CHAPTER THREE: QUANTUM THEORY AND THE ELECTRONIC STRUCTURE OF

ATOMS

A. Energy - Definitions (Section 3.1)

1. Energy = capacity to do work or to transfer heat. 2. Forms of energy = mechanical energy, light energy, electrical energy, and heat

energy, and others. 3. Energy has two essential manifestations: kinetic energy and potential energy.

a. Kinetic energy is energy of motion = mv2/2. � Thermal energy is the energy associated with the random motion of atoms

and molecules.

b. Potential energy is stored energy = the energy an object possesses because of its position, condition, or composition.

� Two major types in chemistry:

o Chemical energy- Stored within the structural units of chemical substances.

o Electrostatic energy- Potential energy resulting from the interaction of charged particles.

4. The Law of Conservation of Energy: Energy can be converted between these two

from, but it cannot be created or destroyed. 5. The SI unit of energy is the joule (J):

EK = �kg� ��� �→ � ∙�

�

�� = J

6. Electrostatic Energy (Section 3.1)

a. Electrostatic energy is proportional to the product of the charges involved (Q)

divided by the distance between them (d):

E�� ∝ Q�Qd

� If the charges are opposite, Eel is negative which leads to an attraction.

� If the charges are the same, Eel is positive which leads to a repulsion.

B. The Wave Nature of Light (Section 3.2) 1. Structure of atom had been established as cloud of electrons around a heavy nuclear

core = the nuclear atom picture. 2. Beyond that, nothing was known of arrangement of the electrons. 3. Final clues came from spectroscopy = interaction of matter with light. 4. Atomic emission spectra = light emitted from excited atoms, analyzed into

wavelength components. 5. Therefore some background facts about light are needed. 6. Light is energy propagating as an oscillating electromagnetic field. 7. The electromagnetic wave:

8. Wavelength λ = distance between two identical points on a wave (peaks or troughs).

� Units in meters (typically nanometers)

9. Frequency ν = number of waves that pass a point during a given time.

� Units of “cycles per second”; s-1, Hz

10. Qualitative Relationship between λ and ν

a. For the wave pictured, there are three complete waves. b. If this happened during a one second, the frequency would be 3 Hz. c. Longer wavelengths mean less waves can be completed in the same amount of

time.

11. Quantitative Relationship between λ and ν a. Mathematically, the wavelength and frequency of a given wave of light are related

by the following equation:

c = λν c = speed of light (2.99792458 x 108 m/s)

λ = wavelength (m)

ν = frequency (s-1)

b. The two values multiplied together equal a constant value.

� The wavelength and frequency of a wave are inversely related.

12. Separation of Light

a. “White” light actually contains all the colors of the visible spectrum.

b. Isaac Newton first separated light into its component wavelengths using a prism.

� Referred to as dispersion.

c. This spectrum is said to be continuous- no dark spots between the colors.

13. The Electromagnetic Spectrum

14. CD/DVD Players

� Using light with smaller wavelengths allows the pits to be smaller, enabling the storage of more data.

15. Additional Aspects of EM Waves

a. Electromagnetic waves are so named because they have both an electrical and magnetic component.

� Equivalent wavelengths, frequencies, and amplitudes, but oscillating

perpendicular (90°) to one another.

b. When passed through successive narrow slits, waves diffract and generate interference patterns.

C. Quantum Effects and Photons (3.3)

1. Max Planck (1900) showed that light also has particle-like characteristics. Comes in particles he called photons.

2. Energy of a photon of light is given as:

E = hν Ε= energy (J)

h = Planck’s Constant (6.626 x 10-34 J∙s

ν = frequency (s-1)

� Using c = λν, could also write: E = �∙�λ

3. Example Problems: FM radio stations typically use signals with a frequency of 1.00x108 s-1, while microwave ovens use radiation with a wavelength of 0.1225 meters. Which has more energy, the radio waves or the microwaves? Heating glass in a flame results in a yellow glow emitted from excited Na atoms in the glass with a wavelength of 590 nm. Calculate:

– The frequency of this light. – The energy of a photon of this light. – The energy of one mole of photons of this light in joules per mole.

