Download Lecture 19 (Slides) October 5

Quantum Mechanics, Wave Functions and
the Hydrogen Atom
• We have seen how wave functions provide
insight into the energy level patterns of atoms
and molecules. In the one-dimensional, twodimensional and three- dimensional PIAB
models we derive very approximate wave
functions and, correspondingly, approximate
energies for electrons in atoms and molecules.
The PIAB models are useful in that they do
“explain” quantization of energy in molecules.
Wave Properties of Small Particles
• The PIAB models also reflect the wave
properties of small/light particles. The waves
have differing amplitudes (dependent on the
values of spatial and time coordinates) and can
exhibit nodes. In quantum mechanics the
probability of finding a particle at a given
point in space is found from the square of the
amplitude of the wave function.
Plots of Ψn(x) and Ψn(x)2 for the OneDimensional PIAB Model
• The graphs on the “next” slide show again the
Ψn(x) versus x plots for the one-dimensional
PIAB. In all cases the range of x values is
defined by 0 ˂ x ˂ L. We note that:
• 1. As the value of the quantum number n
increases, the corresponding energy, En,
increases.
• 2. As n increases the number of nodes seen for
the wave in the box increases.
Plots of Ψn(x) and Ψn(x)2 for the OneDimensional PIAB Model (continued)
• 3. As n increases the frequency of the wave
increases (wavelength decreases). These
results should be compared to the familiar
result for light where EPhoton = hνPhoton.
• Waves exhibit both positive and negative
amplitudes. Nodes are equally spaced.
Particle in Box: Standing Waves,
Quantum Particles, and Wave Functions
ψ, psi, the wave function.
Should correspond to a standing
wave within the boundary of
the system being described.
Particle in a box.

