Ch. 1: Atoms: The Quantum World
CHEM 4A: General Chemistry with
Quantitative Analysis
Fall 2009
Instructor: Dr. Orlando E. Raola
Santa Rosa Junior College
Overview
1.1The nuclear atom
1.2 Characteristics of electromagnetic radiation
1.3 Atomic spectra
1.4 Radiation, quanta, photons
1.5 Wave-particle duality
1.6 Uncertainty principle
The energy of a photon is conserved.
E photon  Ekinetic, electron  Work Function of metal
h 
1
me v 2  
2
frequency velocity
An electron will be ejected when
hν > Φ because Ek,electron will be
non-zero
WARNING
The following material contains heavy
mathematical machinery, including integrals
and differential equations. The purpose is to
show you how scientist arrived at very
important conclusions that will allow you to
understand everyday chemistry. You do not
have to memorize or even attempt to write
down all the numerous mathematical
expressions.
DO NOT RUN AWAY. THEY ARE PERFECTLY
TAME AND BEYOND THIS POINT,
EVERYTHING IS DOWNHILL!!!!
1
2
mev  h  
2
y  mx  b
Diffraction Pattern of Electrons
Constructive
interference
(peak + peak)
Destructive
interference
(peak + trough)
Waves show diffraction…
Small angle x-ray diffraction on colloidal crystal, from http://www.chem.uu.nl/fcc/www/peopleindex/andrei/andrei.htm
Electrons show diffraction…
Electron diffraction taken from a crystalline sample, from http://www.matter.org.uk/diffraction/electron/electron_diffraction.htm
therefore electrons are waves!
h
h


mv p
Heisenberg Uncertainty Principle (1927)
ill defined location
well defined momentum
well defined location
ill defined momentum
Heinsenberg’s Uncertainty Principle
As a result from the analysis of many
experiments and thoughtful theoretical
derivations, Heinsenberg (1927) expressed the
principle that the momentum and the position of
a particle cannot be determined simultaneously
with arbitrary precision. In fact the product of
the uncertainties in these two variables is
always at least as large as Planck constant over
4.
p x 
2
Heisenberg Uncertainty Principle (1927)
In its mathematical expression:
1
p x 
2

Example 1.7
mv x 
2
x 
2mv
-34
1.054571628  10 J  s

3
3
1
2 1.0  10 kg  2.0  10 m  s
2
1.054571628  10 kg  m  s  s

3
3
-1
2 1.0  10 kg  2.0  10 m  s
-34
 2.6  10
29
m
2
The Born interpretation
At a node:
•Ψ2 = 0 (no electron density)
•Ψ passes through 0
electron density
Erwin Schrödinger
Features of the equation:
• Solutions exist for only certain cases.
• The left side is often written as HΨ.
• H is known as the “hamiltonian”.
The Schrödinger equation
d

V(x)  E
2
2m dx
2
H = E
The Particle-in-a-box problem
For the conditions in the box V(x) = 0 everywhere,
energy is only kinetic, and
d 2

 E
2
2m dx
has solutions
 (x)  A sin kx  B cos kx
which gives an expression for E
k 2h2
E
8 2m
The Particle-in-a-box problem
From the boundary conditions
 (0)  0
we get B = 0
the other boundary condition  (L)  0
makes
n
k
L
and the expression for E becomes
k 2h 2
E
8mL2
The Particle-in-a-box problem
To find the constant A, we apply the normalization
condition, since the particle has to be somewhere
inside the box:
 n x 
  (x) dx  A  sin  L  dx  1
L
L
2
0
2
2
0
 2
and then A 
 L 
1
2
and the wavefunction for the particle in a box is
1
2
 2
 n x 
 n    sin 
 L
 L 
n  1,2,3...
Particle in a Box
1
 2  2  nπx 
ψ n ( x)    sin 

L
 L 
n  1, 2,...
values of n
Changing the Box
As L increases:
• energies of levels decrease
• separations between levels
decrease
Lsmall
Llarge
wavefunction (Ψ)
probability density (Ψ2)
lowest density
highest density
Locating Nodes
Ψ passes through 0
Number of nodes = n – 1
Ψ2 = 0
Spherical polar coordinates
colatitude
azimuth
radius
General formula of wavefunctions for the
hydrogen atom
 (r,, )  R(r )Y(, )
For n = 1

