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PROFESSOR CIMA: You'll recall that we talked about the ionic diatomic
molecule, where we have a cation and an anion attracted to one another by
coulombic forces. They reach a equilibrium separation that we
call the bond length. And the energy to separate them is the bond
energy. Now coulombic forces are non-directional.
So there's nothing stopping this molecule from
associating with another. In fact, bringing up an anion here, for instance,
generates some energy. Because it's closer to this cation.
Admittedly, it's also close to an anion. But you can see the separation.
Here's a one dimensional array of anions and cations like I've just
described happening with a single anion. And what we'll do now is sum up the energy
for this one dimensional array. So we'll start here in the center at r equals
0. And add the energy for the two neighboring
anions to this center cation.
The total coulombic energy will be this for these two anions neighboring
the cation. I haven't closed the paren here, because I'm
also going to add in the repulsive potential exhibited between this
cation and this cation as well as this cation and the center cation.
That's going to be positive 2 over 2r. And why the 2r?
It's not at the distance r anymore, but it's 2r.
Now we have an attractive term at 3r. That is, attractive being negative and so
forth. You can see these terms are going to alternate
in sign, because first there's an attraction, then there's a repulsion,
attraction, repulsion, and so forth.
We can factor out the 2 over r in all of these terms to get that the
coulombic energy of this array is just (2 e^2) over (4 pi epsilon_0 r), (-1
+1/2 -1/3 +1/4) and so forth. Well you can see this looks like the sum--
this sum actually converges to zero point minus--
sorry-- 0.693.
So we'd have that the coulombic potential energy of this linear ionic
array is just-- this is per atom.
So if we wanted to do it per mole you would have to multiply
by Avogadro's number. Now what we've neglected in this analysis
is, of course, the repulsive potential between these.
If you do that, you find that the total potential energy is going to be
just (N_A e^2 -1.386) over (4 pi epsilon_0 r).
So there's the Born repulsion that's shown there. Just like we had in the ionic diatomic molecule.
And notice that this energy now is a per mole quantity, because I
multiplied all the terms by Avogadro's number. We can then find r_0, the equilibrium separation,
by taking the derivative with respect to r and setting it equal to
0. Just as we did for the diatomic molecule.
The same thing happens with the 3D crystal. So in other words, this is a linear crystal
but I could take this linear crystal, stack on top of it another one, shift
it just by r. And eventually, I'd end up with a three dimensional
crystal. The resulting potential energy of the three
dimensional crystal ends up looking like the following expression.
Couple things to note about this is that we get this number M here.
It's called the Madelung constant. And it derives from the same origin as this
number. You can see this number just depended on the
arrangement of ions. It was really just a sum here.
I'm going to have a similar sum for a three dimensional crystal.
And it will converge also to a number. But the number, the Madelung constant, will
depend on the crystal structure. The other thing I've substituted in here is
these Q's. These are the numeric charges on the cation
and anion, respectively. For my little model here, I just assumed they
were +1 and -1. This is a more general expression where I've
put in the actual numerical charges.
So this gives me the total energy of the crystal. We call that the cohesive energy.
That's what's holding it together. And of course finally, down here, this is
the equilibrium bond length. Now let's spend a moment thinking about this
Madelung constant. For things like sodium chloride, the Madelung
constant is around 1.748. The lattice, or the crystal arrangement, of
the sodium ions and the chloride ions is actually called the "sodium
chloride crystal." It's so common.
KCL, potassium chloride, also arranges in the sodium chloride crystal.
And it's Madelung constant ends up being exactly the same, because it's
the same crystal arrangement. Silver iodide crystallizes in what's called
the wurtzite Site crystal. And its Madelung constant is 1.763.
Calcium fluoride is in the fluoride crystal structure, and when you do
this infinite sum for that structure you end up with 5.039.
Cuprite, plus one oxidation state of copper, crystallizes
in the cuprite structure. And the Madelung constant is 4.116.
So let's look at some of the behavior of this function.
In the denominator is the equilibrium separation between the anion and
cation in the ionic crystal. It's the sum of the ionic radii.
Shown here is the lattice energy, or the cohesive energy, of a variety of
ionic crystals all with the same crystal structure. So this point here is lithium fluoride, lithium
and fluorine. This one is sodium fluoride and this is potassium
fluoride. These little cartoons shows the relative size
of the ionic radii of the cations and the anions in this chart.
And what you see, of course, is as-- for a given anion--
as the cation gets larger, the cohesive energy gets smaller.
That's because r_0 is the ionic radii of the cation plus the ionic radii of
the anion, approximately. And you can see that as the cation size gets
larger, with this being constant and this vertical r here, you'll
increase the denominator which decreases the cohesive energy.
So in general, as we make the anions and cations larger, the cohesive
energy goes down in an ionic crystal. This chart shows the consequence of that.
Here we see just the sodium salts. So this is sodium fluoride, sodium chloride,
sodium bromide, and sodium iodide.
And what I'm plotting up here is the melting point.
Just like what we saw with the cohesive energy, as the anions get
larger for a constant cation the cohesive energy drops and so does the
melting point. In other words, melting point is sort of an
indirect measurement of the cohesive energy in ionic crystal.