 n
2
sin 
 L
L

x

FIGURE 8-20
•The standing waves of a particle in a one-dimensional box
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General Chemistry: Chapter 8
Slide 6 of 50
Plots of Ψn(x) and Ψn(x)2 for the OneDimensional PIAB Model (continued)
• The graphs on the “next” slide show the Ψn(x)2
versus x plots for the one-dimensional PIAB.
We note that:
• 1. For all n values, Ψn(x)2 is positive. This is
expected since Ψn(x)2 gives the probability of
finding a particle in a given part of the box.
The probability of finding a particle in a
particular part of the box can’t be negative.
Plots of Ψn(x) and Ψn(x)2 for the OneDimensional PIAB Model (continued)
• 2. In subsequent courses we will see that the area
under each of the Ψn(x)2 versus x curves is unity
(= 1). This is equivalent to saying that, for the
entire box, the probability of finding the particle
somewhere is 100%.
• 3. The probability of finding the particle is very
different for different x values. We can still
identify nodes from the Ψn(x)2 versus x plots.
How? Note that the Ψn(x)2 plot for the right half
of the box is the mirror image of the plot for the
left half of the box.
FIGURE 8-21
The probabilities of a particle in a one-dimensional box
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General Chemistry: Chapter 8
Slide 9 of 50
The Hydrogen Atom
• The wave functions for the H atom can be
obtained as solutions to the Schrodinger
equation, HΨ(r,ϴ,φ) = EΨ(r,ϴ,φ). Aside: This
is both an eigenvalue equation and a second
order differential equation which will be
treated in detail in higher level courses. The
use of spherical polar coordinates is often
useful when coulombic interactions are
important.
The Hydrogen Atom (continued)
• The use of spherical polar coordinates also allows
us, in the case of the H atom, to obtain
particularly useful solutions to the Schrodinger
equation, HΨ(r,ϴ,φ) = EΨ(r,ϴ,φ). The wave
functions for the H atom, Ψ(r,ϴ,φ), can be
“factored” to give a function which has no angle
dependence. This function, R(r), is called the
radial wave function. The second part of the total
wave function, Y(ϴ,φ), gives the angular
dependence of the total wave function.
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
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General Chemistry: Chapter 8
Slide 12 of 50
H Atom and Orbitals
• As mentioned last day, we need three quantum
numbers to describe the energies of the various
energy levels in the H atom. Again, these are n,
l and ml. In the H atom each principal energy
level or shell is found (by experiment) to
consist of a number of subshells which are
labelled according to their l values or a letter
(s, p, d, f….). In the H atom a particularly
simple energy level pattern results.
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
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Magnetic quantum number,
ml= - l …-2, -1, 0, 1, 2…+l
General Chemistry: Chapter 8
Slide 14 of 50
Principal Shells and Subshells
FIGURE 8-23
•Shells and subshells of a hydrogen atom
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General Chemistry: Chapter 8
Slide 15 of 50
H Atom and Energy Degeneracy
• The previous slide shows that, for the H atom,
several energy levels can have the same energy
or are degenerate. For the H atom all of the
orbitals making up a subshell have the same
energy. As well, the various subshells which
comprise a shell are degenerate. This is not the
case for multi-electron atoms. However, the
pattern of degeneracies seen for the H atom is
seen for other monatomic one electron species
such as He+ and Li2+.
H Atom Wave Functions. Where are the
Electrons?
• For the PIAB model the PIAB wave functions
are used to locate nodes and to describe the
probability of finding a particle in a particular
part of the box. For the H atom this is more
difficult. Both one-dimensional and threedimensional plots are used to provide insight
into where electrons are most likely be found
when they are “located” in a specific energy
level.
Interpreting and Representing the
Orbitals of the Hydrogen Atom.
• Represent the probability densities of the
orbitals of the hydrogen atom as three
dimensional surfaces.
• Each orbital has a distinctive shape.
• Acquire a broad qualitative understanding.
Copyright © 2011 Pearson
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General Chemistry: Chapter 8
Slide 18 of 50
H Atom Wave Functions – Locating Nodes
• For the H atom wave functions there are radial
nodes and nodal planes – the latter reflecting
the properties of the “angular parts” of the total
wave function. We will locate radial nodes for
a few orbitals in class and, should time permit,
nodal planes as well. The wave functions
specific to the 1s, 2s, 3s… orbitals have no
explicit angle dependence. Thus, there can
only be radial nodes.
Copyright © 2011 Pearson
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General Chemistry: Chapter 8
Slide 20 of 50
H Atom Wave Function Problems
• 1. Using the wave functions on the previous
slide determine the number of radial nodes for
the H atom 1s, 2s and 3s orbitals.
• 2. Locate the position of the radial node for the
2s orbital of (a) a H atom, (b) a He+ atom and
(c) a Li2+ ion.
• 3. Plot R1,0(r) = R(1s) for both the H atom and
the He+ ion. What do these plots tell us about
the “electron distribution” for the lone electron
in the 1s orbital of each species?
Probability Plots for the H Atom
• Chemists often wish to describe the
probability of finding the electron in the H
atom as a function of its position in three
dimensional space. This requires an
evaluation of Ψ2 and three dimensional plots.
Due to the wave like properties of electrons
the maximum value of r that should be used
in such plots is not obvious (there is a small
likelihood that the electron will be found far
from the nucleus).
Orbital Representations
• In practice it is customary to draw a boundary
surface enclosing the smallest volume which
has, say, a 95% probability of containing the
electron. Chemists also speak in using these
plots of electron density. The s orbitals are
again a special case. The wave functions for s
orbitals, the Ψ(r,ϴ,φ), have in this case no
angle dependence – the probability of finding
the electron somewhere in space depends
“only” on the r value.
s orbitals
FIGURE 8-24
•Three representations of the electron probability density for the 1s orbital
Copyright © 2011 Pearson
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General Chemistry: Chapter 8
Slide 24 of 50
2s orbitals
FIGURE 8-24
•Three-dimensional representations of the 95% electron probability density
for the 1s, 2s and 3s orbitals
Copyright © 2011 Pearson
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General Chemistry: Chapter 8
Slide 25 of 50
Test 1 Examples
• 1. (a) Write the balanced chemical equation
corresponding to the molar enthalpy of
formation of copper (II) nitrate hexahydarate,
Cu(NO3)2.6H2O(s).
•
(b) Write the balanced chemical equation
for the complete combustion of benzoic acid,
C6H5COOH(s).
Test 1 Examples
• 2. The atmospheric pressure at the summit of
Mt Everest is 28.9 kPa and the air density is
0.436 kg/m3. Determine the air temperature
assuming an effective molar mass for air of
29.2 g/mol.
Test 1 Examples
• 7. A gas expands at constant temperature from
a volume of 3.50 L to 7.60 L. Find the work
done by the gas if it expands (a) against a
vacuum and (b) against a constant external
pressure of 1.45 atm.