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PROFESSOR: And today
I'll continue with both

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ideal and non-ideal, or
real, liquid mixtures.

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And let me just say the reason
we're in some sense lavishing

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so much attention on this topic
is because, after all, there's

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just an enormous amount of
chemistry that happens

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in liquid mixtures.
 

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Awful lot of biology.
 

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Awful lot of just
processing of stuff.

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Distillation and
everything else.

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So there's so much chemistry
that takes place in liquid

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mixtures that it is really
important to have a sense of

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what the free energy and
chemical potential of each of

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the species is doing in there.
 

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Because of course this is
what directly guides the

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chemistry that happens.
 

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So I just want to start by
finishing up something, a

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couple of things, from the
topics from the last lecture.

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One of them is, I derived an
expression for the lever rule.

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And I just wanted to
make a little more

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explicit the result.
 

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It may not have been
completely clear.

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So, I just want to
go back to that.

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We had looked at what happens
if you start at some particular

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point, labeled one,
and work your way up.

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So we're raising the pressure.
 

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So we're in the all
gas region to start.

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All gas region of
the phase diagram.

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This is xB and yB.
 

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The liquid and gas
mole fractions of B.

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And the idea is that here
is our initial value.

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Now we're only in
the gas phase.

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There is no liquid.
 

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So this is yB of one.
 

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There is no xB at this point.
 

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We raise the pressure.
 

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And at some point, depending on
where we end, we then look to

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both coexistence curves to
determine the compositions of

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both the liquid and
the gas phases.

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So when we look out here,
at the liquid phase.

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That gives us xB.
 

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At where I labeled point
two, so there's two after

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we raise the pressure.
 

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We started at one.
 

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And then over here, we look at
the gas phase coexistence

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curve. so there's our yB
value at two, right?

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And I derived an expression
for the ratio of the

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number of moles in the
gas and liquid phases.

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Because the idea here is,
let's say you're working

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your way up on the curve.
 

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If you just barely get to the
coexistence curve, of course

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then you could see how much
material you've got

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in the two phases.
 

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But there's very little
material still.

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You've still got almost
everything in the gas phase

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when you first reach here.
 

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And as you work your way up
in pressure, you'd have

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more and more liquid.
 

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Of course, if you were to just
keep going, you'd get into

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the pure liquid phase.
 

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Typically, though, you might
stop somewhere in the middle

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and have some reasonable amount
of material in both phases, and

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you want to find out the
composition in each phase.

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And also, want to know
how much material there

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is in each phase.
 

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So the result that I derived
was n gas at two over

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n liquid at two.
 

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The ratio of total moles
in the gas to the liquid.

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What I derived was xA of
two minus yA of one.

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Over yA of one.
 

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Minus yA of two, and I just
want to, in some sense, finish

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up by making the obvious
substitution, which is, of

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course these are
mole fractions.

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So I can just write
xA is one minus xB.

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And yA is one minus yB.
 

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And I just want to put this in
the terms that are written

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here in the phase diagram.
 

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When I got to this result I
pointed out that therefore

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you can use the lever rule.
 

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And I just want to make that
a little bit more explicit.

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So using that relation, then we
can see that ng of two over nl

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of two, total number of moles
in the gas to the liquid is,

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now I'm going to substitute in
to get yB of one minus xB of

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two over yB of two
minus yB of one.

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So all I've done here
is substitute in

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these expressions.
 

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And now I'm going to just
switch both of these.

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00:05:07 --> 00:05:10
And so I'm going to multiply
both sides by negative one.

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So I can write xB of two
minus yB of one over yB

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of one minus yB of two.
 

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So that's the result
that I want.

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Because if I look at the two
segments of this line, on

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either side of the pressure
value I've reached, then what

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00:05:35 --> 00:05:42
I see, of course, is that
over here I've got xB

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of two minus yB of one.
 

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That's this.
 

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And this part is yB of
one minus yB of two.

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That is, it's this part.
 

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And that's here.
 

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And so that's the point, is
that the ratio of moles in the

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gas and liquid phases is just
given by the two segments

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of this on either side.
 

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And so what this allows us to
see very simply is alright,

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I'm going to raise the
pressure at some point.

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And depending on how much
material I feel like I need to

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collect in the new phase, I can
always determine that simply by

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reading off the phase diagram
what the ratio of these

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two segment lengths is.
 

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Now what I want to do.
 

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What what we've done mostly is
just go through these phase

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diagrams, see how to read them.
 

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How to follow events on them
as you change pressure

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in a diagram like this.
 

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Or change temperature in
diagrams like I also

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showed last time.
 

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Now what I'd like to do is go a
little bit further and just

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look at expressions for the
chemical potential and

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the free energies.
 

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So that we can, a little
more quantitatively,

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see what's going on.
 

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So let's just see
what's happening.

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So let's look at the
chemical potentials.

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In ideal liquid mixtures.
 

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So everything is derived from
the fact that when we have any

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of the constituents in both
phases, the chemical potential

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must be equal in both phases.
 

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Right?
 

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So, we can write for A, mu A of
the liquid at some temperature

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00:08:12 --> 00:08:16
and pressure must equal
chemical potential

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00:08:16 --> 00:08:22
of A for the gas.
 

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That's the partial pressure.
 

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00:08:27 --> 00:08:31
So if we have an ideal gas, and
certainly if we're going to

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assume an ideal liquid mixture,
we can safely assume that

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it's an ideal gas above it.
 

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Then we can write
mu A in the gas.

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Is just mu A naught.
 

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That's a function of the
temperature plus RT

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log of pA over p0.
 

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Nothing new here, this is just
our expression for the chemical

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potential in the gas,
with reference to a

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standard potential.
 

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Usually one bar.
 

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At whatever the temperature is.
 

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And, of course, this is how it
varies as the partial pressure

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of A in the gas phase varies.
 

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So then we can just write our
expression for the liquid.

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At our temperature and
pressure, it's given by mu

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naught in the gas phase.
 

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Plus RT log pA over p0.
 

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So it's the same thing.
 

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Because I'm just taking
advantage of this equality.

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But of course, this
is an obvious step,

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having written this.
 

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But is a super important step.
 

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That's what allows us to
do this treatment in such

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a straightforward way.
 

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We know a lot about the
chemical potential of

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00:10:01 --> 00:10:02
something in the gas phase.
 

