flame color tests cause a similar electron excitement and are characteristic for elements. The color says something tells us electronic structure E = h" of the elements and can help identify elements.

The beauty and intriguing color of fireworks arises from the characteristic color of elements in their salts.

How could scientists account for the lines if atomic theory and classical physics were correct?

How does the emission spectrum relatethe atomic and electronic structure of the H atom?

! (nm)for the visible series, n1 = 2 and n2 = 3, 4, 5, ...

The first explanations of the emission spectra were empirical (i.e. found an equation that worked but without any explanation behind it).

= RRydberg equation -

1!

1n2

2

1n1

2

In 1885 Johann Rydberg developed a generalized but empirical mathematical relationship between " and n for the all lines of hydrogen emission spectrum! This is freaky!

In 1885 Balmer developed an empirical mathematical relationship between " and n for the visible lines of hydrogen emission spectrum.

λ (nm) = 364.56�

n2

n2 − 22

�Balmer equation

In 1913, scientist Neils Bohr uses a planetary-quantum mathematical model that explains the emission spectrum of the hydrogen atom.1. e- can only have specific

(quantized) energy values (planetary like orbits)

2. light is emitted as e- moves from one energy level to a lower energy level

3. the energy of a level:

4. the difference in energy between any two levels:

En = -RH( )1n2

RH (Rydberg constant) = 2.18 x 10-18J

#E = RHi f

1n2

1n2 )(

i = initial e- state and f = final e- state.

Bohr postulates that light is emitted when an e- goes from an orbit with a high n value to a lower n value

n =6n =5n =4n =3

n =2

The Bohr model explains the H atomic spectral lines as “quantized jumps” between the various n “energy levels”.

RH (Rydberg constant) = 2.18 x 10-18Ji f

#E = RH( )1n2

1n2

n =6

n

How and why is e- energy quantizedand what physics can explain it?

The Bohr’s model works for hydrogen, but could not explain the emission spectra of other atoms. Classical physics also says that electrons can not accelerate around a nucleus without eventually collapsing into the nucleus (i.e. simple planetary motion of e- around a nucleus fails).

In 1924, Louis DeBroglie reasoned that if light wave could act like a particle then perhaps a particle could act like a wave. He connects Plank’s Law to Einstein’s energy-mass equivalence equation.

! = wavelength (m)v = velocity (m/s)h = Planck’s constant (6.626 $ 10-34 J-s)

Particles with mass m and velocity v are also waves with wavelength! The wave-like properties only manifest when mass is small and high velocity:

e- that move at high speeds.

Louis DeBroglie

Plank’s Law Einstein’s Law of Mass equivalence

Solving for wavelength, substituting v = velocity for c and watching our units.......we get:

λ =h

mv

E = hν = hc

λ= mc2

The de Broglie Wavelengths of Selected Matter

λ =h

mv

Substance Mass (g) Speed (m/s) de Broglie ! (m)

slow electron

fast electron

alpha particle

one-gram mass

baseball

Earth

9 x 10-28

9 x 10-28

6.6 x 10-24

1.0

120

6.0 x 1027

1.0

5.9 x 106

1.5 x 107

0.01

25.0

3.0x104

7x10-4

1x10-10

7x10-15

7x10-29

1.2x10-34

4x10-63

Soon after de Broglie’s theory was put forth it was found that e-

particles (which were thought to have only particle-like properties) could be made to diffract and interfere just like light waves.

x-ray diffraction of aluminum foil electron diffraction of aluminum foil

What is the de Broglie wavelength (in meters) of: a) pitched baseball with a mass of 120. g and a speed of 100 mph or 44.7 m/s and b) of an electron with a speed of 1.00 x 106 m/s (electron mass = 9.11 x 10-31 kg; h = 6.626 x 10-34 kg m2/s).

λ =h

mν

DeBroglie wavelength (m)

Plank’s Constant = h = 6.63 x 10-34 J s

mass of object (kg)

velocity of object (m/s)

λ =h

mν=

6.626× 10−34 kg m2/s

(0.120 kg) (44.7 m/s)= 1.24× 10−34 m

! = 6.626 x 10-34 kg m2/s(9.11x10-31 kg)(1.00x106 m/s)

= 7.27 x 10-10 m

Erwin Schrodinger, an Austrian physicist puts forth an equation that describes position and time dependence of an electron in hydrogen. We call this the Schrodinger wave equation.

Schrodinger equation is a difficult partial differential equation

how % changes in space

constants

total quantized energy of the atomic system

potential energy at x,y,zwave function

−h2

8π2me

�∂2Ψ(x)

∂x2+

∂2Ψ(y)∂y2

+∂2Ψ(z)

∂z2

�− Ze2

rΨ(x, y, z) = EΨ(x, y, z)

Erwin Schrodinger

Schrodinger’s equation is easier to solve if we transform the x,y,z coordinate system to spherical coordinates: r, #, $.

The solution to this equation gives rise to 3 “quantum numbers” associated with an electron’s energy levels called orbitals.

n principalquantum number

l orbitalquantum number

ml magneticquantum number

"(r,#,$) = R(r) P(#) F($)

"(x,y,z) 1) change of coordinates2) solve the differential equation

A solution to the Schrodinger wave equation, "(x,y,z) is called an a wavefunction or an atomic orbital. It’s a mathematical function that describes an electron in an H atom. It has no physical meaning except when it is squared |"(x,y,z)|2 (like the square of electric field in light).

" (!) =2Ln

where n = 1, 2, 3

The wavefunctions are quantized “allowed states” of an electron like modes of a vibrational string of a guitar.

Wave Motion Is Quantized In Space

Guitar strings (a) can only vibrate at certain wavelengths (frequencies). If we apply this situation to to electrons around a nucleus (b) then only certain whole number wavelengths would be allowed--it is quantized. Non-allowed “states” can not be observed.

! =2Ln

where n = 1, 2, 3 n l m #n,m,l

A wavefunction is a solution to the Schrodinger equation and has “3-quantum numbers” that describe electrons in space.

What does the Schrodinger Equation Tell Us?

1. Electrons in atoms, elements and molecules exist in discrete quantized “states” that are mathematically described by a “wavefunction” or “orbital”. The wavefunction can not be observed in a lab!

2. The solutions of the Schrodinger equation for the hydrogen atom is a list of functions called “wavefunctions” or “orbitals” that describe the spatial location of an electron in a hydrogen atom.

3. Each wavefunction solution gives rise to a function having as variables called 3-quantum numbers: n, l, m.

4. The wavefunction "(r,#,$) by itself is not a physically measurable quantity, but the square of the wavefunction is.