Modern Theory of the Atom
Quantum Mechanical Model
Or
Wave Mechanical Model
Or
Schrodinger’s 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.”
1924: De Broglie
• Proposed that if light can show both
particle and wave behavior, maybe matter
can too.
Every wavelength of light has its
own unique frequency and its own
unique energy.
2 kinds of waves
Traveling wave
• Wave is not confined
to a given space
• Travels from one
location to another
• Interrupted by a
boundary or another
wave
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 #1
• Traveling Wave #2
Guitar string
• Standing wave #1
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.
Guitar vs. Electron
• In the guitar string, only multiples of halfwavelengths are allowed.
• For an orbiting electron, only whole numbers of
wavelengths allowed.
 = h/mv
Where h=Planck’s constant, m=mass, v=velocity
Modern Theory
• Electron is treated as a 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. Scienties = Schrodinger.
Schrödinger’s Equation
Ĥ = 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.
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.
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
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
s orbitals (2)
p orbitals
d orbitals
What can orbitals do for us?
• Physical structure of orbitals explains
– Bonding
– Magnetism
– Size of atoms
– Structure of crystals
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)
Energy Level Diagram Energy levels for Polyelectronic atom
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.
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.
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.
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.
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.
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
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.
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-
Prin.En.Lev Sublevels
1
s
2
s
p
3
s
p
4
d
s
p
d
f
# orbitals/sl Total # elec
1
2
1
2
3
6
1
2
3
6
5
1
3
5
7
10
2
6
10
14
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.
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
Diagonal Rule
1s
2s
3s
4s
5s
6s
7s
2p
3p
4p
5p
6p
7p
3d
4d
5d
6d
4f
5f
6f
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.
Hund’s 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.
Electron Configurations
Compare Bohr & Schrodinger
Frequencies in Chemistry
Electron Configuration & P.T.
Principle
Energy
Levels

n = 1,2,3,4
Holds 2n2
Electrons
max

Hold 2
Sublevels  Orbitals 
Electrons
Max
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
s sublevel holds 1 orbital
p sublevel holds 3 orbitals
d sublevel holds 5 orbital
f sublevel holds 7 orbitals