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PROFESSOR: Alright, so last
time we started talking about

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more complicated mixtures.
 

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To get into colligative
properties.

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And the example we gave
was a binary system that

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has two components.
 

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And two phases.
 

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So we have a liquid
phase on the bottom.

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A gas phase on top, and two
components, A and B, where

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compositions xA and xB
are in the liquid phase.

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Those are the mole fractions.
 

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And yA and yB in the gas
phase, the mole fractions

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in the gas phase.
 

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And what we did last time was
to begin to say how many

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degrees of freedom do we have,
how many variables do we need

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to completely describe
this mixture.

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And it turned out that for
this mixture, we only

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needed two variables.
 

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Two degrees of freedom.
 

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We started out with four
variables; the temperature, the

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pressure, and the components in
the composition in the liquid

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phase and the composition
in the gas phase.

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We need one of these, either
xA or xB, and one of these,

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either yA or yB because
they add up to 1.

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So you start with four, but
because you are in a mixture,

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an equilibrium, you
have constraints.

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You have the chemical
potentials that have to

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be equal to each other.
 

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The chemical potential of A in
the gas phase has to be equal

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to the chemical potential
of A in the liquid phase.

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00:02:16 --> 00:02:20
And the chemical potential of B
in the gas phase has to B equal

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00:02:20 --> 00:02:22
to the chemical potential
of B in the liquid phase.

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So you start with
four variables.

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Temperature, pressure, xA, yA,
then you have the constraints

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that mu A, in the liquid phase
has to be equal to mu

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A in the gas phase.
 

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Because you're at equilibrium.
 

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And mu B in the liquid phase
has to be equal to mu

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00:02:46 --> 00:02:48
B in the gas phase.
 

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So four minus two constraints
means you have two

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degrees of freedom.
 

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And so if you want to know
everything about this mixture,

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all you need is the temperature
and the total pressure.

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And that's enough.
 

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It's kind of amazing.
 

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Give me the temperature and
pressure and these two

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components, like water and
alcohol, for instance, and

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I can tell you everything
about the compositions.

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It's very powerful.
 

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And then we said that this
turned out to be a special

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case, or sub-case of the more
general Gibbs phase rule. where

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the Gibbs phase rule tells you
that if you have C components,

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in this case here we have two,
and the general case of C

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components with p phases, the
number of degrees of freedom

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is C minus p plus two.
 

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This is components.
 

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And phases.
 

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And where we're going to start
today is by proving this.

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And then we'll do more
with this problem here.

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So the first thing is for us
to prove Gibbs' phase rule.

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So Gibbs phase rule.
 

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We start out with all the
variables that we have.

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And then we add
the constraints.

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And for the Gibbs phase rule,
then we start with the two

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knobs that we can
turn externally.

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Which are the temperature
and the pressure.

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So we start with temperature
and pressure, as two

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variables that we have.
 

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And then we have a
bunch of phases.

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We have C phases.
 

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And in each phase we have to
describe the composition

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in that phase.
 

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So, for each phase alpha,
we have to describe

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its mole fraction.
 

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So here we needed to
describe xA for phase A.

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And yA for phase A.
 

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For component A in
the gas phase.

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So now we have
alpha components.

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And for each component we have
to describe its mole fraction

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in that particular phase.
 

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So we have to describe x sub i.
 

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But we have a
constraint on that.

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The constraint is that the
sum of all the mole factions

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has to be equal to one.
 

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That's what it is, to
be a mole fraction.

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So the constraint is that
x sub i, i from one to

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alpha, is equal to one.
 

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So the constraint on the number
of the components, instead of

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having, actually, i goes from
one to C, because that's

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the number components.
 

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So instead of having C
different compositions that

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we need to take care of,
we only need C minus one.

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Just like here we only
needed one, just xA.

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We didn't need both xA and
xB, because we knew the

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sum was equal to one.
 

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00:06:00 --> 00:06:05
So that means that we really
need C minus one variables to

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describe the composition
in one of the phases.

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But we have p phases,
so let me fix this.

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We had C components.
 

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We have p phases.
 

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And so for each phase, we have
to describe the composition.

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00:06:24 --> 00:06:27
So that means that we need p
times C minus one variables to

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describe all the components and
all the phases, plus

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these extra two.
 

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So the total number variables
then that we start out with

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00:06:38 --> 00:06:41
before putting the fact that
we're in equilibrium, is

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going to be two plus
p times C minus one.

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So that's a basic
description of the system.

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Then we put in the
same constraints

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that we had up here.
 

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In terms of the
chemical potentials.

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If I take any of the
components, let's take

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the alpha component,
or the i'th component.

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00:07:03 --> 00:07:07
Because it's in equilibrium,
the chemical potential of that

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particular component has to be
the same in all the phases.

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00:07:12 --> 00:07:14
So let's take the
i'th component.

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So yi in phase one has to be
equal to yi in phase two, has

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00:07:21 --> 00:07:24
to be equal to yi in phase
three, has to be equal

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00:07:24 --> 00:07:33
to yi in phase p.
 

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All the chemical potentials
have to be the same across all

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the phases for that component.
 

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And if you count the number of
equations, the number of equals

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signs here, that's p minus one.
p minus one equals signs,

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constraints due to the
fact that I have an

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equilibrium here.
 

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So this gives me p
minus one constraints.

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And this is true for
every single component.

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Every single component has to
have its chemical potential,

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its chemical potential equal
throughout the phases.

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00:08:12 --> 00:08:14
So p minus one's constraint
for one component

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00:08:14 --> 00:08:16
times C components.
 

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00:08:16 --> 00:08:26
So for a total of C times p
minus one total constraints.

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00:08:26 --> 00:08:29
My variables, my constraints.
 

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00:08:29 --> 00:08:36
So the number of degrees of
freedom then becomes p times

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00:08:36 --> 00:08:49
C minus one minus C times p
minus one and then plus two.

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00:08:49 --> 00:08:52
And if you multiply this
out, p times C gets rid

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00:08:52 --> 00:08:53
of the p times C here.
 

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00:08:53 --> 00:08:58
And you get the plus
C minus p plus two.

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Which is the Gibbs
phase rule, right?

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00:09:02 --> 00:09:05
Which I'm going to
bring back up here.

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C minus p plus two.
 

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So just purely in accounting,
just accounting for the

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variables and the constraints.
 

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OK, any questions on
how we did this?

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Great.
 

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OK.
 

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So let's move on, then, to
these ideal solutions.

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Which is this case right here.
 

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And the first thing we're going
to do is look at the case where

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we have A is a solvent,
and it's volatile.

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And B is a solute,
which is non-volatile.

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00:09:50 --> 00:09:53
So let's take the first case.
 

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We have A volatile,
that's a solvent.

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Could be water.
 

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00:10:01 --> 00:10:05
And B is going to
be the solute.

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00:10:05 --> 00:10:09
It's non-volatile.
 

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00:10:09 --> 00:10:11
And let's interchange
solute or non-volatile

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00:10:11 --> 00:10:14
to be consistent here.
 

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So this could be sugar.
 

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A sweet water.
 

