Gen Chem II – Lec 6 – Crystal Lattices of Solid Structures


♫ ‘Blue Hawaii’ ♫
by Alicia Tibbitt [ukulele playing] Leeward Community College
has a beautiful view of Pearl Harbor. I’m standing on the balcony
here at the college, and in the background you can see
several decomissioned military ships that are slowly being stripped
for their parts. We call that the ship graveyard,
here at the college. Those ships are reminiscent
of a time not forgotton, especially for some of the old-timers
here on the island of O’ahu. Because, on December 7, 1941,
Japanese airplanes invaded this island, and those same beautiful waters
were then a fiery inferno. On that day,
order was broken in our country. And for those old timers,
that day is remembered. For me, today, the view is a serene one. And I often come out to this balcony to talk on the phone with my family
back in Virginia, or to read my Bible. Sometimes I may even come out here
to escape students who are knocking at my office door, in order to prepare
for the next class lecture. In today’s lecture, we focus
on the established order of solids in their crystalline form. And I think you’ll find this lecture
quite interesting today. Crystals can be beautiful objects
to percieve. And their structure, often times, contains
many interesting geometrical features. And looking at a crystal
can be a quite captivating experience. But to the scientific mind,
it’s no surprise that this outward expression
of beauty and geometry is a reflection of the internal ordering that’s occurring
among the particles that form the crystal. I’m holding here a piece of quartz crystal
that was lent to me by Roger Kwok, who’s a professor of science
here at Leeward Community College. Roger also lent several other crystals
for this lecture, that I’ll show you here in just a moment. Today we delve into the realm of solids
in their crystalline form, which is the most naturally occurring form
for a solid to be in. Some solids are amorphous,
meaning their structure has no order. The particles are frozen
in whatever configuration they happen to be in
at the time the solid forms. Glass is an example, where the particles
that form the glass have no order in them. However, most solids are crystalline
in nature, meaning the particles that form the solid
have a certain ordering that exists among them. And today we will study crystals
by looking at that order, by focusing in on what is the arrangement
of particles that form the crystal, or form the solid. So to start off, let’s first define what a crystal is
and see how one is formed. A solid is crystalline
when the particles it is composed of possess a certain internal order,
called the crystal lattice. Crystals form, over time, when the forces of interaction
cause the particles to align properly. Conditions need to be just right, and…they have to exist
for a long period of time. And when you have conditions that remain constant
for a long period of time, the pressure and the temperature
remain relatively constant then the forces of interaction
that exist between the particles are also remaining relatively constant. And given enough time, the forces will guide the particles
right into place, perfectly. And that keeps on happening,
basically, off into infinity, or until the crystal runs out of room. So given enough time, the right conditions,
and the right types of particles, a crystal will, basically, form. So there are lots of situations
where conditions remain constant over long periods of time,
with the right types of particles. And that can occur deep within caverns
within the Earth, and so on. So, because there are many types
of conditions out there, that are constant,
and many types of particles as well, you will find lots of different types
of crystals that form in nature. One way to classify crystals
is by the particles from which it is composed. And many crystals are composed of atoms. There are copper crystals out there. And I do have a piece of copper crystal,
that I’ll show you here up close. Now, the structure of this copper crystal
is not as geometrical as the quartz crystal that I showed you a moment ago. But this is crystalline, none the less. The particles do have a certain order
among them, that you would see if you were to break out
your atomic-level magnifying glass. Silver and gold
also come in crystalline form. Now, this gold wedding ring,
which symbolizes the covenant that I made with my beautiful wife, Alicia,
is actually not crystalline gold. This gold has been heated up
and cooled down many times over, and that ordering in the particles,
that crystal lattice, has long since been removed. But if you were to go out
into a cave somewhere and discover a piece of gold up there
in the wall of the cave, not only would you be quite lucky, but you would see gold
in its crystalline form. And you would see that jaggedness
on the surface of the gold. Many crystals are formed from molecules. And you’ve doubt seen water
in its crystalline form, because that’s ice. You’ve probably seen carbon dioxide
in its crystalline form, which is dry ice. In fact, in our previous lecture
we discussed some differences between water and carbon dioxide
