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PROFESSOR: Today I want
to start us off on a

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somewhat new topic.
 

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Which is phase equilibria.
 

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So, recently you've been
looking at chemical equilibria.

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What happens when substances
change their chemical structure

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and interchange, and you've
figured out how to calculate

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where equilibrium lies in the
case of chemical reactions.

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Calculate equilibrium
constants, and so forth.

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Now we'll introduce
phase equilibria.

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So this is a little
bit different.

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You still have a lot of
molecules in one state changing

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to another state, but they're
not changing their own

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molecular structure.
 

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They're changing what phase
of matter they're in.

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From liquid to solid to gas.
 

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And so the treatment is
a little bit different.

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And there's some new results
that are somewhat different

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from what you've seen in the
case of chemical reactions.

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So we'll start with phase
equilibria in just

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one component system.
 

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So in other words, we're going
to start with a single

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substance and see how it goes
from solid to liquid to gas.

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And we'll from there see
what happens in multiple

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component systems.
 

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Where, for example, then
we'll be able to talk about

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solubility and mixtures
of liquids and so forth.

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And into which phase
different constituents

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in a mixture will go.
 

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But let's start here.
 

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So, let's say we just have
two phases at equilibrium.

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And this could just look
like ice and water.

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You could have a little bit
of ice floating in water.

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So in this case you
have solid and liquid.

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That could be in equilibrium.
 

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And this is going to be at some
pressure and temperature.

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Or, you could have liquid
and gas in equilibrium.

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So let's put a lid
on our container.

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Still have some specified
pressure and temperature.

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And now there's a
liquid at the bottom.

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And a gas at the top.
 

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So, you could have equilibrium
between liquid and gas phases.

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And in some cases you have
equilibrium between solid

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and gas phases, depending
on the pressure.

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And and a little piece of
solid could sit at the bottom

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and there's equilibrium
with the gas above it.

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And so, to start, what quantity
are we going to need to look at

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to tell whether the different
phases are in equilibrium, or

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if they're not in equilibrium,
in which direction are

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things going to go?
 

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Are things going to change so
that the ice will melt and

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we'll have water or, so
the water will freeze

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and we'll have ice.
 

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What quantity is going to
tell us about all that?

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Who thinks they can guess?
 

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Give you a hint.
 

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We're specifying pressure
and temperature.

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Nobody really knows?
 

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What quantity is it that you've
seen from which you can

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determine everything.
 

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Yes.
 

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The Gibbs free energy is
going to tell us everything.

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And we're going to frame
the discussion in terms of

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the chemical potential.
 

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That is, the Gibbs free
energy for a mole.

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And since there's only one
constituent, one component at

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this point, we just have n.
 

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We don't have different mole
numbers for different

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constituents, we'll
see that later.

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OK, so what's going to happen
is, the value of the chemical

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potential in the different
phases is going to

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tell us everything.
 

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Because you know that at
equilibrium, the chemical

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potential needs to be the
same in all of these.

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If not, let's say the potential
energy here is lower in the

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solid than in the gas,
nothing's going to stop the

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molecules in the gas phase from
going into the solid phase.

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That's what they'll do.
 

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If the chemical potential of
the water is lower than the

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chemical potential of the ice,
then the ice is going to melt.

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Because the molecules are going
to want to go toward lowest

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possible accessible
chemical potential.

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And there's the
liquid right there.

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They have access to it, so
they can, the ice will melt.

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And you'll have just water.
 

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So the main point here is
that at equilibrium, mu

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is identical everywhere.
 

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By everywhere, I mean in all
phases that are at equilibrium.

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And so if multiple phases are
present, mu has to be the

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same in all the phases.
 

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And if it isn't, then the
molecules will go into the

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phase with the lower
chemical potential.

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And this is what is going
to guide everything in

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our consideration of
phase equilibrium.

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So, OK, what's going to happen?
 

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Let's consider our ice
water equilibrium.

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So of course, if mu of the
solid is equal to mu of the

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liquid, which for water it
should happen at zero

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degrees Centigrade.
 

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And pressure of one bar,
for example, not the only

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place it could happen.
 

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But certainly at
that condition.

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Then the ice and water
will be in equilibrium,

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they'll co-exist.
 

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Now, what happens if you just
want some ice water and you

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take some ice out of the
freezer and put it in the cup,

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and you put some water in?
 

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And you see them both there.
 

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They're sitting there.
 

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Maybe for a while.
 

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And yet it's not at zero
degrees Centigrade, at one bar.

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It may be at one bar, you might
be out in the room, but it

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might be room temperature.
 

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Does that tell us that
this isn't true?

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What's going to happen?
 

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The ice is going to melt.
 

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It may take a while.
 

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This doesn't tell us about
the kinetics of it.

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And it could take a while,
gradually, from the surface

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inward, for the ice to melt.
 

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So the fact that you might see
something sitting there with

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two phases present, that alone
of course, doesn't tell you the

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two phases are in equilibrium.
 

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And it may take time for
equilibrium to occur.

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And what's going to happen
if at some pressure and

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temperature, you have the mu s
greater than mu of the liquid.

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Then which way are
things going to go?

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What's going to happen?
 

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You've got to be able
to figure this out.

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Of course.
 

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The ice is going to melt.
 

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The chemical potential
of the liquid is lower.

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And so molecules are going
to move toward that phase.

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And, of course, if mu solid
less than mu liquid,

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then the water freezes.
 

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All this behavior is something
that we can summarize

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in a phase diagram.
 

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So the phase diagram can tell
us for all pressures and

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temperatures, what phase,
or what phases, will be

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present in equilibrium.
 

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And so typically, phase
diagram might look like this.

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Where here's our solid.
 

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Here's our liquid.
 

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Here's our gas phase.
 

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So let's just look at
what's happening here.

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If we go to really
low temperature, in

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general, what happens?
 

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What phase does stuff enter
when you're at the lowest

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accessible temperatures?
 

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Solid, right?
 

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Everything eventually freezes.
 

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In fact, really I should make
this, leave some room on

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my axis for that effect.
 

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So of course, low
temperature you get solid.

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How about high pressure,
what usually happens?

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You squeeze something
real hard.

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Tends to go into
the solid phase.

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Now let's warm stuff up.
 

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And depending on the
pressure, you might go

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to the liquid phase.
 

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Certainly you'll go to
the gas phase at high

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enough temperature.
 

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And that's what'll happen.
 

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And now, if you start changing
the pressure, let's say you're

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at high temperature and you've
got just the gas present,

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in some big container.
 

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Now you start squeezing it.
 

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You just apply more and more
pressure to the gas at

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constant temperature.
 

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What's going to happen?
 

