Chemistry 417!
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Fall 2012
Chapter 2 Notes
September 3, 2012!
1.
Chapter 2, up to shielding
Atomic Structure in broad terms
a. nucleus and electron cloud
b. nomenclature, so we may communicate
c.
Carbon-12 is the reference for weights
d. isotopes
i.
some elements (P, F for example) are monotopic
e. allotropes
i.
some elements occur in different forms
1. examples are C, O, S, P
2. an example with consequences is Sn (look up “tin pest”)
a. http://www.youtube.com/watch?v=Hb0VoQ-xQhU&NR=1
2. The Bohr atom
a. electrons do not orbit the nucleus as in classical model
i.
why would classical orbits not be viable?
b. experiments came before theory
i.
atoms of hydrogen (and other elements) that were excited by heat or
electric discharge emitted light of specific wavelengths
ii.
an equation to account for the position of the emissions in the
electromagnetic spectrum was empirically developed. This is the Balmer
equation
c.
Balmer equation, extended
d. Paschen, Balmer & Lyman series
e. theorists developed a model of the atom where electrons exist in stationary
states about the nucleus
i.
these states differ in amount of angular momentum of the electron, which is
quantized.
ii.
transitions of the electron between these states is allowed, with release or
absorption of energy corresponding to the difference in energy between the
two states. Gain or loss of arbitrary amounts of energy is not allowed
f.
we treat the electron as having wave properties
g. the heart of the model is the Schrödinger equation (see page 23)
i.
wave function ψ must be single-valued
ii.
wave function and its first derivative must be continuous
iii. wave function must approach zero as r approaches infinity
iv. the integral of the product of the wave function and its complex conjugate
over all space must equal 1
v.
the integral of a wave function with another wave function (two different
orbitals) must be zero; the orbitals are orthogonal.
h.
Particle in a box (your full treatment of this is in 2nd semester P-Chem)
i.
a model so we can have a more sophisticated picture of an orbital other
than “itʼs not an orbit.”
Chemistry 417!
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Fall 2012
you arenʼt responsible for reproducing the derivation on p. 24–25, but
intellectually, you need to be able to follow it.
i.
The square of the wave function (ψ2) does have physical significance. Its
magnitude is proportional to the probability of finding the electron at various
places
3. The quantum numbers (see p. 26)
a. Weʼll develop this the “Hydrogen Atom Wave Functions” and use them as the
basis for understanding multi-electron atoms
b. the quantum number rules
c.
your General Chemistry text will have a nice thorough background on this stuff
4. Angular and Radial Functions
a. A wave function in 3 dimensions can be conveniently thought of as having two
parts (radial function and two angular functions)
i.
the angular function: how does the electron density differ as a function of
position a given distance from the nucleus?
ii.
the radial functions: how does electron density differ as a function of
distance from the nucleus?
b. Angular function. The familiar shapes of s, p, and d orbitals follow straight
from the angular functions for the hydrogen atom
i.
depends on the quantum numbers l and ml
ii.
see Table 2.3 (p. 27), follow the progression across the table to the
rightmost columns
c.
Radial functions.
i.
the presence of nodes within lobes of the orbital (or, for the s orbital, within
the whole orbital itself)
ii.
the radial function tends toward zero farther away from the nucleus
iii. we draw pictures that capture some arbitrary amount (90, 95, or 99%) of the
electron density which are of practical use (knowing that orbitals extend
infinitely with infinitely small electron density is not of practical use).
d. Nodal surfaces.
i.
arise from both the angular and the radial functions
1. see Figure 2.7 on p. 31. Here we see that a node is a place where the
sign of the wave function crosses through zero.
2. In terms of probability of finding the electron, the probability at a node is
zero.
3. Angular nodes are planar or conical. In Figure 2.8, see (b) through (f) for
examples of angular nodes. (c) has both an angular (planar) node and a
radial (spherical) node. The nodes depicted in (d) are conical nodes—
you are looking at a cross-section of the orbital.
4. Radial nodes are spherical. In Figure 2.8, (a) and (c) show radial nodes,
which look like a circle in the drawings because you are looking at a
cross-section of the orbital
5. Aufbau Principle (see also any General Chemistry text)
a. how we fill orbitals in multi-electron atoms
b. paying attention to quantum number rules
c.
fill orbitals according to their energy levels
Chemistry 417!
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ii.
iii.
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Fall 2012
lower n is lower energy
for a given n, lower the l the lower the energy
*caution* sometimes a certain (n, l) combination will be higher in energy
than a different (n+1, l) combination. (the value of l is different in each
case.)
6. Pauli Exclusion Principle
a. no electrons in an atom may have the exact same values for the four quantum
numbers
7. Hundʼs rule of maximum multiplicity
a. see Table 2.6 to make this idea clear
b. diagram bottom p. 35
i.
Coulombic energy—it costs energy to put two electrons in the same orbital
versus different orbitals (this one makes good sense)
ii.
Exchange energy—it costs energy to have two electrons not be strictly
interchangeable
Chemistry 417!
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An alternative, non-outline take on this material:
Notes with Figures from Chapter 2
Balmer (1885) rationalizes the wavelengths of spectral lines of atomic hydrogen.
This can be generalized to:
where l means lower, h means higher
A positive value means energy is emitted; a negative value means that energy was
absorbed.
Very important: This model, though it works very well, only works for hydrogen and
other single-electron systems (He+, Li2+, etc.).
de Broglie asserted that the electron has wave-like properties:
And Heisenbergʼs uncertainty principle tells us that the uncertainties in position and
momentum are intertwined:
Which means for us that although we find energies (and thus momenta) of electrons
with great precision, the consequence is large uncertainty in the position of the electron.
Thus the ideal simplicity of orbits gives way to orbitals:
orbitals are regions that describe the probable location of electrons.
We will talk about probability of finding electrons in a region, and use the term electron
density to represent this.
So, a useful mental picture is given on page 21: electrons “can be thought of as forming
standing waves” but the motion and position cannot be precisely determined.
We use equations to calculate approximations of the electron density in a system. The
most common way is with the Schrödinger equation:
Hψ = Eψ
Chemistry 417!
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Fall 2012
ψ is the wavefunction
E is the energy of the electron
H, the Hamiltonian, operates on the wavefunction, and the result is the wavefunction
back, multiplied by the energy of the electron.
Just what are the wavefunctions?
We develop the idea of the wavefunction by looking at the 1-dimensional model of a
particle in a box.
For us, the point of the particle in a box derivation is to justify the experimental reality
that energy levels of electrons in atoms, ions, and molecules are quantized, and
transitions between levels only occur with discrete energy changes.
Atomic wavefunctions are derived for three dimensions the way that 1-dimensional
wavefunctions are done for the particle in the box.
Particle in a box exists in one dimension, and has 1 quantum number
Orbitals in 3 dimensions have 3 quantum numbers
Page 27 gives a tabular guide to the angular and radial functions
Angular functions (Θ, Φ)
these lead to the shapes (Θ) and orientations (Φ) of orbitals (use quantum numbers l
and ml)
sometimes the two functions are combined into a single angular function Υ
Radial function (R)
these lead to the shapes and energies of orbitals (use quantum numbers n and l)
Considering the radial function, we make an important conceptual distinction between
the radial function (R) and the radial probability function (4πr2R2). The radial probability
function is the square of the radial function multiplied by a factor so that it gives the
probability of finding the electron at a given distance from the nucleus on a very thin
spherical shell. See the following figure derived from Figure 2.7 (page 31)
Chemistry 417!
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Radial functions compared to Radial probability functions
Fall 2012