in Electrons Atoms
Rutherford atom
Electrons are not drawn into nucleus
Petrucci 7.5
Quantum Mechanics
Tiny, fast objects (like electrons around the nucleus) behaves differently
than macroscopic objects
To describe the behavior of such object we need to use quantum mechanics
Wave-Particle duality
Quantum particles move from one point to another as if they are waves
At a detector they always appear as discrete lumps of matter
Heisenbergs Uncertainty principle
Petrucci 7.1
Wave-Particle duality causes fundamental limitations on the simultaneous determination of position and
momentum
Interferance
Diffraction
Petrucci 7.5
Particle-Wave Duality
Matter and energy exhibits both wave-like and particle-like properties
The wavelength of matter is called the
"de Broglie wavelength"
=
m
v
=
h
h
=
p
mu
h
mu
Electron
9.1 1031 kg
1 107 m/s
7 1011 m
Atomic sizes
Baseball
0.10 kg
35 m/s
2 1034 m
<< atom
De Broglie Wavelength
The wavelength of matter is called the "de Broglie wavelength"
=
m
v
=
h
mu
h
h
=
p
mu
Electron
Baseball
31
9.1 10
kg
0.10 kg
1 107 m/s
35 m/s
11
7 10
m 2 1034 m
Atomic sizes
<< atom
Is It a Wave???? Is It a Particle????
The de Broglie relationship assigns wave properties to particles and particle properties to waves! Does this mean that particles convert into waves and back?
2
Its neither a circle nor a triangle, but what you see depends on the type of experiment!
Petrucci 7.1, 7.3
Light
Electromagnetic wave of oscillating electric and magnetic elds that in vacuum travels at the speed of light, c = 3.0 108 m/s
Stream of particles (Photons) each with energy E = h
Plancks constant, h = 6.63 1034 Js
Frequency =
c
(unit s1 = Hz)
Electromagnetic Spectrum
White light consists of all colors
Dispersing light though a prism splits it
into its spectral components
3
7.5 1014 s1
4.3 1014 s1
Petrucci 7.2
Atomic Spectra
Spectra of geseous atoms consists of discrete lines (only specic frequencies)
Electrons can only have discrete energies
Transition between two discrete energy levels involves an energy change,
E = E2 E1
Conservation of energy then requires
absorption or emission of photon with
matching energy
h = E
Petrucci 7.4
The Bohr Atom
Niels Bohr made a semi classical model that successfully explained the experimentally observed lines in the hydrogen atom
"Quantized Planetary Model"
4
En =
RH
n2
n = 1, 2, 3,
RH = 2.179 1018 J
n is the angular quantum number
The higher n, the further away from the
nucleus
Example 1
The Bohr Atom
1. Would a transition from energy level n = 7 to n = 3 require absorption or
emission of a photon?
2. What would be the wavelength of the photon for the above transition?
Petrucci 7.5
Heisenbergs Uncertainty Principle
Wave-particle duality causes fundamental limitation on the simultaneous
determination of position and momentum (p = mv )
xp > h
We cant know initial position and velocity to absolute accuracy
cant predict absolute trajectory
5
Petrucci 7.6
Standing waves
Water standing waves formed with vertical oscillation in a
circular dish
Ray Tomes: http://ray.tomes.biz/cymatics.htm
2L
n = 1, 2, 3,
n
Total number of nodes = n + 1
=
Schrdinger Equation
(r, t) = H (r, t)
t
The Schrdinger equation is an equation that describes how the quantum
state of a physical system changes in time
ih
It is as central to quantum mechanics as Newtons laws are to classical
mechanics
The particle is described by its wave function,
2 probability density
Particle in a Box
6
Electron
=
h
mu
The electron will only "t" in the box if the wave function is a standing
wave.
The possible wave functions of a particle is determined by the potential it
experiences
A simple example is the "particle in a box" problem that can easily be solved analytically
Wave functions are standing 1D waves
n (x) =
2
nx
sin
L
L
=
2L
n
Node: = 0
Total number of nodes = n + 1
Zero Point Energy
Lowest energy possible (n = 1) is called the zero-point energy
Zero-point energy is non-zero!
