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PROFESSOR: So let's start
with some of the things

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we learned last time.
 

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So there are two things
that were important.

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We learned that the chemical
potential for a species is the

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Gibbs free energy for that
species divided by the number

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of moles, this is Gibbs
free energy per mole.

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We learned that the pressure
dependence of the Gibbs free

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energy gives you the
pressure dependence for the

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chemical potential.
 

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That it's equal to the chemical
potential at one bar for an

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ideal gas plus RT log p.
 

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We also learned that a species
will want to go to minimize its

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chemical potential, and we saw
that for the cell bursting in

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salt, in distilled water.
 

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Or an ice cube melting at
a temperature greater

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than 0 degrees.
 

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And what we want to prove right
now is that if I take a species

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A, in a mixture, some
temperature T, some pressure p,

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and I compare its chemical
potential to the same species,

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A, when it's pure, the same
temperature, the same pressure,

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what I want to argue is that
the chemical potential in the

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mixture is always less than the
chemical potential

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when it's pure.
 

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The same conditions of
pressure and temperature.

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And, so we can do a little
thought experiment.

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Let's do a little
thought experiment.

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Let's make a box, and
in our box we're going

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to have a partition.
 

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And a flexible membrane here.
 

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And on one side of the
partition we're going

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to have a gas, B.
 

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A gas, A, on this side here.
 

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And gas A on this side here.
 

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So let me just red chalk for
A, And I don't think I have

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any other colors listed.
 

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Yellow chalk for B here.
 

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And everything's one bar.
 

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Everything's one bar.
 

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So one bar B here.
 

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One bar A here,
one bar A here .

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And this membrane here
only lets A through.

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And this membrane is
deformable, but it's elastic.

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You can't deform it forever.
 

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It has some strength
to it, right?

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So if you push on,
it'll push back.

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There'll be pressure
associated with that.

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So the next thing I do, then,
in my experiment, shouldn't

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have done it here.
 

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Next thing I do in my
experiment is to break

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this partition here.
 

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I'm going to break
the partition.

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And this will cause
A and B to mix.

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So now in my box I have my
partition, my membrane here.

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I've got A at one bar here,
total pressure of one bar.

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And on the other side I have
A plus B, with a total

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pressure of one bar.
 

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That's my initial point now.
 

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What's going to happen?
 

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What's going to happen is that
molecules of A here are going

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to want to go through the
membrane to go in

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this area here.
 

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And there's two ways
to look at it.

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You can look at it the
thermodynamic way, which is

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the way that we're going to
want to be looking at it.

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Which is that, from what we're
going to prove, is that the

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chemical potential in the
mixture is always less than

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for the pure substance.
 

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Here we have a mixture.
 

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One bar.
 

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Here we have the pure
substance at one bar.

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So these molecules are going to
look around and say hey, you

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know, I'm much happier here.
 

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And they're all going to want
to go in this area here.

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As a result, the volume of this
area, if you want to keep the

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same pressure on both
sides, you're going to

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deform the membrane.
 

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The membrane's going
to get deformed.

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It's going to bloat
on that side here.

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It's going to cause an increase
in pressure on that side.

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And the increase in pressure
from the membrane sort of

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deforming and pushing back, is
going to increase until the

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partial pressure of A, here,
equals the pressure of A here.

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At which point the flow of A
from either side is going

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to be the same and you're
going to be in equilibrium.

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And at that point the chemical
potentials of both sides

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are going to be the same.
 

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So I sort of gave you the
other way of looking at it.

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Which is just purely in
terms of pressures.

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At equilibrium, the partial
pressure of A here has to be

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the same as the pressure
on this side here.

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So that the flow of A on either
side of that membrane, going

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from right to left or left
to right is the same.

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And that's going to cause
this mixing to happen.

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If you look at it from partial
pressure perspective.

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But really, it's a
chemical potential idea.

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Now, the chemical potential,
as we saw, was the

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Gibbs free energy.
 

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And the Gibbs free energy, you
can write it as H minus TS.

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So basically, in this process
that I described, the

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enthalpy's not doing anything.
 

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These are ideal gases.
 

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They're not interacting
with each other.

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The only thing that's changing,
that's driving the chemical

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potential, which is basically
Gibbs free energy per mole, the

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only thing that's driving the
chemical potential to be lower

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on this side here, is
that entropy term.

