lecture 17: the hydrogen atom reading: zuhdahl 12.7-12.9 outline –the wavefunction for the h atom...
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Lecture 17: The Hydrogen Atom
• Reading: Zuhdahl 12.7-12.9
• Outline– The wavefunction for the H atom– Quantum numbers and nomenclature– Orbital shapes and energies
Schrodinger Equation• Erwin Schrodinger develops a mathematical
formalism that incorporates the wave nature of matter:
ˆ H E
ˆ H The Hamiltonian:
ˆ p 2
2m (PE)
Kinetic Energy
The Wavefunction:x
E = energy
d2/dx2
Potentials and Quantization (cont.)
• What if the position of the particle is constrained by a potential:
“Particle in a Box”
Potential E
x0
inf.
0 L = 0 for 0 ≤ x ≤ L
= all other x
Potentials and Quantization (cont.)
• What does the energy look like?
Energy is quantized
E
E n2h2
8mL2
n = 1, 2, …
Potentials and Quantization (cont.)• Consider the following dye molecule, the length of which
can be considered the length of the “box” an electron is limited to:
E h2
8mL2 n final2 ninitial
2 h2
8m(8Å)2 22 1 2.8x10 19 J
What wavelength of light corresponds to E from n=1 to n=2?
L = 8 Å
700nm(should be 680 nm)
N
N
+
H-atom wavefunctions
• Recall that the Hamiltonian is composite of kinetic (KE) and potential (PE) energy.
• The hydrogen atom potential energy is given by:
e-
P+r
r0
V (r) e2
r
H-atom wavefunctions (cont.)• The Coulombic potential can be generalized:
e-
P+r
V (r) Ze2
r Z
• Z = atomic number (= 1 for hydrogen)
H-atom wavefunctions (cont.)• The radial dependence of the potential suggests that we should from Cartesian coordinates to spherical polar coordinates.
p+
e-
r = interparticle distance (0 ≤ r ≤ )
= angle from “xy plane” (/2 ≤ ≤ - /2)
= rotation in “xy plane” (0 ≤ ≤ 2)
H-atom wavefunctions (cont.)• If we solve the Schrodinger equation using this potential, we find that the energy levels are quantized:
En Z 2
n2
me4
802h2
2.178x10 18 J
Z 2
n2
• n is the principle quantum number, and ranges from 1 to infinity.
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 on the quantum numbers (n, l, m).
Orbital Shapes• Let’s take a look at the lowest energy orbital, the
“1s” orbital (n = 1, l = 0, m = 0)
1s 1
Z
ao
32
e
Z
a0
r
1
Z
ao
32
e
• a0 is referred to as the Bohr radius, and = 0.529 Å
En 2.178x10 18 JZ 2
n2
2.178x10 18 J
1
1
Orbital Shapes (cont.)• Note that the “1s” wavefunction has no angular
dependence (i.e., and do not appear).
1s 1
Z
ao
32
e
Z
a0
r
1
Z
ao
32
e
*Probability =
• Probability is spherical
Orbital Shapes (cont.)• Radial probability (likelihood of finding the
electron in each spherical shell
1s 1
Z
ao
32
e
Z
a0
r
1
Z
ao
32
e
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 (-l…0…l) is usually “dropped”
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 3
2 3d -2, -1, 0, 1, 2 5
Orbital Shapes (cont.)
• Example: Write down the orbitals associated with n = 4.
Ans: n = 4
l = 0 to (n-1) = 0, 1, 2, and 3 = 4s, 4p, 4d, and 4f
4s (1 ml sublevel)4p (3 ml sublevels)4d (5 ml sublevels4f (7 ml sublevels)
Orbital Shapes (cont.)s (l = 0) orbitals
• r dependence only
• as n increases, orbitals demonstrate n-1 nodes.
Orbital Shapes (cont.)2p (l = 1) orbitals
• not spherical, but lobed.
• labeled with respect to orientation along x, y, and z.
2 pz
1
4 2Z
ao
32
e
2 cos
Orbital Shapes (cont.)3p orbitals
• more nodes as compared to 2p (expected.).
• still can be represented by a “dumbbell” contour.
3 pz
2
81 Z
ao
32
6 2 e 3 cos
Orbital Shapes (cont.)3d (l = 2) orbitals
• labeled as dxz, dyz, dxy, dx2-y2 and dz2.
Orbital Shapes (cont.)4f (l = 3) orbitals
• exceedingly complex probability distributions.
Orbital Energies
• energy increases as 1/n2
• orbitals of same n, but different l are considered to be of equal energy (“degenerate”).
• the “ground” or lowest energy orbital is the 1s.