A whole mole of Na atoms emitting one photon would emit how much energy? Convert to calories: (4.184 J = 1 cal)

Enough energy to raise 4.85 x 104 grams or 48 kg of water by 1° C. 4. The Photoelectric Effect

a. Ejection of an electron from the surface of a metal, or from another material, when light shines on it.

b. Led to confirmation of particle-like nature of light.

c. Observations:

� e- ejected only if light has ν above some threshold frequency νο: ν > νο

� If ν is below νο, no e- ejected no matter how bright (or intense) the light

source.

d. Implication (Einstein - 1905): Electrons in the metal are interacting with individual particles of light

(photons). This means energy of a single photon must be above the threshold energy φ:

E > φ φ = work function of the metal = min. energy required to eject an e-.

D. Emission Spectra (Section 3.4)

1. Electrons in an atom can absorb energy, moving from a lower to a higher energy level.

– Referred to as an excited state.

2. The electron can then release the energy to go back to its original, lowest energy state. – Referred to as the ground state.

3. The light emitted can be dispersed through a prism or diffraction grating into its

component wavelengths (colors). – This is an emission spectrum.

4. Ordinary sunlight or incandescent light gives a continuous spectrum of light.

5. Excited atoms in a vacuum only emit certain characteristic wavelengths of light, called a line spectrum.

6. Pattern of emission is different for every element. These spectra serve as “fingerprints” for identification of elements.

7. Balmer (1885) showed that the wavelengths in the visible spectrum of Hydrogen can be fit by a simple formula:

1

λ= 1.097 × 107m−1 1

22− 1

n2

- Balmer Equation

8. Rydberg found an equation that reproduces the λ of the all lines in Hydrogen emission

spectrum, including ultraviolet and infrared:

1

λ= R

1

n1

2− 1

n2

2

- Rydberg Equation

R∞ = 1.097 x 107 m-1. (“Rydberg constant”) n1, n2 are positive integers and n1 < n2

9. Bohr (1913) explained Rydberg equation by his theory of H atom. a. Electron orbits the nucleus in circular orbits. b. Orbital energy is quantized; i.e., it can have only certain distinct values. electron’s energy: En = - a/n2 (n = 1, 2, 3...)

n = quantum number (tells what “quantum state” the electron inhabits)

Bohr’s energy level diagram:

c. Bohr Frequency Rule: The atom can only absorb or emit light having just the right

energy (and thus frequency or wavelength) to move the e- between these energy levels.

– Can also “jump” multiple levels.

10. Changing energy levels a. The further the electron is from the nucleus, the higher its (potential) energy.

– Due to attractions between the nucleus and the electron.

b. Electrons absorb energy to move further away from the nucleus (higher energy).

c. Electrons release energy to move closer to the nucleus (lower energy).

11. Whole H emission spectrum explained this way:

12. Bohr theory failed to explain atoms bigger than Hydrogen. 13. “Orbit” idea also had important defects on physical grounds:

According to physics an orbiting charge should be continuously emitting radiation and losing energy. Orbit should spiral inward.

14. Orbiting particle idea had to be scrapped! Replaced by a wave picture of the electron.

E. The Wave Nature of Matter. (Section 3.5) 1. 1925 - Louis de Broglie described the phenomenon called the wave/particle duality =

microscopic particles possess wave-like character, and vice versa.

i.e. - just as light usually seems wave-like and yet has particle-like behavior (photons), electrons also have both particle-like and wave-like behavior.

λ = h/mv λ = wavelength (m)

h = Planck's constant (6.626 x 10-34 J∙s m = mass (kg) v = velocity (m/s)

2. De Broglie equation proved by Davidson-Germer experiment two years later. Electrons shown to diffract just like waves.

F. Quantum Mechanical Picture of the Atom. (Section 3.6) 1. 1920’s - Quantum mechanics (wave mechanics) replaced Newtonian mechanics. 2. Treats microscopic particles according to their wave-like character. 3. In classical physics, an object can have any amount of energy.

a. You can stop at any place on a hill.

4. In quantum physics, an object can only have certain amounts of energy. a. You can only stop in specific places on stairs.

5. 1927 - Heisenberg Uncertainty Principle:

It is impossible to measure both the velocity and position of an e- simultaneously to an arbitrarily high degree of precision.

Therefore, cannot view the e- as following a precise trajectory around the nucleus. 6. The more precisely you measure the position, the less precisely you can

simultaneously measure its momentum (or velocity), and vice versa.