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00:10:27 --> 00:10:41
So, Mr. Raoult looked at such
solutions, and found that they

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00:10:41 --> 00:10:46
could be well described by
the following diagram here.

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00:10:46 --> 00:10:51
So if I plot on the x-axis
here, from zero to one, I'm

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going to plot the partial
molar fraction of the solute.

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00:10:57 --> 00:11:00
Partial molar fractions of the
solute, that's going to be xB.

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00:11:00 --> 00:11:02
 


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00:11:02 --> 00:11:06
So when xB equals to zero, I
have a pure water solution.

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00:11:06 --> 00:11:11
When xB is equal to one,
I have pure sugar.

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Then on the y axis, I'm going
to plot the total pressure.

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00:11:17 --> 00:11:20
So when I have a pure water
solution, the total pressure is

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going to be the same as the
vapor pressure of water at

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a particular temperature.
 

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00:11:28 --> 00:11:36
Let's take temperature
fixed at some value.

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00:11:36 --> 00:11:39
Temperature is fixed
at some value.

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00:11:39 --> 00:11:44
I'm going to have the total
pressure is going to be the

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00:11:44 --> 00:11:45
partial pressure of the water.
 

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00:11:45 --> 00:11:49
And clearly, when I don't have
any water left, all I have is

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00:11:49 --> 00:11:51
the sugar, there's no pressure.
 

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00:11:51 --> 00:11:54
We're going to assume that
it's totally non-volatile.

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00:11:54 --> 00:11:56
Obviously there's going to be a
little bit of vapor pressure,

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00:11:56 --> 00:11:58
even at room temperature.
 

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00:11:58 --> 00:12:01
Ten to the minus, I don't
know, 13 torr or something.

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00:12:01 --> 00:12:05
But anyway, it's
basically zero here.

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00:12:05 --> 00:12:07
And what Raoult said is that
while these two points are just

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00:12:07 --> 00:12:13
connected by a straight line.
 

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00:12:13 --> 00:12:19
Simplest way to
connect two points.

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00:12:19 --> 00:12:27
Basically, he said that the
pressure of, the vapor pressure

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00:12:27 --> 00:12:38
of the water in this case, is
equal to xA times pA star.

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00:12:38 --> 00:12:42
I could also plot this with xA,
going from one to zero, since

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00:12:42 --> 00:12:49
xA plus xB is equal to one.
 

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00:12:49 --> 00:12:56
Or I could write this as one
minus xB times pA star.

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00:12:56 --> 00:12:58
Where a star, the star,
means pure water.

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00:12:58 --> 00:13:01
So pure A.
 

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00:13:01 --> 00:13:06
And pA is the vapor
pressure of the mixture.

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00:13:06 --> 00:13:09
Which in this case here is the
total pressure, because I don't

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have anything else
that's volatile.

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00:13:12 --> 00:13:29
And this is Raoult's law.
 

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00:13:29 --> 00:13:32
And the first thing that you
can use this for is to look

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00:13:32 --> 00:13:35
at your first colligative
property which is vapor

207
00:13:35 --> 00:13:45
pressure lowering.
 

208
00:13:45 --> 00:13:48
And let me point out
another thing first.

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00:13:48 --> 00:13:51
So if a solution obeys
that property here, it's

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00:13:51 --> 00:13:53
called an ideal solution.
 

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00:13:53 --> 00:13:56
So if the water-sugar solution
really did behave like a

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00:13:56 --> 00:13:59
straight line like this, as you
increase the amount of sugar,

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00:13:59 --> 00:14:00
then it would be an
ideal solution.

214
00:14:00 --> 00:14:02
Many things are
very close to it.

215
00:14:02 --> 00:14:07
It's not such a bad
approximation.

216
00:14:07 --> 00:14:09
Especially in that
region up here.

217
00:14:09 --> 00:14:12
Around zero, when the amount
of sugar is pretty small.

218
00:14:12 --> 00:14:19
You expect to obey an
ideal solution behavior.

219
00:14:19 --> 00:14:23
Alright, so let's look
now at a glass of water

220
00:14:23 --> 00:14:26
that has sugar in it.
 

221
00:14:26 --> 00:14:28
We've got my glass
of water, H20.

222
00:14:28 --> 00:14:34
With some sugar
dissolved in it.

223
00:14:34 --> 00:14:42
And it's got a certain vapor
pressure on top. pA, or pH2O.

224
00:14:42 --> 00:14:43
And I'm going to ask the
question, what's the difference

225
00:14:43 --> 00:14:49
between the vapor pressure what
it would be without

226
00:14:49 --> 00:14:51
the sugar in it.
 

227
00:14:51 --> 00:14:59
What is pA star minus pA?
 

228
00:14:59 --> 00:15:07
Obeys Raoult's law. pA
star, pA is xA pA star,

229
00:15:07 --> 00:15:09
plug that in here.
 

230
00:15:09 --> 00:15:15
That's one minus xA pA
star, one minus xA is xB.

231
00:15:15 --> 00:15:18
xB pA star.
 

232
00:15:18 --> 00:15:21
Well, the first thing I know is
that this is a positive number.

233
00:15:21 --> 00:15:25
xB is a positive number. pA
star is a positive number.

234
00:15:25 --> 00:15:26
This is greater than zero.
 

235
00:15:26 --> 00:15:34
That means that the act of
putting some solute in my water

236
00:15:34 --> 00:15:40
decreases the vapor pressure of
the water in the gas phase.

237
00:15:40 --> 00:15:41
Pretty simple.
 

238
00:15:41 --> 00:15:47
It's called vapor pressure law.
 

239
00:15:47 --> 00:15:49
And we can actually take
this one step further.

240
00:15:49 --> 00:15:57
We can draw a phase diagram.
 

241
00:15:57 --> 00:16:00
So our usual phase diagram here
with temperature on this axis

242
00:16:00 --> 00:16:04
here and pressure on this
axis here, and we've got our

243
00:16:04 --> 00:16:09
triple point sitting here.
 

244
00:16:09 --> 00:16:13
And our gas-liquid line,
with a critical point here.

245
00:16:13 --> 00:16:16
So this is gas here,
there's a liquid here.

246
00:16:16 --> 00:16:21
And we've got our, this
is going to be water.

247
00:16:21 --> 00:16:23
So it's got a negative slope.
 

248
00:16:23 --> 00:16:26
I think I've got it
right this time.

249
00:16:26 --> 00:16:29
And there's the
solid phase here.

250
00:16:29 --> 00:16:31
OK, this is for the pure stuff.
 

251
00:16:31 --> 00:16:40
So now I can draw a similar
diagram for sugared water.

252
00:16:40 --> 00:16:44
And I just looked at this,
assuming it obeys Raoult's law.

253
00:16:44 --> 00:16:50
What we looked at here was
the gas-liquid coexistence.

254
00:16:50 --> 00:16:55
And we found that the
gas-liquid coexistence point,

255
00:16:55 --> 00:16:58
the vapor pressure of the
water, was less than

256
00:16:58 --> 00:16:59
if it were pure.
 

257
00:16:59 --> 00:17:02
So that means that if I looked
at the gas-liquid coexistence,

258
00:17:02 --> 00:17:05
which is this line right here.
 