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Forget the phase diagram.
 

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You know what's
going to happen.

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You keep squeezing, what
will happen eventually?

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This isn't going to be
part of your grade.

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You can speak up loud, and even
if you're completely wrong,

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nothing bad is going
to happen to you.

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Yeah.
 

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It's going to
liquefy, of course.

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You're going to start
getting liquid to form.

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It's essentially Le Chatelier's
principle, of course.

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The liquid occupies
less volume.

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The density of the
liquid is much higher.

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So as you increase the
pressure, the stuff wants

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to go from the gas phase
to the liquid phase.

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Yeah.
 

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STUDENT: [INAUDIBLE]
 

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PROFESSOR: Well, with water you
know that at ordinary pressure,

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you know where the gas-liquid
coexistence would occur.

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So let's say you're there.
 

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Let's say you're at
the boiling point.

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And now, you already in some
sense know that that's a

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point that's somewhere
on the gas-liquid curve.

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So now what's going to happen?
 

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Well, if you from there keep
that temperature constant.

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You're at the boiling point.
 

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And you could easily do this.
 

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You've got a pot that's
covered, right?

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And it's boiling.
 

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So there's liquid down here
and there's gas here.

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So what'll happen?
 

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As you start squeezing,
literally what'll happen

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is you'll squeeze out all
the room for the gas.

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And eventually you'll
have all liquid.

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And in fact, you know what'll
happen at that point, if you

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keep squeezing and you
keep the heat on.

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The liquid will get hotter
than the boiling point.

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Normally that doesn't happen,
if you're just out at

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00:13:15 --> 00:13:16
atmospheric pressure.
 

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Because it'll boil.
 

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And instead of getting hotter,
molecules will simply leave the

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liquid and go into
the gas phase.

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So that's what'll happen.
 

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If you increase the pressure,
you'll liquefy all the gas.

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00:13:30 --> 00:13:33
And obviously the opposite
will also occur.

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If you just allow the volume to
expand and keep expanding,

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maybe you don't have too much
liquid there, you wouldn't

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00:13:39 --> 00:13:40
have to expand it too much.
 

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00:13:40 --> 00:13:43
You'll run out, all the
liquid will vaporize.

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00:13:43 --> 00:13:48
And you'll just have
gas at 100 degrees C.

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00:13:48 --> 00:13:53
You could heat it more if
you wanted, but even there.

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So in other words, changing the
pressure if you think about

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actually executing that in the
lab, or in this case even in

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your kitchen, you'll get the
result that you should see

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00:14:03 --> 00:14:05
on the phase diagram.
 

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00:14:05 --> 00:14:06
So let's just look at this.
 

233
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So we've already
talked about boiling.

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That's what's happening
along this line.

235
00:14:14 --> 00:14:18
The line, and I'll explain
the end of this in a

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00:14:18 --> 00:14:28
while, all these lines
are coexistence curves.

237
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That is, these are where
you have equilibrium

238
00:14:31 --> 00:14:33
between two phases.
 

239
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They coexist in equilibrium.
and it could happen for water,

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for that case, for 100
degrees C and one bar.

241
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But that's not the only place.
 

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It could happen it a whole
bunch of temperatures

243
00:14:46 --> 00:14:47
and pressure.
 

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00:14:47 --> 00:14:50
Because as you vary the
pressure, you're almost surely

245
00:14:50 --> 00:14:53
familiar with this, the
boiling point changes.

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So the boiling point up in
Denver at high elevation is not

247
00:14:57 --> 00:14:59
the same as the boiling point
here in Massachusetts

248
00:14:59 --> 00:15:01
right at sea level.
 

249
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Because the pressure's
lower up in Denver.

250
00:15:08 --> 00:15:09
Let's just look at the others.
 

251
00:15:09 --> 00:15:12
So, of course, this is going
from solid to liquid.

252
00:15:12 --> 00:15:16
So we're looking at melting.
 

253
00:15:16 --> 00:15:19
And then there is solid-gas
equilibria in many cases.

254
00:15:19 --> 00:15:29
And this is sublimation.
 

255
00:15:29 --> 00:15:38
And this endpoint is called
the critical point.

256
00:15:38 --> 00:15:43
And there's one point where
all three phases are in

257
00:15:43 --> 00:15:51
equilibrium, called
the triple point.

258
00:15:51 --> 00:15:54
So let's look at
the various cases.

259
00:15:54 --> 00:15:56
For example, let's look
at the melting line.

260
00:15:56 --> 00:15:58
It could be water, it
could be water and ice.

261
00:15:58 --> 00:16:02
Solid going to a liquid.
 

262
00:16:02 --> 00:16:06
Let's just look at
them one at a time.

263
00:16:06 --> 00:16:09
What you have there is
mu of the solid at some

264
00:16:09 --> 00:16:10
temperature and pressure.
 

265
00:16:10 --> 00:16:16
Is equal to mu of the liquid
at the same temperature

266
00:16:16 --> 00:16:18
and pressure.
 

267
00:16:18 --> 00:16:22
Now, just if we back off and
look at this essentially

268
00:16:22 --> 00:16:25
mathematically, there's
an equation here.

269
00:16:25 --> 00:16:26
And there are two
free variables.

270
00:16:26 --> 00:16:28
Temperature and pressure.
 

271
00:16:28 --> 00:16:32
The equation puts a
constraint on them.

272
00:16:32 --> 00:16:34
So there are going to be a
bunch of solutions along

273
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some line or some curve.
 

274
00:16:37 --> 00:16:44
So in other words, it's one
equation in two unknowns.

275
00:16:44 --> 00:16:53
Temperature and pressure.
 

276
00:16:53 --> 00:17:01
So the solution is
a line or a curve.

277
00:17:01 --> 00:17:08
In other words, we could solve
for pressure as a function of

278
00:17:08 --> 00:17:13
temperature, or temperatures
as a function of pressure,

279
00:17:13 --> 00:17:18
along the coexistence curve.
 

280
00:17:18 --> 00:17:24
And we'll do that shortly.
 

281
00:17:24 --> 00:17:32
Now, if we're in one of
these regions, then there

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00:17:32 --> 00:17:33
aren't any equalities.
 

283
00:17:33 --> 00:17:36
There aren't any constraints.
 

284
00:17:36 --> 00:17:40
Because we don't have multiple
chemical potentials that

285
00:17:40 --> 00:17:45
are equal to each other.
 

286
00:17:45 --> 00:17:49
Right here we have three phases
in chemical equilibrium.

287
00:17:49 --> 00:17:53
Solid, liquid and gas.
 

288
00:17:53 --> 00:18:00
So let's just look at
that triple point.