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It's the entropy of mixing.
 

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So entropy of mixing is
really super important.

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When we're talking about
systems where you have

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multiple components.
 

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That's going to drive
a lot of things.

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And in fact that's going to
drive equilibrium, as we're

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going to see a little
bit later today.

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Alright, let's quickly
go through the math

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and prove this here.
 

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So our goal, then, is to have a
mixture, chemical potential of

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the mixture on one side, and
the chemical potential of the

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pure material on
the other side.

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And so we're going to start by
sort of a similar thing here.

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We're going to have a box, let
me redo my box here, we're

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going to have a box with, let
me get rid of these

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one bars here.
 

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We're going to put our
membrane in the box.

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And we're going to have
a pressure, pA, on

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this side here.
 

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And we're going to have a
pressure pA prime for the

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partial pressure of A.
 

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And pB prime for the
partial pressure of B.

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00:08:02 --> 00:08:12
And the p total is going to
be pA prime plus pB prime.

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And I'm going to see at
equilibrium, I'm going to

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write everything I know
about equilibrium.

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At equilibrium, I know that the
partial pressure of A on that

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side here has to be equal
to the pressure of A here.

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The partial pressure, the
pressure is basically the force

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of these molecules hitting that
membrane per unit time,

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times the number of
molecules hitting.

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So we have the same flux of
molecules going this way, is

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equal to the flux of molecules
going the other way.

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So at equilibrium,
pA prime equals pA.

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So.
 

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What else can I say?
 

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At equilibrium, in terms of the
chemical potentials, I know

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that the chemical potential of
the mixture, mu A in the

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mixture, temperature
under pressure p total.

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Is equal to the chemical
potential of the pure system,

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same temperature under pressure
is p sub a on that side.

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These are the two things
that I know at equilibrium.

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So let's start to
turn the crank.

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And see if we can come up
with, well, we basically

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already have this.
 

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If we can massage the right
side of our equation so that

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the pressure term,
pA, is p total.

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And then we'll have an equation
that will compare the chemical

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potential of the mixture under
the same conditions as the

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chemical potential of
A in the pure state.

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So, what can we use?
 

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We can use Dalton's law here,
which tells us that pA prime

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is equal to xA p total.
 

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That's from Dalton.
 

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And now, pA prime
is equal to pA.

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So this is also just pA.
 

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And we can plug that in here,
and suddenly we've got p

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00:10:18 --> 00:10:22
total included in here.
 

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Let's just pass these out.
 

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If you want to, thank you.
 

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Thank you.
 

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So, this is equal to mu A
pure temperature xA p total.

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So what else do I know?
 

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I know that I've written
something here, that mu is

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equal to mu naught, to temp,
so this is at one bar.

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00:10:59 --> 00:11:02
RT log p.
 

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00:11:02 --> 00:11:12
So I'm going to rewrite this
as mu A pure temperature

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00:11:12 --> 00:11:14
T at one bar.
 

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Plus RT log xA p total.
 

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Now, the log of xA times
p total is the log of xA

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plus the log of p total.
 

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It's equal to plus RT log
p total plus RT log xA.

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So I can lump these
two things together.

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I have mu A pure
T plus RT log p.

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Well, that's just the chemical
potential of A, in the pure

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state, at temperature
T and pressure pT.

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00:11:55 --> 00:11:58
 


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That's the equation here to
relate pressure, at some

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00:12:01 --> 00:12:04
variable pressure p, to what
it would be at one bar.

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Which is that.
 

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Then we have the plus RT
log xA sitting here.

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Plus RT log xA.
 

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So we're done.
 

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We're done because on that side
here I have mu A, the chemical

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00:12:17 --> 00:12:19
potential of A in the
mixture, temperature

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T, pressure p total.
 

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00:12:23 --> 00:12:26
I've got mu A pure temperature
T pressure p total,

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00:12:26 --> 00:12:29
plus a term.
 

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00:12:29 --> 00:12:32
Plus RT log xA.
 

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00:12:32 --> 00:12:36
Can xA be bigger than one?
xA's the mole fraction.

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xA's always less than one.
 

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Log xA is always
less than zero.

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This term here is always
less than or equal to zero.

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00:12:48 --> 00:12:54
Therefore, this term is
always less than that term.

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Therefore the chemical
potential in the mixture

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00:12:57 --> 00:13:03
is always less than the
chemical potential inside

210
00:13:03 --> 00:13:09
the pure material.
 

