18

Quantum Theory and Atomic Structure

19 Test Content from AP Chemistry Course Description I. Structure of Matter (20%)

A. Atomic theory and atomic structure 1. Evidence for the atomic theory 2. Atomic masses; determination by chemical and physical means 3. Atomic number and mass number; isotopes 4. Electron energy levels: atomic spectra, quantum numbers, atomic orbitals 5. Periodic relationships including, for example, atomic radii, ionization

energies, electron affinities, oxidation states THE NATURE OF LIGHT Light is a crossed electric and magnetic field that is oscillating in time. Changing electric field creates a magnetic field and a changing magnetic field creates an electric field, etc…

B Et

∂= −∇×

∂ ( )2E c B

t∂

= ∇×∂

Light is self-propagating electromagnetic field. Electric field: ( )0E E sin kz t= −ω + δ

Magnetic field: ( )0H H sin kz t= −ω + δ y

xz

- Light is very peculiar in that it is a particle and a wave at the same time. - Our physical intuition tells us that this is impossible. - Waves are spread out as in ocean waves - Particles are in one place (localized) as in a bowling ball. Q: How can something be spread out and localized at the same time? A: Who knows? It is a mystery of nature.

Supplemental Material

20 THE WAVE NATURE OF LIGHT - Light moves at a constant speed through a particular medium. - c – speed of light in a vacuum (≈ air) - c = 2.997 x 108 m/s (670,000,000 mi/hr) - Waves have two components. - wavelength - λ (Greek lambda) - distance between wave crests - frequency - ν (Greek nu) - how often wave crest moves up and down at a single point - number of beats per second: Hertz – Hz - 1 Hz = 1 /s = 1 s-1 - think of a boat bouncing up and down on waves - Frequency and wavelength are related

ν ⋅ λ = c - if we know ν, we can calculate λ - if we know λ, we can calculate ν

1

1

sin x( )

18.850 x0 5 10 15 20

1

0.5

0

0.5

1

λ

21 THE PARTICLE NATURE OF LIGHT - light comes as particles called photons - energy of a photon is proportional to frequency

E = h ⋅ ν - h = Planck’s constant h = 6.626 x 10-34 J ⋅ s Photoelectric Effect Light shining on a metal surface may cause electrons to be ejected from the surface.

e-

Frequency of light needs to be above threshold frequency to induce photoelectric emission. Kinetic energy of electrons is proportional to frequency of incident radiation. Kinetic energy of electrons is independent of light intensity. - I.e., microwave laser will not induce photoelectron emission. This independence contradicts wave nature of light. - According to wave nature, energy of electrons should be proportional to the

intensity.

22 WAVE-PARTICLE DUALITY (Supplemental Material) Diffraction of Waves Single-slit diffraction Light is a wave The wave relationship between frequency and wavelength is true for light. c = νλ

More evidence that light is a wave exists since it can be reflected, refracted and diffracted. When a traveling wave hits a hole (slit) that is approximately the size of the wave’s wavelength, the wave “expands” as it goes through the hole.

Double-slit diffraction

When a wave is incident on two slits relatively close to each other, a diffraction pattern appears when the waves constructively and destructively interfere with each other.

Comments about diffraction Maximum intensity is directly behind barrier between two slits. Intensity pattern comes from interference of two wavefronts. Diffraction confirms that light is a wave.

Direction of propagation

screen

constructive interference

destructive interference

23 ELECTROMAGNETIC SPECTRUM Name Wavelength Frequency (Hz) Radio 300 km to 0.3 m 103 – 109 Microwave 30 cm to 1 mm 109 – 3 x 1011 Infrared 1.0 mm to 780 nm 3 x 1011 – 4 x 1014 Visible 780 nm to 390 nm 4 x 1014 – 8 x 1014 Ultraviolet 390 nm to 1 nm 8 x 1014 – 3 x 1017 X-ray 10 Å to 0.06 Å 3 x 1017 – 5 x 1019 Gamma 1.5 Å to 0.3 ym 2 x 1018 - 1033 LINE SPECTRA OF THE ELEMENTS The light emitted by pure elements has specific energies. - Therefore light of only specific wavelengths can be seen. (i. e., different

colors can be seen) - This emitted light is called a line spectrum. (pl. spectra) Line spectra tell us that atoms can only have certain energy levels. - The atoms cannot have any arbitrary value of energy. ATOMIC STRUCTURE HISTORY (Review) Thomson

- discovered “cathode rays”, i.e., electrons Milliken

- found charges come in discrete units, charge of electron is fundamental.