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00:10:02 --> 00:10:05
Since the gas and liquid are in
equilibrium, therefore we know

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the chemical potential in
the liquid phase too.

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So we can rewrite this by
recognizing that if we just go

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to the limit where we only have
pure liquid A, so

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pure liquid A.
 

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Well, in that case, mu A naught
or sorry, mu A in the liquid

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phase, mu A star is mu A
naught in the gas phase.

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Plus RT log pA over p0.
 

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And so for the mixture, now
all I'm going to do is just

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add and subtract terms.
 

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So we can write mu A in the
liquid at T and p, right?

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So this is in the case of the
limit of the pure liquid.

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Now we're going to the mixture.
 

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So it's just mu A naught plus
RT log pA star over p0 minus

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RT log of pA star over p0.
 

181
00:11:54 --> 00:11:59
 


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Wait a minute, lost a term.
 

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Plus RT log pA over p0.
 

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And all I want to do
now is combine terms.

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00:12:15 --> 00:12:22
To write mu A star liquid
temperature and pressure.

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Plus RT log of pA over pA star.
 

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00:12:30 --> 00:12:35
So this is a very
convenient form for it.

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00:12:35 --> 00:12:40
And then, of course, we have
an expression for pA, right?

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00:12:40 --> 00:12:50
From Raoult's law. pA is
just the mole fraction

190
00:12:50 --> 00:12:55
xA times pA star.
 

191
00:12:55 --> 00:12:59
That's just true for
the ideal mixture.

192
00:12:59 --> 00:13:08
So now, finally, we can write
that mu A in the liquid at T

193
00:13:08 --> 00:13:16
and p is just given
by mu A star.

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00:13:16 --> 00:13:18
So that's for the
pure liquid at that

195
00:13:18 --> 00:13:19
temperature and pressure.
 

196
00:13:19 --> 00:13:25
Plus RT log of the
mole fraction of A.

197
00:13:25 --> 00:13:31
So that's a very simple
expression for the

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00:13:31 --> 00:13:33
chemical potential of A.
 

199
00:13:33 --> 00:13:35
And of course the analogous
expression will hold for

200
00:13:35 --> 00:13:37
any of the constituents.
 

201
00:13:37 --> 00:13:43
In an ideal liquid mixture.
 

202
00:13:43 --> 00:13:48
By the way, it's convenient
because it looks just like

203
00:13:48 --> 00:13:51
the chemical potential in
a mixture of ideal gases.

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Except that we have liquid
instead of gases, right?

205
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Now, of course, since this is
a mole fraction, it's always

206
00:14:03 --> 00:14:05
between zero and one.
 

207
00:14:05 --> 00:14:09
That means this is always
a negative number.

208
00:14:09 --> 00:14:16
So what that means is that the
chemical potential in the

209
00:14:16 --> 00:14:20
solution is always lower than
the chemical potential

210
00:14:20 --> 00:14:57
of the pure liquid.
 

211
00:14:57 --> 00:15:00
Very important result.
 

212
00:15:00 --> 00:15:03
So it has all sorts of
implications that we'll see.

213
00:15:03 --> 00:15:05
One of them is
osmotic pressure.

214
00:15:05 --> 00:15:11
It means that if I have a
biological cell, or some

215
00:15:11 --> 00:15:14
container with a membrane
through which one of the

216
00:15:14 --> 00:15:17
constituents might pass.
 

217
00:15:17 --> 00:15:19
So in other words, the
let's say, component A,

218
00:15:19 --> 00:15:21
maybe it's just water.
 

219
00:15:21 --> 00:15:25
It can pass freely, let's say,
from the outside to the inside

220
00:15:25 --> 00:15:29
of the cell, or from one side
to another of a membrane.

221
00:15:29 --> 00:15:31
And on one side I've
just got pure water.

222
00:15:31 --> 00:15:34
And on the other I've got
saline solution or whatever

223
00:15:34 --> 00:15:36
is in the cells, right?
 

224
00:15:36 --> 00:15:40
What's the water going to
do in that situation?

225
00:15:40 --> 00:15:42
What's going to happen?
 

226
00:15:42 --> 00:15:44
Anybody know?
 

227
00:15:44 --> 00:15:49
So I've got, I just take fresh
cells and plunk them into

228
00:15:49 --> 00:15:52
pure water solution.
 

229
00:15:52 --> 00:15:56
What happens?
 

230
00:15:56 --> 00:15:56
Yeah.
 

231
00:15:56 --> 00:15:58
They burst.
 

232
00:15:58 --> 00:16:00
Water rushes in.
 

233
00:16:00 --> 00:16:03
Because the chemical
potential is lower inside.

234
00:16:03 --> 00:16:07
It's always lower in
solution than outside.

235
00:16:07 --> 00:16:09
So water rushes in, and
the membrane expands.

236
00:16:09 --> 00:16:11
Now, if the membrane is
strong enough, at some

237
00:16:11 --> 00:16:14
point it may not burst.
 

238
00:16:14 --> 00:16:17
And the pressure might
start to go up.

239
00:16:17 --> 00:16:19
Let's say the membrane
is strong enough to

240
00:16:19 --> 00:16:22
resist and not burst.
 

241
00:16:22 --> 00:16:26
Eventually, the water won't
keep going in indefinitely.

242
00:16:26 --> 00:16:29
That's because it won't be
at the same temperature

243
00:16:29 --> 00:16:30
and pressure any more.
 

244
00:16:30 --> 00:16:34
In particular, the pressure
will have changed.

245
00:16:34 --> 00:16:36
So you can build up some high
pressure, what's called

246
00:16:36 --> 00:16:38
osmotic pressure.
 

247
00:16:38 --> 00:16:42
Because of the fact that at the
same pressure the chemical

248
00:16:42 --> 00:16:45
potential of the water's lower
inside the cell or inside the

249
00:16:45 --> 00:16:48
enclosure with the membrane.
 

250
00:16:48 --> 00:16:50
So water will keep filling.
 

251
00:16:50 --> 00:16:52
And at some point the
chemical potentials will

252
00:16:52 --> 00:16:54
equalize because of the
change in pressure.

253
00:16:54 --> 00:17:00
And at that point there'll be
an equilibrium established.

