Modern Theory of the Atom

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Modern Theory of the Atom

• Quantum Mechanical Model
• Or
• Wave Mechanical Model
• Or
• Schrodingers Model

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Recap of Bohr Model

• Electrons treated as particles moving in circular
  orbits. Specify speed, position, energy.
• Quantization of energy levels is imposed.
• Ground state electrons close to nucleus
• Electron transitions between energy levels can
  occur. Higher energy levels are farther from
  nucleus.
• Moving up, electron absorbs energy
• Moving down, electron emits light energy
• Wavelengths of light in H spectrum can be
  predicted. Depend on energy difference of 2
  levels involved in transition.

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Problems with Bohr Model

• Only worked for 1-electron systems.
• Quantization of energy levels had to be
  imposed.

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By the end of the 1800s

• Physicists thought they were done!
• Light or energy was waves. Could come in any
  frequency or wavelength. No mass. Delocalized.
• Matter was particles. Have mass. Specify
  position.
• Nice packages, everything done. Matter Energy
  were DIFFERENT!
• But there was trouble ahead!

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Two problems

• Black body radiation
• Photoelectric effect

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Energy of a photon

• Einstein proposed that light comes in discrete
  packages called photons.
• Einstein Ephoton h?
• All radiation is quantized!
• Each energy of radiation will have its own
  frequency.

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Photoelectric Effect
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Photons

• One photon can give all its energy to one
  electron. One electron can only accept one
  photon.
• So if the photon is energetic enough, the
  electron will escape from the metal. If the
  photon is low energy, the electron cant escape.
  (Threshold)
• Brighter light means more photons so more
  electrons can escape. (Number)

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Wave-particle Duality

• Light cant be forced into categories like
  everyday objects.
• In some situations, light exhibits interference
  phenomena, like water waves.
• In some situations, light shows energy transfers
  like particles in a collision.

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E mc2

• Ephoton h? hc/? mc2
• So m hc/? ? c2
• m h/?c for a photon.

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Electrons as Waves

• 1924 Louis de Broglie Dual nature of MATTER
• Suggested that since light can act like a
  particle, maybe a particle, the electron, can act
  like a wave.
• m h/?c for a photon.
• Electrons are not electromagnetic waves. They
  are matter waves.
• m h/?v where v velocity of the particle.

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2 kinds of waves

• Standing wave
• Confined to a given space. (Ends pinned.)
• Interference between incident reflected waves.
• At certain frequencies, certain points seem to be
  standing still.
• Other points, displacement changes in a regular
  way.

• Traveling wave
• Wave is not confined to a given space
• Travels from one location to another
• Interrupted by a boundary or another wave

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Standing waves in music
Applet with violin string
So boundary conditions result in only some
wavelengths that will fit on the string
quantization
Wave reflection of a pulse at a fixed end
Interference of two waves traveling through a
medium
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DeBroglie Electron-Wave
The wavelength describing an electron depends on
the energy of the electron. At certain energies,
electron waves make standing waves in the atom.
The wave does not represent electron path.
Only certain wavelengths will fit.
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DeBroglie Electron Waves
These wavelengths will work.
This wavelength will not work.
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Bright spots result from constructive
interference of waves. The waves are in phase
(peaks match). Dark areas result from
destructive interference of waves. Waves are out
of phase. Only waves make diffraction patterns.
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Modern Theory

• Electron is treated as a standing wave.
• Cannot specify both position velocity of
  electron.
• Can determine probability of locating the
  electron in a given region of space.
• Quantized energy levels arise naturally out of
  wave treatment.
• Also called Quantum Mechanics or Wave mechanics.
  Scientist Schrodinger.

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Bohr Model vs. Modern Theory

• Electron particle
• Orbit
• Holds 2n2 electrons
• Spherical
• Each orbit has a specific energy
• Can find position, speed

• Electron Wave
• Orbital
• Holds 2 electrons
• Not necessarily spherical
• Each orbital has a specific energy
• Probable location

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Schrödingers Equation

• H? E?
• Solve for ?, the wave functions.
• ?2 gives the probability of finding an electron
  near a particular point in space.
• Represented as probability distribution or
  electron density map.

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Heisenberg uncertainty principle

• Fundamentally impossible to know the velocity and
  position of a particle at the same time.
• Impossible to make an observation without
  influencing the system.
• A photon colliding with an electron will knock it
  off its path.

