In this screencast we're going to determine the radius of a lead atom and the density
of lead. So the first question we have to ask is what kind of unit cell is lead? And
if we look at this picture on the left lead is face centered cubic. In other words, it
has atoms on all of the corners of its unit cells as well as atoms on each face. So the
lattice constant A is equal to 495 pm which is 495*10^-12 m. So this lattice constant
A goes from here to here. And if you take a look at this picture and look at a diagonal
that goes through there you will notice that it is 4 times the radius of an atom. So we
can rewrite this looking at a right triangle. This is 4 times r and here A and A are our
lattice constants. So we can write this using the Pythagorean Theorem using r = a squared
which is our lattice constant, divided by 8 and take the square root of that. So this
is going to equal to the square root of 495 pm squared divided by 8, or 175 pm. So that's
the radius of a lead atom. Now we have to find the density. The density is the mass
divided by the volume. So we're going to look at our unit cell. And the volume of our unit
cell is just a, the lattice constant, cubed. To find the mass in our unit cell we first
have to find the mass of one lead atom and then multiply it by the number of atoms in
our unit cell. So how do we find the mass of one atom? That's going to equal to the
molar mass of lead, 207.2 g/mol of atoms and then we divide that by Avogadro's number,
6.022 times 10 to the 23 atoms per mole. And what we're left with is 3.4 times 10 to the
minus 22 grams per atom of lead. Now we have to figure out how many atoms there are in
our unit cell. So let's go back and take a look at our unit cell. And you can see in
each face there is one half of an atom and there are 6 faces. So we have one half of
an atom per face and we have 6 faces. Now let's go back and take a look at the corners.
At each one of these corners right here we have 1/8 of an atom and we have 8 corners.
So we add 1/8 of an atom per corner and we multiply that by 8 corners. And so what we're
left with is 4 atoms in our unit cell. So we multiply 4 atoms of lead by 3.4 times 10
to the minus 22 grams per atom of lead, and that's our mass. And we divide this by our
volume which is a cubed or 4.95 times 10 to the minus 8 cm cubed. And we're going to end
up with a density in grams per cm cubed of 11.3 g/cm cubed.