211
00:13:09 --> 00:13:12
This is going to be important
for the next part of

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the presentation.
 

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00:13:14 --> 00:13:17
Any questions?
 

214
00:13:17 --> 00:13:20
This is what drives the
death of saltwater fish

215
00:13:20 --> 00:13:22
in fresh water, right?
 

216
00:13:22 --> 00:13:25
Because osmotic pressure
is basically given by

217
00:13:25 --> 00:13:27
this basic idea here.
 

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00:13:27 --> 00:13:38
And it's all driven by
entropy of mixing.

219
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Sure you don't have
any questions?

220
00:13:38 --> 00:13:44
Speak up.
 

221
00:13:44 --> 00:13:47
So now we have all the
tools we need to look at

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00:13:47 --> 00:13:53
chemical equilibrium.
 

223
00:13:53 --> 00:13:57
When we have a mixture,
and we're going to

224
00:13:57 --> 00:13:58
start with ideal gases.
 

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But everything I'm going to say
about equilibrium and ideal

226
00:14:01 --> 00:14:05
gases is valid for solutions.
 

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00:14:05 --> 00:14:07
An ideal gas, and we're
going to be talking

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00:14:07 --> 00:14:08
about ideal solutions.
 

229
00:14:08 --> 00:14:11
And molecules in an ideal
solution, an ideal solvent,

230
00:14:11 --> 00:14:13
are not very different than
molecules in an ideal gas.

231
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They don't interact
with each other.

232
00:14:15 --> 00:14:18
You use concentration instead
of partial pressures, but

233
00:14:18 --> 00:14:20
pretty much all the
ideas are the same.

234
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All the concepts are the same.
 

235
00:14:22 --> 00:14:24
The equations are
basically the same.

236
00:14:24 --> 00:14:27
You just do a little bit of
replacement of variables.

237
00:14:27 --> 00:14:28
But it pretty much
is the same thing.

238
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It's easier to think about it,
to learn first in terms of the

239
00:14:31 --> 00:14:36
ideal gas, but it applies
equally well to what you're

240
00:14:36 --> 00:14:37
more likely to use.
 

241
00:14:37 --> 00:14:42
Which is solutions.
 

242
00:14:42 --> 00:14:48
So let's look at the
prototypical gas phase reaction

243
00:14:48 --> 00:14:51
that everybody writes down when
they first do this

244
00:14:51 --> 00:14:52
sort of problems.
 

245
00:14:52 --> 00:14:55
The Haber process.
 

246
00:14:55 --> 00:14:59
Gas, temperature T,
you take nitrogen.

247
00:14:59 --> 00:15:05
You react it with hydrogen,
gas, temperature T, p.

248
00:15:05 --> 00:15:09
And you make ammonia.
 

249
00:15:09 --> 00:15:13
NH3 gas T, p.
 

250
00:15:13 --> 00:15:18
And we're going to ask the
question, so I take nitrogen,

251
00:15:18 --> 00:15:21
I take some, hydrogen, I mix
them together in a container.

252
00:15:21 --> 00:15:22
And I make ammonia.
 

253
00:15:22 --> 00:15:27
And I'm going to ask, after I
reach equilibrium, what is the

254
00:15:27 --> 00:15:36
partial pressure of nitrogen
hydrogen and ammonia?

255
00:15:36 --> 00:15:40
Standard equilibrium problem.
 

256
00:15:40 --> 00:15:42
You all, I'm sure you've
all seen equations

257
00:15:42 --> 00:15:43
about equilibrium.
 

258
00:15:43 --> 00:15:47
Equilibrium constant K, log
delta G of the reaction,

259
00:15:47 --> 00:15:48
et cetera, et cetera.
 

260
00:15:48 --> 00:15:51
But you probably don't have
a really good intuition as

261
00:15:51 --> 00:15:54
to why it is what it is.
 

262
00:15:54 --> 00:15:59
So the point of the class here
is not to relearn log K is

263
00:15:59 --> 00:16:02
equal to minus delta G,
reaction divided by RT.

264
00:16:02 --> 00:16:05
The point is to learn
how we get there.

265
00:16:05 --> 00:16:07
How we get to that equation.
 

266
00:16:07 --> 00:16:09
And what parts are important.
 

