Quantum Theory

Chang Chapter 7

Bylikin et al. Chapter 2

Chapter 7 Outline

• Review of Classical Physics• Quantization of Energy• Emission Spectra• Particle/Wave Duality• Quantum Mechanics

• Heisenberg• Schrodinger• Quantum Numbers• Electron Configurations

Review of Classical Physics

• Waves: vibrating disturbance that transmits energy

– Speed of a wave = wavelength (l) ∙ frequency (n)Frequency = number of waves that pass through a particular point in 1 second (units = Hz)

Wavelength has units of nanometers (nm)

UNIT FACTOR: 1 nm = 1x10-9m

Review of Classical Physics

• Radiation: Emission and transmission of energy through space via Electromagnetic (EM) waves– EM radiation = radiant energy = light– Speed of light= 3.00x108 m/s = l∙ = n c

Review of Classical Physics

• Example: The yellow light given off by a sodium vapor lamp used for public lighting has a wavelength of 589 nm. What is the frequency of this radiation?

Review of Classical Physics

Quantization of Energy

• Before 1900: energy is continuous, any amount of energy can be released or absorbed

• In 1900 Max Plank proposed energy is quantized–Quantum= smallest amount of EM energy

that can be absorbed or released

Quantization of Energy

• Energy of ONE quantum = E = h∙n• h = Plank’s Constant = 6.626x10-34 J∙s

• Energy that can be released or absorbed• E = n∙h∙n (where n = 1, 2, 3…)• “energy can be absorbed or released only in

integer units of quantum”

Quantization of Energy

• Example: Calculate the energy of one photon of yellow light with wavelength 589 nm. Repeat for 1 mol of photons.

Emission Spectra

• Continuous or line spectra of radiation emitted by substances–Energize via heat or high-voltage discharge–Element line spectra will emit only specific l–Every element has a unique spectrum

• Example: Hydrogen

Emission Spectra

• In 1913 Bohr proposed an explanation for the emission spectrum of H–Proposed a new model of atom

• Massive proton at center • Light electron orbit around

– The electron can only occupy certain orbits– Each orbit has a distinct energy (quantized)

Particle/Wave Duality

• Why can the H e- only orbit at certain distances?–De Broglie proposed that e- in atoms

behaved like standing waves• The length of the wave must fit the circumference

of the orbit exactly

Particle/Wave Duality

• De Broglie’s reasoning led to the conclusion that waves can behave like particles and particles can have wave-like properties–Deduced that the particle-wave properties

are related by:

l = h_

m∙u

l = wavelength (m)h = Plank’s constant (J∙s)m = particle mass (kg)u = particle velocity (m/s)

Particle/Wave Duality

• Example: What is the wavelength of an electron moving with a speed of 5.97x106m/s?

Quantum Mechanics

• Bohr’s model only works for one electron species (H, He+, Li+2…)–The problem is with location

• Particles have locations• Waves do not have locations

– If an electron acts as a wave, it’s location must be uncertain…

Quantum Mechanics

• Heisenberg Uncertainty principal– It is impossible to know both the position (x)

and the momentum (p) of an e- at the same time with unlimited precision.

–Therefore the well-defined orbits of the Bohr Model are incorrect

Quantum Mechanics

• Schrödinger Equation (1926)–Advanced mathematical description of the

behavior and energies of submicroscopic particles

HY = EY–Solution yields

• Energy levels available to a particular e-

• Wave functions (Y) that correspond to those energies

–Cannot be solved exactly for atoms with more than one e-

Quantum Mechanics

• Y2 = e- density

= probability of finding the e- in a given region of space

–Atomic Orbital = the wave function of an electron in an atom (no longer use “orbit”)

Quantum Mechanics

• Quantum Numbers–Used to describe the distribution of electrons

in atoms• Derived from the Schrödinger equation for H

–Each electron can be described by a unique set of 4 numbers• (1) principal quantum number, n• (2) angular momentum quantum number, l• (3) magnetic quantum number, ml

• (4) electron spin quantum number, ms

Quantum Mechanics

• Principal Quantum Number (n)–n = 1, 2, 3…–n is related to the average distance between

the e- and the nucleus–For the H atom, n specifies the energy of the

orbital

Quantum Mechanics

• Angular Momentum Quantum Number (l)– l = 0, 1, 2, 3, …, (n-1)– l indicates the shape of the orbital– Indicate the value of l with letters:

–Example: What are the values of l if n=3?• l can be: 0 (s), 1 (p) or 2 (d)

l 0 1 2 3

Orbital type s p d f

Quantum Mechanics

• Magnetic Quantum Number (ml)–ml describes the orientation of the orbital in

space–For a given values of l, (2l+1) values

possible for ml

–ml= -l, (-l+1), (-l+2)…0…(l-2), (l-1), l

–Example: If l=2, what are the values of ml?• ml = [-2, -1, 0, 1, 2]

Quantum Mechanics

• Electron Spin Quantum Number–ms indicates the spin on the e-

–ms = +1/2 or -1/2

–Spinning e- acts like tiny magnets and generate magnetic fields

Quantum Mechanics

• Quantum Number Summary

Quantum Mechanics

• Atomic Orbitals – s– l=0, ml=0–s orbitals have spherical shapes

Quantum Mechanics

• Atomic orbitals – p– l=1, ml=-1,0,1

Quantum Mechanics

• Atomic Orbitals – d– l= 2, ml=-2,-1,0,1,2

Quantum Mechanics

• Electron Configurations–How the electrons are distributed among the

various atomic orbitals• described by the quantum numbers

–The four quantum numbers allow us to label any electron in any orbital (like an address)• Example: Write the four quantum numbers for

an electron in a 4p orbital

Quantum Mechanics

• Orbital Energies• In hydrogen, energy of the orbital determined by

n only

Quantum Mechanics

• Orbital Energies– In atoms w/ multiple electrons, energy of the

orbital determined by n & l

Quantum Mechanics

• Electron configuration for H

1s1

• Orbital Diagram (shows spin)

__

1s

Principal Quantum Number

AngularMomentumQuantumNumber

Number ofElectrons inthe Orbital or Subshell

Quantum Mechanics

• For atoms with more than one electrons, we need more tools to determine how electrons fill their orbitals

–Aufbau Principle- as protons are added one by one to the nucleus to build up the elements, electrons are similarly added to the atomic orbitals

Quantum Mechanics

–Pauli Exclusionary Principle: no two electrons in the same atom can have the same set of quantum numbers

–Hund’s Rule: the most stable arrangement of electrons in subshells is the one with the maximum numbers of parallel electron spins

Quantum Mechanics

Quantum Mechanics

• Electron Configuration of N (Z=7)

• Orbital Diagram

• Energy Level Diagram

Quantum Mechanics

• Example: Show the electron configuration, the orbital diagram, and the energy level diagram for Al

Quantum Mechanics

• Transition Metals–some give rise to unusual electron

configurations• Example: Chromium-

[Ar] 4s13d5 instead of [Ar] 4s23d4

• Example: Copper-

[Ar] 4s13d10 instead of [Ar] 4s23d9

–This occurs because the 4s and 3d orbitals are so close in energy