Rutherford - found that atom has a very small, yet very heavy nucleus.

24 BOHR’S MODEL OF THE HYDROGEN ATOM History - Scientists before Bohr knew atom was made of nucleus and electrons. - They didn’t know where the electrons were or how they behaved. - They also knew each element had a distinct line spectrum.

1913 – Model - Bohr assumed electrons traveled in orbits around nucleus.

- The centripetal force of the electron balances with its electrical attraction to the nucleus.

2 2

20

mv 1 er 4 r

=πε

- Bohr also assumed that electrons could only have specific orbits. - Angular momentum of orbits was quantized.

mvr n= h

- Specific orbits were labeled with a quantum number, n

2 22 2 2 22 2 2 2 2 0

2 20 0 0

4 nmv 1 e me r me rm v r n rr 4 r 4 4 me

πε= ⇒ = ⇒ = ⇒ =

πε πε πεh

h

- Energies of orbits are

2 2 2

n H2 2 20 0 0

1 e e me 1E R4 r 4 4 n n

= − = − = − ⋅πε πε πε h

RH = Rydberg constant (for hydrogen) = 2.18 x 10-18 J - Won Nobel Prize – 1922 SCHEMATIC OF BOHR MODEL

+

Orbits are quantized. Electrons cannot exist between defined orbits.

Supplemental Material

25 Using the Bohr Model to Calculate Spectra Spectra result from light being emitted or absorbed when atom changes energy, i. e.

when electron goes to different orbit. - emission – energy of atom decreases, i. e. electron orbits closer to nucleus - absorption – energy of atom increases, i. e., electron orbits further from

nucleus - ground state – lowest possible energy state - excited state – any other state Bohr model - when energy in an atom changes, we can calculate the change as ΔE = Ef – Ei

ΔE Rn

Rn

Rn nH

fH

iH

i f= − ⋅ − − ⋅

⎛⎝⎜

⎞⎠⎟ = ⋅ −

⎛⎝⎜

⎞⎠⎟

1 1 1 12 2 2 2

- also consider that since ΔE is the energy of the photon emitted or absorbed.

Ephoton = ΔEatom ⇒ Ef – Ei = h ⋅ν

- an equation for the frequency of the photon can be written as

h Rn n

Rh n nH

i f

H

i f⋅ = ⋅ −

⎛⎝⎜

⎞⎠⎟ ⇒ = ⋅ −

⎛⎝⎜

⎞⎠⎟ν ν

1 1 1 12 2 2 2

Example: What frequency of light is emitted when the H atom changes its electron

energy level from n = 6 to n = 3?

ν = ⋅ −⎛⎝⎜

⎞⎠⎟ =

×× ⋅

−⎛⎝⎜

⎞⎠⎟ = × −

⎛⎝⎜

⎞⎠⎟

= × ⋅−

= − ×

−

−

Rh n n

JJ s

Hz

Hz Hz

H

i f

1 1 2 18 106 626 10

16

13

3 29 101

3619

3 29 103

362 74 10

2 2

18

34 2 215

15 15

..

.

. .

Note: Negative sign relates that radiation was emitted. (Frequencies usually reported as positive.)

THE DUAL NATURE OF MATTER - Bad News: The Bohr model is wrong! - Electrons don’t behave like planets. - Electrons have a wave nature that makes them spread out. We have seen that light can behave as a wave and a particle. This dual nature of light is also true for matter. *All matter behaves as a particle and a wave.* - I. e., an electron, an atom or a baseball all behave like a wave.