with the help of phase diagrams. Silicon dioxide, which is
that quartz crystal that I showed you. I’ll show you that one again
up a little bit more closely. Beautiful geometric features in the quartz. So here’s another piece of quartz
little bit smaller. You’ve also probably seen C 12 H 22 O 11,
because that’s sucrose, or table sugar. And when you were a kid,
you may have even made rock candy, which is a very large sucrose crystal. Sulfur comes in a molecular form
as S 8 molecules, 8 sulfur atoms attached together
in the shape of a ring. And these ring-shaped molecules
are the particles that form the crystal. I do have a piece of sulfur crystal here. You can find sulfur crystals near volcanos. Beautiful yellow crystal. You can probably see
some of the geometric features in the sulfur crystal. Many crystals are composed of ions. And you’ve also probably seen
sodium chloride, which is table salt, but in the crystal community,
it’s known as halite. You’ve…may have seen calcium carbonate,
which is calcite. Calcite forms these really nice crystals. And I’ll show you a piece
of calcite right here. An almost perfect piece of calcite,
where the edges, or the faces are almost perfect parallelograms. Now this piece of calcite
was actually cleaved, meaning it came from a larger rock,
and that rock was struck, with a hammer or something,
and this is the piece that came off. So very interesting
how you can strike a rock and have something like this break off. Calcium fluoride, or fluorite,
forms these nice green crystals, and I’ll show you a piece of fluorite. If you look closely, you might be able to see
some box-shaped…portions of this crystal. It’s like a bunch of boxes
that are stuck together. Really nice-looking green crystal. And my favorite up here
is iron sulfide, or pyrite. Pyrite forms these nice
cubical-shaped crystals. Very metallic looking.
An almost perfect cube of pyrite here. Now pyrite is also known as fool’s gold. And whereas this
is a more perfect piece of pyrite, it probably had a lot of time to form,
that’s why it formed so perfectly, if you see a less perfect piece of pyrite,
it might look like this. And maybe you can see some of those cubes
kind of stuck together. So, this is known as fool’s gold,
and if you were to see a piece like this on the wall of a cave, you might
mistakenly think you’ve struck it rich. So there are many different types
of crystals out there, some of them are quite beautiful. They contain certain features in common. And the most common feature
is that the particles have a certain order among them called the crystal lattice. Now, the various crystals
contain different crystal lattices, but by studying one particular type
of crystal lattice we’ll be able to understand something
about all of them. And I’ve drawn here
a representative lattice for you to see. What the lattice is an arrangement
of particles in 3-dimensional space, and it’s a repeating pattern of units. So, here is a lattice, which shows
a repeating pattern that occurs over and over and over again,
basically off into 3-dimensional space. And the unit that’s being repeated here
is the shape of a box, or a cube. So, I’ve drawn here actually 27 cubes, stacked 3 cubes tall, 3 cubes wide
and 3 cubes thick. And I’ve taken one out,
kind of zoomed it up for you to get a better look at it. You can see the particle at each
of the eight corners of the cube. Now many crystals
do take on the cubic lattice, and have a smallest repeating unit,
or unit cell, which is in the shape of a cube. It might not be a big surprise
that that piece of pyrite here is formed from that cubic lattice. And if you were to break out
your atomic-level magnifying glass, you would see the particles…
in the shape of a cube, or basically arranged in a cubic form, which is repeated
over and over and over again. Now, it might be more of a surprise, but that copper is also formed
from a cubic lattice. Not as obvious here as it is right here. So sometimes the large shape
of the crystal reflects exactly the small shape of the unit cell
from which the crystal is composed of. But other times it’s not as obvious. Another crystal that is formed
from the cubic lattice is that piece of fluorite right here. So, you can probably see
some of the cubic features in the fluorite. This sulfur that I showed you is formed from a closely related
type of lattice, which is composed
of these box-shaped unit cells stacked over and over and over again. So, little bit closely related to the cube,
but not quite a perfect cube. The piece of calcite
that I showed you right here, it’s very interesting that that calcite
is formed from a lattice that is composed of this shape,
which is called the rhomb. So it’s almost an identical shape, it actually is identical
to the large shape of the crystal. This rhomb right here is the same unit cell
from which quartz is composed of, but not as obvious here
as it is for the calcite. So all crystals
have their own crystal lattice, which is composed from a certain unit cell. And the way we’re gonna study crystals
is by focusing on one type of lattice and one type of unit cell,
which the simplest type, also probably the most interesting type,