267
00:16:09 --> 00:16:12
Specifically, how entropy
of mixing really becomes

268
00:16:12 --> 00:16:15
key to equilibrium.
 

269
00:16:15 --> 00:16:17
But before we get there, let
me give you a little bit of

270
00:16:17 --> 00:16:20
history as to why this is
such an important reaction.

271
00:16:20 --> 00:16:25
This reaction, which you see
in every chemistry class,

272
00:16:25 --> 00:16:28
was developed by Mr.
Haber and Mr. Bosch.

273
00:16:28 --> 00:16:33
Mr. Bosch made it
large-scale around 1910.

274
00:16:33 --> 00:16:38
And it's the reaction that
you use to make fertilizer.

275
00:16:38 --> 00:16:40
Ammonia is the feed stock
to make nitrogen-based

276
00:16:40 --> 00:16:44
fertilizers.
 

277
00:16:44 --> 00:16:47
So today, there are a hundred
million tons of fertilizer, of

278
00:16:47 --> 00:16:50
nitrogen fertilizer made,
using, essentially,

279
00:16:50 --> 00:16:51
the Haber process.
 

280
00:16:51 --> 00:16:53
It's a huge, huge, commodity.
 

281
00:16:53 --> 00:17:00
1% of the world's energy is
taken up to make this reaction.

282
00:17:00 --> 00:17:06
Almost 1% is about 3/4 of the
world's energy is used on this.

283
00:17:06 --> 00:17:11
In World War 1, Germany
was making explosives out

284
00:17:11 --> 00:17:13
of nitrogen feed stock.
 

285
00:17:13 --> 00:17:17
And it was getting its
feed stock from Chile.

286
00:17:17 --> 00:17:18
From saltpeter mines in Chile.
 

287
00:17:18 --> 00:17:21
Chile was under British
hands at the time.

288
00:17:21 --> 00:17:24
Well, the British
didn't let Chile sell

289
00:17:24 --> 00:17:26
saltpeter to Germany.
 

290
00:17:26 --> 00:17:30
Germany had to find another
way to make ammunitions.

291
00:17:30 --> 00:17:34
And the Haber process, which
had just been invented by

292
00:17:34 --> 00:17:43
Bosch-Haber, became the way
that Germany made explosives.

293
00:17:43 --> 00:17:48
Without this, Germany would
have stopped the war in 1916.

294
00:17:48 --> 00:17:49
Or even before then.
 

295
00:17:49 --> 00:17:55
Way before then.
 

296
00:17:55 --> 00:18:03
And this process was basically,
and Haber and Bosch got the

297
00:18:03 --> 00:18:05
Nobel Prize, essentially,
for this process.

298
00:18:05 --> 00:18:08
For showing how to take
chemistry, and doing chemistry

299
00:18:08 --> 00:18:10
and large scale processes under
high pressure and high

300
00:18:10 --> 00:18:12
temperature conditions.
 

301
00:18:12 --> 00:18:17
Haber got the Nobel Prize in
1918 and Bosch got it in 1931.

302
00:18:17 --> 00:18:21
This process, arguably, you
could say, was the birth of

303
00:18:21 --> 00:18:24
the dominance of Germany
as a chemical industry.

304
00:18:24 --> 00:18:27
Based on that, and the fact
that they had to make

305
00:18:27 --> 00:18:30
liquid fuels out of coal.
 

306
00:18:30 --> 00:18:34
The syn-gas, or the
syn-fuel process.

307
00:18:34 --> 00:18:38
Also invented during World War
1, where they had to use, they

308
00:18:38 --> 00:18:43
couldn't find any oil and they
had to use their coal mines.

309
00:18:43 --> 00:18:46
And obviously this process, the
syn-fuel process, is coming

310
00:18:46 --> 00:18:50
back in vogue with the
energy crisis around now.

311
00:18:50 --> 00:18:56
So they developed a lot of
knowledge about how to do

312
00:18:56 --> 00:18:58
large-scale chemical reactions.
 

313
00:18:58 --> 00:19:02
And that was the birth of, or
the explosion of, companies

314
00:19:02 --> 00:19:08
like Merck and Bayer and Bosch,
and BASF, and all these German

315
00:19:08 --> 00:19:10
companies that dominate,
basically, the

316
00:19:10 --> 00:19:12
chemical industry.
 