26 DE BROGLIE WAVES (MATTER WAVES) All moving particles have a wavelength. Wavelength of particle is inversely proportional to particle’s momentum. (De Broglie’s Relation) note: smaller p implies higher λ higher p implies smaller λ

Picture of a de Broglie wave - note that the wave is localized somewhat - as wavelength decreases wave becomes more localized (Note: momentum has

increased.)

In the macroscopic world, objects do not have large enough wavelengths to exhibit wave-like behavior. Only in the microscopic world (as in the atom) do objects exhibit wave-like behavior. Let us illustrate with a couple of examples. Example: Calculate the wavelength of an electron when the electron is

moving 2.18 x 106 m/s. me = 9.109 x 10-31 kg h = 6.626 x 10-34 J⋅s

λ = =×

=⋅

×

=⋅⋅

=⋅ ⋅

⋅

= =

−

−

− −

−

hp

hm v

x J sx kg x m s

xJ s

kg mx

kgms

s

kg mx m

6 626 109 109 10 2 18 10

3 34 10 3 34 10

3 34 10 3 34

34

31 6

102

10

2

22

10

.. . /

. .

. . Å

- Note: The wavelength is about the same as the size of the atom.

Recall: Momentum is defined as mass × velocity

0.981

0.981

f x( )

2.992.99 x3 2 1 0 1 2 3

1

0.5

0

0.5

1

0.976

0.976

g x( )

2.9992.999 x3 2 1 0 1 2 3

1

0.5

0

0.5

1

hp

λ =

27 Example: Calculate the wavelength of a baseball moving 60 mi/hr (26.8 m/s). The

mass of baseball is 0.14 kg. 34 2

34 34h h 6.626 x10 J s J s1.8 x10 1.8x10 mp m v 0.14 kg 26.8m / s kg m

−− −⋅ ⋅

λ = = = = =× × ⋅

- Note: This is an extremely small wavelength, especially when compared to the size of the baseball.

To summarize: Microscopic objects when moving become wave-like. Macroscopic objects have a wave-like nature when moving, but the wave nature is insignificant compared to its particle nature. Experimental confirmation for wave-like properties has been found for electron, proton, neutron, hydrogen atom, sodium atom, bucky balls, et al. - matter waves interfere with each other just like light waves (or any wave) WELCHER-WEG EXPERIMENT (Supplemental Material) Electrons have wave-like and particle-like properties simultaneously. Whether wave-like or particle-like properties are observed depends on the experimental apparatus used. If the apparatus measures wave-like properties, then particle-like properties will be absent, and vice versa. In the welcher weg (German for which way) experiment, an electron gun is fired with an intensity such that one electron is in a double-slit diffraction apparatus at one time. The cumulative effect of the electrons striking the screen is an interference pattern. For an interference pattern to result, the electron must pass through both slits. However, the electron is a “particle”. Thus the curious mind would like to know through which slit the particle went. When such detection is done, the inference pattern disappears. When the detection is not done, the interference pattern returns. The moral of the story is: when the electron is measured as a wave, it behaves like a wave, when it is measured like a particle, it behaves like a particle. Diffraction apparatus Welcher weg apparatus

28 INTRODUCTION TO THE SCHRODINGER EQUATION (Supplemental Material) The Schrödinger equation is the fundamental equation of quantum chemistry. The equation has the form of an eigenvalue/eigenvector problem. Classically, the total energy of a system is written in a system’s Hamiltonian, H. A Hamiltonian is a classical quantity that is defined as H = T + V. T – kinetic energy V – potential energy Kinetic Energy Operator

2 2 22

2

1 p 1 p mp pT mv p mv v T m2 m 2 m 2m 2m

⎛ ⎞= ⇒ = ⇒ = ⇒ = = =⎜ ⎟⎝ ⎠

Thus, the Hamiltonian can be written as2p V

2m= +H

Schrödinger in 1926 replaced p with p̂ and used the Hamiltonian as an energy operator.

