chapter 12 handouts

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Chapter 2: Quantum Mechanics and Atomic Theory

Chapter 12: Phenomena

Phenomena: Different wavelengths of electromagnetic radiation were directedonto two different metal sample (see picture). Scientist then record if anyparticles were ejected and if so what type of particle as well as the speed of theparticle. What patterns do you see in the results that were collected?

KExp.

Wavelength of LightDirected atSample

IntensityWhere Particles

EjectedEjectedParticle

Speed ofEjected Particle

1 5.410-7 m Medium Yes e- 4.9104

2 3.310-8 m High Yes e- 3.5106

3 2.0 m High No N/A N/A

4 3.310-8 m Low Yes e- 3.5106

5 2.0 m Medium No N/A N/A

6 6.110-12 m High Yes e- 2.7108

7 5.5104 m High No N/A N/A

FeExp.

Wavelength of LightDirected atSample

Where ParticlesEjected

EjectedParticle

Speed ofEjected Particle

1 5.410-7 m Medium No N/A N/A

2 3.410-11 m High Yes e- 1.1108

3 3.9103 m Medium No N/A N/A

4 2.410-8 m High Yes e- 4.1106

5 3.9103 m Low No N/A N/A

6 2.610-7 m Low Yes e- 1.1105

7 3.410-11 m Low Yes e- 1.1108

EjectedParticle

ElectromagneticRadiation

Chapter 12:QuantumMechanics andAtomic Theory

o Electromagnetic

Radiation

o Quantum Theory

o Particle in a Box

o The Hydrogen Atom

o Quantum Numbers

o Orbitals

o Many Electron Atoms

o Periodic Trends

2

Big Idea: The structure of atomsmust be explainedusing quantummechanics, a theory in

which the properties of

particles and wavesmerge together.


Electromagnetic Radiation

Electromagnetic Radiation:Consists of oscillating(time-varying)electric and magnetic fields thattravel through space at 2.998 108

(c = speed

of light) or just over 660 million mph.

3

Note All forms of radiation transfer energy from one region of space into another.

Examples of Electromagnetic Radiation

Visible light

Radio Waves

X-Rays



Wavelength (): Is thepeak-to-peak distance.

Amplitude: Determinesbrightness of theradiation.

Frequency (): Thenumber of cycles persecond (1 1

).

4

Note

Changing the wavelength changes the

region of the spectrum (i.e. x-ray to visible).



5Chapter 2: Quantum Mechanics and Atomic Theory

Quantum Theory

If white light is passedthrough a prism, acontinuous spectrumof light is found.However, when the

light emitted byexcited hydrogenatoms is passedthrough a prism theradiation is found toconsist of a numberof components orspectra lines.

6


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Quantum Theory

Black Body: An object that absorbs and emits allfrequencies of radiation without favor.

Heated Black Bodies

. .

7

Note max is the most prevalent wavelength, not the longest wavelength.


Quantum Theory

Theory

Scientists used classical physics arguments toderive an expression for the energy density

(Rayleigh Jens Law).

The Problem With Explaining Black Body RadiationUsing Classical Mechanics

Classical mechanics put

no limits on minimum

wavelength therefore at

very small , would be a

huge value even for low temperatures.

8


Quantum Theory

Max Plank: Proposed that the exchangeof energy between matter and radiationoccurs in quanta, or packets of energy.His central idea was that a chargedparticle oscillating at a frequency , canexchange energy with its surroundings bygenerating or absorbing electromagneticradiation only in discrete packets ofenergy.

9

Quanta: Packet of energy (implies minimumenergy that can be emitted).

Charged particles oscillating at a frequency, ,can only exchange energy with theirsurroundings in discrete packets of energy.


Quantum Theory

Photoelectric Effect:Ejection of electrons from ametal when its surface is exposed to ultra violetradiation.

10


Quantum Theory

Findings of the Photoelectric Effect

1) No electrons are ejected unless the radiation has afrequency above a certain threshold valuecharacteristic of the metal.

2) Electrons are ejected immediately, however low theintensity of the radiation.

3) The kinetic energy of the ejected electronsincreases linearly with the frequency of the incidentradiation.

Albert Einstein

Proposed that electromagnetic radiation consistsof massless particles (photons)

Photons are packets of energy

Energy of a single photon =


Quantum Theory

Photoelectric Effect Electromagnetic radiation consists of streams of photons

traveling with frequency, , and energy .

Photons collide with metal.

If the photons have enough energy, electrons will be

removed from the metal.

Work Function ():The minimum amount ofenergy required to remove an electron from thesurface of a metal.

If no electron ejected

If electron will be ejected

12

Note This relationship is not intensity dependent.


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Quantum Theory

With what speed will the fastest electronsbe emitted from a surface whose

threshold wavelength is 600. nm, whenthe surface is illuminated with l ight ofwavelength 4.010

7?


