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The atmosphere, part 1
Ch. 8
Midterm … No lecture
The atmosphere, part 2
Light, blackbodies, Bohr
Postulates of QM, p-in-a-box
Ch. 8
Ch. 9
Ch. 9
Hydrogen atom
Ch. 9
Multi-electron atoms
Periodic properties
Periodic properties
Valence-bond; Lewis structures
Hybrid orbitals; VSEPR
VSEPR
MO theory
MO theory
Review for exam
Ch.10
Ch. 10
Ch. 10
Ch. 11
Ch. 11, 12
Ch. 12
Ch. 12
Ch. 12
PARTICLE IN A BOX
(x)  A sin kx
ENERGY
d 
 2
 E
2
8 m dx
2
h
2
(0) = 0
BOUNDARY
CONDITION
(L) = 0
0
x
L
PARTICLE IN A BOX
2 2
2 2
ENERGY
n=3
h 3
E3 
2
8 mL
2 2
n=2
n =1
h 2
E2 
2
8 mL
2
h
E1 
2
8 mL
3
hn
En 
2
8 mL
2
1
QUANTIZED
PARTICLE IN A BOX
The application of the BOUNDARY CONDITIONS
Gives a series of QUANTIZED ENERGY LEVELS
ONLY CERTAIN ENERGIES ALLOWED!
DETERMINED BY THE
NUMBER n
2 2
hn
En 
2
8 mL
n  1, 2, 3.......
n is a QUANTUM NUMBER
THE WAVEFUNCTIONS
n(x) 
2
n
sin
x
L
L
ENERGY
n=3
3
2
n=2
n =1
1
A NODE

CHANGES
SIGN
THE NUMBER OF
NODES IS GIVEN BY
N-1 ….. INCREASING
NUMBER OF NODES
AS THE ENERGY
INCREASES
What does  Mean?????
The answer lies in
WAVE-PARTICLE DUALITY
Electrons have both wavelike and particle like
properties.
Because of the wavelike character of electron
we CANNOT say that an electron
WILL be found at certain point in an atom!
THE HEISENBERG UNCERTAINTY PRINCIPLE
He postulated that if...
Dx is the uncertainty in the particle’s position
Dp is the uncertainty in the particle’s momentum
h
DxDp 
4
For a particle like an electron, we cannot know
both the position and velocity
to any meaningful precision simultaneously.
PARTICLE IN A BOX
FOR ONE DIMENSION
SCHRODINGER EQUATION
d n ( x )
 2
 En n ( x )
2
8 m dx
2
h
2
2 2
hn
En 
2
8 mL
1-D REQUIRES
ONE QUANTUM NUMBER!
THREE DIMENSIONS
SCHRODINGER EQUATION

2
h
{
2
2
2
d
d
d
+
+
n ( x , y , z )  Enx ny nz nx ny nz ( x , y , z )
2
2
2
2
8 m dx
dy
dz
}
2
2
nx
2
ny
2
nz
+
h
+
Enxnynz 
8m Lx Ly Lz
(
)
3-D REQUIRES THREE QUANTUM NUMBERS!!!
2
2
nx
2
ny
2
nz
+
h
+
Enxnynz 
8m Lx Ly Lz
Note that when Lx=Ly=Lz
E2,3,4 = E 3,2,4 = E4,2,3
These levels are said to be degenerate – this
means they are different wavefunctions, but
have the same energy
z
HYDROGEN ATOM
y
Electron at (x,y,z)
Proton at (0,0,0)
x
The hydrogen atom is
composed of a proton (+1)
and an electron (-1). If the
proton is located at the origin,
the electron is at (x, y, z)
• We want to obtain the energy of the hydrogen atom
system. We will do this the same way as we got it for
the particle-in-a-box: by performing the “energy
operation” on the wavefunction which describes the
H atom system.
H  E
• Remember that this equation is called the
Schrodinger wave equation (SWE)
• H = KE operator + PE operator
H=
(h2 /
82m)
2
+V
• H  E
(h2 /
{
Where
82m)
2
2
+ V }   E
= {d2/dx2 + d2/dy2 +d2/dz2}
• Now there is a difference from the particlein-a-box problem: here there is a potential
energy involved ….
• Coulombic attraction between the proton
and electron
• V = Ze2 / r
Since the potential energy depends
on the separation between the
proton and the electron, it is more
convenient to think about the problem
using a different co-ordinate system:
(x,y,z)  (r,q,j)
z
y
Electron at (r, q, j)
Proton at (0,0,0)
x
After the transformation we still
have the Schrodinger equation
2
2
2
{ (h / 8 m) + V }   E
2
Where now
has terms in
{d2/dr2 ; d2/dq2 ; d2/dj2}
and V = Ze2 / r
• The result of solving the Schrodinger
equation this way is that we can split the
hydrogen wavefunction into two:
(x,y,z)  (r,q,j) = R(r) x Y(q,j)
Depends on r only
Depends on angular variables
• The solutions have the same features we
have seen already:
– Energy is quantized
• En =  R Z2 / n2
=  2.178 x 10-18 Z2 / n2 J [ n = 1,2,3 …]
– Wavefunctions have shapes which depend on
the quantum numbers
– There are (n-1) nodes in the wavefunctions
• Because we have 3 spatial dimensions, we end up with 3
quantum numbers:
n, l, ml
• n = 1,2,3, …; l = 0,1,2 … (n1); ml = l, l+1, …0…l1, l
• n is the principal quantum number – gives energy and
level
• l is the orbital angular momentum quantum number – it
gives the shape of the wavefunction
• ml is the magnetic quantum number – it distinguishes the
various degenerate wavefunctions with the same n and l
n
l
ml
1
0 (s)
0
2
0 (s)
1 (p)
0
-1, 0, 1
3
0 (s)
1 (p)
2 (d)
0
-1, 0, 1
-2, -1, 0, 1, 2
•En =  R Z2 / n2
=  2.178 x 10-18 Z2 / n2 J [ n = 1,2,3 …]
… degenerate
The result (after a lot of math!)
Node at s = 2!!
Probability Distribution
for the 1s wavefunction:
3/2
1  1  -r/a0
100 
e


  a0 
Maximum probability at nucleus
A more interesting way to look at
things is by using the radial probability
distribution, which gives probabilities of
finding the electron within an annulus
at distance r (think of onion skins)
max. away from nucleus
90% boundary:
Inside this lies
90% of the probability
nodes