254
00:17:00 --> 00:17:05
We'll see that quantitatively
a little bit later.

255
00:17:05 --> 00:17:06
OK.
 

256
00:17:06 --> 00:17:13
The other thing to notice is,
this is familiar from the

257
00:17:13 --> 00:17:15
expression for a gas mixture.
 

258
00:17:15 --> 00:17:18
What drives the gas mixture?
 

259
00:17:18 --> 00:17:22
Remember back when we discussed
mixing of gases and the fact

260
00:17:22 --> 00:17:24
that they would mix at all.
 

261
00:17:24 --> 00:17:25
What makes that happen?
 

262
00:17:25 --> 00:17:28
What's driving it?
 

263
00:17:28 --> 00:17:30
Yeah, it's entropy, right?
 

264
00:17:30 --> 00:17:31
And that's what's
happening here too.

265
00:17:31 --> 00:17:35
Of course, in the case of the
ideal liquid mixture, there's

266
00:17:35 --> 00:17:37
no energetic interaction.
 

267
00:17:37 --> 00:17:40
The molecules are
non-interacting in this case.

268
00:17:40 --> 00:17:43
But of course, entropy is
going to want them to mix.

269
00:17:43 --> 00:17:45
And that's what's resulting
in the decrease in

270
00:17:45 --> 00:17:50
chemical potential.
 

271
00:17:50 --> 00:17:53
Let's see that a little more
explicitly by just calculating

272
00:17:53 --> 00:17:56
out the free energy
change of mixing.

273
00:17:56 --> 00:18:02
So, delta G of mixing.
 

274
00:18:02 --> 00:18:15
So we're going to start with
two separated liquids.

275
00:18:15 --> 00:18:18
And then we'll
remove the barrier.

276
00:18:18 --> 00:18:26
And we'll have the
two mixed together.

277
00:18:26 --> 00:18:31
So of course, the free energy
in either case is just the sum

278
00:18:31 --> 00:18:34
of the number of moles of each
times the chemical

279
00:18:34 --> 00:18:35
potential of each.
 

280
00:18:35 --> 00:18:37
We have expressions for that.
 

281
00:18:37 --> 00:18:42
So starting G1 in this case.
 

282
00:18:42 --> 00:18:45
It's n of A, number
of moles of A.

283
00:18:45 --> 00:18:49
Times the chemical potential.
 

284
00:18:49 --> 00:18:58
So it's xA mu A star
in the liquid.

285
00:18:58 --> 00:19:08
Plus nB xB B mu B
star of the liquid.

286
00:19:08 --> 00:19:12
Here, G of two.
 

287
00:19:12 --> 00:19:14
After we let them mix.
 

288
00:19:14 --> 00:19:21
Then it's n -- wait a minute.
 

289
00:19:21 --> 00:19:26
It's just n.
 

290
00:19:26 --> 00:19:29
I think I've got that
wrong in the notes also.

291
00:19:29 --> 00:19:31
Right, it's just a number of
moles times the chemical

292
00:19:31 --> 00:19:33
potential in each case.
 

293
00:19:33 --> 00:19:37
So it's the number
of moles times xA.

294
00:19:37 --> 00:19:43
Times mu A in the mixture.
 

295
00:19:43 --> 00:19:52
Plus n xB mu B in the mixture.
 

296
00:19:52 --> 00:19:55
But we've just figured it out
our expressions for mu A and mu

297
00:19:55 --> 00:19:58
B, our chemical potentials of
each constituent in the

298
00:19:58 --> 00:20:02
ideal liquid mixture.
 

299
00:20:02 --> 00:20:11
So this is just n xA.
 

300
00:20:11 --> 00:20:13
And here's our expression.
 

301
00:20:13 --> 00:20:15
And of course these
are going to cancel.

302
00:20:15 --> 00:20:16
Let's write it out.
 

303
00:20:16 --> 00:20:27
So it's mu A star
plus RT log xA.

304
00:20:27 --> 00:20:37
And then, mu B star
plus RT log xB.

305
00:20:37 --> 00:20:39
So our delta G of mixing
is just the difference

306
00:20:39 --> 00:20:44
between these two.
 

307
00:20:44 --> 00:20:58
So it's n RT xA log
xA plus xB log xB.

308
00:20:58 --> 00:21:03
Where have you
seen that before?

309
00:21:03 --> 00:21:03
Yeah.
 

310
00:21:03 --> 00:21:07
The same thing that you had
for the delta G of mixing

311
00:21:07 --> 00:21:10
for an ideal gas mixture.
 

312
00:21:10 --> 00:21:10
Why?
 

313
00:21:10 --> 00:21:13
Because we're not accounting
for any interactions

314
00:21:13 --> 00:21:16
between the molecules in
either of the phases.

315
00:21:16 --> 00:21:22
And if the molecules aren't
interacting, it's all entropy.

316
00:21:22 --> 00:21:26
And the entropy term has the
same form in either case.

317
00:21:26 --> 00:21:29
In microscopic terms, it's just
measuring the fact that there's

318
00:21:29 --> 00:21:32
more disorder in the mixture
than in the pure liquids.

319
00:21:32 --> 00:21:45
Just the same it was
in the gas phase.

320
00:21:45 --> 00:21:49
OK.
 

321
00:21:49 --> 00:21:55
We can see this even more
explicitly if we just recall

322
00:21:55 --> 00:22:00
that G is V dp minus S dT.
 

323
00:22:00 --> 00:22:02
So we can just explicitly
calculate the

324
00:22:02 --> 00:22:05
entropy of mixing.
 

325
00:22:05 --> 00:22:13
Delta S of mixing is just the
partial of delta G of mixing.

326
00:22:13 --> 00:22:17
Negative partial.
 

327
00:22:17 --> 00:22:19
With respect to temperature.
 

328
00:22:19 --> 00:22:20
At constant pressure.
 

329
00:22:20 --> 00:22:30
So it's just minus n R xA
log xA plus xB log xB.

330
00:22:30 --> 00:22:32
 


331
00:22:32 --> 00:22:36
So that's our
entropy of mixing.

332
00:22:36 --> 00:22:41
We can calculate our
enthalpy of mixing.

333
00:22:41 --> 00:22:44
Just delta G of mixing.
 

334
00:22:44 --> 00:22:48
Plus T delta S of mixing.
 