259
00:17:05 --> 00:17:08
In the presence of sugar,
that whole line is

260
00:17:08 --> 00:17:11
going to get lowered.
 

261
00:17:11 --> 00:17:14
The pressure, the vapor
pressure, of the water is

262
00:17:14 --> 00:17:17
going to be lower when
the sugar is there.

263
00:17:17 --> 00:17:26
So the whole thing is
going to get lower.

264
00:17:26 --> 00:17:32
Now, if I have a solid, if the
water is a solid, it's going

265
00:17:32 --> 00:17:34
to crystallize the water,
the sugar is going to be

266
00:17:34 --> 00:17:36
separated from the water.
 

267
00:17:36 --> 00:17:37
The sugar and the water
are not going to really

268
00:17:37 --> 00:17:40
know about each other.
 

269
00:17:40 --> 00:17:44
They're just going to have big
chunks of pure water, with

270
00:17:44 --> 00:17:49
every now and then a molecule
of sugar that's in there.

271
00:17:49 --> 00:17:53
So you don't really expect to
see any difference between the

272
00:17:53 --> 00:17:58
pure solid water, in terms
of the gas-solid interface,

273
00:17:58 --> 00:18:02
and the pure liquid water.
 

274
00:18:02 --> 00:18:06
There's no sugar
in the gas phase.

275
00:18:06 --> 00:18:10
And as far as the gas phase
is concerned, that solid

276
00:18:10 --> 00:18:11
phase is pure water.
 

277
00:18:11 --> 00:18:14
It's just seeing pure ice
crystals on the surface.

278
00:18:14 --> 00:18:17
Every now and then there's
a molecule of sugar.

279
00:18:17 --> 00:18:22
So you expect this to
be pretty close here.

280
00:18:22 --> 00:18:24
So you expect the triple
point to go down.

281
00:18:24 --> 00:18:29
And if I now, I go up to the
solid-liquid coexistence,

282
00:18:29 --> 00:18:33
and go up like this, I can
make a prediction about

283
00:18:33 --> 00:18:36
solid-liquid coexistence.
 

284
00:18:36 --> 00:18:41
I can predict that if I'm
looking at, let's put

285
00:18:41 --> 00:18:49
one bar here somewhere.
 

286
00:18:49 --> 00:18:52
One bar.
 

287
00:18:52 --> 00:18:53
Room temperature,
sitting here somewhere.

288
00:18:53 --> 00:18:57
One bar, room temperature,
water is all liquid.

289
00:18:57 --> 00:19:03
This is what it would usually,
solid, liquid, that would

290
00:19:03 --> 00:19:06
be 273 degrees Kelvin.
 

291
00:19:06 --> 00:19:10
That's the melting
point of water.

292
00:19:10 --> 00:19:14
So now if I put sugar in the
water, and I follow my diagram

293
00:19:14 --> 00:19:21
here, what I find is that the
new melting point of water is

294
00:19:21 --> 00:19:29
some new temperature T, T1,
where T1 is less than

295
00:19:29 --> 00:19:30
273 degrees Kelvin.
 

296
00:19:30 --> 00:19:31
You all know this.
 

297
00:19:31 --> 00:19:34
When you put something, like an
impurity, a salt or sugar in

298
00:19:34 --> 00:19:37
the water, the melting
point gets depressed.

299
00:19:37 --> 00:19:43
It goes down.
 

300
00:19:43 --> 00:19:44
Straight from here.
 

301
00:19:44 --> 00:19:46
You can build it up
straight from here.

302
00:19:46 --> 00:19:51
When we get to the colligative
properties, we'll attack

303
00:19:51 --> 00:19:55
it from the point of
chemical potentials.

304
00:19:55 --> 00:19:57
We'll equate chemical
potentials and we'll look at

305
00:19:57 --> 00:19:59
the change in the chemical
potential in the water, and

306
00:19:59 --> 00:20:02
the presence of the sugar
and we'll do it rigorously.

307
00:20:02 --> 00:20:05
But this is sort of the first
indication that these

308
00:20:05 --> 00:20:08
colligative properties are
all connected to each other.

309
00:20:08 --> 00:20:13
They're connected to
this diagram here.

310
00:20:13 --> 00:20:14
OK.
 

311
00:20:14 --> 00:20:19
So when I was a kid, and I
knew that this was the case.

312
00:20:19 --> 00:20:21
Everybody knows this
is the case, right?

313
00:20:21 --> 00:20:24
You put salt in the water
and you expect all sorts

314
00:20:24 --> 00:20:25
of things to happen.
 

315
00:20:25 --> 00:20:27
You expect the vapor
pressure to go down.

316
00:20:27 --> 00:20:31
And certainly you know that
you add salt to the roads and

317
00:20:31 --> 00:20:33
the water melts, et cetera.
 

318
00:20:33 --> 00:20:36
And my mother always told me
that the reason why you added

319
00:20:36 --> 00:20:42
salt to the water was so that
the temperature of the water

320
00:20:42 --> 00:20:46
would get higher when
you try to boil pasta.

321
00:20:46 --> 00:20:50
And for, I don't know, fifteen
years I believed her.

322
00:20:50 --> 00:20:53
Until I taught thermodynamics.
 

323
00:20:53 --> 00:20:57
And I actually calculated the
change in the temperature

324
00:20:57 --> 00:21:01
by adding a few pinches
of salt to the water.

325
00:21:01 --> 00:21:04
And you know how small that
temperature change is?

326
00:21:04 --> 00:21:05
You should calculate it.
 

327
00:21:05 --> 00:21:07
It's really tiny.
 

328
00:21:07 --> 00:21:09
It won't make any
difference for the water.

329
00:21:09 --> 00:21:12
Probably the difference in the
temperature of the water will

330
00:21:12 --> 00:21:16
be the same as if I cooked here
versus halfway up Mount

331
00:21:16 --> 00:21:18
Washington in New
Hampshire or something.

332
00:21:18 --> 00:21:19
Probably even less.
 

333
00:21:19 --> 00:21:20
There's no difference.
 

334
00:21:20 --> 00:21:23
The reason why you add salt
is so that it tastes better.

335
00:21:23 --> 00:21:24
That's the only reason.
 

336
00:21:24 --> 00:21:28
It has nothing to do
with thermodynamics.

337
00:21:28 --> 00:21:35
Alright, any questions on this?
 

338
00:21:35 --> 00:21:37
So we going to make it a little
bit more complicated now,

339
00:21:37 --> 00:21:38
if there's no questions.
 

340
00:21:38 --> 00:21:43
We're going to now have,
instead of having one

341
00:21:43 --> 00:21:47
non-volatile solute, use we're
going to have two, we're going

342
00:21:47 --> 00:21:47
to have this mixture.
 

343
00:21:47 --> 00:21:58
Where both components
are volatile.

344
00:21:58 --> 00:22:03
Can we erase this?
 

345
00:22:03 --> 00:22:03
Actually, I may not erase this.
 

346
00:22:03 --> 00:22:12
Because I want to keep
Raoult's law here.