xp̂i x∂

=∂

h

In one dimension, the kinetic energy operator is

2 22 2 2 2

2

p 1T2m 2m i x 2m x x 2m x

∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= = = − = −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠h h h

In three dimensions, the kinetic energy operator is

2 2 2 2

2 2 2T2m x y z

⎛ ⎞∂ ∂ ∂= − + +⎜ ⎟∂ ∂ ∂⎝ ⎠

h

The del squared operator, 2∇ , is a shorthand notation for the second derivative. In Cartesian coordinates, the del squared operator is

2 2 22

2 2 2x y z∂ ∂ ∂

∇ = + +∂ ∂ ∂

Note: In other coordinate systems, 2∇ is much more complicated. Overall, the quantum mechanical energy operator (Hamiltonian) can be written as

22ˆ V

2m= − ∇ +

hH

When the Hamiltonian is applied to an eigenfunction of the Hamiltonian, the energy eigenvalue results. Thus, the Schrödinger equation can be written as

ˆ Eψ = ψH

29 Where ψ is called the wavefunction. It is the mathematical description of the electron in an orbital. The hydrogen atom wavefunctions are functions of the variables (r, θ, φ)

( ) 0

1Zr3 2a

30

Z1s ea

−⎛ ⎞ψ = ⎜ ⎟π⎝ ⎠

( ) 0

3Zr22a

0 0

1 Z Zr2s 2 ea a4 2

−⎛ ⎞ ⎛ ⎞ψ = −⎜ ⎟ ⎜ ⎟

π ⎝ ⎠ ⎝ ⎠

( ) 0

3 2 Zr23a2

0 0 0

1 Z Z Z3s 27 18 r 2 r ea a a81 3

−⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟ψ = − +⎜ ⎟ ⎜ ⎟⎜ ⎟π ⎝ ⎠ ⎝ ⎠⎝ ⎠

( ) 0

3 2 3 Zr24a2 3

0 0 0 0

3 Z Z Z Z4s 192 144 r 24 r r ea a a a

−⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ψ = − + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟π ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

( ) 0

51 2 Zr22 3a2x

0 0 0

4 1 Z Z Z3p 6 r r e sin cos81 8 a a a

−⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎜ ⎟ψ = − θ φ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟π⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

Note that for the “s” wavefunctions, the terms of the “r” polynomial increase as “n” increases. This increase leads to an increase in the number of roots of the polynomial which leads to an increase in the number of radial nodes. Learning objective 1.13 Given information about a particular model of the atom, the student is able to determine if the model is consistent with specified evidence. (See SP 5.3; Essential Knowledge 1.D.1) ATOMIC ORBITALS The wave nature of the electron expresses itself within an atom as an orbital (electron cloud). ATOMIC ORBITALS AND QUANTUM NUMBERS - Bohr model has one quantum number. - Orbital model has three quantum numbers. 1. Principal quantum number – n a) has values of n = 1, 2, 3, 4, … b) related to distance away from nucleus c) nearly same as n quantum number in Bohr model d) all the energy levels in an atom with the same n value are called a shell

30 2. Azimuthal quantum number – l a) has values of l = 0, 1, 2 ,…, n-1 b) For a given shell, all the energy levels in an atom with the same l are called a

subshell. - subshell is labeled with n quantum number and letter for l quantum number. c) Letters are used to represent the value of l.

l 0 1 2 3 4 5, … subshell s p d f g h, …

d) related to the shape of the orbital (more in a little bit.) orbital – an electron cloud in an atom with specific quantum numbers, i.e., same n, l, ml

e) Note for a given shell, n possible number of subshells exist.