which is the cubic unit cell. So, before we do that,
let’s first take a look at how crystals are studied,
how this crystal lattice is observed, and that’s by X-ray crystallography. The structure of a crystal lattice
can be found by X-ray crystallography. In this method, passing an X-ray beam
through a crystal creates an interference pattern
on the other side of the crystal. And this interference pattern is based
on the particle positions within the crystal. So, to see how this method works, and to understand
what an interference pattern is, let’s first examine the interference pattern
for ocean waves that are hitting the shore. Suppose you have a set of ocean waves
that are traveling towards shore, and they hit some wall that has
a couple of holes in the wall, that are separated by some distance. And further, suppose that the separation
of the holes is comparable in size to the wavelength
of waves that are hitting the shore. Now, what happens, is at each hole, another wave is emanated on the other side
of the hole in a circular fashion. And this is a well-known phenomenon. You can see this happen all the time
if you study ocean waves. So, if you have a circular wave emanating
from this hole, and another one from that hole
at the same time, then these two circular waves will interact
with eachother at certain locations in the water,
and they’ll kind of double-up and create a really tall wave. That’s called constructive interference. Well, at other locations in the water
the waves will sort of cancel off, and the water will be flat,
and that’s called destructive interference. So if you observe this situation
from some further location, you will see that interference pattern, where the waves will constructively interfere
and destructively interfere. And that interference pattern
can be studied and used to predict the separation of the two holes
from where the waves first emanated from. So just by looking
at the interference pattern, you can predict how far the holes
were separated from one another. Well, that’s exactly what is done
in X-ray crystallography. In X-ray crystallography,
you have a source of X-rays, which are really high-energy light waves. And the wavelength of X-rays
are comparable to the size of an atom or a molecule or particle within a crystal. So what you do is you shoot
a beam of X-rays, which are a bunch of light waves,
traveling at the same time, and you pass that through a crystal. And these X-ray waves interact
with the particles in the crystal, and are then deflected. And on the other side of the crystal, all of those deflected X-rays
will interact with eachother, and will also generate
an interference pattern. And you call this a defraction pattern. And that defraction pattern
can be collected and studied and used to determine the positions
of the particles that are in the crystal. And, many crystals have been studied
by X-ray crystallography, and many crystal lattices have been solved. And so those crystal lattices,
being built up by their small unit cell, we have found many types of unit cells
that occur within the different crystal lattices
out there. And so I’ve drawn several examples
of unit cells. Here is a cubic unit cell,
where all of the edge lengths are equal, and it’s all right angles. So many particles are found to take on
the cubic arrangement in crystals. Other crystals have their particles
arranged in a sort-of elongated box, called the orthorhomb. Another boxy type of unit cell
is the tetragonal arrangement. But there are various unit cells
that are not boxy in shape. One example is the rhombohedral arrangement, which is actually the same arrangement
as that piece of cleaved calcite that I showed you in the previous slide. So, in this lecture, we’ll focus
on one type of lattice, the one that’s built up
from the cubic unit cell. In the cubic lattice system particles arrangement can be divided
into three main classes, primitive, also known as simple,
face-centered and body-centered. In this discussion, the particles
that we’re working with are atoms. It’s much easier to deal with atoms
versus molecules because an atom is a spherical object. The primitive lattice, or simple,
cubic lattice is taken on by the element polonium. And this lattice is represented right here
in this 2-dimensional rendition, where the particles are circles. And you can see the particles
are lined up perfectly. This line right here is exactly lined up
with that line and that line and the following line, and you can imagine
this going off into all directions. The 3-dimensional system
will have particles coming out of the board and particles going back
into the board as well. But you can get a good understanding
of this system by looking at the 2-dimensional lattice. Now one thing you might notice
is that these particles in this line are not packed in very well
with the particles in the adjoining lines. They’re not fitting in to the crevices
of the next line over. So it’s not a very efficient way
to pack particles. And you would describe that
by the packing efficiency. The packing efficiency
of the primitive lattice is only 52%. So in the 3-dimensional space,