347
00:22:12 --> 00:22:13
OK, we're going to
have a mixture of two

348
00:22:13 --> 00:22:16
volatile components.
 

349
00:22:16 --> 00:22:19
For instance, water
and alcohol.

350
00:22:19 --> 00:22:25
Your your typical
martini type situation.

351
00:22:25 --> 00:22:26
Or margarita.
 

352
00:22:26 --> 00:22:40
Whatever's your favorite drink.
 

353
00:22:40 --> 00:22:47
So now we have xA, we have
xB, we have yA, we have yB.

354
00:22:47 --> 00:23:00
And we're going to assume that
both obey Raoult's law with

355
00:23:00 --> 00:23:02
respect to each other.
 

356
00:23:02 --> 00:23:08
So now if I draw a diagram, and
I just focus on, so there's the

357
00:23:08 --> 00:23:12
total pressure sitting here.
 

358
00:23:12 --> 00:23:15
And I just focus on one
component and ignore the other.

359
00:23:15 --> 00:23:18
Suppose I focus on A.
 

360
00:23:18 --> 00:23:22
And I know that A in the
presence of B, the partial

361
00:23:22 --> 00:23:25
pressure of A is going
to obey Raoult's law.

362
00:23:25 --> 00:23:28
And let's take A to be the
more volatile component

363
00:23:28 --> 00:23:30
in this case here.
 

364
00:23:30 --> 00:23:35
So pA star is bigger
than pB star.

365
00:23:35 --> 00:23:39
The partial pressure, pure A,
had a particular temperature,

366
00:23:39 --> 00:23:45
T fixed is bigger than
pB star, it's the more

367
00:23:45 --> 00:23:48
volatile of the two.
 

368
00:23:48 --> 00:23:57
So if I were to plot pA as a
function of xB, going from zero

369
00:23:57 --> 00:23:59
to one, so it'll be Raoult's
law, I'm going to

370
00:23:59 --> 00:24:04
start at pA star.
 

371
00:24:04 --> 00:24:09
And I'm going to go down
linearly to xB equals one.

372
00:24:09 --> 00:24:12
Now I can do the same thing for
the partial pressure of B.

373
00:24:12 --> 00:24:19
And assume that in the presence
of A, just looking at this

374
00:24:19 --> 00:24:21
component B, now, and I'm
looking at the partial pressure

375
00:24:21 --> 00:24:24
of B, and it's got its
impurity in it, A.

376
00:24:24 --> 00:24:27
B obeys Raoult's law
with A in there.

377
00:24:27 --> 00:24:34
But now if I want to keep the
same x axis here, I can look

378
00:24:34 --> 00:24:36
at xA here, going
from zero to one.

379
00:24:36 --> 00:24:40
So I just basically
flip the thing over.

380
00:24:40 --> 00:24:44
So at xA equals zero
that's pure B. pB star

381
00:24:44 --> 00:24:45
is less than pA star.
 

382
00:24:45 --> 00:24:50
I'm starting here, on that side
here. pB star, and I'm going

383
00:24:50 --> 00:25:00
down in a straight line to
zero. pB is equal to xB pB

384
00:25:00 --> 00:25:02
star, straight line all the way
to zero when xB is

385
00:25:02 --> 00:25:05
equal to zero.
 

386
00:25:05 --> 00:25:08
So that's pB.
 

387
00:25:08 --> 00:25:10
And that's pA.
 

388
00:25:10 --> 00:25:17
Now, the total pressure is the
sum of the two. pA plus pB.

389
00:25:17 --> 00:25:21
So the total pressure in this
system here, that I measure

390
00:25:21 --> 00:25:23
here, is just the sum of
these two straight lines.

391
00:25:23 --> 00:25:25
Let me do it in blue.
 

392
00:25:25 --> 00:25:30
The total pressure, if you
add up these two straight

393
00:25:30 --> 00:25:32
lines together, you got
another straight line.

394
00:25:32 --> 00:25:37
Two straight lines make a
straight line. p is pA

395
00:25:37 --> 00:25:48
plus pB is equal to xA pA
star plus xB pB star.

396
00:25:48 --> 00:25:54
Now, xA and xB are related.
xB is one minus xA.

397
00:25:54 --> 00:25:57
And so really, this is only a
function of one variable here.

398
00:25:57 --> 00:26:02
Linearly, with respect
to one variable.

399
00:26:02 --> 00:26:04
OK, what does this
diagram tell me?

400
00:26:04 --> 00:26:13
So now, let's ignore
how we built it.

401
00:26:13 --> 00:26:19
And let's take a look at it.
 

402
00:26:19 --> 00:26:23
So I have, going to take
xB on the bottom here.

403
00:26:23 --> 00:26:25
From zero to one.
 

404
00:26:25 --> 00:26:29
We could choose either one, and
I've got the straight line for

405
00:26:29 --> 00:26:36
the total pressure going
from pA star to pB star.

406
00:26:36 --> 00:26:38
T is fixed.
 

407
00:26:38 --> 00:26:42
I need to tell you that T is
fixed because I have two

408
00:26:42 --> 00:26:43
degrees of freedom, right?
 

409
00:26:43 --> 00:26:45
I have two degrees of freedom.
 

410
00:26:45 --> 00:26:48
My degrees of freedom are the
temperature, which I'm fixing.

411
00:26:48 --> 00:26:52
And the composition in the
liquid phase, which I'm fixing.

412
00:26:52 --> 00:26:59
So my two variables
are xB and T.

413
00:26:59 --> 00:27:00
Those are my two
degrees of freedom.

414
00:27:00 --> 00:27:03
And I'm telling you what
the total pressure is.

415
00:27:03 --> 00:27:08
I'm giving you total pressure
as a function of xB and T.

416
00:27:08 --> 00:27:11
By fixing T, I'm really telling
you what the pressure is

417
00:27:11 --> 00:27:13
of the function of xB.
 

418
00:27:13 --> 00:27:16
So the Gibbs phase rule then
tells me that the pressure

419
00:27:16 --> 00:27:20
should be only a function
of xB if T is fixed.

420
00:27:20 --> 00:27:21
Should be a line.
 

421
00:27:21 --> 00:27:22
Not necessarily straight,
but it should be a

422
00:27:22 --> 00:27:26
line in this diagram.
 

423
00:27:26 --> 00:27:28
To have coexistence.
 

424
00:27:28 --> 00:27:34
So what this line is, then,
this line is the line of points

425
00:27:34 --> 00:27:37
that tells me when I have
coexistence between the gas

426
00:27:37 --> 00:27:41
phase and the liquid phase.
 

427
00:27:41 --> 00:27:41
The coexistence line.
 

428
00:27:41 --> 00:27:45
This is the gas-liquids
coexistence line

429
00:27:45 --> 00:27:47
for the mixture.
 

430
00:27:47 --> 00:27:55
It's not so different than this
coexistence line up here.

431
00:27:55 --> 00:27:59
So you can think of this as
a phase diagram, kind of

432
00:27:59 --> 00:28:02
like you thought, you know
this as a phase diagram.

433
00:28:02 --> 00:28:05
It's got a coexistence line.
 