3. Magnetic quantum number – ml a) has values of ml = l, l-1, l-2, …, -( l-1), -l - note dependence on l b) related to orientation of orbital c) Note # of orbitals in a subshell is 2l + 1 Shapes of the orbitals s orbital - spherical shape - electron cloud is densest at nucleus p orbital - “dumbbell” shape - electron cloud is not densest at nucleus - orbital has a node at the nucleus where e- density is zero - p orbitals have 1 radial node d orbital - “cloverleaf shape - electron cloud is not densest at nucleus - d orbitals have 2 radial nodes f orbitals - various shapes - f orbitals have 3 radial nodes

4f , ml = 0 4f , ml = 1 4f , ml = 2 4f , ml = 3 X Y, Z,( ) X Y, Z,( ) X Y, Z,( ) X Y, Z,( )

31 Orientation of orbitals - The ml quantum number indicates the orientation of the orbital within the atom. - Number of possible ml values indicates number of possible orientations. - Consider the orbitals of the 2p subshell as an example. Note: values of ml have been assigned arbitrarily. EXCULSION STATEMENT Essential Knowledge 1.C.2 Assignment of quantum numbers to electrons is beyond the scope of this course and the AP Exam. ENERGY LEVELS OF THE ORBITALS The three quantum numbers, n, l, ml, not only describe the size, shape and orientation of an orbital. More important, the quantum numbers describe the energy of electrons in the orbital Notes about energy levels of orbitals - bigger change in energy is between shells - lesser change in energy between subshells - no change in energy between orbitals in same subshell (orbitals are degenerate) - note: energy of 4s orbital is less than energy of 3d orbitals

y

x

z

y

x

z

y

x

z

2py ⇒ ml = -1 2pz ⇒ ml = 1 2px ⇒ ml = 0

1s

2s

3s

4s

2p

3p

3d

32

Hydrogenic Wavefunctions [Supplemental Material]

1s 2s 2p 3s

3p 3d

4p 4d 5p Note: orbital not to scale Note: l – indicates number of angular nodes

33 Learning objective 1.5 The student is able to explain the distribution of electrons in an atom or ion based upon data. (See SP 1.5, 6.2; Essential Knowledge 1.B.1) Learning objective 1.7 The student is able to describe the electronic structure of the atom, using PES data, ionization energy data, and/or Coulomb’s law to construct explanations of how the energies of electrons within shells in atoms vary. (See SP 5.1, 6.2; Essential Knowledge 1.B.2) Learning objective 1.13 Given information about a particular model of the atom, the student is able to determine if the model is consistent with specified evidence. (See SP 5.3; Essential Knowledge 1.D.1) PHOTOELECTRON SPECTROSCOPY - Technique used to measure the energy (and number) of electrons in an atom

(molecule) - Sample is bombarded with x-ray (or ultraviolet light) - X-rays collide with electrons in sample ejecting them - Ejected electron interacts with a detector which measures its energy. - Thousands of electrons are measured and their measurements are gathered

together to form a photoelectron spectrum - Position of peaks on the x-axis yields the energy of the electrons - Height of the peaks yields information of the relative number of electrons with

a particular energy (i.e. within a particular subshell)

Inte

nsity

Energy (eV) 87048.4721.61

Photoelectron Spe ctrum of Neon

Inte

nsity

Energy (eV)40320.3314.53

Photoelectron Spectrum of Boron

Compare the photoelectron spectra of Boron and Neon

- The 870 peak of Ne and the 403 peak of B are 1s electrons - The 48.47 peak of Ne and the 20.33 peak of B are 2s electrons - The 21.61 peak of Ne and the 14.53 peak of B are 2p electrons - The relative intensity of the 2p peak revealing the relative number of

electrons in the subshell (6 for Ne vs. 1 for B) - The how the nuclear charge of Ne has increased the energies

34 HEISENBERG UNCERTAINTY PRINCIPLE The Heisenberg uncertainty principle limits the precision of measuring quantities in the microscopic world. - This limitation is a fundamental principle of physics and not just a technology

problem. To understand measuring in the microscopic world, let us concentrate first on measuring in the macroscopic world. Macroscopic Measurement Measuring the velocity a ball rolling on the floor, we need to measure at least the position of the ball at two different times.