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Orbital Modern Theory

• Orbital term used to describe region where an
  electron might be.
• Each orbital has a specific energy and a
  specific shape. Each holds 2 electrons.
• Described by 4 parameters in the wave function
  quantum numbers n, l, m, s like an address

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s orbitals (?2)
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p orbitals
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d orbitals
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What can orbitals do for us?

• Physical structure of orbitals explains
• Bonding
• Magnetism
• Size of atoms
• Structure of crystals

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Quantum Numbers

• Each electron in an atom has a set of 4 quantum
  numbers like an address.
• 3 quantum numbers describe the orbital
• 1 quantum number gives the electron spin
• No two electrons can have all 4 quantum numbers
  the same. (Pauli exclusion principle)

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Energy level diagram for orbitals of Hydrogen
atom. H? E? can be solved exactly Only 1
electron.
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Energy levels for Polyelectronic atom
Energy Level Diagram
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n principal quantum number

• Related to size and energy of orbital
• n has integral values 1, 2, 3, 4,
• As n increases, the orbital becomes larger the
  electron spends more time farther from the
  nucleus, which also means higher energy.

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l angular momentum quantum number

• Related to shape of orbital.
• l has integral values from 0 to n -1 for each
  value of n.
• Orbitals with different shapes have slightly
  different energies. Each type of orbital resides
  on a different sublevel of the principle energy
  level.

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l angular momentum quantum number

• Principal energy levels are made up of sublevels.
• The number of sublevels depends on the principal
  energy level.
• 1st principal energy level has 1 sublevel
• 2nd 2
  
• 3rd
  3
• 4th
  4 , etc.

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Naming sublevels

• Sublevels are usually labeled s, p, d, or f
  instead of using more numbers.
• If l 0, call it an s orbital.
• If l 1, call it a p orbital.
• If l 2, call it a d orbital.
• If l 3, call it an f orbital.

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ml magnetic quantum number

• ml related to orientation of orbital in space
  relative to other orbitals in the atom.
• ml has integral values between l and -l,
  including 0.
• For n 1, l 0 and ml 0.
• For n 2, l 0 or 1.
• If l 0 then ml 0
• If l 1, then ml -1, 0, or 1.

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orbitals

• Sublevels are made up of orbitals
• Each kind of sublevel has a specific of orbitals

Sublevel of orbitals
s 1
p 3
d 5
f 7
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Spin quantum number, ms

• ms describes the spin state of the electron in
  the orbital.
• ms has two possible values ½ and ½
• Pauli exclusion principle No two electrons in
  the same atom can have all 4 quantum numbers the
  same. So each orbital can hold only two
  electrons.

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Orbitals

• Each orbital can hold two electrons with opposite
  spins.
• s sublevels, 1 orbital 2 e- max capacity
• p sublevels, 3 orbitals 6 e-
• d sublevels, 5 orbitals 10 e-
• f sublevels, 7 orbitals 14 e-

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Prin.En.Lev Sublevels orbitals/sl Total elec
1 s 1 2
2 s 1 2
p 3 6
3 s 1 2
p 3 6
d 5 10
4 s 1 2
p 3 6
d 5 10
f 7 14
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3rd principal energy level, 3 sublevels
2nd principal energy level, 2 sublevels s p
1st principal energy level, 1 sublevel s
Each box represents an orbital and holds 2
electrons.
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Order of fill Aufbau principle

• Each electron occupies the lowest orbital
  available
• Learn sequence of orbitals from lowest to highest
  energy
• Is some overlap between sublevels of different
  principal energy levels

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Diagonal Rule
Sequence of orbitals 1s, 2s, 2p, 3s, 3p, 4s, 3d,
4p, 5s, 4d, Follow the arrows Exceptions do
occur half-filled orbitals have extra stability.

• 1s
• 2s 2p
• 3s 3p 3d
• 4s 4p 4d 4f
• 5s 5p 5d 5f
• 6s 6p 6d 6f
• 7s 7p

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Hunds Rule

• Distribution of electrons in equal energy
  orbitals Spread them out as much as possible!
• Also, all electrons in singly occupied orbitals
  must have the same spin state.

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Electron Configurations
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Compare Bohr Schrodinger
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Frequencies in Chemistry
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Electron Configuration P.T.
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Principle Energy Levels
Hold 2 Electrons Max
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Sublevels
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Orbitals
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1st energy level has 1 sublevel s 2nd
2 sublevels s and p 3rd
3 s, p, and d 4th
4 s, p, d, and f
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n 1,2,3,4 Holds 2n2 Electrons max
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s sublevel holds 1 orbital p sublevel holds 3
orbitals d sublevel holds 5 orbital f sublevel
holds 7 orbitals