434
00:28:05 --> 00:28:08
And then if I'm above this
line, if I'm at a certain

435
00:28:08 --> 00:28:14
pressure here, I'm at this
point here, this pressure well,

436
00:28:14 --> 00:28:18
let's go to a slightly higher
different pressure here.

437
00:28:18 --> 00:28:22
And this composition here.
 

438
00:28:22 --> 00:28:24
I'm not on the line.
 

439
00:28:24 --> 00:28:27
That means that I don't
have coexistence

440
00:28:27 --> 00:28:29
between liquid and gas.
 

441
00:28:29 --> 00:28:30
I'm above the line.
 

442
00:28:30 --> 00:28:35
I'm at a pressure higher then
where I would get coexistence.

443
00:28:35 --> 00:28:38
The pressure is higher,
means I probably don't

444
00:28:38 --> 00:28:42
have a gas any more.
 

445
00:28:42 --> 00:28:44
Probably means that the total
pressure pushing down on

446
00:28:44 --> 00:28:48
my mixture is too high to
have a gas phase around.

447
00:28:48 --> 00:28:55
It means that the top part
here is a liquid phase.

448
00:28:55 --> 00:29:00
So I'm in the liquid phase if
I'm above this line and I have

449
00:29:00 --> 00:29:03
a coexistence between the gas
and the liquid if

450
00:29:03 --> 00:29:06
I'm on the line.
 

451
00:29:06 --> 00:29:13
Now, if I'm below this line
here, well, I get in trouble.

452
00:29:13 --> 00:29:17
I get in trouble because my
first instinct would be,

453
00:29:17 --> 00:29:19
well this is just gas.
 

454
00:29:19 --> 00:29:22
I go from liquid, and
then I have coexistence.

455
00:29:22 --> 00:29:24
Then I go into the gas phase.
 

456
00:29:24 --> 00:29:27
But my x-axis here is
the composition in

457
00:29:27 --> 00:29:28
the liquid phase.
 

458
00:29:28 --> 00:29:31
There's no liquid around.
 

459
00:29:31 --> 00:29:35
So I really can't
say anything here.

460
00:29:35 --> 00:29:37
This diagram is a little
bit meaningless down here.

461
00:29:37 --> 00:29:40
Because my x-axis is describing
a phase which doesn't

462
00:29:40 --> 00:29:45
exist on the diagram.
 

463
00:29:45 --> 00:29:48
On this diagram, here.
 

464
00:29:48 --> 00:29:51
So when you see it, this
diagram here with Raoult's law

465
00:29:51 --> 00:29:54
on it, you've got to remember
that the part that's really

466
00:29:54 --> 00:30:00
interesting is above the line
and at the line itself.

467
00:30:00 --> 00:30:14
So you could do an experiment,
then, on this line here.

468
00:30:14 --> 00:30:14
T fixed.
 

469
00:30:14 --> 00:30:20
This is the liquid phase.
 

470
00:30:20 --> 00:30:22
So suppose I start with
some high pressure

471
00:30:22 --> 00:30:23
up here somewhere.
 

472
00:30:23 --> 00:30:29
At this point, let's
call this point one.

473
00:30:29 --> 00:30:36
We've got my container, my
piston here. p is equal to p1.

474
00:30:36 --> 00:30:38
Again, I put the point
in an awkward place.

475
00:30:38 --> 00:30:49
Let's put it right there.
p1, and composition

476
00:30:49 --> 00:30:56
xB. p1 and xB here.
 

477
00:30:56 --> 00:31:02
And I'm going to slowly
decrease the pressure.

478
00:31:02 --> 00:31:03
I start in this composition.
 

479
00:31:03 --> 00:31:05
The composition doesn't change.
 

480
00:31:05 --> 00:31:07
I'm just decreasing
the pressure.

481
00:31:07 --> 00:31:08
Decrease the pressure, decrease
the pressure, decrease

482
00:31:08 --> 00:31:09
the pressure.
 

483
00:31:09 --> 00:31:14
And at some point, I
get to that line.

484
00:31:14 --> 00:31:18
Get to that line, what happens
when I get to that line?

485
00:31:18 --> 00:31:21
I start to see bubbles
coming out of the liquid.

486
00:31:21 --> 00:31:22
Vapor starts to form, because
it wants to be in the

487
00:31:22 --> 00:31:24
coexistence line.
 

488
00:31:24 --> 00:31:25
So I get to that line.
 

489
00:31:25 --> 00:31:29
And I get a little bit
of bubbles forming.

490
00:31:29 --> 00:31:32
And a little bit of vapor.
 

491
00:31:32 --> 00:31:33
That's the liquid here.
 

492
00:31:33 --> 00:31:36
Little bit of gas.
 

493
00:31:36 --> 00:31:44
And that's why this line here
is called the bubble line.

494
00:31:44 --> 00:31:48
So I've gone down,
decreased the pressure.

495
00:31:48 --> 00:31:50
So p has gone down now.
 

496
00:31:50 --> 00:31:52
Decreased the pressure.
 

497
00:31:52 --> 00:31:58
And if I keep on decreasing
pressure, well, I'm not

498
00:31:58 --> 00:32:00
going to jump into this
area of the diagram.

499
00:32:00 --> 00:32:02
Because this area of the
diagram means there's

500
00:32:02 --> 00:32:05
no liquid, it's a
meaningless area.

501
00:32:05 --> 00:32:08
If I keep decreasing the
pressure, I'm still going

502
00:32:08 --> 00:32:09
to be in coexistence.
 

503
00:32:09 --> 00:32:12
I'm just going to
make more gas.

504
00:32:12 --> 00:32:13
More vapor.
 

505
00:32:13 --> 00:32:15
I'm going to transfer
more of the liquid

506
00:32:15 --> 00:32:17
into the vapor phase.
 

507
00:32:17 --> 00:32:21
I'm going to ride
down that line.

508
00:32:21 --> 00:32:22
I'm going to ride
down the line.

509
00:32:22 --> 00:32:23
I'm decreasing the pressure.
 

510
00:32:23 --> 00:32:24
I'm not going to
skip into here.

511
00:32:24 --> 00:32:29
I'm going to ride
down the line here.

512
00:32:29 --> 00:32:33
We're going to keep decreasing
the pressure even more.

513
00:32:33 --> 00:32:34
p goes down even more.
 

514
00:32:34 --> 00:32:37
Now, I make even more vapor.
 

515
00:32:37 --> 00:32:41
There's my piston right there.
 

516
00:32:41 --> 00:32:43
There's the gas phase.
 

517
00:32:43 --> 00:32:46
There's the liquid phase.
 

518
00:32:46 --> 00:32:51
Now I've transferred material
from the liquid phase

519
00:32:51 --> 00:32:53
to the gas phase.
 

520
00:32:53 --> 00:32:57
But I also changed the
composition of my liquid phase.

521
00:32:57 --> 00:33:02
And if I read my new
composition, this is what

522
00:33:02 --> 00:33:05
I started out with. xB.
 

523
00:33:05 --> 00:33:09
My new composition in the
liquid phase is going to be,

524
00:33:09 --> 00:33:15
let's call it xB prime.
 