335
00:22:48 --> 00:22:50
But it's immediately apparent
that these are just

336
00:22:50 --> 00:22:53
going to cancel when we
multiply this by T.

337
00:22:53 --> 00:22:55
So there's no
enthalpy of mixing.

338
00:22:55 --> 00:22:58
Just as we expect when
there are no energetic

339
00:22:58 --> 00:22:59
terms involved.
 

340
00:22:59 --> 00:23:06
It's all entropy that's
driving the mixture.

341
00:23:06 --> 00:23:08
One more detail, it's
straightforward to see that

342
00:23:08 --> 00:23:14
there's no volume change.
 

343
00:23:14 --> 00:23:17
That is, if we take the
derivative of delta G of mixing

344
00:23:17 --> 00:23:22
the partial derivative with
respect to pressure, at

345
00:23:22 --> 00:23:24
constant temperature, of
course, there's no explicit

346
00:23:24 --> 00:23:26
pressure dependence.
 

347
00:23:26 --> 00:23:27
This is zero.
 

348
00:23:27 --> 00:23:32
So again, in the ideal liquid
mixture case, their molecules

349
00:23:32 --> 00:23:33
aren't interacting.
 

350
00:23:33 --> 00:23:36
So there's no reason, when I
open that barrier, that the

351
00:23:36 --> 00:23:46
amount of volume they occupy
altogether is going to change.

352
00:23:46 --> 00:24:00
Any questions, so far?
 

353
00:24:00 --> 00:24:01
OK.
 

354
00:24:01 --> 00:24:26
Then, let's move on to
non-ideal solutions.

355
00:24:26 --> 00:24:30
By the way, just to return
briefly to this topic of

356
00:24:30 --> 00:24:34
osmotic pressure, I just want
to emphasize that result didn't

357
00:24:34 --> 00:24:36
need any kind of energy
of mixing, either, right?

358
00:24:36 --> 00:24:39
Just from the entropy
term you would burst the

359
00:24:39 --> 00:24:41
cell or do whatever.
 

360
00:24:41 --> 00:24:43
You end up with a
larger pressure.

361
00:24:43 --> 00:24:53
So in other words, if you
have a situation where,

362
00:24:53 --> 00:24:57
here is A, pure liquid.
 

363
00:24:57 --> 00:25:08
And here is A plus B liquid, so
mu A star is greater than mu A.

364
00:25:08 --> 00:25:14
We know that in all cases at
the same pressure, the chemical

365
00:25:14 --> 00:25:16
potential of the mixture is
lower than the chemical

366
00:25:16 --> 00:25:18
potential of the pure liquid.
 

367
00:25:18 --> 00:25:19
What's going to happen?
 

368
00:25:19 --> 00:25:26
Well, A is going to,
oh, sorry, A over here

369
00:25:26 --> 00:25:28
is going to rush in.
 

370
00:25:28 --> 00:25:31
It's going to get through this,
if I've got a semi-permeable

371
00:25:31 --> 00:25:34
membrane, it's permeable
to A but not to B.

372
00:25:34 --> 00:25:36
Common situation, of course.
 

373
00:25:36 --> 00:25:38
That's certainly the case
for the cell membrane.

374
00:25:38 --> 00:25:42
Water may flow easily in
through the membrane, but all

375
00:25:42 --> 00:25:45
of the stuff, all the salt and
everything else that's inside

376
00:25:45 --> 00:25:48
the cell generally won't.
 

377
00:25:48 --> 00:25:52
So of course A then flows
through this, what's called

378
00:25:52 --> 00:26:01
a semi-permeable membrane.
 

379
00:26:01 --> 00:26:09
And you'll wind up with either
a burst membrane, or something

380
00:26:09 --> 00:26:10
that looks like this.
 

381
00:26:10 --> 00:26:18
Where now there is
additional pressure here.

382
00:26:18 --> 00:26:20
And you'll have some different
mole fraction than you

383
00:26:20 --> 00:26:29
started with before.
 

384
00:26:29 --> 00:26:54
Now let's go over to
non-ideal solutions.

385
00:26:54 --> 00:26:58
So let's just think
microscopically for a

386
00:26:58 --> 00:27:01
moment about how this
is going to work.

387
00:27:01 --> 00:27:05
You know, normally, always,
there are interactions between

388
00:27:05 --> 00:27:06
the molecules and the liquid.
 

389
00:27:06 --> 00:27:08
The liquid is a
condensed phase.

390
00:27:08 --> 00:27:11
The molecules are in immediate
proximity to each other.

391
00:27:11 --> 00:27:14
So it's very different from the
gas phase, where it can be a

392
00:27:14 --> 00:27:17
pretty realistic approximation
to say, well, the molecules are

393
00:27:17 --> 00:27:19
essentially non-interacting.
 

394
00:27:19 --> 00:27:22
The ideal gas law may
turn out to be a very

395
00:27:22 --> 00:27:24
good approximation.
 

396
00:27:24 --> 00:27:26
In the liquid, really, it's
never the case that you

397
00:27:26 --> 00:27:28
don't have interactions.
 

398
00:27:28 --> 00:27:43
So if we just sketch that.
 

399
00:27:43 --> 00:27:50
If we start with a bunch
of molecules in A,

400
00:27:50 --> 00:27:53
they're interacting.
 

401
00:27:53 --> 00:27:58
So there's some
interaction energy.

402
00:27:58 --> 00:28:01
I'll call it AA, it's
the interaction between

403
00:28:01 --> 00:28:04
two molecules of A.
 

404
00:28:04 --> 00:28:09
In most cases it'll
be less than zero.

405
00:28:09 --> 00:28:13
That is, there'd be
a weak attraction.

406
00:28:13 --> 00:28:26
Same thing between
molecules of B.

407
00:28:26 --> 00:28:29
So that's my situation in
the separated liquids.

408
00:28:29 --> 00:28:32
I've got molecules
in each container.

409
00:28:32 --> 00:28:32
They're interacting.
 

410
00:28:32 --> 00:28:38
There's some energy
associated with that.

411
00:28:38 --> 00:28:50
So here's the pure
separated liquids.

412
00:28:50 --> 00:28:53
And now, I'll open up
the barrier and let

413
00:28:53 --> 00:28:54
the liquids mix.
 