 (r, , ) 
2e
r
a0
3
2
0
a


1
2
1
2

e
r
a0
 a 
1
3 2
0
a0 
4 0
mee
2
2
General formula of wavefunctions for the
hydrogen atom
 (r,, )  R(r )Y(, )
1
For n = 2 and E 2   h
4
 (r, , ) 
1
1
2 6
5
2
0
a

re
r
2a0
1
2
r



 3 
1
1
2a0

sin

cos


r
e
sin  cos 

5

4  2 a0 
 4 
Quantum numbers
n: principal quantum number
determines the energy
indicates the size of the orbital
: angular momentum quantum number,
relates to the shape of the orbital
m : magnetic quantum number, possible
orientations of the angular momentum
around an arbitrary axis.
magnetic
quantum number
principal
quantum number
orbital angular momentum
quantum number
Electron probability in the
ground-state H atom.
Radial probability distribution
Allowable Combinations of Quantum Numbers
l = 0, 1, …, (n – 1)
ml = l, (l – 1), ..., -l
No two electrons in the same atom have the
same four quantum numbers.
Higher probability
of finding an
electron
Lower probability
of finding an
electron
most probable radii
The most probable
radius increases as n
increases.
boundary surface
• 90% likelihood of finding
electron within
radial
nodes
Wavefunction (Ψ) is nonzero
at the nucleus (r = 0).
For an s-orbital, there is a nonzero
probability density (Ψ2) at the nucleus.
radial nodes
n=1
l=0
no radial nodes
n=2
l=0
1 radial node
n=3
l=0
2 radial nodes
2p-orbital
n=2
l = 1, 0, or -1
no radial nodes
1 nodal plane
Plot of
wavefunction is
for yellow lobe
along blue arrow
axis.
The three p-orbitals
nodal planes
The labels “x”, “y”, and “z” do not correspond
directly to ml values (-1, 0, 1).
The five d-orbitals
n = 3, 4, …
dark orange (+)
l = 2, 1, 0, -1, -2
light orange (–)
nodal planes
The seven f-orbitals
n = 4, 5, …
dark purple (+)
l = 3, 2, 1, 0, -1, -2, -3
light purple (–)
Allowed orbitals
Allowed subshells
2 electrons
per orbital
Maximum of
32 electrons
for n = 4 shell
Stern and Gerlach Experiment: Electron Spin
Atoms with
one type of
electron spin
Atoms with
other type of
electron spin
Silver atoms
(with one unpaired electron)
Spin States of an Electron
Spin magnetic quantum number (ms) has two possible values:
Relative Energies of Orbitals in a Multi-electron Atom
Z is the atomic number.
After Z = 20, 4s orbitals have
higher energies than 3d orbitals.
Probability maxmima
for orbitals within a
given shell are close
together.
A 3s-electron has a
greater probability of
being found near the
nucleus than 3p- and
3d-electrons due to
contribution of peaks
located closer to the
nucleus.
Paired spins
Lower energy
Parallel spins
Higher energy
Electron Configurations: H and He
1s electron (n, l, ml, ms)
• 1, 0, 0, (+½ or –½)
1s electrons (n, l, ml, ms)
• 1, 0, 0, +½
• 1, 0, 0, –½)
Electron Configurations: Li and Be
1s electrons (n, l, ml, ms)
• 1, 0, 0, +½
• 1, 0, 0, –½
1s electrons (n, l, ml, ms)
• 1, 0, 0, +½
• 1, 0, 0, –½
2s electron*
• 2, 0, 0, +½
2s electrons
• 2, 0, 0, +½
• 2, 0, 0, –½
* one possible assignment
Electron Configurations: B and C
1s electrons (n, l, ml, ms)
• 1, 0, 0, +½
• 1, 0, 0, –½
1s electrons (n, l, ml, ms)
• 1, 0, 0, +½
• 1, 0, 0, –½
2s electrons
• 2, 0, 0, +½
• 2, 0, 0, –½
2s electrons
• 2, 0, 0, +½
• 2, 0, 0, –½
2p electron*
• 2, 1, +1, +½
2p electrons*
• 2, 1, +1, +½
• 2, 1, 0, +½
* one possible assignment
* one possible assignment
Filling order for orbitals
subshell being filled
maximum number of electrons in subshell
The Hydrogen atom: atomic orbitals
The potential in a hydrogen atom can be
expressed as
2
V(x)  
e
4 0r
Schrödinger (1927) found that the exact
solutions for his equation give expression for
the energy as
h
E 2
n