289
00:18:00 --> 00:18:12
Mu solid equals mu liquid
equals mu of the gas.

290
00:18:12 --> 00:18:14
We have two unknowns,
or two variables,

291
00:18:14 --> 00:18:15
temperature and pressure.
 

292
00:18:15 --> 00:18:18
And we have two equations.
 

293
00:18:18 --> 00:18:21
What that means is there's only
one solution, and that's what

294
00:18:21 --> 00:18:23
the phase diagram shows.
 

295
00:18:23 --> 00:18:27
There's a unique solution, the
triple point temperature and

296
00:18:27 --> 00:18:30
the triple point pressure.
 

297
00:18:30 --> 00:18:48
So this has only one solution.
 

298
00:18:48 --> 00:18:52
If we summarize, how many
degrees of freedom, or how

299
00:18:52 --> 00:18:57
many free variables, we have,
anywhere here, we can do

300
00:18:57 --> 00:18:58
that in the following way.
 

301
00:18:58 --> 00:19:03
We can write F is 3 minus p.
 

302
00:19:03 --> 00:19:16
This is the number of
degrees of freedom.

303
00:19:16 --> 00:19:22
Or independent variables.
 

304
00:19:22 --> 00:19:33
And this is the number of
phases in equilibrium.

305
00:19:33 --> 00:19:37
So, what's it telling us?
 

306
00:19:37 --> 00:19:41
If we have just one phase in
equilibrium, or way over in the

307
00:19:41 --> 00:19:43
solid or just gas or just
liquid part of the

308
00:19:43 --> 00:19:45
phase diagram.
 

309
00:19:45 --> 00:19:46
This is one.
 

310
00:19:46 --> 00:19:49
We have two independent
variables.

311
00:19:49 --> 00:19:51
Which is exactly what we
know to be the case.

312
00:19:51 --> 00:19:54
In other words, in those
regions, temperature and

313
00:19:54 --> 00:19:59
pressure can vary freely
without constraints.

314
00:19:59 --> 00:20:04
If we go to any of the
coexistence curves, now we have

315
00:20:04 --> 00:20:07
two phases in equilibrium.
 

316
00:20:07 --> 00:20:11
And then only one variable
can be changed freely.

317
00:20:11 --> 00:20:15
If we want to stay on the
coexistence curve, let's

318
00:20:15 --> 00:20:19
say we've got gas and
liquid in equilibrium.

319
00:20:19 --> 00:20:21
It's boiling.
 

320
00:20:21 --> 00:20:24
We could change the pressure
some, but if we want to keep

321
00:20:24 --> 00:20:27
them in equilibrium, if we've
changed the pressure, we'd

322
00:20:27 --> 00:20:29
better change the temperature
accordingly to stay on

323
00:20:29 --> 00:20:30
the coexistence curve.
 

324
00:20:30 --> 00:20:33
Otherwise everything will
go up into the liquid

325
00:20:33 --> 00:20:35
or into the gas phase.
 

326
00:20:35 --> 00:20:38
Because you'll go off of the
coexistence curve if you were

327
00:20:38 --> 00:20:42
to change one of these
variables and not the other.

328
00:20:42 --> 00:20:44
In other words,
there's a constraint.

329
00:20:44 --> 00:20:46
So only one can vary freely.
 

330
00:20:46 --> 00:20:48
And if all three phases are in
equilibrium at the triple

331
00:20:48 --> 00:20:51
point, then you have no
degrees of freedom.

332
00:20:51 --> 00:20:53
Nothing can vary and
there's only one place

333
00:20:53 --> 00:20:59
where that happens.
 

334
00:20:59 --> 00:21:04
Now let's see if we can
understand also, why does the

335
00:21:04 --> 00:21:07
whole thing have the overall
structure that it does.

336
00:21:07 --> 00:21:12
Why are some of the slopes
generally steeper than others.

337
00:21:12 --> 00:21:15
Turns out, we can get a
qualitative understanding

338
00:21:15 --> 00:21:20
of phase equilibria in
a pretty simple way.

339
00:21:20 --> 00:21:41
So in other words, can we
understand how p and T change

340
00:21:41 --> 00:21:44
with respect to each other
along any one of these

341
00:21:44 --> 00:21:46
coexistence curves?
 

342
00:21:46 --> 00:21:49
Can we understand the
qualitative features of

343
00:21:49 --> 00:21:54
the phase diagrams?
 

344
00:21:54 --> 00:21:59
And what we'll see is, we
can get an equation for

345
00:21:59 --> 00:22:04
dp/dT at coexistence.
 

346
00:22:04 --> 00:22:07
That's, in a sense, our
objective here in order

347
00:22:07 --> 00:22:14
to get the sort of
qualitative picture.

348
00:22:14 --> 00:22:16
So let's say we are
somewhere here.

349
00:22:16 --> 00:22:19
It could be the gas-solid part.
 

350
00:22:19 --> 00:22:23
And we have some T.
 

351
00:22:23 --> 00:22:30
And now we go to T plus dT,
and out here is pressure.

352
00:22:30 --> 00:22:37
How much does the pressure
change when we stay on

353
00:22:37 --> 00:22:44
the coexistence curve?
 

354
00:22:44 --> 00:22:46
Well, let's go through this.
 

355
00:22:46 --> 00:22:48
Let's just consider any two
phases, alpha and beta.

356
00:22:48 --> 00:23:04
They could be solid,
liquid, gas.

357
00:23:04 --> 00:23:07
And the main condition, if
we're going to be on the

358
00:23:07 --> 00:23:10
coexistence curve, is that
the chemical potentials

359
00:23:10 --> 00:23:22
have to be equal.
 

360
00:23:22 --> 00:23:25
Well, that's fine.
 

361
00:23:25 --> 00:23:45
Now let's change the
conditions a little bit.

362
00:23:45 --> 00:23:48
Staying on the
coexistence curve.

363
00:23:48 --> 00:23:56
So, let's start again. mu s, or
mu alpha of T and p is equal

364
00:23:56 --> 00:24:04
to mu beta at T and p.
 

365
00:24:04 --> 00:24:09
Now let's let temperature
go to T plus dT, and

366
00:24:09 --> 00:24:15
pressure go to p plus dp.
 

367
00:24:15 --> 00:24:19
But since we're staying on the
coexistence curve still, at the

368
00:24:19 --> 00:24:21
new temperature and pressure,
mu alpha and beta have to

369
00:24:21 --> 00:24:23
be equal to each other.
 

370
00:24:23 --> 00:24:30
So mu alpha is going to go to
mu alpha plus d mu alpha.

371
00:24:30 --> 00:24:33
There'll be some change in it.
 