Quantum Theory

H atoms only emit/absorb certainfrequencies.

What is causing these properties?

Electrons can only exist withcertain energies.

The spectral lines are transitionsfrom one allowed energy level toanother.

14


Quantum Theory

Finding energy levels of H atom

3.29 10

(Found experimentally)

3.29 10

6.626 10

3.29 10

-2.178 10

=

Energ y leve l of H ato m

2.178 10

n= 1, 2, 3,

Energ y level of othe r 1e-system s

2.178 10

n= 1, 2, 3,

15

Note In this equation n2 is always the large number which corresponds to the

higher energy level.

Note

The negative sign appeared because a negative sign was taken out of

parenthesis.


Student Question

Quantum Theory

What is the wavelength of the radiationemitted by a 2

atom when an electron

transitions between 4 to the 2 levels?

Help ful Hint: c = 2.998108

a ) -1.6641024m

b ) -5.39910-8m

c ) 1.62110-7m

d ) 4.86410-7m

e )None o f t he Above

16


Quantum Theory

ConstructiveInterference: When thepeaks of wavescoincide, the amplitudeof the resulting wave is

increased.

Destructive Interference:When the peak of onewave coincides with thetrough of another wavethe resulting wave isdecreased.


Quantum Theory

When light passes

though a pair of closely

spaced slits, circularwaves are generated at

each slit. These waves

interfere with each

other.

Where they interfereconstructively, a bright

line is seen on the screen

behind the slits; where

the interference is

destructive, the screen is

dark.

18


4/10


Student Question

Quantum Theory

What is the wavelength of an electron travelingat

the speed of light?

Helpful informa t ion: 9.109 10 a n dh = 6.62610-34Js

a ) 2.4210-9m

b ) 2.4210-12m

c ) 4.121011m

d )None o f t he Above


Quantum Theory

Particles have wave like properties, therefore,classical mechanics are incorrect.

Classical Mechanics:

Particles have a defined trajectory.

Location and linear momentum can bespecified at every moment.

Quantum Mechanics:

Particles behave like waves.

Cannot specify the precise location of aparticle.

20

Note For the hydrogen atom, the duality means that we are not going to be able

to know the speed of an electron orbiting the nucleus in a definite trajectory.


Quantum Theory

Wavefunction ():A solution of the Schrdingerequation; the probability amplitude.

Probability Density ():A function that, whenmultiplied by volume of the region, gives theprobability that the particle will be found in thatregion of space.

21

Note This gives you a representation of the particles trajectory.

Examples

sin

Note This value must be between 0 and 1


Particle in a Box

Particle in a 1-D Box

Boundary Conditions

0 0

0

What is the wavefunction?

sin sin0 0 sin 0

22

Note The term

is there for normalization in order to get 1

Note n is a quantum number. Quantum numbers are integers or integers

that label a wavefunction.


Take 1st derivative of

Take 2nd derivative of

Particle in a Box

Get the allowed energy levels from Schrdingers equation

23

What are the allowed energy levels?

Note V(x)=0 for the particle in a box


Particle in a Box

Plug 2nd derivative back into wavefunction

sin

sin

Cancel out

Simplify

24


5/10


The Hydrogen Atom

Atomic Orbitals:A regionof space in which there isa high probability offinding an electron in an

atom.

The wavefunction of an electron is given inspherical polar coordinates

= distance from the center of the atom

= the angle from the positive z-axis, which can bethought of as playing the role of the geographicallatitude

= the angle about the z-axis, the geographicallongitude

25

Note The wavefunctions of electrons

are the atomic orbitals.


The Hydrogen Atom

Wavefunction of an electron in a1-electron atom

,, ,

Radial Wavefunction

, Angular Wavefunction

26


The Hydrogen Atom

General wavefunction expression for a hydrogenatom:

27

+1

-1

0

+2

-1

+1

-2

0


The Hydrogen Atom

Schrdinger Equation for the H-atom

sin

Columbic potential energy between the protonand electron

Reduced mass (proton and electron)

Scientists have solved for the energy of the H-atom

.

3.290 10

principle quantum number can be 1, 2, 3,

planks constant 6.6260810-34Js

28


The Hydrogen Atom

Pictured: Thepermitted energylevels of ahydrogen atom.

The levels arelabeled with thequantum number which rangesfrom 1 (groundstate) to .


Quantum Numbers

Principle Quantum Number

Rela ted To: Size and Energy

Al low ed Values: 1,2,3, (shells)

30

Note The larger the , the larger the size, the larger the orbital, and the larger theenergy.

Note When identifying the orbital, the value comes in front of the orbital type.

Example

3p 3


6/10


Quantum Numbers

Angular MomentumQuantum Number ()

Relate d To:Shape

Al low ed Values: 0,1,2, , 1 (subshells)

What are the possible valuesgiven the following values?

