Quantum or Wave Mechanics
Quantum or Wave Mechanics
de
de Broglie
Broglie (1924)
(1924) proposed
proposed
that
that all
all moving
moving objects
objects
have
wave
properties.
have wave properties.
For
For light:
light: EE == mc
mc22
EE == hν
ν == hc
hhν
hc // λλ
L. de Broglie
(1892-1987)
Therefore,
Therefore, mc
mc == hh // λλ
and
for
particles
and for particles
(mass)(velocity)
(mass)(velocity) == hh // λλ
WAVE FUNCTIONS, Ψ
Experimental proof of wave
properties of electrons
e- with velocity =
1.9 x 108 cm/sec
EQUATION
Solution gives set of math
expressions called WAVE
E. Schrodinger
FUNCTIONS, Ψ
1887-1961
Each describes an allowed energy
state of an eQuantization introduced naturally.
λ = 0.388 nm
Uncertainty Principle
angles.
— the region of space within which an
electron is found.
location of the electron.
He developed the WAVE
λ = 1.3 x 10 -32 cm
• Each Ψ corresponds to an ORBITAL
• Ψ2 is proportional to the probability of
Schrodinger applied idea of ebehaving as a wave to the
problem of electrons in atoms.
Baseball (115 g) at
100 mph
• Ψ is a function of distance and two
• Ψ does NOT describe the exact
Quantum or Wave Mechanics
W. Heisenberg
1901-1976
Problem
Problem of
of defining
defining nature
nature
of
of electrons
electrons in
in atoms
atoms
solved
Heisenberg
solved by
by W.
W. Heisenberg.
Heisenberg..
Cannot
Cannot simultaneously
simultaneously
define
define the
the position
position and
and
momentum
momentum (=
(= m•v)
m•v) of
of an
an
electron.
electron.
We
We define
define ee- energy
energy exactly
exactly
but
but accept
accept limitation
limitation that
that
we
we do
do not
not know
know exact
exact
position.
position.
finding an e- at a given point.
Types of
Orbitals
s orbital
p orbital
d orbital
Page 1
Orbitals
• No more than 2 e- assigned to an
orbital
• Orbitals grouped in s, p, d (and f)
subshells
s orbitals
p orbitals
d orbitals
Subshells & Shells
n=1
n=2
n=3
n=4
s orbitals
Subshells & Shells
p orbitals
d orbitals
s orbitals
p orbitals
d orbitals
No.
orbs.
1
3
5
No.
e-
2
6
10
• Subshells grouped in shells.
• Each shell has a number called
the PRINCIPAL QUANTUM
NUMBER, n
• The principal quantum number
of the shell is the number of the
period or row of the periodic
table where that shell begins.
QUANTUM NUMBERS
Each orbital is a function of 3 quantum
numbers:
n (major)
--->
l (angular) --->
ml (magnetic) --->
shell
subshell
designates an orbital
within a subshell
Page 2
QUANTUM NUMBERS
Symbol
Values
n (major)
1, 2, 3, ..
Description
Orbital size
and energy
where E = -R(1/n 2)
l (angular)
0, 1, 2, .. n-1
Orbital shape
or type
(subshell)
subshell)
ml (magnetic) -l..0..+l
Orbital
orientation
# of orbitals in subshell = 2 l + 1
Shells
Shells and
and Subshells
Subshells
When n = 1, then l = 0 and m l = 0
Therefore, in n = 1, there is 1 type of
subshell
and that subshell has a single orbital
(ml has a single value ---> 1 orbital)
ss Orbitals
Orbitals
1s Orbital
All s orbitals are spherical in shape .
This subshell is labeled s (“ess
”)
(“ess”)
Each shell has 1 orbital labeled s,
and it is SPHERICAL in shape.
2s Orbital
See
See Figure
Figure 7.14
7.14 on
on page
page 319
319 and
and
Screens
Screens 7.10
7.10 and
and 7.11.
7.11.
3s Orbital
p Orbitals
Typical
Typical pp orbital
orbital
When
When nn == 2,
2, then
then ll == 00 and
and 11
Therefore,
Therefore, in
in nn == 22 shell
shell
there
there are
are 22 types
types of
of
planar
orbitals
planar node
node
orbitals —
— 22 subshells
subshells
For
For ll == 00 m
mll == 00
When l = 1, there is
this
a
this is
is aa ss subshell
subshell
PLANAR NODE
For
For ll == 11 m
mll == -1,
-1, 0,
0, +1
+1
thru
this
this is
is aa p
p subshell
subshell
the nucleus.
with
3
orbitals
with 3 orbitals
See Screens 7.11 and 7.13
Page 3
pp Orbitals
Orbitals
2px Orbital
2py Orbital
3px Orbital
3py Orbital
pz
90o
A p orbital
px
py
The three p
orbitals lie 90o
apart in space
2pz Orbital
Page 4
dd Orbitals
Orbitals
3pz Orbital
When n = 3, what are the values of l?
l = 0, 1, 2
and so there are 3 subshells in the shell.
For l = 0, ml = 0
---> s subshell with single orbital
For l = 1, ml = -1, 0, +1
---> p subshell with 3 orbitals
For l = 2, ml = -2, -1, 0, +1, +2
--->
3dxy Orbital
d subshell with 5 orbitals
3dxz Orbital
Page 5
dd Orbitals
Orbitals
typical d orbital
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
planar node
planar node
See
See Figure
Figure 7.16
7.16
3dyz Orbital
3dz2 Orbital
3dx2- y2 Orbital
ff Orbitals
Orbitals
When n = 4, l = 0, 1, 2, 3 so there are 4
subshells in the shell.
For l = 0, ml = 0
---> s subshell with single orbital
For l = 1, ml = -1, 0, +1
---> p subshell with 3 orbitals
For l = 2, ml = -2, -1, 0, +1, +2
---> d subshell with 5 orbitals
For l = 3, m l = -3, -2, -1, 0, +1, +2, +3
---> f subshell with 7 orbitals
Shell Principal Quantum Number, n
1
2
3
Relate to n
No.
Subshells
No.
Orbitals
1
2
3
=n
1
4
9
= n2
No. e-
2
8
18
= 2 n2
Page 6