372
00:24:33 --> 00:24:40
Mu beta is going to go to
mu beta plus d mu beta.

373
00:24:40 --> 00:24:43
But we already have that mu
alpha and mu beta at the

374
00:24:43 --> 00:24:46
beginning were equal
to each other.

375
00:24:46 --> 00:24:48
And these have to be
equal to each other.

376
00:24:48 --> 00:24:49
Because we've specified
that we're staying on

377
00:24:49 --> 00:24:52
the coexistence curve.
 

378
00:24:52 --> 00:25:03
So what that says is d mu
alpha equals d mu beta.

379
00:25:03 --> 00:25:12
And now, remember in general
d mu is dG per mole, so our

380
00:25:12 --> 00:25:18
fundamental equation for G,
it's minus S per mole dT

381
00:25:18 --> 00:25:22
plus molar volume dp.
 

382
00:25:22 --> 00:25:25
 


383
00:25:25 --> 00:25:38
So this equality is telling us
that minus S alpha dT plus V

384
00:25:38 --> 00:25:51
alpha dp is equal to minus
S beta dT plus V beta dp.

385
00:25:51 --> 00:25:55
 


386
00:25:55 --> 00:25:58
So we're just applying the
fundamental relation to the

387
00:25:58 --> 00:26:00
changes in each of the phases.
 

388
00:26:00 --> 00:26:07
But since the changes in mu are
equal, so are these changes.

389
00:26:07 --> 00:26:09
And now I'm just going
to rearrange these.

390
00:26:09 --> 00:26:18
In other words, S beta minus
S alpha dT is equal to V

391
00:26:18 --> 00:26:22
theta minus V alpha dp.
 

392
00:26:22 --> 00:26:25
 


393
00:26:25 --> 00:26:28
And that immediately is
going to give us our

394
00:26:28 --> 00:26:29
derivative dp/dT.
 

395
00:26:29 --> 00:26:45
 


396
00:26:45 --> 00:26:56
So, dp/dT, staying the
coexistence curve, is just

397
00:26:56 --> 00:27:04
S beta minus S alpha over
V beta minus V alpha.

398
00:27:04 --> 00:27:13
Which is to say it's just, we
can say delta S over delta V.

399
00:27:13 --> 00:27:18
Going from alpha to beta.
 

400
00:27:18 --> 00:27:21
So we have a result that can
guide our thinking into what

401
00:27:21 --> 00:27:25
the slopes of those coexistence
lines are going to be.

402
00:27:25 --> 00:27:30
And it's pretty useful
to look at that.

403
00:27:30 --> 00:27:34
By the way, we can actually
express this in another

404
00:27:34 --> 00:27:35
way if we want.
 

405
00:27:35 --> 00:27:38
And in some cases it
can be advantageous.

406
00:27:38 --> 00:27:46
And that is, we know that
of course G is H minus TS.

407
00:27:46 --> 00:27:49
 


408
00:27:49 --> 00:27:56
So our mu alpha equals mu beta
condition, or a d mu alpha is d

409
00:27:56 --> 00:28:00
mu beta condition, can
be applied that way.

410
00:28:00 --> 00:28:11
That is, H alpha minus TS alpha
equals H beta minus TS beta.

411
00:28:11 --> 00:28:13
So we can do the same
rearrangements.

412
00:28:13 --> 00:28:23
H beta minus H alpha equals
T S beta minus S alpha.

413
00:28:23 --> 00:28:25
And what that's saying, in
other words, is that we can

414
00:28:25 --> 00:28:31
substitute for delta
S, delta H over T.

415
00:28:31 --> 00:28:36
So we can write our relation
for dp/dT in two ways.

416
00:28:36 --> 00:28:44
We can also write dp/dT at
coexistence is equal to

417
00:28:44 --> 00:28:50
delta H over T delta V.
 

418
00:28:50 --> 00:29:00
Going from alpha to beta.
 

419
00:29:00 --> 00:29:11
These are both forms of what's
called the Clapeyron equation.

420
00:29:11 --> 00:29:12
And these are always valid.
 

421
00:29:12 --> 00:29:14
We haven't made any
approximations.

422
00:29:14 --> 00:29:16
Yet.
 

423
00:29:16 --> 00:29:18
Later, in going to the gas
phase, we'll make some

424
00:29:18 --> 00:29:20
approximations about
having an ideal gas.

425
00:29:20 --> 00:29:24
We'll be able to treat this
delta V in a simple way

426
00:29:24 --> 00:29:26
using the ideal gas law.
 

427
00:29:26 --> 00:29:29
But for now, these are exact.
 

428
00:29:29 --> 00:29:31
OK, so now let's look
at what it tells us.

429
00:29:31 --> 00:29:42
Let's look at the different
pieces of the phase diagram

430
00:29:42 --> 00:29:56
and see what we can conclude.
 

431
00:29:56 --> 00:30:00
Essentially we should be able
to construct qualitatively,

432
00:30:00 --> 00:30:05
the features of the phase
diagram using this relation.

433
00:30:05 --> 00:30:10
So let's try to do that.
 

434
00:30:10 --> 00:30:11
Here's p and T.
 

435
00:30:11 --> 00:30:13
We're going to start
with an empty slate.

436
00:30:13 --> 00:30:20
And now let's look at
the liquid gas case.

437
00:30:20 --> 00:30:25
So, let's look at our
delta S delta V, going

438
00:30:25 --> 00:30:28
from liquid to gas.
 

439
00:30:28 --> 00:30:35
Well, one thing you know, when
you go from liquid to gas, what

440
00:30:35 --> 00:30:40
happens to the molar volume?
 

441
00:30:40 --> 00:30:42
It increases.
 

442
00:30:42 --> 00:30:45
By a little, by a lot?
 

443
00:30:45 --> 00:30:46
By a lot, right?
 

444
00:30:46 --> 00:30:48
The molar volume in the gas
is enormous compared to the

445
00:30:48 --> 00:30:51
molar volume in the liquid.
 

446
00:30:51 --> 00:30:56
So delta V is positive
and very big.

447
00:30:56 --> 00:31:00
Now let's think about delta S.
 

448
00:31:00 --> 00:31:04
What's it going to do,
going from liquid to gas?

449
00:31:04 --> 00:31:05
It's going to increase?
 

450
00:31:05 --> 00:31:07
The disorder increases.
 

451
00:31:07 --> 00:31:10
Now, it increases
significantly.

452
00:31:10 --> 00:31:12
But the liquid already
is a disordered phase.

453
00:31:12 --> 00:31:14
And the molecules and the
liquid already have

454
00:31:14 --> 00:31:16
considerable freedom.
 

