### Lecture 18 (Slides) October 4

```PIAB - Notes on the Ψn’s
• The PIAB wave functions exhibit nodes. As
we move to higher energy (higher n) states the
number of nodes increases. As well, as one
moves to higher n values the characteristic
wavelength decreases. (This is reminiscent of
light where, again, the energy of a photon
increases as the wavelength of the light
decreases).The wave functions can have both
positive and negative amplitude.
Amplitudes and Probabilities
• In quantum mechanics the probability of
finding a particle at a particular point in space
is proportional to the square of the amplitude
of the wave function at that particular point.
Probabilities must always be positive and the
squares of the amplitudes for the first few one
dimensional PIAB wave functions are shown
on the next slide.
FIGURE 8-21
The probabilities of a particle in a one-dimensional box
General Chemistry: Chapter 8
Slide 3 of 50
Probabilities of Finding an e- ?
• The graphs on the previous slide describe the
probability of finding a particle (electrons are
especially of interest in chemistry) in different
parts of the “box”. In all cases there is a 50%
probability of finding a particle in the left half
of the box. Calculus (or a graphing calculator)
can tell us the probability of finding the
particle in any arbitrarily chosen part of the
box.
Probabilities – Class Examples
• We will use Slide 12 to determine when
calculus must be used to calculate the
probability of finding a particle in a given part
of the box and when simpler “symmetry
arguments” can be used. We’ll consider a
number of cases where symmetry arguments
can be used to specify exactly the probability
of finding a particle in a certain part of the
box.
PIAB Energies – Mass & Particle Size
• The PIAB model allows us to calculate
energies for a particle in a particular quantum
state as a function of quantum number (n),
particle mass (m) and box length (L). As
values of n increase energies increase. As
particle mass increase and/or box size
increases energy level spacings decrease
rapidly. For the limiting case as m → ∞ and
L → ∞ the energies vary continuously
(familiar ground?).
PIAB – 3 Dimensions
• The Schrodinger Eqtn. for the 3-dimensional
PIAB can also be solved.
h2
n12
n22
n 32
• E=
+ 2 + 2
2
8m
a
c
b
• Here a, b and c are the lengths of the 3 sides of
the box and n1, n2 and n3 are a set of three
quantum numbers needed for this higher
dimensionality case. We will need three
quantum numbers when we discuss atoms!
{
}
One, Two and Three Dimensional
PIAB Models
• The one dimensional PIAB model does
account “qualitatively” for the electronic
energy level patterns seen in some conjugated
polyenes (nearly linear carbon chains). Most
molecules are not linear. More complex PIAB
models provide more realistic models for
planar and three dimensional molecules. We
will consider very briefly two PIAB models
which serve to introduce the concept of energy
degeneracy (degenerate energy levels).
Varying Dimension PIAB Models –
Energies and Quantum Numbers
•
Square PIAB Model – Doubly
Degenerate Energies (Energy Levels)
nx Value
1
2
1
2
3
1
ny Value
1
1
2
2
1
3
nx2 + ny2
(Relative Energy)
2
5
5
8
10
10
Square PIAB Model – Doubly
Degenerate Energy Levels
• In the previous slide relative energies are
shown for clarity – this makes the
degeneracies easier to spot. The precise
energies can be calculated by multiplying the
relative energies by h2/(8mL2). Similar
arguments apply to the relative energies of
the cubic PIAB box shown on the next slide
(for the lowest energies!). Here triply
degenerate energy levels are seen.
Cubic PIAB Model – Triply
Degenerate Energies (Energy Levels)
nx Value
ny Value
ny Value
1
2
1
1
1
1
3
6
1
1
2
1
1
2
6
6
2
3
1
2
1
3
2
1
1
12
11
11
1
1
3
11
nx2 + ny2 +ny2
(Relative Energy)
Symmetry in Chemistry
• Symmetry is important throughout chemistry.
The energy expression for the 2d square PIAB
is simpler than for the 2d rectangular box.
More importantly, due to molecular symmetry,
the H2O molecule has only one bending
vibration whereas the CO2 molecule has two
degenerate bending vibrations. (Chemistry
2302 and 3500).
Spherical Polar Coordinates
• When we move from the one dimensional
PIAB model to the wave functions for an
electron in a H atom it’s simplest to use
spherical polar coordinates when we
construct the H atom wave functions. The
correspondence between Cartesian and
spherical polar coordinates is shown on the
next slide.