∆x ⋅ ∆px ≥ h

2π

∆x ⋅ ∆vx ≥ h

2πm

7. Basic Postulates of Quantum Mechanics:

a. Atoms and molecules can exist only in certain energy states. b. When they change energy, they must absorb or emit precisely the required energy

to place them in the new energy state.

As the differences between the steps get smaller, these become

the same thing!

c. The allowed energy states are indexed by sets of numbers called quantum numbers.

d. Electron in an atom is treated as a standing wave (not a traveling wave, like

light). e. It’s position is prescribed probabilistically according to a wave function ψ(x,y,z). f. ψ(x,y,z) is a mathematical function of spatial coordinates (x,y,z) which is found

by solving the Schrodinger equation (1926): −h2

8π2m

∂2ψ∂x2

+ ∂2ψ∂y2

+ ∂2ψ∂z2

+ Vψ = Eψ

g. This equation has as many solutions as the atom has quantum states. h. The solutions are indexed by four quantum numbers: n, l, mllll, ms.

i. The wave function Ψ�,ℓ,�ℓ,��

tells the size and shape of the region of space where the probability of finding the electron is high.

ψ2 ≈ atomic orbitals

G. Quantum Numbers (Section 3.7)

1. The principal quantum number, n, specifies energy level or shell an e- occupies. Determines the energy and size of the orbital. Allowed values:

n = 1, 2, 3, 4, ...

K, L, M, N… shell

2. The angular momentum (azimuthal) quantum number, l, specifies the sublevel or subshell an electron occupies, determines the shape of the region in space an electron occupies. Allowed values:

l = 0, 1, 2, ..., (n - 1)

We give a letter notation to each value of l. Each letter corresponds to a different kind

of atomic orbital.

l = 0, 1, 2, 3, ..., (n - 1) s p d f sublevel 3. The magnetic quantum number, ml, designates the spatial orientation of an atomic

orbital. Allowed values:

ml = (-l ), ... 0, ..., (+l ) 4. The spin quantum number, ms, refers to the spin of an electron and the orientation of

the magnetic field produced by this spin. Allowed values:

ms = ± 1/2

Example from everyday life: At Tech football games a person sits in Section J, Row

21, Seat 8. It takes 3 numbers to fully specify where the person is sitting!

H. Atomic Orbital Representations (Section 3.8)

1. An AO is specified by n, l, ml l l l . 2. AO is a probability distribution function for finding the electron in space.

– Represents an area where the electron is likely to be found most of the time.

3. As the value of n increases, the size of the orbital increases – On average, the electron spends a larger amount of time further from the

nucleus.

� Higher energy!

Radial Probability Distribution (Figure 3.18)

4. Effect of l on orbitals

a. There are four main values of l to consider:

– l = 0, s (sharp) – l = 1, p (principal) – l = 2, d (diffuse) – l = 3, f (fundamental) – l = 4, g; l = 5, h; etc.

b. The value of l determines the subshell.

– The orbitals of a given subshell have a characteristic shape determined by the value of l.

5. Effect of ml on orbitals

a. The value of ml determines the spatial orientation of the orbital in the subshell: – For l = 0, ml = 0 (n=1)

� s orbitals are spherical, so there can’t be more than one on the same set of axes

– For l = 1, ml = -1, 0, +1 (excite to n = 1) � Three p orbitals in a subshell, each oriented along a different

Cartesian axis. – For l = 2, ml = -2, -1, 0, +1, +2 (excite to n = 3)

� The d subshell has five orbitals!

6. Energy Levels a. As n increases, the energy increases.

b. Energy also increases as l increases, but to a lesser extent.

– In a hydrogen atom (or any single electron ion) the l value does not affect

the energy.

7. To place the electron (↑) in one of these orbitals with spin ms = +1/2 (↑) or ms = - 1/2 (↓) is the last step in fully specifying which quantum state the electron occupies.

8. Quantum Numbers- Example Problems

• How many electrons can have the following quantum numbers?

– n = 5

– n = 4, ℓ = 3

– n = 6, ℓ = 2, mℓ = -2

– n = 7, ℓ = 5, mℓ = -3, ms = +1/2

– n = 3, mℓ = 0

• What is the maximum number of electrons allowed in:

– A 6d orbital?

– A 4f subshell?