455
00:31:16 --> 00:31:18
Certainly not near as much as
they have in the gas phase.

456
00:31:18 --> 00:31:19
But still considerable.
 

457
00:31:19 --> 00:31:22
So, to be sure,
delta S positive.

458
00:31:22 --> 00:31:28
And it's not small.
 

459
00:31:28 --> 00:31:32
So there's your condition.
 

460
00:31:32 --> 00:31:37
It's not nearly as big as delta
V is going to turn out to be.

461
00:31:37 --> 00:31:40
So what's going to happen here
is, if we look at the slope

462
00:31:40 --> 00:31:45
going from liquid to
gas, it's positive.

463
00:31:45 --> 00:31:46
But it's not enormous.
 

464
00:31:46 --> 00:31:48
It's not very steep.
 

465
00:31:48 --> 00:32:02
So dp/dT, for liquid-gas
coexistence right, greater

466
00:32:02 --> 00:32:07
than zero, but small.
 

467
00:32:07 --> 00:32:14
Not steep slope.
 

468
00:32:14 --> 00:32:19
So somewhere, we're going to
put our liquid-gas curve.

469
00:32:19 --> 00:32:22
And I, where exactly we
can worry about it.

470
00:32:22 --> 00:32:24
We haven't put any
numbers on this scale.

471
00:32:24 --> 00:32:26
But the point is, wherever
it is, it's not going to

472
00:32:26 --> 00:32:30
be a very steep slope.
 

473
00:32:30 --> 00:32:35
How about solid to gas?
 

474
00:32:35 --> 00:32:42
Well, so what's delta V going
from the solid to the gas?

475
00:32:42 --> 00:32:49
What happens to
the molar volume?

476
00:32:49 --> 00:32:54
What happens to the molar
volume, going from solid?

477
00:32:54 --> 00:32:56
Of course, by a
lot, by a little?

478
00:32:56 --> 00:32:57
By a lot.
 

479
00:32:57 --> 00:32:57
It's an enormous increase.
 

480
00:32:57 --> 00:33:01
Even bigger than before.
 

481
00:33:01 --> 00:33:06
How about delta S,
the entropy change?

482
00:33:06 --> 00:33:07
Yeah, it increases.
 

483
00:33:07 --> 00:33:09
And also by a lot, because now
you're going from an ordered

484
00:33:09 --> 00:33:13
crystalline solid, usually, so
you have a very, very,

485
00:33:13 --> 00:33:15
low entropy there.
 

486
00:33:15 --> 00:33:17
Into a disordered phase.
 

487
00:33:17 --> 00:33:19
Not just the liquid,
but the gas phase.

488
00:33:19 --> 00:33:23
So this is also very big.
 

489
00:33:23 --> 00:33:27
Well, ok, so now we've got
delta S over delta V.

490
00:33:27 --> 00:33:30
There's going to be
a steeper slope.

491
00:33:30 --> 00:33:40
Because this is bigger
then for liquid to gas.

492
00:33:40 --> 00:33:46
So typically, the solid-liquid
part will have a steeper slope

493
00:33:46 --> 00:33:55
then the liquid to gas part.
 

494
00:33:55 --> 00:33:57
Now, wait.
 

495
00:33:57 --> 00:33:59
Sorry, sorry, this isn't right.
 

496
00:33:59 --> 00:34:06
And that's because I've really
badly misplaced things here.

497
00:34:06 --> 00:34:08
That was solid to gas
that we just discussed.

498
00:34:08 --> 00:34:13
It has the steepest slope
relative to liquid-gas.

499
00:34:13 --> 00:34:23
OK, now let's think
about solid to liquid.

500
00:34:23 --> 00:34:30
Delta V for going from
solid to a liquid.

501
00:34:30 --> 00:34:31
What is it?
 

502
00:34:31 --> 00:34:35
What sign is it?
 

503
00:34:35 --> 00:34:36
How big is it?
 

504
00:34:36 --> 00:34:38
Let's forgot about the
sign for a minute.

505
00:34:38 --> 00:34:41
How much does the volume,
molar volume, change going

506
00:34:41 --> 00:34:42
from solid to liquid?
 

507
00:34:42 --> 00:34:44
Yeah, not a very lot, right?
 

508
00:34:44 --> 00:34:45
You melt the solid.
 

509
00:34:45 --> 00:34:48
There's still a little
piece of stuff down there.

510
00:34:48 --> 00:34:50
Didn't increase in
volume enormously.

511
00:34:50 --> 00:34:52
Usually, it increases.
 

512
00:34:52 --> 00:35:00
So generally delta V is
greater than, or sort

513
00:35:00 --> 00:35:01
of equal to, zero.
 

514
00:35:01 --> 00:35:05
It's not a big amount.
 

515
00:35:05 --> 00:35:06
How about entropy?
 

516
00:35:06 --> 00:35:10
Going from solid to liquid?
 

517
00:35:10 --> 00:35:11
It increases, by a good amount.
 

518
00:35:11 --> 00:35:14
You're going from an ordered
to a disordered phase.

519
00:35:14 --> 00:35:19
So delta S is
greater than zero.

520
00:35:19 --> 00:35:23
And so the result here is, now,
if you look at dS/dV, or delta

521
00:35:23 --> 00:35:27
S over delta V, delta S
is positive and you a

522
00:35:27 --> 00:35:28
pretty good amount.
 

523
00:35:28 --> 00:35:31
Delta V is small.
 

524
00:35:31 --> 00:35:33
The molar volume
hardly changed.

525
00:35:33 --> 00:35:35
And so what that's going
to tell us is the slope

526
00:35:35 --> 00:35:37
is going to be steep.
 

527
00:35:37 --> 00:35:39
And generally positive.
 

528
00:35:39 --> 00:35:47
So now, we're going to
really be on the increase.

529
00:35:47 --> 00:35:51
Typical kind of phase
diagram and individual

530
00:35:51 --> 00:35:59
coexistence curve.
 

531
00:35:59 --> 00:36:02
And all these things, again,
you could understand in terms

532
00:36:02 --> 00:36:04
of Le Chatelier's principle.
 

533
00:36:04 --> 00:36:09
In terms of going from, say,
solid to liquid if you know

534
00:36:09 --> 00:36:15
think about two pressures.
 

535
00:36:15 --> 00:36:18
Go from p1 to p2, some
higher pressure.

536
00:36:18 --> 00:36:19
Where are you?
 

537
00:36:19 --> 00:36:23
Well, if you want to stay on
the coexistence curve you need

538
00:36:23 --> 00:36:29
to raise the temperature quite
a bit in order to do that.