occupied by a primitive crystal lattice, only 52% of that space is actually occupied
by particles. The other 48% is empty space,
which is the crevices between the particles. Now, the coordination number
in this lattice is six. And what the coordination number is, is how many other particles each one
is in contact with. So if you look at, say,
this particle right here and you ask, ‘How many other particles
is it in contact with?’. Well, in this single 2-dimensional layer
you can count four other particles. But in three dimensions,
there’s gonna be one more particle coming out of the board
that it’s in contact with, and one more particle going back
into the board. So this particle is actually in contact
with six other particles. Coorination number 6. And since every particle
has an identical environment, the coordination number is 6 for all
of the particles. So it’s a property of the lattice,
in general. Now…this lattice can be divided up
into individual repeating unit cells. And one way to draw the unit cell
is as follows. The unit cells in this 2-dimensional
lattice are represented as squares. And you can see that the square starts
from the beginning of this particle, and it goes over to the middle
of that particle. So, this square can be repeated
over and over and over again, and all of the squares look the exact same. So this is a valid unit cell. Now, to get a better understanding
of the 3-dimensional unit cell, I’ve popped this one out
into 3 dimensions for you. And you can see the front face
of this 3-dimensional cubic unit cell looks the exact same
as that 2-dimensional one, where the face goes
right through the middle of these front four particles. Now in the 3-dimensional unit cell,
you have… another set of four particles
behind these front four particles. And so, there’s actually eight particles
that are in contact with this single unit cell. Now you might ask, inside this unit cell,
‘How many particles are there?’. Well, the answer there is the equivalent
of one particle. Now that might be kind of difficult to see, because you don’t see an entire particle
inside this box. What you see are eight pieces of a particle
inside the box. And so, what do those pieces add up to? Well the answer is they add up
to the equivalent of one particle. And to better see that,
I’ve drawn four unit cells here, where a particle has been immersed
into those four unit cells. So one half of this blue particle
is immersed into those four unit cells, and the other half of the blue particle
is immersed into the upper four unit cells that are not drawn. So, if this particle is sitting
right at the corner of these four lower unit cells, then what fraction of the particle
is inside this cell right here? Well the answer is 1/8 of the particle. Anytime you have a particle that’s centered
at a corner, 1/8 of the particle is inside the box and the other 7/8 of the particle
are in the other 7 adjoining unit cells. So here, where you have this particle
that’s sitting, centered right at this corner, 1/8 of the particle is actually
inside this box, and the other 7/8
are in the other 7 unit cells that are in contact at that corner. And so, this corner,
1/8 of the particle is inside the box. And the other corners also have
1/8 of the particle inside the same box. So since there are eight corners,
you can count them all up, eight eighths, means the equivalent of one particle
completely inside this unit cell. Now, you can ask, ‘How long is this edge length,
l, of the unit cell?’. Well it’s pretty simple to determine that
for the primitive system. You can see that the edge length
is twice the radius. Because going form this corner,
which is at the center of this particle, over to that corner,
which is at the center of the next particle you pass through two radii. So, the edge-length is ‘2r’ here. Now, the simple, or primitive, system,
is kind of a weak lattice, because there’s not a lot
of particle-particle contact here. The efficiency is pretty low,
and so because there’s not a lot of particle-particle contact,
the coordination is only six, it’s a pretty easy lattice to break up. Other crystal lattices have
a higher coordination number, and more particle-particle contact,
and are therefore stronger. If you take a look
at the face-centered lattice, this is actually the most efficient way
to pack particles. And…the face-centered lattice
is taken on by several elements, gold and aluminum included. I’ve drawn here three layers
of the face-centered lattice. The pink particles are in the back,
the yellow particles are in the middle, and the blue particles are furthest out
of the board. Now these particles are
all of the same type — they’re just color-coded to help us see
which ones are which. So I think you’ll get
the best understanding by focusing on the yellow particles that are
in the middle layer. Now there are also 16 yellow particles
drawn here, just like there are 16 particles
drawn over there. But these yellow particles,
whereas they’re all lined up right here, these four particles are fitting
into the crevices of the particles in the next line,
which are fitting into the crevices of the particles in the next line.