525
00:33:15 --> 00:33:17
There's going to be
more solute in there.

526
00:33:17 --> 00:33:24
Or more B in there, than
what I started out with.

527
00:33:24 --> 00:33:25
Now which one was the
more volatile one?

528
00:33:25 --> 00:33:27
A or B?
 

529
00:33:27 --> 00:33:29
A was more volatile, right?
 

530
00:33:29 --> 00:33:34
So as I decreased the pressure
and I started bubbling off the

531
00:33:34 --> 00:33:38
material, which one's going to
bubble off preferentially?

532
00:33:38 --> 00:33:39
A is going to.
 

533
00:33:39 --> 00:33:47
So it makes sense that I would
concentrate B in there.

534
00:33:47 --> 00:33:49
Decrease the pressure.
 

535
00:33:49 --> 00:33:51
Bubbles that are
rich in A come out.

536
00:33:51 --> 00:33:52
Both A and
 

537
00:33:52 --> 00:33:52
B come out.
 

538
00:33:52 --> 00:33:55
But more A comes out than B.
 

539
00:33:55 --> 00:33:58
And what's left over is rich
and then more than the less

540
00:33:58 --> 00:34:01
volatile material which is B.
 

541
00:34:01 --> 00:34:05
So we're beginning to see how
distillation comes about.

542
00:34:05 --> 00:34:07
This is the first step in it.
 

543
00:34:07 --> 00:34:18
In a distillation process.
 

544
00:34:18 --> 00:34:21
So let's make this a little
bit more complicated now.

545
00:34:21 --> 00:34:30
Any questions, first?
 

546
00:34:30 --> 00:34:32
Alright, now, suppose
I wanted to know.

547
00:34:32 --> 00:34:35
So I found out where
the composition in the

548
00:34:35 --> 00:34:38
liquid phase is here.
 

549
00:34:38 --> 00:34:41
And I know that should tell
me immediately whether the

550
00:34:41 --> 00:34:43
composition in the
gas phase is here.

551
00:34:43 --> 00:34:47
So I have all, I have
everything I need to know to

552
00:34:47 --> 00:34:52
calculate what the composition
in the gas phase is.

553
00:34:52 --> 00:34:55
So I have everything I need,
then, to calculate and draw a

554
00:34:55 --> 00:34:58
diagram that looks
just like this.

555
00:34:58 --> 00:35:04
Except where my x-axis is
the y's instead of the x's.

556
00:35:04 --> 00:35:08
Going to make the same diagram,
except now I'm going to use

557
00:35:08 --> 00:35:10
the gas phase as my
reference point.

558
00:35:10 --> 00:35:15
The composition in
the gas phase.

559
00:35:15 --> 00:35:22
So I need to get, so here I
got p as a function of xB.

560
00:35:22 --> 00:35:30
I want p, total pressure,
as a function of yB.

561
00:35:30 --> 00:35:33
So I can draw a diagram that
looks just like this except

562
00:35:33 --> 00:35:41
now with the x-axis being
the gas phase composition.

563
00:35:41 --> 00:35:41
Let's turn the crank.
 

564
00:35:41 --> 00:35:45
What do I know?
 

565
00:35:45 --> 00:35:55
I know Dalton's law. pA is
yA times total pressure.

566
00:35:55 --> 00:35:59
I'm trying mix up the partial
pressures and the total

567
00:35:59 --> 00:36:02
pressure in all ways that
I know how to write it.

568
00:36:02 --> 00:36:05
And then hope that something's
going to come out that's

569
00:36:05 --> 00:36:06
going to be helpful.
 

570
00:36:06 --> 00:36:10
So that's Dalton's law.
 

571
00:36:10 --> 00:36:17
I know Raoult's law. pA is
xA star times pA star.

572
00:36:17 --> 00:36:22
And that's Raoult.
 

573
00:36:22 --> 00:36:25
And I can write Raoult's
law in a different way.

574
00:36:25 --> 00:36:30
I can write as pB is equal
to xB pB star, that's

575
00:36:30 --> 00:36:35
just Raoult's law for B.
 

576
00:36:35 --> 00:36:41
One minus xA pB star.
 

577
00:36:41 --> 00:36:45
And I'm looking for this.
 

578
00:36:45 --> 00:36:48
I'm looking for the
composition in the gas phase.

579
00:36:48 --> 00:36:50
Or the total pressure as a
function of the composition

580
00:36:50 --> 00:36:51
in the gas phase.
 

581
00:36:51 --> 00:36:54
Let's start by finding the
composition in the gas phase as

582
00:36:54 --> 00:36:58
a function of the composition
in the liquid phase.

583
00:36:58 --> 00:37:01
And the total pressure.
 

584
00:37:01 --> 00:37:08
Rewrite yA is equal
to pA over P.

585
00:37:08 --> 00:37:11
Total pressure is pA plus pB.
 

586
00:37:11 --> 00:37:16
Sum of the two partial
pressures. pA over pA plus pB.

587
00:37:16 --> 00:37:19
And I'm trying to get yA
in terms of the xA's.

588
00:37:19 --> 00:37:21
Or xB's, in terms
of composition in

589
00:37:21 --> 00:37:22
the liquid phase.
 

590
00:37:22 --> 00:37:25
And I know how to relate
that to a constant.

591
00:37:25 --> 00:37:29
Which is the vapor pressure of
the pure material times the,

592
00:37:29 --> 00:37:33
and the composition of the
liquid phase, Raoult's law.

593
00:37:33 --> 00:37:44
This is xA pA star divided by
xA pA star plus xB pB star.

594
00:37:44 --> 00:37:46
And I know that xB
is one minus xA.

595
00:37:46 --> 00:37:58
So this is xA pA star times
pB star plus pA star

596
00:37:58 --> 00:38:02
minus pB star times xA.
 

597
00:38:02 --> 00:38:07
Where all I did was replace
xB with one minus xA.

598
00:38:07 --> 00:38:10
And rearrange my
equation on the bottom.

599
00:38:10 --> 00:38:15
So I've gotten the composition
in the gas phase in terms of

600
00:38:15 --> 00:38:17
the composition in
the liquid phase.

601
00:38:17 --> 00:38:18
They're not independent.
 

602
00:38:18 --> 00:38:20
I knew they weren't
independent.

603
00:38:20 --> 00:38:24
And this just shows
me that math works.

604
00:38:24 --> 00:38:24
OK.
 

605
00:38:24 --> 00:38:25
So I've got that.
 

606
00:38:25 --> 00:38:28
And I can invert this to get
the composition in the liquid

607
00:38:28 --> 00:38:31
phase in terms of the
composition in the gas phase.

608
00:38:31 --> 00:38:35
It's not so straightforward,
but you can get xA as a

609
00:38:35 --> 00:38:37
function of yA, as well.
 

610
00:38:37 --> 00:38:40
You've got xA, if you
reverse this you get xA

611
00:38:40 --> 00:38:44
is equal to yA pB star.
 

612
00:38:44 --> 00:38:54
It looks kind of the same. pB
star plus pA star plus pB star

613
00:38:54 --> 00:39:01
minus pA star times yA.
xA as a function of yA.