317
00:19:12 --> 00:19:18
And when Germany lost the war,
the US government confiscated

318
00:19:18 --> 00:19:23
the American divisions of
these German companies.

319
00:19:23 --> 00:19:26
And I'm sure you've heard
of Merck as the company.

320
00:19:26 --> 00:19:29
And you think of Merck
as an American company

321
00:19:29 --> 00:19:30
that makes drugs.
 

322
00:19:30 --> 00:19:31
Well, that's true.
 

323
00:19:31 --> 00:19:34
It's a very large American
company that makes drugs.

324
00:19:34 --> 00:19:37
But there's another Merck,
which is the German Merck.

325
00:19:37 --> 00:19:39
Which also makes drugs.
 

326
00:19:39 --> 00:19:41
But it makes liquid crystals.
 

327
00:19:41 --> 00:19:43
And liquid crystals is
where it makes all of

328
00:19:43 --> 00:19:45
its money right now.
 

329
00:19:45 --> 00:19:48
And it's very confusing
because they're both Merck.

330
00:19:48 --> 00:19:52
And they both came from the
Merck family from the 1600s

331
00:19:52 --> 00:19:54
that were pharmacists.
 

332
00:19:54 --> 00:20:02
But after World War 1, Merck
Germany was allowed to use the

333
00:20:02 --> 00:20:07
name Merck everywhere in the
world except for the US.

334
00:20:07 --> 00:20:09
In the US it's called EMD.
 

335
00:20:09 --> 00:20:13
Merck USA is allowed to use the
name Merck in the US and

336
00:20:13 --> 00:20:16
outside of the US it's called,
let me remind myself,

337
00:20:16 --> 00:20:18
it's called MSD.
 

338
00:20:18 --> 00:20:19
It's very confusing.
 

339
00:20:19 --> 00:20:20
Bayer is the same way.
 

340
00:20:20 --> 00:20:22
So Bayer was split up.
 

341
00:20:22 --> 00:20:26
And there was Buyer in Germany
and there was Bayer in the US.

342
00:20:26 --> 00:20:30
You've got Bayer Aspirin
from the US term Bayer.

343
00:20:30 --> 00:20:32
And you've got all the
other pharmaceuticals

344
00:20:32 --> 00:20:33
from Bayer Germany.
 

345
00:20:33 --> 00:20:35
All the chemicals.
 

346
00:20:35 --> 00:20:39
And a few years ago, Buyer
bought Bayer, and now we have

347
00:20:39 --> 00:20:44
one Buyer-Bayer, depending
on where you live.

348
00:20:44 --> 00:20:47
And so it's very interesting to
see the history of all these

349
00:20:47 --> 00:20:48
big chemical companies.
 

350
00:20:48 --> 00:20:51
And that's why Switzerland and
Germany are still the home of,

351
00:20:51 --> 00:20:53
it's all due to that reaction.
 

352
00:20:53 --> 00:20:55
That's why you always
see that reaction.

353
00:20:55 --> 00:20:58
Fertilizers, at the birth of
the chemical company, and

354
00:20:58 --> 00:21:06
it's a great example
for equilibrium.

355
00:21:06 --> 00:21:07
So that's the example.
 

356
00:21:07 --> 00:21:08
Let's generalize it.
 

357
00:21:08 --> 00:21:10
We're actually not
going to work on this

358
00:21:10 --> 00:21:11
until the very end.
 

359
00:21:11 --> 00:21:12
And we'll see it again.
 

360
00:21:12 --> 00:21:14
But let's generalize.
 

361
00:21:14 --> 00:21:19
Let's just take a mixture of
gases with stoichiometries nu

362
00:21:19 --> 00:21:29
A, a gas, pressure, temperature
plus nu B, B, gas, pressure,

363
00:21:29 --> 00:21:36
temperature, goes to nu sub C
moles of C, it's a gas, with a

364
00:21:36 --> 00:21:39
temperature and pressure, plus
nu sub D moles of D, which is

365
00:21:39 --> 00:21:41
a gas, temperature, pressure.
 

366
00:21:41 --> 00:21:43
All ideal gases.
 

367
00:21:43 --> 00:21:48
For now.
 

368
00:21:48 --> 00:21:57
So this the setup here.
 