539
00:36:29 --> 00:36:33
Because, basically, to keep on
there, you know what's going

540
00:36:33 --> 00:36:36
to happen if you have,
say, solid and liquid.

541
00:36:36 --> 00:36:37
It's ice and water.
 

542
00:36:37 --> 00:36:39
You know the freezing point,
it does depend on the

543
00:36:39 --> 00:36:44
pressure, of course.
 

544
00:36:44 --> 00:36:49
But since the change in the
volume isn't that big, what

545
00:36:49 --> 00:36:54
happens is, a modest change, if
you change the temperature by a

546
00:36:54 --> 00:36:56
little bit, maybe easier to
think about it that way, you've

547
00:36:56 --> 00:36:59
got to change the pressure by a
lot to stay on the

548
00:36:59 --> 00:37:01
coexistence curve.
 

549
00:37:01 --> 00:37:04
Otherwise it'll have a great
tendency to simply go into

550
00:37:04 --> 00:37:10
one phase or the other.
 

551
00:37:10 --> 00:37:14
Now we've, in a sense,
reconstructed the coexistence

552
00:37:14 --> 00:37:21
curve by our consideration of
the Clapeyron equation.

553
00:37:21 --> 00:37:24
There's another way we can
usefully understand the

554
00:37:24 --> 00:37:28
qualitative features
of the phase diagram.

555
00:37:28 --> 00:37:30
And that's just by
looking at mu itself.

556
00:37:30 --> 00:37:33
Looking at the free energy
itself, at different

557
00:37:33 --> 00:37:36
temperature.
 

558
00:37:36 --> 00:37:37
So let's just do that.
 

559
00:37:37 --> 00:37:49
Let's just consider mu of
T at some fixed pressure.

560
00:37:49 --> 00:37:51
Just qualitatively,
what will happen?

561
00:37:51 --> 00:38:02
So let's go back to d mu
is minus S dT plus V dp.

562
00:38:02 --> 00:38:05
And just look at
the derivatives.

563
00:38:05 --> 00:38:11
So d mu / dT at constant
pressure is minus s.

564
00:38:11 --> 00:38:13
Minus the molar volume.
 

565
00:38:13 --> 00:38:15
This is the one we're
really going to look at.

566
00:38:15 --> 00:38:21
Of course, we also could look
at d mu / dp at constant

567
00:38:21 --> 00:38:24
temperature, and just
the molar volume.

568
00:38:24 --> 00:38:29
Let's just look at this.
 

569
00:38:29 --> 00:38:32
We know that entropy is always
a positive number, right?

570
00:38:32 --> 00:38:36
Entropy at the lowest, in a
perfect crystal, at zero

571
00:38:36 --> 00:38:40
Kelvin, could be what value?
 

572
00:38:40 --> 00:38:44
Entropy of a perfect pure
crystal at zero degrees Kelvin.

573
00:38:44 --> 00:38:45
Zero, right?
 

574
00:38:45 --> 00:38:49
And it's only going
up from there.

575
00:38:49 --> 00:38:59
So entropy is always positive.
 

576
00:38:59 --> 00:39:06
So what that tells us is that,
then, if we just look at mu

577
00:39:06 --> 00:39:10
versus T, it's negative
S, for whatever phase.

578
00:39:10 --> 00:39:24
So it's always negative.
 

579
00:39:24 --> 00:39:26
So in some sense, you say why?
 

580
00:39:26 --> 00:39:29
Well, it's because we're
keeping pressure constant and

581
00:39:29 --> 00:39:33
we're raising the temperature
and essentially the effect of

582
00:39:33 --> 00:39:35
the entropy is more important.
 

583
00:39:35 --> 00:39:37
Is going up.
 

584
00:39:37 --> 00:39:41
You could see it just by
saying well, G is H minus TS.

585
00:39:41 --> 00:39:46
And as you raise temperature,
the product TS is increasing.

586
00:39:46 --> 00:39:48
Which means the G is
decreasing, because

587
00:39:48 --> 00:39:49
it's H minus TS.
 

588
00:39:49 --> 00:39:52
 


589
00:39:52 --> 00:39:54
So it's always a
negative slope.

590
00:39:54 --> 00:39:56
Let's look at it in
the different phases.

591
00:39:56 --> 00:40:00
So we have entropy of the
gas, that's greater than

592
00:40:00 --> 00:40:01
the entropy of the liquid.
 

593
00:40:01 --> 00:40:05
Which is greater than the
entropy of the solid.

594
00:40:05 --> 00:40:08
So that immediately tells
us the relative slopes

595
00:40:08 --> 00:40:11
for the three phases.
 

596
00:40:11 --> 00:40:25
So d mu gas / dT at constant
pressure is minus S gas. d mu

597
00:40:25 --> 00:40:41
liquid / dT is minus S of
the liquid. d mu solid / dT

598
00:40:41 --> 00:40:46
is minus S of the solid.
 

599
00:40:46 --> 00:40:48
And the magnitude is much
bigger in the gas than the

600
00:40:48 --> 00:40:50
liquid than the solid.
 

601
00:40:50 --> 00:40:54
And so the same goes
for the slope.

602
00:40:54 --> 00:41:10
So now let's sketch this.
 

603
00:41:10 --> 00:41:12
There's mu.
 

604
00:41:12 --> 00:41:13
And there's temperature.
 

605
00:41:13 --> 00:41:17
So what we've seen is for
the gas, first of all, the

606
00:41:17 --> 00:41:20
slope's always negative.
 

607
00:41:20 --> 00:41:24
And it's steepest by far,
or steepest substantially,

608
00:41:24 --> 00:41:27
for the gas.
 

609
00:41:27 --> 00:41:35
Next steepest for the liquid.
 

610
00:41:35 --> 00:41:46
And least steep of
all for the solid.

611
00:41:46 --> 00:41:48
Now, there can be variation.
 

612
00:41:48 --> 00:41:49
Because, of course,
these do change as a

613
00:41:49 --> 00:41:51
function of pressure.
 

614
00:41:51 --> 00:41:55
But this, certainly the slope,
the magnitudes of the slopes

615
00:41:55 --> 00:41:56
are always going
to be like this.

616
00:41:56 --> 00:41:58
What could change as a
function of pressure is

617
00:41:58 --> 00:42:01
where these happen to lie.
 

618
00:42:01 --> 00:42:02
But this is certainly typical.
 

619
00:42:02 --> 00:42:04
So what does it mean?
 

620
00:42:04 --> 00:42:08
So, it means, remember, the
stuff is always going to be

621
00:42:08 --> 00:42:16
in the phase with the
lowest value of mu.

622
00:42:16 --> 00:42:20
So at low temperature, down
here, that's where the solid

623
00:42:20 --> 00:42:32
phase has the lowest
chemical potential.