414
00:28:54 --> 00:28:57
And now suddenly, of course, A
and B are going to interact

415
00:28:57 --> 00:29:00
with each other as well as
other molecules of

416
00:29:00 --> 00:29:02
their own kind.
 

417
00:29:02 --> 00:29:10
So now, suddenly, there's
going to be some

418
00:29:10 --> 00:29:12
interaction energy. uAB.
 

419
00:29:12 --> 00:29:31
 


420
00:29:31 --> 00:29:36
And very simplistically, we can
envision that for some of the

421
00:29:36 --> 00:29:40
molecules, essentially, there
would be exchange of this sort.

422
00:29:40 --> 00:29:44
So, one neighbor of this
molecule and one neighbor of

423
00:29:44 --> 00:29:47
this molecule will wind up
exchanged for some pairs of

424
00:29:47 --> 00:29:51
molecules with the
unlike species.

425
00:29:51 --> 00:29:56
And essentially, likes
interactions will be replaced

426
00:29:56 --> 00:30:00
by unlike interactions.
 

427
00:30:00 --> 00:30:04
So there's some now change
in energy involved.

428
00:30:04 --> 00:30:08
There's a delta u.
 

429
00:30:08 --> 00:30:11
And as outlined in this
simple picture, it's just

430
00:30:11 --> 00:30:18
2 uAB minus uAA plus uBB.
 

431
00:30:18 --> 00:30:23
 


432
00:30:23 --> 00:30:26
Real liquids are
very complicated.

433
00:30:26 --> 00:30:31
For liquids with relatively
simple non-directional

434
00:30:31 --> 00:30:33
interactions, things like
organic liquids that

435
00:30:33 --> 00:30:37
might interact van der
Waal's interactions.

436
00:30:37 --> 00:30:42
This might be a reasonable
starting point.

437
00:30:42 --> 00:30:46
And in general, if we're going
to deal with relatively small

438
00:30:46 --> 00:30:51
deviations from the ideal
liquid mixture, the ideal

439
00:30:51 --> 00:30:56
solution, then we can start
with a model like this.

440
00:30:56 --> 00:31:00
And this difference is what's
going to determine how far the

441
00:31:00 --> 00:31:05
liquid mixture will deviate
from the ideal solution case.

442
00:31:05 --> 00:31:08
Of course, we could always
just write a term like this.

443
00:31:08 --> 00:31:13
It may or may not be easily
expressed in this sort of way.

444
00:31:13 --> 00:31:16
So let's just think about
what the possibilities are.

445
00:31:16 --> 00:31:18
Of course, simply,
there are two.

446
00:31:18 --> 00:31:21
The sign could be
positive or negative.

447
00:31:21 --> 00:31:25
And what that means is, you
could have situations where

448
00:31:25 --> 00:31:28
it's positive because basically
there's an energy of

449
00:31:28 --> 00:31:30
mixing now in these cases.
 

450
00:31:30 --> 00:31:35
If that's not zero, unlike the
ideal case, now unlike before,

451
00:31:35 --> 00:31:38
where we just had entropy
driving the mixture, now

452
00:31:38 --> 00:31:40
there's an energy of mixing.
 

453
00:31:40 --> 00:31:43
Could be positive or negative.
 

454
00:31:43 --> 00:31:47
If it's positive, that means
energetically speaking, the

455
00:31:47 --> 00:31:49
mixture is unfavorable.
 

456
00:31:49 --> 00:31:54
In other words, the
intermolecular interactions

457
00:31:54 --> 00:32:00
between like molecules are more
favorable then the interactions

458
00:32:00 --> 00:32:11
between the unlike
molecules, in that case.

459
00:32:11 --> 00:32:25
So, delta u is
greater than zero.

460
00:32:25 --> 00:32:34
And our delta H of mixing
is going to be close

461
00:32:34 --> 00:32:35
to delta u of mixing.
 

462
00:32:35 --> 00:32:38
That is, we're going to figure
that delta pV is not going

463
00:32:38 --> 00:32:43
to be something large.
 

464
00:32:43 --> 00:32:52
And in that case, we can say
delta G of mixing is 1/4 n.

465
00:32:52 --> 00:32:57
And that's only because there
are 4 species involved here.

466
00:32:57 --> 00:32:58
Times delta u.
 

467
00:32:58 --> 00:33:02
In other words, I'm multiplying
this molecular delta u.

468
00:33:02 --> 00:33:05
This is the change in
energy for this collection

469
00:33:05 --> 00:33:07
of four molecules.
 

470
00:33:07 --> 00:33:11
Or of four molecular pairs.
 

471
00:33:11 --> 00:33:15
Plus the other stuff that we've
seen for the ideal case.

472
00:33:15 --> 00:33:18
That is, the entropy term,
of course, is still there.

473
00:33:18 --> 00:33:29
xA log xA plus xB log xB.
 

474
00:33:29 --> 00:33:33
And because we're treating the
case where this is positive,

475
00:33:33 --> 00:33:47
that means this is bigger than
in the ideal solution case.

476
00:33:47 --> 00:33:50
So of course there are lots of
examples of each of these.

477
00:33:50 --> 00:33:53
In fact, this is the
more common deviation.

478
00:33:53 --> 00:33:55
So, and lots of examples.
 

479
00:33:55 --> 00:33:59
An easy one is acetone.
 

480
00:33:59 --> 00:34:02
And carbon disulfide.
 

481
00:34:02 --> 00:34:10
So, if we look at that, and
just see what the phase diagram

482
00:34:10 --> 00:34:38
is going to look like,
it's the following.

483
00:34:38 --> 00:34:43
So let's start with
the ideal case.

484
00:34:43 --> 00:34:47
So I'm going to make B the
more volatile component.

485
00:34:47 --> 00:34:49
That is, B is going to be CS2.
 

486
00:34:49 --> 00:34:59
So here's p star CS2,
which I'll call p star B.

487
00:34:59 --> 00:35:05
And somewhere down here
will be p star for A, that

488
00:35:05 --> 00:35:15
is, p star for acetone.
 

489
00:35:15 --> 00:35:18
And this is going to be
the mole fraction of CS2.

490
00:35:18 --> 00:35:24
Or xB.
 

491
00:35:24 --> 00:35:29
So let's start with
the ideal case.

492
00:35:29 --> 00:35:32
Not steep enough.
 