1 1

2 2

3 3

What type of orbital is associated withthe following quantum numbers?

1 1 and 0

2 2 and 13 3 and 0

31

Valueof l

OrbitalType

0 s

1 p

2 d

3 f


Quantum Numbers

Magnetic Quantum Number ()

Rela te d To: Orientation in space

(specifies exactly which orbital [ex: ])Al low ed Va lues: , 1, ,0 , , (orbitals)

What are the possible values of given thefollowing and values:

1 1 and 0

2 2 and 1

3 2 and 0

32


Quantum Numbers

33

Note The quantum number ml is an alternative label for the individual orbitals: inchemistry, it is more common to use x, y, and z instead.


Quantum Numbers

A summary of the arrangements of shells,subshells, and orbitals in an atom and the

corresponding quantum numbers.

34


Student Question

Quantum Numbers

The following set of quantum numbers is notallowed: 3, 0, 2. Assuming thatthe and values are correct, change the value to an allowable combination.

a) 1

b) 2

c) 3

d) 4

e) None of the Above


Quantum Numbers

Electron Spin Quantum Number()

Rela te d To:Spin of the electron

Al low ed Va lues: (spin up ) or - (spindown )

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7/10


Orbitals

What does an s-orbital ( 0) look like?

1 and 0

(radial wavefunction)

,

(angular wavefuction)

, ,

, ,


Orbitals

The three s-orbitals ( ) of lowest energy

The simplest way ofdrawing an atomic

orbital is as a boundarysurface, a surface within which there is a high probability

(typically 90%) of finding the electron. The darker theshaded region within the boundary surfaces, the larger the

probability of the electron being found there.

38


Orbitals

What do p-orbitals ( 1) look like?


Orbitals

What do d-orbitals ( 2) look like?

40


Many Electron Atoms

The electrons in many-electron atomsoccupy orbitals like those of hydrogenbut the energy of the orbitals differ.

Differences between the many-electronorbital energies and hydrogen atomorbital energies:

The nucleus of the many-electron orbitals havemore positive charge attracting the electrons,thus lowering the energy.

The electrons repel each other raising theenergy.


Many Electron Atoms

The number of electrons in an atom affects theproperties of the atom.

42

Note H with 1 electron has no electron-electron repulsion and the electron will have

the same energy if is the 2s or 2p orbital (all orbitals within one shell degenerate).

Potential Energy for He atom (2 p and 2 e-)

Attraction of

electron 1 to

the nucleus

Attraction of

electron 2 to

the nucleus

Repulsion

between the

two electrons

2

4

2

4

4

1 = distance between electron 1 and the nucleus

2 = distance between electron 2 and the nucleus

12 = distance between the two electrons


8/10


Many Electron Atoms

Why do the energies change?

Shielding: The repulsion experienced byan electron in an atom that arises from

the other electrons present andopposes the attraction exerted by thenucleus.

Effective Nuclear Charge (Zeff):The netnuclear charge after taking intoaccount the shielding caused by otherelectrons in the atom.

43

Amount of Shielding

Distance from nucleus

Note s electrons can penetrate through the nucleus while p and

d electrons cannot.

S P D F


Many Electron Atoms

44

Order of Orbital Filling

1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p

7s 7p 7d 7f

6s 6p 6d 6f

5s 5p 5d 5f

4s 4p 4d 4f

3s 3p 3d

2s 2p

1s


Many Electron Atoms

Electron Configuration: A list of an atomsoccupied orbitals with the number of electronsthat each contains.

In the ground state the electrons occupy atomic orbitals

in such a way that the total energy of the atom is aminimum.

We might expect that atoms would have all their

electrons in the 1s orbital however this is only true for Hand He.

Pauli Exclusion Principle: No more than twoelectrons may occupy any given orbital. Whentwo electrons occupy one orbital, the spins mustbe paired.

45

Note Paired spins means and electron ()


Many Electron Atoms

Writing Electron Configurations

Step 1:Determine the number of electrons theatom has.

Step 2: Fill atomic orbitals starting with lowestenergy orbitals first and proceeding to higherenergy orbitals.

Step 3:Determine how electrons fill the orbitals.

46

Subshell lSmallest

ns 0 1p 1 2d 2 3f 3 4


Many Electron Atoms

Electron Configurations: He Step 1:

Step 2:

Step 3:

Be Step 1:

Step 2:

Step 3:


Many Electron Atoms

Electron Configurations: C Step 1:

Step 2:

Step 3: __ __ __ 2p __ __ __ 2p __ __ __ 2p

2s 2s 2s

1s 1s 1s

48

Note Electrons that are farther apart repel each other less.

Note Electrons with parallel spins tend to avoid each other more.


9/10


Periodic Trends

Effective NuclearCharge:(Zeff) The netnuclear charge aftertaking into accountthe shielding causedby other electrons inthe atom.