624
00:42:32 --> 00:42:34
Now, let's keep going up.
 

625
00:42:34 --> 00:42:35
Eventually they cross.
 

626
00:42:35 --> 00:42:40
This is where, for this
particular pressure, this is

627
00:42:40 --> 00:42:42
where the liquid and the
solid are in equilibrium.

628
00:42:42 --> 00:42:44
In other words, we've started,
we've got the solid and

629
00:42:44 --> 00:42:44
we're raising the pressure.
 

630
00:42:44 --> 00:42:46
Now it's going to melt.
 

631
00:42:46 --> 00:42:51
Here's our melting point at
this particular pressure.

632
00:42:51 --> 00:42:54
And now we've got, if we keep
raising the temperature, we're

633
00:42:54 --> 00:42:58
no longer in equilibrium, and
we'll be in the one phase

634
00:42:58 --> 00:42:59
part of the phase diagram.
 

635
00:42:59 --> 00:43:04
It's just the liquid.
 

636
00:43:04 --> 00:43:07
Because that's the lowest
chemical potential.

637
00:43:07 --> 00:43:09
Let's keep raising
the temperature.

638
00:43:09 --> 00:43:11
Down here, of course, the
chemical potential of the

639
00:43:11 --> 00:43:14
gas is way up there, right?
 

640
00:43:14 --> 00:43:17
So it's not going to come into
play now, though they're

641
00:43:17 --> 00:43:18
going to be in equilibrium.
 

642
00:43:18 --> 00:43:24
Here is the boiling point.
 

643
00:43:24 --> 00:43:26
So now let's keep raising
the temperature.

644
00:43:26 --> 00:43:34
And we'll be in just the
gas part of the curve.

645
00:43:34 --> 00:43:40
And of course that'll
go on forever.

646
00:43:40 --> 00:43:43
So that can just help us
understand, in addition to the

647
00:43:43 --> 00:43:47
slopes, the positions, the
relative positions, of where

648
00:43:47 --> 00:43:50
these equilibria occur.
 

649
00:43:50 --> 00:43:54
Now if we start changing the
pressure in particular,

650
00:43:54 --> 00:43:55
we could move this.
 

651
00:43:55 --> 00:43:58
So that you could have a
sublimation point where the

652
00:43:58 --> 00:44:00
gas and the solid are in
equilibrium because this

653
00:44:00 --> 00:44:06
has moved over to here.
 

654
00:44:06 --> 00:44:11
Any questions about the overall
structure of the phase diagrams

655
00:44:11 --> 00:44:15
and these phase equilibria?
 

656
00:44:15 --> 00:44:20
OK, let me just end with a few
comments about one part of

657
00:44:20 --> 00:44:26
the phase diagram that we
haven't talked about yet.

658
00:44:26 --> 00:44:27
This.
 

659
00:44:27 --> 00:44:31
What's going on there?
 

660
00:44:31 --> 00:44:34
If we were to change, of
course, I've got a finite

661
00:44:34 --> 00:44:37
amount of space on the
blackboard, so I haven't drawn

662
00:44:37 --> 00:44:39
T and p out to infinity.
 

663
00:44:39 --> 00:44:46
But if I could, this line
would keep going forever.

664
00:44:46 --> 00:44:48
There would always be some
pressure and temperature at

665
00:44:48 --> 00:44:50
which you would have
equilibrium between two

666
00:44:50 --> 00:44:54
different, distinct solid
and liquid phases.

667
00:44:54 --> 00:44:59
Turns out, that's not true
for the gas and the liquid.

668
00:44:59 --> 00:45:00
And there are a bunch
of ways to see this.

669
00:45:00 --> 00:45:04
But one is you could think
of the symmetry difference.

670
00:45:04 --> 00:45:08
When you go from a crystal, a
crystal and solid, to a liquid

671
00:45:08 --> 00:45:11
or a gas, there's no question
that you've changed phase.

672
00:45:11 --> 00:45:13
Because there's a symmetry
in the solid that's no

673
00:45:13 --> 00:45:18
longer maintained in the
isotropic liquid or gas.

674
00:45:18 --> 00:45:20
So clearly there's never going
to be a point where these two

675
00:45:20 --> 00:45:23
phases become the same thing.
 

676
00:45:23 --> 00:45:25
Become somehow
indistinguishable.

677
00:45:25 --> 00:45:29
But the gas and the liquid,
I don't know, what's the

678
00:45:29 --> 00:45:30
difference between them?
 

679
00:45:30 --> 00:45:31
The density, right?
 

680
00:45:31 --> 00:45:33
The gas is more dense.
 

681
00:45:33 --> 00:45:34
It's condensed.
 

682
00:45:34 --> 00:45:36
Sorry, the liquid is condensed.
 

683
00:45:36 --> 00:45:39
The gas isn't.
 

684
00:45:39 --> 00:45:44
But that starts to change when
you adjust the pressure and

685
00:45:44 --> 00:45:46
temperature accordingly.
 

686
00:45:46 --> 00:45:48
And you can sort
of see it coming.

687
00:45:48 --> 00:45:54
What'll happen is, you change
the temperature, the molar

688
00:45:54 --> 00:45:57
volumes of the liquid and
the gas start to get

689
00:45:57 --> 00:45:59
equal to each other.
 

690
00:45:59 --> 00:46:01
That's what'll end
up happening.

691
00:46:01 --> 00:46:05
And at that point, what's the
difference between them?

692
00:46:05 --> 00:46:06
There is no difference.
 

693
00:46:06 --> 00:46:07
There's no symmetry difference.
 

694
00:46:07 --> 00:46:11
So you can actually, instead
of going through a phase

695
00:46:11 --> 00:46:13
transition, where there's a
meniscus, you can see

696
00:46:13 --> 00:46:15
the top of the liquid
when you heat it up.

697
00:46:15 --> 00:46:19
There's boiling and
then there's the gas.

698
00:46:19 --> 00:46:23
You can actually go from the
gas, let's go the other way,

699
00:46:23 --> 00:46:28
from the liquid to the gas
phase, without ever boiling.

700
00:46:28 --> 00:46:30
If you change the
pressure as well.

701
00:46:30 --> 00:46:33
So that you can actually
walk all the way around

702
00:46:33 --> 00:46:34
the critical point.
 

703
00:46:34 --> 00:46:37
What happens, is up here you
don't have two distinguishable

704
00:46:37 --> 00:46:39
phases any more.
 

705
00:46:39 --> 00:46:44
And actually, material out at
this region has a bunch of

706
00:46:44 --> 00:46:45
very unusual behaviors.
 

