handout lecture18
TRANSCRIPT
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Lecture 18: The Hydrogen Atom Reading: Zuhdahl 12.7-12.9
Outline The wavefunction for the H atom Quantum numbers and nomenclature
Orbital shapes and energies
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H-atom wavefunctions
Recall from the previous lecture that theHamiltonian is composite of kinetic (KE) andpotential (PE) energy.
The hydrogen atom potential energy is given by:
e-
P+r
r0
V(r) =-e
2
r
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H-atom wavefunctions (cont.) The Coulombic potential can be generalized:
e-
P+r
V(r) =-Ze
2
r Z
Z = atomic number (= 1 for hydrogen)
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H-atom wavefunctions (cont.) The radial dependence of the potential suggests
that we should from Cartesian coordinates to sphericalpolar coordinates.
p+
e-
r = interparticle distance(0 r )
q = angle from xy plane
(p/2 q - p/2)
f = rotation in xy plane(0 f 2p)
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H-atom wavefunctions (cont.) If we solve the Schrodinger equation using this
potential, we find that the energy levels arequantized:
En= -
Z2
n2
me4
8e0
2h
2
= -2.178x10
-18J
Z2
n2
n is the principle quantum number, and ranges
from 1 to infinity.
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H-atom wavefunctions (cont.) In solving the Schrodinger Equation, two other
quantum numbers become evident:
l, the orbital angular momentum quantum number.Ranges in value from 0 to (n-1).
ml, the z component of orbital angular momentum.
Ranges in value from -l to 0 to l.
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H-atom wavefunctions (cont.) In solving the Schrodinger Equation, two other
quantum numbers become evident:
l, the orbital angular momentum quantum number.Ranges in value from 0 to (n-1).
m, the z component of orbital angular momentum.
Ranges in value from -l to 0 to l.
We can then characterize the wavefunctions based onthe quantum numbers (n, l, m).
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Orbital Shapes Lets take a look at the lowest energy orbital, the
1s orbital (n = 1, l = 0, m = 0)
y1s =1
p
Z
ao
32
e
-Z
a0
r
=1
p
Z
ao
32
e-s
a0 is referred to as the Bohr radius, and = 0.529
En= -2.178x10-18J
Z2
n2
= -2.178x10
-18J
1
1
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Orbital Shapes (cont.) Note that the 1s wavefunction has no angular
dependence (i.e., Q and F do not appear).
y1s = 1
pZa
o
32
e
-Z
a0
r
= 1p
Za
o
32
e-s
y*yProbability =
Probability is spherical
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Orbital Shapes (cont.) Naming orbitals is done as follows
n is simply referred to by the quantum number l (0 to (n-1)) is given a letter value as follows:
0 = s 1 = p 2 = d 3 = f
- ml (-l0l) is usually dropped
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Orbital Shapes (cont.)
Table 12.3: Quantum Numbers and Orbitals
n l Orbital ml # of Orb.
1 0 1s 0 12 0 2s 0 1
1 2p -1, 0, 1 33 0 3s 0 1
1 3p -1, 0, 1 32 3d -2, -1, 0, 1, 2 5
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Orbital Shapes (cont.)
Example: Write down the orbitals associated with n = 4.
Ans: n = 4l = 0 to (n-1)= 0, 1, 2, and 3= 4s, 4p, 4d, and 4f
4s (1 ml sublevel)4p (3 ml sublevels)4d (5 m
l
sublevels4f (7 ml sublevels)
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Orbital Shapes (cont.)s (l = 0) orbitals
r dependence only
as n increases, orbitalsdemonstrate n-1 nodes.
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Orbital Shapes (cont.)2p (l = 1) orbitals
not spherical, but lobed.
labeled with respect to orientation along x, y, and z.
y2pz
=1
4 2p
Z
ao
32
se-s
2 cosq
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Orbital Shapes (cont.)3p orbitals
more nodes as compared to 2p (expected.).
still can be represented by a dumbbell contour.
y3pz
=2
81 p
Z
ao
32
6s-s2( )e-s
3 cosq
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Orbital Shapes (cont.)
3d (l = 2) orbitals
labeled as dxz, dyz, dxy, dx2-y2 and dz2.
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Orbital Shapes (cont.)
3d (l = 2) orbitals
dxy dx2-y2
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Orbital Shapes (cont.)
3d (l = 2) orbitals
dz2
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Orbital Shapes (cont.)
4f (l = 3) orbitals
exceedingly complex probability distributions.
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Orbital Energies
energy increases as 1/n2
orbitals of same n, but differentl are considered to be of equalenergy (degenerage).
the ground or lowest energyorbital is the 1s.
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Spin Further experiments
demonstrated the need
for one more quantumnumber.
Specifically, someparticles (electrons inparticular)demonstrated inherentangular momentum.
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Spin (cont.) The new quantum
number is ms
(analagous to ml).
For the electron, mshas two values:
+1/2 and -1/2
ms = 1/2
ms = -1/2