Wave Functions of the Hydrogen Atom
Schrödinger, 1927
Eψ = H ψ
H (x,y,z) or H (r,θ,φ)
ψ(r,θ,φ) = R(r) Y(θ,φ)
R(r) is the radial wave function.
Y(θ,φ) is the angular wave function.
FIGURE 8-22
•The relationship between spherical polar coordinates and Cartesian coordinates
General Chemistry: Chapter 8
Slide 15 of 50
H Atom Wave Functions
• The previous slide states that the H atom wave
functions, determined again by solving the
Schrodinger equation, can be factored into an
angular and a radial part if we employ
spherical polar coordinates. The use of these
coordinates makes it especially easy to locate
nodes (regions of zero “electron density”) and
to represent 3 dimensional probabilities (i.e.
represent in 3 dimensions the probability of
finding an electron in space/in an atom).
Subatomic Particles Can Move!
• Like larger objects subatomic particles can
move. In this course the motion of electrons
will be most important. When removed from
an atom electrons can be focused and
accelerated to high velocities using an electric
field. Inside an atom, electrons move rapidly
around the nucleus (electrons have orbital
angular momentum!) and can also “spin”.
Subatomic Particles Can be Charged!
• Two common subatomic particles, the electron
and the proton, are charged. Charged particles
in motion can exhibit magnetic properties (act
as small magnets). Such charged particles in
motion will have their energies changed if they
are placed in an “external” magnetic field. This
can serve to remove or lift energy degeneracies
by slightly changing, for example, the energies
of electrons.
Energy Levels – H and More Complex
Atoms
• The observed energies levels of a H atom can
normally be described using the Bohr equation
(which applies to other one electron species
such as He+). For more complex atoms (extra
electrons) more quantum numbers are required
to describe energies. We’d expect to need
three! In all atoms we will encounter
quantization of energy and quantization of
angular momentum.
8-7 Quantum Numbers and Electron
Orbitals
• Principle quantum number, n = 1, 2, 3…
• Angular momentum quantum number,
l = 0, 1, 2…(n-1)
l = 0, s
l = 1, p
l = 2, d
l = 3, f
Magnetic quantum number,
ml= - l …-2, -1, 0, 1, 2…+l
General Chemistry: Chapter 8
Slide 20 of 50
H Atom Spectra and Energies
• The next two slides show the energy lelevl
patterns seen for the H atom – a particularly
simple case. Here degeneracies are very
important. Subshells with the same n value but
different l values have the same energy in the
H atom. We say that these energy levels are
degenerate. This result follows in the first
instance from experiments.
-RH
ΔE = Ef – Ei =
nf2
1
= RH ( 2
ni
–
–
-RH
n i2
1
) = h = hc/λ
2
nf
FIGURE 8-14
Energy-level diagram for the hydrogen atom
General Chemistry: Chapter 8
Slide 22 of 50
Principal Shells and Subshells
FIGURE 8-23
•Shells and subshells of a hydrogen atom
General Chemistry: Chapter 8
Slide 23 of 50
Atoms Beyond H (“Many” e-)
• For many e- atoms many energy level
degeneracies seen for H disappear. Subshells
with the same n but different l values are no
longer degenerate (have the same energy).
This follows 1st from experiment but we will
gain some insight into this after we consider
simple atomic wave functions and orbitals.
General Chemistry: Chapter 8
Slide 24 of 50
FIGURE 8-36
Orbital energy-level diagram for the first three electronic
shells
General Chemistry: Chapter 8
Slide 25 of 50
Atomic Emission/Absorption Spectra
• The atomic absorption and emission spectra
seen for H are very simple. Rather more
complex spectra are seen for atoms with a
number of electrons. How does this follow
from the previous slide?
Test Example
• 3. The interstellar medium is comprised
almost entirely of H atoms at a concentration
of about 1.4 atom/cm3. (a) Find the gas
pressure in outer space. (b) Find the volume in
liters that contains 2.5 g of H atoms. Assume
that the temperature of the interstellar medium
is 3.5 K.
Test Example
• 7. Calculate the amount of pressure volume
work done when 35.5 g of tin are reacted with
excess acid at 25 OC and 1.02 atm.
```