And so the lines are sort of staggered. And these yellow particles cannot be packed
together any more tightly than they are. So this yellow layer
is the most efficient way to pack those yellow particle. And similarly, the pink particles
in the layer behind that are also packed in the most efficient…manner as well.
Identical staggering in the pink particles. And the blue particles as well.
I’ve only drawn a few blue particles. If you wanted to add more blue particles
in this outer layer, you would put one right here over this hole
that is formed between the yellow particles and the pink particles behind it. Put one right over that hole.
One over this hole. This one. And that one. So these blue particles are placed
over the holes that are formed between the yellow and the pink particles
behind the yellow ones. Now there’s actually another way
that the blue particles can be placed on top of the yellow layer. And instead of putting them over the holes,
they could have been placed over… the pink particles,
directly on top of the pink particles, so, the blue particles could have placed,
instead of right here, it could have been placed,
here, here and here. And then again
right there, there and there. Now that would have also been
an efficient way to pack the blue particles together right on top
of the yellow particles. But if you did that,
the blue particles would be lined up perfectly with all of the pink particles, and you wouldn’t be able to see
the pink particles behind that. And that type of lattice is actually
not a cubic lattice. There is no way to draw a cubic unit cell
with that arrangement, even though it’s also a very close way
to pack particles, it’s a different type of lattice. This type of lattice,
is a cubic arrangement. And in order to see
the actual cubic unit cell, which is rather difficult…
I’ve done my best here. Now this cubic unit cell
has been color-coded as well, [background noise]
and you can see this blue particle has been drawn right there. These six yellow particles,
along the white triangle, are drawn right here,
one, two, three, four, five, six. And then the pink particles,
this one, this one and that one, are actually,
this, this and that one right there. Now…the face-centered unit cell
is somewhat different than the primitive unit cell. You still have a particle at each
of the eight corners. But in addition to that, you also have
a particle located at each of the six faces. Now when a particle is at a face,
one half of the particle is inside the box, and the other half is outside the box. And you can see that by looking
at this little drawing that shows a particle, sitting right on a face, centered on a face. Half of it is inside this unit cell
and half of it is in the other unit cell that’s not drawn. Now… how many atoms are inside
are inside this face-centered unit cell? Well, the answer is four. And the way you can determine that
is by looking at the eight corners. Eight corners, and in each corner,
1/8 of the particle is inside the box, so that is one particle inside the box. But then you have six faces, and at each face you have one half
of the particle in the box. So that’s six halves as well.
So eight eighths and six halves add up to the equivalent of four particles
inside this unit cell. Now…the packing efficiency
for the face-centered lattice is a lot greater
than the primitive lattice. This is the maximum packing efficiency
for packing particles, and it’s 74%. So, there’s a lot of
particle-particle contact here. And the coordination number is actually 12. So you remember
what the coordination number is. It’s how many other particles
each one is in contact with. So one way see the coordination number is to look at maybe one
of these yellow particles. And you can see that this yellow particle is touching six other yellow particles
in the same layer. So how many particles is it touching
in the layer behind it? Well, it’s touching the other three
pink particles in the layer behind it. And then, in the layer in front of it,
it would be touching three blue particles. So, the coordination number
of this one is 12. And you could see that each yellow particle
would be touching three blue particles in the layer in front. So a coordination number of 12 means
a lot of particle-particle contact. So this is a rather strong lattice. Now, the edge length
of the face-centered crystal lattice is a little more complicated to get to
than the primitive lattice. The length of the edge of this box
is actually 2 times root 2 times the atomic radius. So, it’s no longer twice the radius
because there’s a space between these two corner particles.
