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In this screencast were going to
determine the radius of a lead atom
and
the density on lead.
So the first question we have to ask is
what kind the unit cell is lead?
And if we look at this picture on the left
lead is face-centered cubic
and other words and it has atoms
on all the corners of its unit cells
as well as atoms on
each face, so the lattice constant
"a" is equal to 495
picometers which is 495
times 10 to the minus 12 (=495^10-12)
meters. 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 at a right triangle
this is four times "r"
and here "a" and "a"
are our lattice constants.
So we can write this using the Pythagorean theory
as "r" equals "a" squared
which is our lattice constant
divided by 8 and take the square root
of that. Sothis is going to equal
the square root of 495
picometers.
squared divided by 8
or 175
picometers. So that's the radius
of a lead atom and now we have to find
the density and the density by definition
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. So now to find the mass
in our unit 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? now it's going to equal the molar mass
of lead, 207.2
grams per mole of atoms
and then we divide that by Avogrado's number
6.22 x 10^23
atoms
per mole. And what we are left with is
3.4 times 10
to the -22 grams
per atom of lead. Now we have to figure out how many
atoms there are in our unit cells. Now so let's go back
and take a look at our unit cell
and you can see in each face
there is 1/2 of an atom and there are six faces
so we have 1/2 of an
atom and we have
six faces. So now let's go back
and take a look at the corners and at each one of these corners
right here we have a 1/8
of an atom and we have eight corners
so we add to this 1/8 of an
atom per corner and we multiply that by
8 corners and so what we're left with
is four atoms
in our unit cell. So we multiply
four atoms of lead
by 3.4 x 10 ^-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 centimeters
cubed and we're gonna end up with the density
in grams per centimeter cubed
of 11.3 grams per
centimeter cubed.