Particle cannot be at rest (Heisenberg)
De Broglie
Kinetic energy
h
mu
h
u=
m
=
1
Ek = mu2
2
h2 2
=
n
8mL2
7
=
h
m(2L/n)
1
=m
2
hn
2Lm
2
Petrucci 7.7
Multi Dimensional Standing Waves
3D Wave Functions Quantum Numbers
The electron moves in three dimensions and we therefore need three quantum
numbers
Principal quantum number, n n = 1, 2, 3, 4, ... Relates to the energy and
most probable distance of an electron from the nucleus
Angular momentum quantum number,
= 1, 2, 3, ..., n 1 Determines
angular distribution
Magnetic quantum number, m m = , + 1, ..., 1, 0, 1, ..., 1,
Determines orientation
Transition
Metals
Lanthanides
(3f ) & Actinides (4f )
Petrucci 7.8
d( = 2)
f ( = 3)
group
IIIA
VIIA
group
IA & IIA
Atomic Orbitals
s( = 0) p( = 1)
n=1
n=2
n=3
n=4
8
Petrucci 7.9
Fourth Quantum Number: Spin
Two electrons can never exist in the exact same quantum state
Each electron is characterized by its four quantum numbers:
Spatial quantum numbers: n, , and m
Spin quantum number, ms
Two electrons in each orbital with opposite spins
1
Electrons can either have spin up (ms = 2 ) or spin
down (ms = 1 )
2
Moving charge results in magnetic eld
Petrucci 7.12
Responsible for magnetic properties of atoms
Petrucci 7.8
Hydrogen Orbitals
The Schrdinger equation can be solved analytically for the hydrogen atom
(n, , m ) = radial function(n, ) angular function( , m )
Angular and radial wave functions can be found in table 7.1
Angular distribution independent of n
The orbitals are named according to the value of :
= 0 s-orbital
= 2 d-orbital
= 1 p-orbital
= 3 f -orbital
9
Angular and Radial Nodes
Points with = 0 are called nodes
For the hydrogen atom we distinguish between angular and radial nodes
depending on whether the angular or radial wave function is zero
The number of angular nodes is equal the to angular momentum quantum
number
# angular nodes =
The total number of nodes (angular and radial) for a hydrogen orbital is
always one lower than the principle quantum number
total # nodes = n 1
s Orbitals
Orbitals with angular momentum = 0 are called s orbitals
Only one possible value for m = 0
n=1
n=2
n=3
n=4
0 radial nodes
1 radial node
2 radial nodes
3 radial nodes
The higher n, the further the electron is from the core
p orbitals
Orbitals with angular momentum = 1 are called p orbitals
Three possible values for m = 1, 0, 1
Lowest p orbital is 2p
10
2py
2px
Zero probability of nding
electron in yz plane
Zero probability of nding
electron in xz plane
The pz orbital: zero
probability of nding
electron in xy plane
2px
3px
4px
1 angular node
1 angular node
1 angular node
0 radial nodes
1 radial node
2 radial nodes
Number of angular nodes equals
Total number of nodes equals n 1
d orbitals
Orbitals with angular momentum = 2 are called d orbitals
Five possible values for m = 2, 1, 0, 1, 2
Lowest d orbital is 3d
11
3dxz
3dyz
3dxy
Zero probability of nding
electron in yz and xy
planes
Zero probability of nding
electron in xz and xy
planes
Zero probability of nding
electron in xz and yz
planes
3dz2
3dx2 y2
2pz
3dz2
Number of angular nodal planes =
Total number of nodes equals n 1
Example 2
Identify Orbital
What orbital is depicted here?
12
4fz3
Example 3
Identify Orbital
What are the values of n and for the orbital depicted here?
http://cat.sckans.edu/physics/hydrogenic.htm
Example 4
Shapes of Orbitals
Identify the orbitals (be as specic as possible):
1. No radial nodes and 2 angular nodes
2. 4 radial nodes and one angular node in the xy -plane.