So it’s more than twice the radius. And the way you can find the edge length
is to draw a triangle from one corner all the way
to the other corner and then back. So if you look at this triangle right here, this is the edge length ‘l’,
that is the edge length ‘l’ and the length of the diagonal
is four times the radius — it’s going right through these atoms,
so it’s four times the radius. So I’ve taken this triangle,
and I’ve put it right there. So you have an edge length,
the same edge length, and then four times the radius. And because this is a right triangle,
you can use the Pythagorean theorem to solve for ‘l’,
and when you solve for ‘l’, it turns out that ‘l’
is 2 times root 2 times ‘r’. So, the last type of cubic unit cell,
or cubic arrangement, is the body-centered cubic, which is taken on by elements
molybdenum and barium. And the body-centered arrangement
is more obviously a cubic arrangement, as opposed to the face-centered arrangement. The body-centered arrangement,
we can see that the particles sort of line up in squares. Now, what you have here
in the body-centered arrangement is you have one layer staggered relative
to the other layer. So the staggering kind of alternates
back and forth. You can imagine these white particles
that are in the layer behind, there would be another equivalent layer
directly in front of these clear particles, that would kind of line up
with the white particles. So, the packing efficiency
in the body-centered cubic is 68%, which is in the middle of the primitive
and face-centered. The coordination number for body-centered
arrangement is eight. And this is rather easy to see, I think. Because if you look at one particle,
say this white one right here, it’s in contact
with these four clear particles in front, and another four particles
in the equivalent layer behind it. So a coordination number of eight
for all of the particles. Now, I’ve drawn here several unit cells,
represented by squares, and if we pop one out, you can see
the four clear particles right here. And then the white particle behind it. And then the other four clear particles
are behind that. So again, in this unit cell, you have
a particle at each of the eight corners, but now you have one
completely inside the box. So if you add up all of those portions
of particles as well as the one inside the box,
you get the equivalent of two particles inside this unit cell. So, the edge length
of the body-center unit cell is more complicated than the face-centered. The edge length turns out to be 4 times
the radius over the square root of 3. Now, rather than show you the derivation,
I think it’s a good exercise if you go ahead and derive this yourself. So, in order to do that,
you’ll wanna find the the triangle that needs to be drawn, and again,
you’ll use the Pythagorean theorem. But what triangle are you going to use? It’s not the same one that you would use
for the face-centered lattice. So try to find that triangle,
and go ahead and derive this formula for the edge length,
in terms of the atomic radius. The groundwork that we just laid out
on the cubic crystal lattices will allow us to predict certain properties,
which crystallize in those lattices. And the properties that we’ll be examining
are the intensive properties, meaning they do not depend
on how much material. So, for an intensive property,
it’s the same whether you have this much material, or that much. An example is the density. So the density of a substance
doesn’t depend on how much material. The density is the same
as if it’s that much versus that much, versus even a tiny unit cell’s worth
of material. And that’s the approach
that we’ll be taking, is to focus in on just a unit cell. And the property of that unit cell
will be the same as the property of the large material, in general. So, the first example asks us to find
the fraction of empty space in the body-centered cubic lattice. Now the fraction of empty space is closely
related to the packing efficiency. You remember the packing efficiency,
which is 68% for the body-centered lattice, is the percentage of filled space
within the lattice. Now, if the percentage of filled space
in the body-centered cubic lattice is 68%, then the percentage of empty space in the body-centered cubic lattice
would be 32%. So, if you convert that percentage
of empty space to fraction of empty space, we should expect about 0.32
for the fraction of empty space. So let’s see what we get. We can calculate the fraction
of empty space, as 1 minus the fraction of filled space. The fractions add up to 1. So if you take 1 minus
the fraction of filled space, and the fraction of filled space
would be the volume of the atoms inside the unit cell,
divided by the volume of the unit cell. So that the fraction of filled space
in the unit cell. So 1 minus the fraction of filled space
would be the fraction of empty space. Now, we can rewrite this fraction — 1 minus the volume of the atoms
would be the volume of 1 atom, which is 4/3 times π r cubed,
where r is the atomic radius, times 2, because there are 2 atoms
in the BCC unit cell. So, the volume of the atoms
in the BCC unit cell divided by the entire volume of the cell,
which is the edge length cubed. And you remember,
for the body-centered cubic unit cell, the edge length is
4r over square root of 3. So this right here is the expression,
and if you reduce it on down, there will be a lot of cancellation occur.