614
00:39:01 --> 00:39:05
 


615
00:39:05 --> 00:39:05
Alright.
 

616
00:39:05 --> 00:39:11
So now you can actually
put everything together,

617
00:39:11 --> 00:39:13
starting from Dalton's law.
 

618
00:39:13 --> 00:39:15
Because really what we want
is the total pressure as a

619
00:39:15 --> 00:39:19
function of the composition
in the gas phase.

620
00:39:19 --> 00:39:20
So there's our total
pressure here.

621
00:39:20 --> 00:39:22
There's the composition
in the gas phase.

622
00:39:22 --> 00:39:24
There's pA.
 

623
00:39:24 --> 00:39:26
We've found composition in
the gas phase in terms

624
00:39:26 --> 00:39:30
of composition in
the liquid phase.

625
00:39:30 --> 00:39:38
P is equal to yA over pA.
 

626
00:39:38 --> 00:39:45
Which is equal to yA
over xA pA star.

627
00:39:45 --> 00:39:48
So if we could get rid of
this here, in terms of

628
00:39:48 --> 00:39:49
yA, we'd be all set.
 

629
00:39:49 --> 00:39:52
We'd get the total pressure as
a function of the composition

630
00:39:52 --> 00:39:54
in the gas phase.
 

631
00:39:54 --> 00:39:57
And there's what we need.
 

632
00:39:57 --> 00:40:00
So we plug that in here.
 

633
00:40:00 --> 00:40:06
And now we have the total
pressure in terms of constants

634
00:40:06 --> 00:40:12
like these pure vapor pressures
and the composition of

635
00:40:12 --> 00:40:18
A in the gas phase.
 

636
00:40:18 --> 00:40:21
So let me find a way to
write that somewhere.

637
00:40:21 --> 00:40:24
Let me write it right here
in this little box here.

638
00:40:24 --> 00:40:29
Plug it in, you get something
that's not a straight line.

639
00:40:29 --> 00:40:31
Unfortunately.
 

640
00:40:31 --> 00:40:38
Doesn't look like Raoult's law.
pA star pB star. pA star plus

641
00:40:38 --> 00:40:48
pB star minus pA star times yA.
 

642
00:40:48 --> 00:40:49
It's not a straight line.
 

643
00:40:49 --> 00:40:57
It's a more
complicated function.

644
00:40:57 --> 00:41:16
What does it look like?
 

645
00:41:16 --> 00:41:17
Looks like this.
 

646
00:41:17 --> 00:41:23
I'm going to use the
same sort of plot.

647
00:41:23 --> 00:41:24
I going to plot the total
pressure on this axis

648
00:41:24 --> 00:41:25
here, on the y axis.
 

649
00:41:25 --> 00:41:28
I'm going to keep the T fixed.
 

650
00:41:28 --> 00:41:31
Temperature is fixed.
 

651
00:41:31 --> 00:41:34
So I'm going to find the total
pressure as a function of

652
00:41:34 --> 00:41:35
the composition in the gas
 

653
00:41:35 --> 00:41:35
phase.
 

654
00:41:35 --> 00:41:42
I'm going to put yB here.
yB going from zero to one.

655
00:41:42 --> 00:41:50
So it's kind of looking
like what I had before.

656
00:41:50 --> 00:41:51
Total pressure is going
to look the same.

657
00:41:51 --> 00:41:54
If you want to do it in terms
of B, you just have everywhere

658
00:41:54 --> 00:41:55
you see B replace it with A.
 

659
00:41:55 --> 00:41:57
And everywhere you see
A replace it with B.

660
00:41:57 --> 00:42:02
You just need to
change the two.

661
00:42:02 --> 00:42:06
You know that if yB is equal to
zero, that you have pure A.

662
00:42:06 --> 00:42:12
So you know that this is
going to be pA star here.

663
00:42:12 --> 00:42:14
You know if you have yB equals
to one in the gas phase, you

664
00:42:14 --> 00:42:16
have pure B in the gas phase.
 

665
00:42:16 --> 00:42:20
You know that what you have is
the vapor pressure of the pure

666
00:42:20 --> 00:42:26
B, in this case the pure
water. pB star, OK?

667
00:42:26 --> 00:42:31
Where A is more
volatile then B.

668
00:42:31 --> 00:42:35
A would be the ethanol and
B would be the water.

669
00:42:35 --> 00:42:37
And it's not a straight line.
 

670
00:42:37 --> 00:42:39
If it were Raoult, he would
say just connect these

671
00:42:39 --> 00:42:40
two with a straight line.
 

672
00:42:40 --> 00:42:41
But it's not.
 

673
00:42:41 --> 00:42:44
It's not, we just figured
out with an equation here.

674
00:42:44 --> 00:42:49
And it turns out to be a
line that looks like this.

675
00:42:49 --> 00:42:55
It's a curved line that
looks a bit like this.

676
00:42:55 --> 00:42:57
OK, what does this line mean?
 

677
00:42:57 --> 00:43:01
This line is also a
coexistence line.

678
00:43:01 --> 00:43:02
That's what we
started out with.

679
00:43:02 --> 00:43:05
We started out with Raoult.
 

680
00:43:05 --> 00:43:07
That was part of
our derivation.

681
00:43:07 --> 00:43:15
Raoult's law tells you the
composition at coexistence.

682
00:43:15 --> 00:43:18
Tells you the pressure of, the
total pressure, or partial

683
00:43:18 --> 00:43:21
pressure at coexistence
and the composition.

684
00:43:21 --> 00:43:26
So this is a coexistence line
between the liquid and the gas.

685
00:43:26 --> 00:43:28
Coexistence between
liquid and gas.

686
00:43:28 --> 00:43:30
But now, it's not as a
function of the composition

687
00:43:30 --> 00:43:30
at the liquid phase.
 

688
00:43:30 --> 00:43:34
It's a function of the
composition in the gas phase.

689
00:43:34 --> 00:43:38
Which was missing before.
 

690
00:43:38 --> 00:43:44
So now if I do my experiment,
and I'm, so what

691
00:43:44 --> 00:43:44
does this mean?
 

692
00:43:44 --> 00:43:48
So if I have a point, if I have
a pressure which is lower than

693
00:43:48 --> 00:43:51
the line here, then
I have pure gas.

694
00:43:51 --> 00:43:52
Pressure, low pressure,
I have pure gas.

695
00:43:52 --> 00:43:53
Don't have any liquids.
 

696
00:43:53 --> 00:43:57
And that's fine, because I can,
I know what that point is.

697
00:43:57 --> 00:44:02
If I'm sitting below the
line here somewhere,

698
00:44:02 --> 00:44:06
so I'm sitting here.
 

699
00:44:06 --> 00:44:09
Some composition in the gas
phase, a certain pressure.

700
00:44:09 --> 00:44:11
That's fine.
 

701
00:44:11 --> 00:44:15
It's a meaningful point.
 

702
00:44:15 --> 00:44:18
So my container here
has a piston on it.

703
00:44:18 --> 00:44:26
And I'm starting out with
this pressure, p1 here.

704
00:44:26 --> 00:44:27
p1 pressing down.
 