493
00:35:32 --> 00:35:33
Pretty good.
 

494
00:35:33 --> 00:35:45
OK.
 

495
00:35:45 --> 00:35:47
And now let's look at
what's going to happen

496
00:35:47 --> 00:35:49
in the real mixture.
 

497
00:35:49 --> 00:35:54
Well, because we've got a
positive deviation, what's

498
00:35:54 --> 00:35:58
going to happen is the
mixing is unfavorable.

499
00:35:58 --> 00:36:00
The molecules aren't that
happy any more about

500
00:36:00 --> 00:36:02
being in solution.
 

501
00:36:02 --> 00:36:07
So what that means is, relative
to the ideal solution, they'd

502
00:36:07 --> 00:36:09
rather go up into
the gas phase.

503
00:36:09 --> 00:36:12
They don't like being
in the mixture.

504
00:36:12 --> 00:36:16
Compared to what they
would feel in the ideal

505
00:36:16 --> 00:36:17
solution, where there
are no interactions.

506
00:36:17 --> 00:36:20
Because the interactions
between unlike molecules

507
00:36:20 --> 00:36:22
are unfavorable.
 

508
00:36:22 --> 00:36:27
So what's going to
happen, then, is the

509
00:36:27 --> 00:36:33
pressure is higher.
 

510
00:36:33 --> 00:36:36
You have deviations from
the ideal case, in

511
00:36:36 --> 00:36:38
the total pressure.
 

512
00:36:38 --> 00:36:53
And in each of the individual
partial pressures.

513
00:36:53 --> 00:36:58
That's the total
pressure. pA plus pB.

514
00:36:58 --> 00:37:03
In the ideal case it's
this straight line.

515
00:37:03 --> 00:37:06
In the non-ideal case,
though, it's bigger

516
00:37:06 --> 00:37:07
than the straight line.
 

517
00:37:07 --> 00:37:10
Because more stuff wants to
get up into the gas phase.

518
00:37:10 --> 00:37:12
And it's the same for each
individual component.

519
00:37:12 --> 00:37:18
So here's pA.
 

520
00:37:18 --> 00:37:21
 


521
00:37:21 --> 00:37:22
And there's pB.
 

522
00:37:22 --> 00:37:27
 


523
00:37:27 --> 00:37:32
Any questions?
 

524
00:37:32 --> 00:37:36
OK, so it's obvious just from
looking at this that each of

525
00:37:36 --> 00:37:40
the partial pressures is bigger
than it is in the ideal case.

526
00:37:40 --> 00:37:43
And so, so is the
total pressure.

527
00:37:43 --> 00:37:46
So in other words, if I say
what's the partial pressure of

528
00:37:46 --> 00:37:50
CS2, well, since it's not the
simple ideal case, actually I

529
00:37:50 --> 00:37:51
don't know what it
is in general.

530
00:37:51 --> 00:37:55
It's not easy to calculate a
priori what it's going to do

531
00:37:55 --> 00:37:58
all the way across the phase
diagram for any mole fraction.

532
00:37:58 --> 00:38:01
What I do know, though, is it's
bigger than it would be for

533
00:38:01 --> 00:38:03
the ideal solution case.
 

534
00:38:03 --> 00:38:10
In other words, it's bigger
than x CS2 times p star

535
00:38:10 --> 00:38:15
CS2, the pressure over
the pure liquid CS2.

536
00:38:15 --> 00:38:19
Same for acetone.
 

537
00:38:19 --> 00:38:29
It's bigger than x
acetone p star acetone.

538
00:38:29 --> 00:38:31
And since both partial
pressures are bigger, the total

539
00:38:31 --> 00:38:32
pressure has to be bigger.
 

540
00:38:32 --> 00:38:34
And of course, that's something
you can read right off

541
00:38:34 --> 00:38:36
the diagram as well.
 

542
00:38:36 --> 00:38:49
In other words, the total
pressure is bigger than in

543
00:38:49 --> 00:38:55
the ideal solution case.
 

544
00:38:55 --> 00:39:00
Any questions?
 

545
00:39:00 --> 00:39:02
OK.
 

546
00:39:02 --> 00:39:06
So the negative case,
of course, is exactly

547
00:39:06 --> 00:39:07
the opposite.
 

548
00:39:07 --> 00:39:17
So I'll just go
through it quickly.

549
00:39:17 --> 00:39:23
So, remember positive
deviations came, let's

550
00:39:23 --> 00:39:27
put this back up there.
 

551
00:39:27 --> 00:39:31
That came from the case where
the unlike intermolecular

552
00:39:31 --> 00:39:34
interactions are
not as favorable.

553
00:39:34 --> 00:39:38
There's less attraction, more
repulsion than in this case.

554
00:39:38 --> 00:39:41
Of course, there are plenty of
cases of the opposite sort.

555
00:39:41 --> 00:39:45
Where the unlike intermolecular
interactions are attractive.

556
00:39:45 --> 00:39:50
More than the interactions
between like molecules.

557
00:39:50 --> 00:39:54
And in that case, you have
the opposite result.

558
00:39:54 --> 00:40:05
So you can have
negative deviations.

559
00:40:05 --> 00:40:09
In other words, delta
u is less than zero.

560
00:40:09 --> 00:40:10
Lots of examples.
 

561
00:40:10 --> 00:40:14
I've got one in the notes, it's
just acetone and chloroform.

562
00:40:14 --> 00:40:22
So if we get rid of the carbon
disulfide and put CHCl3 in

563
00:40:22 --> 00:40:25
there, that's more favorable.
 

564
00:40:25 --> 00:40:27
Because there's weak hydrogen
bonding between the

565
00:40:27 --> 00:40:29
unlike species.
 

566
00:40:29 --> 00:40:31
And not between
the like species.

567
00:40:31 --> 00:40:39
So what happens is, you
have acetone, it's this.

568
00:40:39 --> 00:40:47
And you have chloroform.
 

569
00:40:47 --> 00:40:51
And there's weak hydrogen
bonding between them.

570
00:40:51 --> 00:40:52
There's an attraction there.
 

571
00:40:52 --> 00:40:57
That doesn't exist between
either case of the

572
00:40:57 --> 00:40:59
like molecules.
 

