When we looked at ordered solids,
we saw this beautiful picture of some salt crystals,
and we said that many solids are organized--
the atoms are organized in some sort of a cubic arrangement.
And that's the case for salt, so let's look at salt. Often,
it's actually called rock salt is
the common name that's given to this structure we're
going to cover.
And as you may know, salt is table salt,
rock salt is made from sodium and chlorine.
And something we notice from the ratio or from the formula
there is it's a one to one ratio of sodium to chlorine.
So you have to keep that in mind when
we start to build our structure, and at the end of the day,
make sure that there's the same number of sodiums
as there are chlorines inside the unit cell.
So what we can do is we can position atoms at the corners.
I'm going to be as careful as I can
to show this time the fraction of an atom at each corner,
so I'm trying to show that at the corner is only 1/8
of an atom inside the unit cell.
And these are the chlorine atoms,
although you'll see you could actually
create the same structure later by reversing
the positions of sodium and chlorine, its equivalent.
Largely by convention, the chlorines
are usually represented at the corners.
So there's some atoms at the corners,
but again, that's not all of them.
There's also atoms on the center of each face,
and I've actually drawn in the back here of the cube,
so you get the full three-dimensional picture.
So there's also an atom of chlorine centered in each
of the face-centered positions.
We have top, right side, front side,
and you've got to be paying attention here.
As careful as I'm trying to be, it's going to look messy.
Back face, bottom face, left side face.
And then let me draw in that one on the back, the bottom
left corner as well.
So we can see those positions, and you're probably
fairly comfortable with that, because you've already
seen the position of atoms in aluminum,
and so you may notice that this is similar.
This seems to be chlorine atoms in face-centered positions.
In fact, it is.
It's not exactly face-centered.
We'll cover that later, but these atoms of chlorine
are indeed in face-centered positions,
so you can transfer that knowledge.
You already know face-centered cubic.
Consider this chlorine atom's in the same position.
So now what we need to do is we need
to position the same number of atoms of sodium,
and without even having to count,
if you know face-centered cubic really well,
and we know that chlorine is in face-centered cubic positions,
you know that there's four chlorines.
So we need to position now four sodium atoms.
And it turns out that they actually
reside in these edge positions.
So right on the center of each edge is a chlorine.
I'm not actually drawing the direction
of contact between atoms yet.
We'll cover that a little bit later.
But in the center of each edge, so the front face,
the top to bottom, left to right, bottom there,
and dash in these ones in the back-- is a sodium atom--
and in that corner there.
So there's 12 edges, and what fraction of a sphere
is represented there on the edge.
We've cut, essentially, a sphere into two planes.
So you're left with quarters, little wedges of an orange,
you can imagine, perhaps.
So there's a quarter of an atom in each
of those edge positions.
So let's just tally that up.
In the edges, there's one quarter,
and how many edges are there?
There's 12, so that gives us 3.
Uh-oh.
There's a problem.
There's only three there, and we know there
needs to be four chlorines.
And so the missing one is in the very, very center,
right in the very center of this cube.
Floating out in the middle of the cube
is the final sodium atom.
So now what we can do is I'm going
to go back to what we're really after here,
which is the density.
So what's the density of sodium chloride going to be?
Well, it's going to be the number of sodium atoms
times the atomic weight of sodium,
or the molar mass of sodium, plus the number of chlorine
atoms, times the atomic weight of chlorine.
That's how much the unit cell weighs,
and then you've got the volume of the unit cell.
And again, to make the units work out,
we've got Avogadro's number, because atomic weight
is represented in moles.
And that's not bad.
But we can generalize this a little bit
further, because we can realize that there's
a ratio between the number of sodiums
and the number of chlorines.
And for this particular example, it's one to one.
So what we could actually do is we could say, well, in order
to form this unit cell here, in order to form this unit cell,
if I could go to the store and buy
formulated formulas of sodium chloride,
I could buy one sphere of sodium and one sphere of chlorine.
And they came in pairs like that.
Well, how many pairs of sodium and chlorine
would I have to buy or obtain to form this unit cell?
And I think the answer should come to you fairly quickly when
you realize, well, there's four chlorines.
There has to be four sodiums.
We worked that out.
We know that there's four sodiums, four chlorines.
So we could instead of just using n sub A and n sub CL,
we could be a little bit more general
for any type of ceramic, and we could say, OK.
The theoretical density is going to be some number,
but the number has to represent the number of multiples
of the formula.
That's the formula.
And in this case, it was 4.
In a general case, we're just going to say it's n.
We had n though for a metal, so let's call this n prime,
so it's a different number, and that represents the number
of multiples of the chemical formula that
go into the unit cell.
Then times the mass of the atoms in the formula,
so that would be the sum of the atomic weights of element 1
plus the sum of the atomic weights
of element 2 in the formula itself,
in the chemical formula.
In our case, sodium and chlorine is 1.
And then divide it by the volume of the unit cell
and Avogadro's number.
That's a U. That's a C. They look like-- I'm sorry.
That's kind of messy.
All right.
So let's summarize this equation here now, and label everything
again.
That's Avogadro's number.
OK.
That's the volume of the unit cell.
This term in the parentheses here
represents the mass of the formula, the molar
mass of the formula, and then this
is the number of repeats of the formula.
Sometimes this is called the number of formula units in case
you come across that.
But it's the number of times you repeat the formula
to create a unit cell.