705
00:44:27 --> 00:44:30
And I got yB in this here.
 

706
00:44:30 --> 00:44:33
And I increase the pressure,
pressure goes up.

707
00:44:33 --> 00:44:35
Composition doesn't change.
 

708
00:44:35 --> 00:44:38
I'm just increasing
the pressure.

709
00:44:38 --> 00:44:39
Increasing the pressure,
increasing the pressure.

710
00:44:39 --> 00:44:42
Now, at some point the pressure
becomes big enough that I

711
00:44:42 --> 00:44:44
start to see liquid forming.
 

712
00:44:44 --> 00:44:54
Little drops of liquid
forming in my container.

713
00:44:54 --> 00:44:58
So now I have little drops of
liquid that are forming in my

714
00:44:58 --> 00:45:01
container, that are going to
start to rain down and form a

715
00:45:01 --> 00:45:05
little bit of liquid pooling at
the bottom of the container.

716
00:45:05 --> 00:45:10
And that's called the dew line.
 

717
00:45:10 --> 00:45:12
The other one was called
the bubble line, this

718
00:45:12 --> 00:45:15
is called the dew line.
 

719
00:45:15 --> 00:45:19
So when I reach the dew line,
I start forming liquid.

720
00:45:19 --> 00:45:21
And again, just like in the
previous case, if I keep

721
00:45:21 --> 00:45:24
increasing the pressure, I
don't go into this area here.

722
00:45:24 --> 00:45:25
First of all, this
area's meaningless for

723
00:45:25 --> 00:45:27
this diagram here.
 

724
00:45:27 --> 00:45:31
Because this area's
pure liquid.

725
00:45:31 --> 00:45:33
But the x-axis here
doesn't tell me about the

726
00:45:33 --> 00:45:34
composition of pure liquid.
 

727
00:45:34 --> 00:45:35
It's all yB.
 

728
00:45:35 --> 00:45:38
It's all about the composition
in the gas -- yes.

729
00:45:38 --> 00:45:40
STUDENT: [INAUDIBLE]
 

730
00:45:40 --> 00:45:43
PROFESSOR: Between the dew
line and the bubble line?

731
00:45:43 --> 00:45:43
STUDENT: Yeah.
 

732
00:45:43 --> 00:45:44
PROFESSOR: This
never, never world.

733
00:45:44 --> 00:45:47
Where you're not
allowed to inhabit.

734
00:45:47 --> 00:45:54
We're going to go
into that next time.

735
00:45:54 --> 00:45:57
On how to use those
two curves together.

736
00:45:57 --> 00:45:59
OK, but the same
thing happens here.

737
00:45:59 --> 00:46:01
If I increase the pressure,
I'm going to ride the

738
00:46:01 --> 00:46:01
coexistence line up.
 

739
00:46:01 --> 00:46:12
I ride it up, and
keep riding it up.

740
00:46:12 --> 00:46:14
And I get liquid
forming in here.

741
00:46:14 --> 00:46:16
And I can find out the
composition in the

742
00:46:16 --> 00:46:17
gas phase now.
 

743
00:46:17 --> 00:46:20
The composition in the gas
phase, as I increase the

744
00:46:20 --> 00:46:25
pressure, I get yB prime,
there's yB that I

745
00:46:25 --> 00:46:26
started out with with.
 

746
00:46:26 --> 00:46:30
With yB prime is less than yB.
 

747
00:46:30 --> 00:46:37
And remember, B was the
less volatile of the two.

748
00:46:37 --> 00:46:42
As I increase the pressure,
I am forming liquid.

749
00:46:42 --> 00:46:46
The composition in the
gas phase changes.

750
00:46:46 --> 00:46:50
You decrease the mole fraction
of the less volatile

751
00:46:50 --> 00:46:51
material in the gas phase.
 

752
00:46:51 --> 00:46:55
The more volatile A
becomes more prevalent

753
00:46:55 --> 00:46:57
in the gas phase.
 

754
00:46:57 --> 00:47:00
And what's coming down
is mostly, then,

755
00:47:00 --> 00:47:02
it's enriched in B.
 

756
00:47:02 --> 00:47:06
You've done an enrichment
of the less volatile

757
00:47:06 --> 00:47:10
material down in here.
 

758
00:47:10 --> 00:47:12
And now you can put the
two together, as the

759
00:47:12 --> 00:47:19
question was asked.
 

760
00:47:19 --> 00:47:21
Now we can have, we
mix the two together.

761
00:47:21 --> 00:47:27
We mix our Raoult diagram
with our dew line diagram.

762
00:47:27 --> 00:47:31
We mix the bubble line and
the dew line together.

763
00:47:31 --> 00:47:32
In one diagram, which is
going to be a powerful

764
00:47:32 --> 00:47:48
diagram to use.
 

765
00:47:48 --> 00:47:51
Total pressure.
 

766
00:47:51 --> 00:47:54
Temperature fixed.
 

767
00:47:54 --> 00:48:02
You've got two materials, pA
star is bigger than pB star.

768
00:48:02 --> 00:48:08
So I'm going to have pA star
here, pB star down here.

769
00:48:08 --> 00:48:13
And if I choose, as my x-axis,
to have xB going from zero to

770
00:48:13 --> 00:48:18
one, I can draw a coexistence
line on this axis.

771
00:48:18 --> 00:48:20
On this diagram.
 

772
00:48:20 --> 00:48:24
That's the coexistence
line in terms of xB.

773
00:48:24 --> 00:48:29
And any point up here is
well-described by this xB

774
00:48:29 --> 00:48:32
value, and the p value.
 

775
00:48:32 --> 00:48:36
And as long as I have xB here,
I'm not allowed to touch the

776
00:48:36 --> 00:48:38
bottom part of this diagram.
 

777
00:48:38 --> 00:48:43
But now if I add another
x-axis, I'm going to add yB.

778
00:48:43 --> 00:48:46
 


779
00:48:46 --> 00:48:50
From zero to one.
 

780
00:48:50 --> 00:48:53
I draw a coexistence
line in terms of yB.

781
00:48:53 --> 00:48:55
 


782
00:48:55 --> 00:48:56
Like this.
 

783
00:48:56 --> 00:49:01
So this line is p as a function
of yB coexistence line.

784
00:49:01 --> 00:49:08
This line is total pressure
of function of xB.

785
00:49:08 --> 00:49:10
And as long as I'm on this
line or below this line,

786
00:49:10 --> 00:49:15
then this axis makes sense.
 

787
00:49:15 --> 00:49:17
That's how we build
this diagram here.

788
00:49:17 --> 00:49:21
It's got a bubble line on top,
and the dew line on the bottom.

789
00:49:21 --> 00:49:26
So I'm up here going down, I'm
going to hit the bubble line.

790
00:49:26 --> 00:49:28
If I'm down here and go up
in pressure, I'm going

791
00:49:28 --> 00:49:29
to hit the dew line.
 

792
00:49:29 --> 00:49:32
And next time -- well, next
time we have an exam.

793
00:49:32 --> 00:49:38
But on Friday, we'll figure
out how to use this diagram.

794
00:49:38 --> 00:49:39