573
00:40:59 --> 00:41:03
So now there's stronger
attraction between them than

574
00:41:03 --> 00:41:06
there is between molecules in
either of the pure

575
00:41:06 --> 00:41:08
liquid cases.
 

576
00:41:08 --> 00:41:11
So, of course, you know
that the opposite result

577
00:41:11 --> 00:41:12
is going to happen.
 

578
00:41:12 --> 00:41:18
In this case, now you mix them
together, relative to the ideal

579
00:41:18 --> 00:41:21
solution where there are no
interactions, now they

580
00:41:21 --> 00:41:22
want to be together.
 

581
00:41:22 --> 00:41:27
They're clinging to each other
through these hydrogen bonds.

582
00:41:27 --> 00:41:30
What's that mean for the
pressure up in the gas

583
00:41:30 --> 00:41:32
phase, in the solution?
 

584
00:41:32 --> 00:41:35
They don't want to
go there any more.

585
00:41:35 --> 00:41:38
Stuff in the gas phase
now wants to condense

586
00:41:38 --> 00:41:39
down into the liquid.
 

587
00:41:39 --> 00:41:43
Because there they have
favorable interactions.

588
00:41:43 --> 00:41:45
So that's, of course,
what'll happen.

589
00:41:45 --> 00:41:50
And so then you just see the
opposite sort of deviation from

590
00:41:50 --> 00:41:56
what I just illustrated before.
 

591
00:41:56 --> 00:42:08
So then if we look at p
star CHCl3, where are

592
00:42:08 --> 00:42:13
these lines going?
 

593
00:42:13 --> 00:42:18
Well, may have been
more artistic.

594
00:42:18 --> 00:42:27
But I'm afraid I'm going to
have to go with reality.

595
00:42:27 --> 00:42:28
p star for acetone.
 

596
00:42:28 --> 00:42:35
There's chloroform.
 

597
00:42:35 --> 00:42:38
And now our deviation
is going to be in the

598
00:42:38 --> 00:42:39
opposite direction.
 

599
00:42:39 --> 00:42:42
So all the, both of the partial
pressures and the total

600
00:42:42 --> 00:42:55
pressure are going to go
lower than indicated here.

601
00:42:55 --> 00:43:03
So p total is now less than,
for the real case, is less than

602
00:43:03 --> 00:43:11
p total for the ideal case.
 

603
00:43:11 --> 00:43:14
In the situation where you
have attractive interactions

604
00:43:14 --> 00:43:17
between the unlike molecules.
 

605
00:43:17 --> 00:43:20
Any questions?
 

606
00:43:20 --> 00:43:22
OK.
 

607
00:43:22 --> 00:43:27
Now, in general, this
is complicated.

608
00:43:27 --> 00:43:31
To describe quantitatively
throughout the entire

609
00:43:31 --> 00:43:33
phase diagram.
 

610
00:43:33 --> 00:43:36
The interactions
are complicated.

611
00:43:36 --> 00:43:38
Lots of times, molecules
interact with more

612
00:43:38 --> 00:43:40
than one neighbor.
 

613
00:43:40 --> 00:43:45
So in general, we don't have a
simple analytical expression

614
00:43:45 --> 00:43:49
for what the pressure is going
to do as a function of mole

615
00:43:49 --> 00:43:52
fraction all the way
from zero to one.

616
00:43:52 --> 00:43:54
Of course, we have it in
very simple form for

617
00:43:54 --> 00:43:55
the ideal solution.
 

618
00:43:55 --> 00:43:58
Because everything's linear.
 

619
00:43:58 --> 00:44:01
So it's very simple.
 

620
00:44:01 --> 00:44:03
These are replaced by
equalities and we

621
00:44:03 --> 00:44:04
know everything.
 

622
00:44:04 --> 00:44:05
Now we don't.
 

623
00:44:05 --> 00:44:08
But we can at least make
some progress by looking

624
00:44:08 --> 00:44:10
at the limiting cases.
 

625
00:44:10 --> 00:44:13
Where you have just
very dilute solutions.

626
00:44:13 --> 00:44:17
So in that case, you
can anticipate what's

627
00:44:17 --> 00:44:18
going to happen.
 

628
00:44:18 --> 00:44:22
So let's say we mix A and B.
 

629
00:44:22 --> 00:44:26
And one of them is in very
dilute concentration.

630
00:44:26 --> 00:44:31
For the other one, for B,
most molecules of B are just

631
00:44:31 --> 00:44:33
going to see themselves.
 

632
00:44:33 --> 00:44:34
They hardly know that
you've mixed any

633
00:44:34 --> 00:44:37
molecules of A in there.
 

634
00:44:37 --> 00:44:42
For them, it's pretty
much an ideal solution.

635
00:44:42 --> 00:44:44
They think they're in
an ideal solution.

636
00:44:44 --> 00:44:47
Which is to say there's an
entropy change in mixing.

637
00:44:47 --> 00:44:51
But hardly any of them
experience interactions

638
00:44:51 --> 00:44:54
with the unlike molecules.
 

639
00:44:54 --> 00:44:58
For the dilute molecules,
there is a change.

640
00:44:58 --> 00:45:01
You can't use Raoult's
law any more.

641
00:45:01 --> 00:45:06
But at least we can look of the
slope of the curve and say, OK,

642
00:45:06 --> 00:45:10
at least in the dilute solution
case for a little while

643
00:45:10 --> 00:45:11
it's going to be linear.
 

644
00:45:11 --> 00:45:13
We can understand it.
 

645
00:45:13 --> 00:45:16
We can tabulate it.
 

646
00:45:16 --> 00:45:41
So let's look at
how that works.

647
00:45:41 --> 00:45:43
Alright.
 

648
00:45:43 --> 00:45:46
That limit is something that
has a name, that's called

649
00:45:46 --> 00:45:48
the ideal dilute solution.
 

650
00:45:48 --> 00:45:50
Not to be confused with
the ideal solution.

651
00:45:50 --> 00:45:53
Where both constituents
follow Raoult's law,

652
00:45:53 --> 00:45:55
as we've seen before.
 

653
00:45:55 --> 00:46:07
So ideal dilute solution.
 

654
00:46:07 --> 00:46:09
We're going to just
look at these limits.

655
00:46:09 --> 00:46:17
So, if we start with the ideal
case, so let's imagine we're

656
00:46:17 --> 00:46:22
going to look at carbon
disulfide again.