3. d orbital with xy and yz nodal planes
4. s orbital with 2 radial nodes
Specify the number of angular and radial nodes for the following orbitals (specify
nodal planes if possible)
1. 5py
2. 4f
Example 5
Quantum Numbers and Electron Orbitals
How many electrons can have the following sets of quantum numbers:
1. n = 2? 8
n
0
2
1
m
0
2s
1 2px
0 2py
1 2pz
13
ms
+1
2
1
2
2. n = 2, and ms = 1 ? 4
2
3. n = 2, and m = 0? 4
4. n = 2, and m = 1, and ms = 1 ? 2
2
1
5. n = 2, m = 1, and ms = 2 ? 1
6. n = 2, = 0, m = 1, and ms = 1 ? 0
2
Petrucci 7.10
Multielectron Atoms
For atoms with atomic number Z > 1
More protons in core
Electrons repel each other
The orbitals have same angular shape as hydrogen orbitals
Radial distribution is different
Energy levels are different
In hydrogen all orbitals within a shell are degenerate (have same energy), for example 2s and 2p
In multielectron atoms only orbitals within a sub-shell are degenerate
Shielding
For a given electron:
Electrons in shells with lower principle quantum number shield with a value
of 1
Electrons in same shell shield with a value between 01, electrons with
lower shield more than electrons with higher because they are closer to
the core
14
Energy Levels in Multielectron Atoms
Petrucci 7.11
Trends in Filling Orbitals
1. Electrons occupy orbitals in a way that minimizes the overall energy of the
atom
Sub shells with lower energy occupied rst (see next slide)
Caution: The energy levels are not xed values independent of number
of electrons!
2. The Pauli exclusion principle: No two electrons can have all four quantum
numbers alike
For two electrons to occupy same orbital they MUST have opposite
spins
3. Hunds rule: Every orbital in a subshell is singly occupied with one electron before any one orbital is doubly occupied, and all electrons in singly
occupied orbitals have the same spin.
It "costs" energy to pair electrons in same orbital (pairing energy)
15
Representing Electron Congurations
Examples:
[Nobel Gas Cong]nlx
Lithium (Z = 3)
Locate element in P.T
[He]2s1
Locate previous noble gas
Zinc (Z = 30)
Identify period (n)
[Ar]4s2 3d1 0
Count!
Bromine (Z = 35)
[Ar]4s2 3d1 04p5
Example 6
Electronic Congurations
Write the electronic conguration for each of the following elements. How many
unpaired electrons does each element have?
1. Chlorine?
2. Iron?
3. Tin?
4. Iridium?
Symmetric Electron Distributions are Stabilized
Exceptions to the general rules
No rules (almost) without exceptions!
Full and half-full orbitals are particularly stable
For the heavier elements where the energy levels are close it is sometimes
more favorable (energetically) to have a combination of full and half-lled
orbitals rather than following the general rules
Examples:
16
Element
Chromium
Copper
# valence electrons
6
11
Following Rules
[Ar]4s2 3d4
[Ar]4s2 3d9
Actual
[Ar]4s1 3d5
[Ar]4s1 3d10
Explanation
Two half full
Half full + Full
Silver and gold follow same trend as copper.
Molybdenum follows the same rule as chromium, but tungsten DOES NOT
Cant be predicted without detailed calculations
Concepts
Interference
Standing wave
Photon
Wave function,
Wave-particle duality
Orbital
Heisenbergs uncertainty principle
Shell and subshell
De Broglie wavelength
Shielding and penetration
Atomic spectrum
Effective charge
Absorption and emission
Aufbau Principle
The Bohr atom
Hunds rule
Quantum number
Unpaired electron
Degenerate
You should be able to
Calculate energy of a photon from wavelength or frequency and vice verss
Determine wavelength of a photon needed for a specic transition in the
Bohr atom (for example from n = 5 8) and the transition requires
absorption or emission of a photon
Draw wave functions for a particle in a box for a given n and specify where
and how many nodes it has
Draw angular shapes of s, p, and d orbitals and specify location of nodal
plane(s)
Specify number of angular nodal planes and radial nodes for any orbital
given n and l and vice versa
Write electronic congurations for any of the main group elements and for
transition metals that follow the aufbau principle and Hunds rule and specify how many unpaired electrons an element has
17