Why:Going across the periodic table, the numberof core electrons stays the same but the number

of protons increases. The core electrons are

responsible for most of the shielding, therefore the

Zeff gets larger as you go across a period.

Although going down a group adds more coreelectrons it also adds more protons therefore Zeff is

pretty much constant going down a group.


Periodic Trends

Atomic Radii:Half the distancebetween the centers

of neighboring atomsin a solid of ahomonuclearmolecule.

Why:Going across a period the Zeff increases,therefore the pull on the electrons increases andthe atomic radii decrease. Going down a group,

more atomic shells are added and the radii

increase.

50


Periodic Trends

First Ionization Energy:The minimum energyrequired to remove thefirst electron from theground state of agaseous atom,molecule, or ion.

X(g) X+(g) +e-

Why:Going across a period the Zeff increasestherefore it is harder to remove an electron and

the first ionization energy increases. However,

going down a group the electrons are located

farther from the nucleus and they can be

removed easier.


Periodic Trends

1st Ionization Energy Data

Why dont B and Al fit the 1st ionization energy trend?

52

H

He

Li

BeB

C

NO

F

Ne

Na

MgAl

Si

P S

Cl

Ar

KCa

0

500

1000

1500

2000

2500

0 1 2 3 4 5 6 7 8 9 1 0 11 12 13 14 15 16 17 18 19 20

Ionizationenergy,kJmol-1

Atomic Number

Ionization Energies of the First 20 Elements


Periodic Trends

Electron Affinity: (Eea)The energy releasedwhen an electron isadded to agas-phase atom.

X(g) + e- X-(g)

Why:Going across a period the Zeff increasestherefore the atom has a larger positive charge

and releases more energy when an electron is

added to the atom. Going down a group theelectron is added farther from the nucleus and the

electron affinity decreases. This trend does not holdtrue for the noble gases.

53

Note This trends has the most atoms that do not obey it.


Periodic Trends

54

H

He

Li

BeB

C

N

O

F

Ne

Na

Mg

Al

Si

P

S

Cl

Ar

K

Ca

-50

0

50

100

150

200

250

300

350

400

0 1 2 3 4 5 6 7 8 9 1 0 11 12 13 14 15 16 17 18 19 20

EA(kJ/mole)

Atomic Number

Electron Affinity of the First 20 Elements


10/10


Take Away From Chapter 12

Big Idea: The structure of atoms must be explainedusing quantum mechanics, a theory in which theproperties of particles and waves merge together.

Electromagnetic Radiation (22)

Be able to change between frequency and wavelength

Be able to assign what area of the electromagneticspectrum radiation comes from.

55

Numbers correspond to end of chapter questions.



Quantum Theory

Know that electromagnetic radiation has both wave andmatter properties. (30)

Photo electric effect (particle property) (27, 28, 29)

Know what the work function of a material is Diffraction (wave property)

Be able to calculate the energy carried in an

electromagnetic wave (21, 25)

Know that matter has both wave and matter properties.

Be able to calculate the wavelength of matter. (32, 34, 35)

Know that the Heisenberg uncertainty principle determines

the accuracy in which we can measure momentum andposition.

56




Particle in a Box

Know that the wavefunction of different systems can be

found using boundary conditions.

Particle in a box wavefunction

sin

Know that the square of the wavefunction tells us the

probability of the particle being in a certain location

Know the most likely location of a particle in a box (ex: n = 1

particle most likely in center of box, n = 2 particle most likely onsides of box)

Know that the Schrdinger equation allows us to matchwavefunctions with allowed energy values.

Particle in a box

(53, 54, 55, 57)

The Hydrogen Atom

Know that when quantum mechanics is applied to 1

electron systems the observed energy and the theoretical

energies match. (38, 40, 41, 42, 44, 46, 47, 141) 2.178 10

57




Quantum Number (58,62,64,65,66, 74,75) Know that the quantum number gives the size and

energy

Allowed values of 1,2,

Know that the quantum number l gives the orbital type(shape)

Allowed values of 0 ,1 ,2 , 1

Know that the quantum number gives the orbitalorientation (ex: ) Allowed values of ,0,

Know that a fourth quantum number was needed to have theory

match experiment, Allowed values of

58




Orbitals Know that when atomic orbitals are drawn they represent the

area in which e- have a 90% chance of being in. (67)

Know the what s, p, and d orbitals look like.

Many Electron Atoms

Be able to write both the full and shorthand electronconfigurations of atoms.(76, 77, 80, 84, 92)

Be able to predict the number of unpaired electrons in asystem.(90, 91)

Periodic Trends Know the trends of effective nuclear charge, atomic radii,

ionization energy, and electron affinity. (13, 14, 96, 97, 98, 100, 106,

111, 112, 133, 136)

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