707
00:46:45 --> 00:46:47
Very useful
behaviors, actually.

708
00:46:47 --> 00:46:50
Chemical behavior can be really
very, very, different in what's

709
00:46:50 --> 00:46:53
called supercritical, the
supercritical region beyond

710
00:46:53 --> 00:46:56
the critical point.
 

711
00:46:56 --> 00:46:58
Water behaves really
unusually there.

712
00:46:58 --> 00:47:02
Or it turns out, in that
region, it turns out to

713
00:47:02 --> 00:47:05
be a terrific solvent
for organic stuff.

714
00:47:05 --> 00:47:07
Organic molecules which
normally don't dissolve

715
00:47:07 --> 00:47:09
well in water.
 

716
00:47:09 --> 00:47:13
And it doesn't dissolve
salts very well.

717
00:47:13 --> 00:47:15
That's pretty unusual.
 

718
00:47:15 --> 00:47:20
Carbon dioxide, CO2, turns out
to be a pretty good solvent

719
00:47:20 --> 00:47:22
for organics in the
supercritical region.

720
00:47:22 --> 00:47:25
That's what dry cleaning is.
 

721
00:47:25 --> 00:47:30
Very nice, because know
CO2, apart from its slow

722
00:47:30 --> 00:47:32
effect on our atmosphere,
it's not poisonous.

723
00:47:32 --> 00:47:33
It's not toxic.
 

724
00:47:33 --> 00:47:38
So you can do dry cleaning
without organic solvents.

725
00:47:38 --> 00:47:41
All sorts of unusual
behavior happens above

726
00:47:41 --> 00:47:42
the critical point.
 

727
00:47:42 --> 00:47:44
Beyond the critical point.
 

728
00:47:44 --> 00:47:49
There's also a whole range of
behavior at the critical point.

729
00:47:49 --> 00:47:53
Right at the critical point
that's extremely unusual.

730
00:47:53 --> 00:47:56
That you could think of as a
consequence of the fact that

731
00:47:56 --> 00:47:58
the molar volumes are getting
to be almost equal

732
00:47:58 --> 00:48:00
to each other.
 

733
00:48:00 --> 00:48:03
That's unusual in
a lot of ways.

734
00:48:03 --> 00:48:06
And what ends up happening is,
you can see if you go online,

735
00:48:06 --> 00:48:11
actually, if you just Google
critical point, you can see

736
00:48:11 --> 00:48:13
amazing movies of stuff that's
at the critical point.

737
00:48:13 --> 00:48:16
Because what happens is,
so you've got the liquid.

738
00:48:16 --> 00:48:18
And you've got a meniscus,
and you've got the gas.

739
00:48:18 --> 00:48:18
And you can tell.
 

740
00:48:18 --> 00:48:19
You look at it.
 

741
00:48:19 --> 00:48:20
There's the liquid.
 

742
00:48:20 --> 00:48:22
There's the gas.
 

743
00:48:22 --> 00:48:25
But if you go real close to
the critical point, the

744
00:48:25 --> 00:48:27
stuff doesn't really
know what phase it's in.

745
00:48:27 --> 00:48:30
Because it's losing the
distinction between

746
00:48:30 --> 00:48:30
the two phases.
 

747
00:48:30 --> 00:48:34
You start seeing globules
of liquid up there

748
00:48:34 --> 00:48:34
in the gas phase.
 

749
00:48:34 --> 00:48:37
And globules of the gas
down in the liquid.

750
00:48:37 --> 00:48:39
Because their densities - I
mean, the only reason the

751
00:48:39 --> 00:48:41
liquid's at the bottom
normally is because

752
00:48:41 --> 00:48:43
the density is higher.
 

753
00:48:43 --> 00:48:48
If that stops being the case,
it has no more right to occupy

754
00:48:48 --> 00:48:51
the bottom part than the gas.
 

755
00:48:51 --> 00:48:54
And so they sort of
start interchanging.

756
00:48:54 --> 00:48:57
And if you get even closer, of
course, what eventually happens

757
00:48:57 --> 00:48:59
is, there's just one phase.
 

758
00:48:59 --> 00:49:02
You don't see distinct
meniscus's any more.

759
00:49:02 --> 00:49:08
But right near there, you see
enormous fluctuations in the

760
00:49:08 --> 00:49:11
local density and
in the location.

761
00:49:11 --> 00:49:14
The physical locations
of different phases.

762
00:49:14 --> 00:49:17
And we'll see that come in
another context later.

763
00:49:17 --> 00:49:20
Because when we talk about two
phase equilibria solutions.

764
00:49:20 --> 00:49:24
You get this with you oil and
water, you get critical

765
00:49:24 --> 00:49:28
mixtures also, where right at
the special point, the meniscus

766
00:49:28 --> 00:49:31
goes away and the oil and
water start interchanging.

767
00:49:31 --> 00:49:36
And in those situations, a
tiny amount of force can give

768
00:49:36 --> 00:49:38
you a really big response.
 

769
00:49:38 --> 00:49:41
Because the two things are just
teetering in the balance.

770
00:49:41 --> 00:49:45
So a little bit of force pushes
things one way or the other.

771
00:49:45 --> 00:49:47
That's actually useful in a
whole bunch of different cases.

772
00:49:47 --> 00:49:51
Because that critical behavior
happens in all kinds

773
00:49:51 --> 00:49:52
of phase transitions.
 

774
00:49:52 --> 00:49:54
Magnetic phase transitions.
 

775
00:49:54 --> 00:49:55
Ferroelectric phase
transitions.

776
00:49:55 --> 00:49:59
And in those situations, a tiny
little magnetic field, for

777
00:49:59 --> 00:50:02
example, gives you a really big
change in the magnetization.

778
00:50:02 --> 00:50:05
If the alignment of spins.
 

779
00:50:05 --> 00:50:06
And that's very useful.
 

780
00:50:06 --> 00:50:09
You can get a very big
material response and change.

781
00:50:09 --> 00:50:11
You can switch something,
for example, with a really

782
00:50:11 --> 00:50:14
small external field.
 

783
00:50:14 --> 00:50:18
And in these cases, you can
change the density really a

784
00:50:18 --> 00:50:21
lot by only a small
change in pressure.

785
00:50:21 --> 00:50:25
OK, next time we'll extend the
Clapeyron equation and also

786
00:50:25 --> 00:50:27
go to two component systems.
 

787
00:50:27 --> 00:50:30
So you actually, Professor
Blendi will take over

788
00:50:30 --> 00:50:31
again on Friday.
 

789
00:50:31 --> 00:50:33
And we'll see you then.
 

790
00:50:33 --> 00:50:33