657
00:46:22 --> 00:46:32
Here's p star CS2,
which is our pB star.

658
00:46:32 --> 00:46:36
And this is x CS2, or xB.
 

659
00:46:36 --> 00:46:40
 


660
00:46:40 --> 00:46:50
OK, well, what happens if
we've got almost pure

661
00:46:50 --> 00:46:53
carbon disulfide.
 

662
00:46:53 --> 00:46:58
Well, what's going to happen
is in this limit, like I was

663
00:46:58 --> 00:47:02
mentioning, up here the CS2
pretty much sees other CS2.

664
00:47:02 --> 00:47:06
It thinks it's in
an ideal solution.

665
00:47:06 --> 00:47:11
It's going to look like that.
 

666
00:47:11 --> 00:47:15
Now, let's go to
the other limit.

667
00:47:15 --> 00:47:30
So this is the limit x CS2
equals xB approaches one.

668
00:47:30 --> 00:47:33
Raoult's law.
 

669
00:47:33 --> 00:47:38
Two, the other limit. x
CS2 approaches zero.

670
00:47:38 --> 00:47:39
We're down here.
 

671
00:47:39 --> 00:47:43
So it's definitely not
going to be the same.

672
00:47:43 --> 00:47:47
Because now, every molecule,
every CS2 molecule, is

673
00:47:47 --> 00:47:50
completely surrounded
by acetone.

674
00:47:50 --> 00:47:54
So they all feel the
energy of mixing.

675
00:47:54 --> 00:47:55
Because there are hardly
any of them there.

676
00:47:55 --> 00:47:58
So every one of them is, it's
very rare for one to find a

677
00:47:58 --> 00:48:00
neighbor of its own kind.
 

678
00:48:00 --> 00:48:03
And this is a case where
there's a positive

679
00:48:03 --> 00:48:04
deviation from ideality.
 

680
00:48:04 --> 00:48:07
That is, the interactions
are less favorable

681
00:48:07 --> 00:48:09
between unlike species.
 

682
00:48:09 --> 00:48:13
So, it's up here.
 

683
00:48:13 --> 00:48:14
Doesn't like that.
 

684
00:48:14 --> 00:48:18
It says, get me out here, I'd
much rather be in the gas

685
00:48:18 --> 00:48:22
phase, compared to the ideal
case, then surrounded by all

686
00:48:22 --> 00:48:26
these acetone molecules, right?
 

687
00:48:26 --> 00:48:27
So that's what'll happen.
 

688
00:48:27 --> 00:48:35
And now it's up to the artist.
 

689
00:48:35 --> 00:48:39
But in some way there's
going to be a set of

690
00:48:39 --> 00:48:41
values in between.
 

691
00:48:41 --> 00:48:44
And the ideal dilute solution
by itself isn't going

692
00:48:44 --> 00:48:45
to tell us about that.
 

693
00:48:45 --> 00:48:48
It's telling us about
the two limits.

694
00:48:48 --> 00:48:51
Now, if we follow this,
and I can see I'm going

695
00:48:51 --> 00:48:52
to be in trouble.
 

696
00:48:52 --> 00:48:58
But I can do a little creative
bending of the line here.

697
00:48:58 --> 00:49:01
If I follow this slope
up to here, it'll

698
00:49:01 --> 00:49:03
intercept somewhere.
 

699
00:49:03 --> 00:49:04
That has a name.
 

700
00:49:04 --> 00:49:07
It's called the
Henry's law constant.

701
00:49:07 --> 00:49:10
I can write it as K CS2, or KB.
 

702
00:49:10 --> 00:49:14
 


703
00:49:14 --> 00:49:18
So the point is, that at
least for some range

704
00:49:18 --> 00:49:21
of concentrations, the
variation is linear.

705
00:49:21 --> 00:49:25
And what that means is, if
I know this Henry's law

706
00:49:25 --> 00:49:31
constant, for CS2 mixed
with acetone specifically,

707
00:49:31 --> 00:49:32
it would be different.
 

708
00:49:32 --> 00:49:34
If it's mixed with
something else, right?

709
00:49:34 --> 00:49:36
Other kinds of interactions
with different energies will

710
00:49:36 --> 00:49:38
have a different slope.
 

711
00:49:38 --> 00:49:41
But if I know that number,
it's still useful.

712
00:49:41 --> 00:49:47
Because at least for that pair
of constituents, I know the

713
00:49:47 --> 00:49:52
dependents for some range
of concentrations.

714
00:49:52 --> 00:50:15
So this is Henry's law. p CS --
Oops, CS2 is x CS2 times K CS2.

715
00:50:15 --> 00:50:27
Not quite as convenient
as Raoult's law.

716
00:50:27 --> 00:50:29
That number, I don't
need to know what the

717
00:50:29 --> 00:50:31
other constituent is.
 

718
00:50:31 --> 00:50:33
It's just a property of CS2.
 

719
00:50:33 --> 00:50:36
It depends on the temperature,
but not the other

720
00:50:36 --> 00:50:37
member of the mixture.
 

721
00:50:37 --> 00:50:41
This one does, but still
there's at least a

722
00:50:41 --> 00:50:43
substantial number of
these things tabulated.

723
00:50:43 --> 00:50:46
Especially for
important mixtures.

724
00:50:46 --> 00:50:49
So it becomes useful
for dilute solutions.

725
00:50:49 --> 00:50:51
And there's an awful
lot of chemistry.

726
00:50:51 --> 00:50:53
There's an awful lot of biology
that happens in relatively

727
00:50:53 --> 00:50:56
dilute solutions.
 

728
00:50:56 --> 00:51:02
OK.
 

729
00:51:02 --> 00:51:04
I'll leave it at this point.
 

730
00:51:04 --> 00:51:06
Next time we'll look at the
total phase diagram, just like

731
00:51:06 --> 00:51:10
we did last time looking at
both the liquid and gas

732
00:51:10 --> 00:51:13
coexistence curves, and seeing
what the behavior is as a

733
00:51:13 --> 00:51:15
function of mole fractions.
 

734
00:51:15 --> 00:51:17
And we'll talk about
distillation and so forth

735
00:51:17 --> 00:51:19
in non-ideal liquids.
 

736
00:51:19 --> 00:51:21