The r cubed will cancel off, and some other stuff
will cancel off as well. You’ll be left with 1 minus square root of 3
times π divided by 8. So if you calculate this, it’s
1 minus 0.68, which is where that 68% comes from,
and a bunch more decimal places. And so you end up with
0.3198252384… So you can continue this on
if you prefer, because all of these digits
are significant. That’s because all of these numbers that you use to calculate it
are exact numbers. So quite a lot of significant figures,
as many as you need. The next example. Gold crystallizes
in the face-centered cubic… arrangement, or lattice,
and has an atomic radius of 1.44 angstroms. Find the density.
So density is another intensive property. Now, we’ll just calculate the density
of the unit cell, and that will be the same
as the density of gold. So the density of the unit cell
will be the mass inside the unit cell divided by the volume of the unit cell. So the mass inside the unit cell
will be the mass of four atoms, because there are four atoms
inside a face-centered cubic unit cell. So the way you calculate
the mass of four atoms is as follows. We know for gold, the atomic mass is
196.98 grams per mole. So, you can convert this to grams per atom
by the following conversion factor, 1 mole is Avogadro’s number of atoms,
so the moles cancel off, and you’ll be left with grams per atom, then when you multiply by 4 atoms,
you get grams up here in the top. So that’s how many total grams
are in one unit cell. And you divide by the volume.
For the face-centered cubic unit cell, the volume of this cell
is the edge length cubed, which is 2 times root 2 times r,
all cubed. And if you plug that in,
2 times root 2 times the atomic radius, 1.44 angstroms…
now we’ll convert that atomic radius to centimeters, 100 centimeters is the same
as 10 to the 10 power angstroms. So the angstroms cancels,
and then you have 2 times the square root of 2
times the atomic radius (in centimeters), all cubed. And if you do this calculation,
you get 19.3346 grams per centimeter cubed, but it’s a 3 sig fig number
because of the atomic radius, and so it’s 19.3 grams per centimeter cubed. Now if you want to compare this
to the density of water, which is 1 gram per centimeter cubed,
you can see this is more than 19 times heavier than the water.
Gold’s quite heavy. The last example. Polonium crystallizes
in the primitive lattice, and has a density
of 9.196 grams per centimeter cubed. Find the atomic radius of polonium. Now the equation that we’ll be using
is very similar to the previous equation in example 2,
because we’re given the density and asked to find the radius here,
whereas in the previous example, we were given the radius
and asked to find the density. So the equation will be very similar —
we’ll just be working backwards. And if you want to compare the two,
you’ll notice very few differences. The only differences are in the value
for the density, this is unknown,
and we know the density here, the atomic mass for polonium
is 209 grams per mole. And there’s only one atom in the unit cell
of the primitive lattice. And in the denominator,
the volume is the edge length cubed, but the edge length for the primitive
lattice is 2 times the radius. So here, the radius is the unknown.
And with the same conversion factor, if we solve for the radius,
we’ll get the answer in angstroms. So doing that, we calculate 1.20 angstroms
for polonium, which is close to the known value. So it’s very interesting how we
can calculate these intensive properties, or use the intensive properties
to calculate certain atomic properties, with just the knowledge
of the crystal lattice. I hope you have enjoyed this lecture.
I have really enjoyed putting it together. And this does close up the first phase
of our course on phases of matter. In the next portion of our course,
we’ll start to cover solutions. So in the next several lectures,
we’ll get into various aspects of solutions. So I hope you join me for that.
Aloha. ♫ ‘Blue Hawaii’ ♫
by Alicia Tibbitt ♫ Night and you, ♫ ♫ in blue Hawaii, ♫ ♫ the night is heavenly… ♫

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