• For the following invalid sets of quantum numbers, determine which value makes the set

invalid, and make the set valid by changing only one of the four numbers:

– n = 7, ℓ = 3, mℓ = -2, ms = +1

– n = -2, ℓ = 1, mℓ = 0, ms = -1/2

– n = 3, ℓ = 3, mℓ = 2, ms = +1/2

– n = 3, ℓ = 2, mℓ = -3, ms = -1/2

I. Electron Configurations of Multi- Electron Atoms (Section 3.9)

1. Electron configuration is shorthand notation for what AO the electron occupies: Example - The ground state of H atom (lowest energy state):

2. Atoms bigger than H are treated by placing additional electrons into H-like orbitals:

3. Note that maximum of 2 electrons can go into an atomic orbital with opposite spins, consequence of Pauli Principle.

4. Pauli Exclusion Principle = no two e- in an atom can have identical set of 4 quantum

numbers.

5. P.E.P. also implies that for 3 e- atom like Li, would have to start to fill 2s AO:

6. Aufbau Principle (Section 8.2) = “Building-up principle” = electron configuration of multi e- atoms built up by addition of electrons to H-like AO to give the lowest total energy for the atom.

filling order = 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p...

7. Orbital energies are approximated by (n + l)

a. The lower the value of (n + l), the lower the energy.

b. When (n + l) for two orbitals are the same, the one with the lower n value is lower in energy.

• (n + ℓ) = 1; 1s

• (n + ℓ) = 2; 2s

• (n + ℓ) = 3; 2p, 3s

• 2p fills first because lower n.

• (n + ℓ) = 4; 3p, 4s

• (n + ℓ) = 5; 3d, 4p, 5s

• (n + ℓ) = 6; 4d, 5p, 6s

• (n + ℓ) = 7; 4f, 5d, 6p, 7s

• etc.

8. In multi-electron atoms the energies of AO’s follow this design:

9. Note that 4s is lower in energy than 3d, so 4s fills first.

n = 1 1s

n = 2 2s

2p

n = 3

3s

3p

3d

n = 4

4d

4f

4s

4p

10. Now let’s write some electron configurations.

11. Hund’s Rule = when filling a sublevel having more than one AO (such as 2p sublevel)

one places electrons singly in separate orbitals before pairing begins. These unpaired e- have parallel spins. This is lower energy.

12. Let’s continue:

13. Note at Neon, 2p sublevel is filled, and also n = 2 level is filled. At He n = 1 level is

filled. These are stable, chemically inert configurations. 14. Let’s do Row 3:

15. Note similarity between F and Cl, a look ahead! same outer electron configuration but different n level 16. Row 4 and 5, the d sublevels start filling:

J. Magnetic Properties of Atoms (Section 3.9)

1. Electron in an atom behaves like a tiny magnet and orients in a magnetic field.

2. Magnetic field from two electrons with paired spins (one up and one down) cancels

itself out. No net magnetism. 3. Atoms with all paired-up electrons are called diamagnetic - they are not attracted by

an external magnetic field, but actually slightly repelled.

Example: Hg vapor Why? 4. Unpaired electrons in an atom impart an overall magnetism to the atom and these are

called paramagnetic. They are attracted by a magnetic field.

Example: Na vapor Why? 5. This behavior proves Hund’s Rule is in effect. For example, the electronic

configuration of Carbon is:

K. The Periodic Table of Electron Configurations (Section 3.10)

1. Electron configurations explain the shape of the Periodic Table, and vice versa, we can

use Periodic Table to figure out electron configuration.

2. For the main group elements, the period number tells you the value of n.

– The d subshells are n - 1

– The f subshells are n – 2

3. We can use this information to simplify electron configurations.

Se: 1s2

2s2

2p6

3s2

3p6

4s2

3d10

4p4

� [Ar]4s2

3d10

4p4

4. Ground and Excited States

a. The electron configurations we’ve discussed so far have all been ground states.

– Ground state – Lowest energy electron configuration for an element.

b. Any other valid configuration of electrons for a given element represents an excited

state.

– An element will have only one ground state, but it can have many excited

states.

___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 1s 2s 2p 1s 2s 2p

Ground State Excited State

___ ___ ___ ___ ___ 1s 2s 2p

Invalid Configuration

5. Electron Configuration Example Problems

• Write the electron configurations for the following elements:

– Mg

– Ga

– Tc

– Ir

• For each of the following electron configurations for neutral atoms, determine what

element it corresponds to and if it is a ground or excited state:

– 1s22s22p63s23p5

– [Kr]5s14d3