vr rt t

rt

=−−

=2 1

2 1

ΔΔ

How do we measure the position at both times? We see the light reflected off the ball at t1 and note the position r1. Then we see the light reflected off the ball at t2 and note the position r2. Key Point: We need the light to measure the position accurately. We can’t

measure the position in the dark. Microscopic Measurement To measure the velocity of an electron, we need to measure its position at two different times by seeing light reflected off the electron. The problem (and the crux of the argument) is that when a photon hits the electron, it moves the electron. Thus the act of measuring disturbs the system. Note that measuring the macroscopic object did not disturb the system. Heisenberg’s Uncertainty Principle tells us we are not able to measure the position of an object and its velocity with infinite precision. I.e., there is inherent fuzziness in measuring certain quantities at the same time. As an equation, Heisenberg’s Uncertainty Principle can be stated as

Δr Δp ≥ h/4π

Δr – uncertainty of position measurement Δp – uncertainty of momentum (velocity) measurement Uncertainty principle also relates uncertainty of time and energy.

( )( )E t2

Δ Δ ≥h

- very important to understand the width of spectral lines

35 Consequences of Heisenberg’s Uncertainty Principle - There is always a trade-off of precision when measuring the position of an

electron and its velocity. - If we want to know the position with total precision, we will have no idea how

fast it is going. - If we want to know the velocity of the electron, we will have no idea where it is. MULTI-ELECTRON ATOMS Electron Spin - electron behaves as if it is spinning - electron spin has only two orientations - spin quantum number – ms - two values -½, +½ - by convention +½ is spin up -½ is spin down - Therefore, four quantum numbers fully describe where electrons are within an

atom. - two electrons with opposite spins per orbital - The energies of the two electrons within an orbital are degenerate. Pauli Exclusion Principle - In an atom, electrons can’t share same set of quantum numbers. - I. e., two electrons can’t be in the same place at the same time. - This may seem obvious, however two photons can be in the same place at the

same time. Aufbau Principle - Electrons in ground state atom are in lowest possible energy. - Electrons “fill” into orbitals from low energy to high energy. Hund’s Principle - Electrons fill into degenerate orbitals as to maximize the total spin of the

electrons in the atom. - I. e., electrons would rather go to another orbital rather than pair with another

electron. ORBITAL DIAGRAMS AND ELECTRONIC CONFIGURATIONS Electronic Configuration is a list of subshells and the number of electrons within them. Orbital Diagrams are energy level diagrams that indicate the occupation of the orbitals and the spins of the electrons. Noble gas abbreviations When only the valence (outermost) electrons need to be considered, the core

(innermost) electrons can be represented by the elemental symbol of the closest noble gas preceding the atom that is being examined.

36 Learning objective 1.12 The student is able to explain why a given set of data suggests, or does not suggest, the need to refine the atomic model from a classical shell model with the quantum mechanical model. (See SP 6.3; Essential Knowledge 1.C.2) ELECTRONIC CONFIGURATIONS AND THE PERIODIC TABLE Periodic table tells us subshell of valence electrons. Example: Tellurium is an element that is added to rubber to make it more

resistant to oil. Give the full and abbreviated electronic configuration of the ground state tellurium atom.

Full: 1s22s22p63s23p64s23d104p65s24d105p4 Abbr: [Kr]5s24d105p4 Example: Vanadium is the crucial element in Damascus steel used to make the

highest quality edged weapons in the Middle Ages. What is the orbital diagram and electron configuration of the ground state vanadium atom?

3d 4s 3p 3s 2p 2s 1s Elec. Config. 1s22s22p63s23p64s23d3 or [Ar]4s23d3

d valence e- p valence e-

f valence e-

37 MC Question: Which of the following sets of quantum numbers (n, l, ml, ms) best

describes the valence electron of highest energy in a ground-state gallium atom (atomic number 31)?

(A) 4, 0, 0, ½ (B) 4, 0, 1, ½ (C) 4, 1, 1, ½ (D) 4, 1, 2, ½ (E) 4, 2, 0, ½ EXCLUSION STATEMENT Essential Knowledge 1.C.1 Memorization of exceptions to the Aufbau principle is beyond the scope of this course and the AP Exam.