mee
4
8h 
3 2
0
n  1,2,3....
Quantum Numbers and Atomic Orbitals
An atomic orbital is specified by three
quantum numbers.
n the principal quantum number
- a positive integer
ℓ the angular momentum quantum number
- an integer from 0 to n-1
mℓ the magnetic moment quantum number
- an integer from -ℓ to +ℓ
Quantum Numbers
1.Principal (n = 1, 2, 3, . . .) - related to size and
energy of the orbital.
2.Angular Momentum (ℓ = 0 to n  1) - relates to
shape of the orbital.
3.Magnetic (mℓ = ℓ to ℓ) - relates to orientation
of the orbital in space relative to other orbitals.
4.Electron Spin (ms = +1/2, 1/2) - relates to the
spin states of the electrons.
Table 7.2 The Hierarchy of Quantum Numbers for Atomic Orbitals
Name, Symbol
(Property)
Allowed Values
Principal, n
Positive integer
(size, energy)
(1, 2, 3, ...)
Quantum Numbers
1
2
3
Angular
momentum, ℓ
(shape)
0 to n-1
Magnetic, mℓ
-ℓ,…,0,…,+ℓ
(orientation)
0
0
0
0
1
0
1
2
0
-1 0 +1
-1 0 +1
-2
-1
0
+1 +2
Sample Problem 7.5
Determining Quantum Numbers for an Energy Level
PROBLEM: What values of the angular momentum (ℓ) and magnetic (m )
ℓ
quantum numbers are allowed for a principal quantum number (n) of
3? How many orbitals are allowed for n = 3?
PLAN: Follow the rules for allowable quantum numbers found in the text.
l values can be integers from 0 to n-1; mℓ can be integers from -ℓ
through 0 to + ℓ.
SOLUTION: For n = 3, ℓ = 0, 1, 2
For ℓ = 0 mℓ = 0
For ℓ = 1 mℓ = -1, 0, or +1
For ℓ= 2 mℓ = -2, -1, 0, +1, or +2
There are 9 mℓ values and therefore 9 orbitals with n = 3.
Sample Problem 7.6
Determining Sublevel Names and Orbital Quantum
Numbers
PROBLEM: Give the name, magnetic quantum numbers, and number of orbitals
for each sublevel with the following quantum numbers:
(a) n = 3, ℓ = 2
(b) n = 2 ℓ= 0
(c) n = 5, ℓ = 1 (d) n = 4, ℓ = 3
PLAN: Combine the n value and ℓ designation to name the sublevel.
Knowing ℓ, we can find mℓ and the number of orbitals.
SOLUTION:
n
ℓ
(a)
3
2
3d
-2, -1, 0, 1, 2
5
(b)
2
0
2s
0
1
(c)
5
1
5p
-1, 0, 1
3
(d)
4
3
4f
-3, -2, -1, 0, 1, 2, 3
7
sublevel name possible mℓ values # of orbitals
1s
2s
3s
The 2p orbitals.
Representation of the 1s, 2s and 3s orbitals in
the hydrogen atom
Representation of the 2p orbitals of the
hydrogen atom
Representation of the 3d orbitals
Representation of the 4f orbitals
Types of
Atomic
Orbitals
Levels and sublevels
When n = 1, then ℓ = 0 and mℓ = 0
Therefore, in n = 1, there is 1 type of sublevel
and that sublevel has a single orbital
(mℓ has a single value  1 orbital)
This sublevel is labeled s (“ess”)
Each level has 1 orbital labeled s, and it is
SPHERICAL in shape.
s orbital are spherical
Dot picture of
electron cloud in
1s orbital.
Surface
density
4πr2 versus
distance
Surface of 90%
probability
sphere
1s orbital
2s orbitals
3s orbital
p orbitals
When n = 2, then ℓ = 0 and 1
Therefore, in n = 2 levell there are
2 types of orbitals
— 2 sublevels
For ℓ = 0
mℓ = 0
this is a s sublevel
For ℓ = 1 mℓ = -1, 0, +1
this is a p sublevel
with 3 orbitals
When l = 1, there is
a PLANAR
NODE through the
nucleus
p Orbitals
The three p orbitals lie 90o apart in space
2px Orbital
3px Orbital
d Orbitals
When n = 3, what are the values of ℓ?
ℓ = 0, 1, 2
and so there are 3 sublevels in level n=3.
For ℓ = 0, mℓ = 0  s sublevel with single orbital
For ℓ = 1, mℓ = -1, 0, +1  p sublevel with 3 orbitals
For ℓ = 2, mℓ = -2, -1, 0, +1, +2
d
sublevel with 5 orbitals
d Orbitals
s orbitals have no planar
node (ℓ = 0) and so are
spherical.
p orbitals have ℓ = 1, and
have 1 planar node,
and so are “dumbbell”
shaped.
This means d orbitals
(with ℓ = 2) have 2 planar
nodes
3dxy Orbital
3dxz Orbital
3dyz Orbital
3dx2- y2 Orbital
3dz2 Orbital
f — Orbitals
One of 7 possible f
orbitals.
All have 3 planar
surfaces.
Can you find the 3
surfaces here?
f — Orbitals
Spherical Nodes
2 s orbital
•Orbitals also have spherical
nodes
•Number of spherical nodes
=n-l-1
•For a 2s orbital:
No. of nodes = 2 - 0 - 1 = 1
Summary of Quantum Numbers of Electrons in Atoms
Name
Symbol
Permitted Values
Property
principal
n
positive integers(1,2,3,…) orbital energy (size)
angular
momentum
ℓ
integers from 0 to n-1
magnetic
mℓ
integers from -ℓ to 0 to +ℓ
orbital shape (The ℓ values
0, 1, 2, and 3 correspond to
s, p, d, and f orbitals,
respectively.)
orbital orientation
spin
ms
+1/2 or -1/2
direction of e- spin
The 3d orbitals
One of the seven
possible 4f orbitals.
Schematic representation of the
energy levels of the hydrogen atom