369
00:21:57 --> 00:22:04
Now, when we write that
reaction up here, what we're

370
00:22:04 --> 00:22:11
really writing is, in terms of
the process, is take a

371
00:22:11 --> 00:22:16
container that contains A, plus
a container that contains

372
00:22:16 --> 00:22:20
B, and go to a container
that contains C, plus a

373
00:22:20 --> 00:22:24
container that contains D.
 

374
00:22:24 --> 00:22:32
And when we write delta G for
that reaction here, the process

375
00:22:32 --> 00:22:35
that we're talking about
is taking the reactions

376
00:22:35 --> 00:22:40
separately from each other.
 

377
00:22:40 --> 00:22:41
That's the initial state.
 

378
00:22:41 --> 00:22:43
And the final state
is the product.

379
00:22:43 --> 00:22:48
Separated in their own
containers from each other.

380
00:22:48 --> 00:22:51
And when we say delta G is less
than zero, for this process,

381
00:22:51 --> 00:22:54
which means it's spontaneous,
we mean the process to go from

382
00:22:54 --> 00:22:59
the separated reactions to
the separated products.

383
00:22:59 --> 00:23:00
But in reality, that's
not what you do when

384
00:23:00 --> 00:23:02
you do an experiment.
 

385
00:23:02 --> 00:23:04
In reality, you take these
two containers and you

386
00:23:04 --> 00:23:08
mix them together.
 

387
00:23:08 --> 00:23:09
A plus B.
 

388
00:23:09 --> 00:23:11
And you let that react.
 

389
00:23:11 --> 00:23:14
And at the end of the day, you
have a big container with A

390
00:23:14 --> 00:23:20
plus B plus C plus D inside,
all mixed up together

391
00:23:20 --> 00:23:23
in equilibrium.
 

392
00:23:23 --> 00:23:30
And this is not
the same as that.

393
00:23:30 --> 00:23:32
It's not the same as that.
 

394
00:23:32 --> 00:23:35
Because you've got entropy
of mixing happening

395
00:23:35 --> 00:23:37
in all of this.
 

396
00:23:37 --> 00:23:40
Forget about A and B
interacting with each other.

397
00:23:40 --> 00:23:43
Entropy of mixing is going
to dominate equilibrium.

398
00:23:43 --> 00:23:51
You've got this
problem to deal with.

399
00:23:51 --> 00:23:54
So that means that we're going
to have to worry about, if

400
00:23:54 --> 00:23:59
we're going to want to know at
which state the process is in

401
00:23:59 --> 00:24:00
equilibrium, you're going
to have to worry about

402
00:24:00 --> 00:24:03
this issue right here.
 

403
00:24:03 --> 00:24:06
It's not enough to know what
delta G of the reaction is.

404
00:24:06 --> 00:24:10
So for instance, if I plot,
as function of the reaction,

405
00:24:10 --> 00:24:14
I've got the reactants
on that side here.

406
00:24:14 --> 00:24:18
And the products on
this side here.

407
00:24:18 --> 00:24:22
And I want to plot delta G as
a function of the reaction.

408
00:24:22 --> 00:24:28
Well, my initial delta G that I
would write, to calculate delta

409
00:24:28 --> 00:24:35
G of the reaction, which is the
delta G of the products, minus

410
00:24:35 --> 00:24:41
the delta G of the reactants,
so initially I have delta

411
00:24:41 --> 00:24:43
G of the reactants.
 

412
00:24:43 --> 00:24:45
That's, in those two
boxes, that aren't mixed.

413
00:24:45 --> 00:24:47
So I'm up here somewhere.
 

414
00:24:47 --> 00:24:55
Delta G of the reactants.
 

415
00:24:55 --> 00:24:57
And at the end of the process,
in these two boxes, when I

416
00:24:57 --> 00:25:00
calculate this delta G naught
reaction, I'm sitting

417
00:25:00 --> 00:25:01
somewhere here.
 

418
00:25:01 --> 00:25:03
Let's say it's a
downhill process.

419
00:25:03 --> 00:25:06
Delta G naught of the products.
 

420
00:25:06 --> 00:25:09
Well, the first thing that
happens, when I take these two

421
00:25:09 --> 00:25:13
boxes and mix them together, is
delta G naught of the

422
00:25:13 --> 00:25:15
reaction's going to go up or
going to go down,

423
00:25:15 --> 00:25:19
or stay the same.
 

424
00:25:19 --> 00:25:24
Anybody want to guess?
 

425
00:25:24 --> 00:25:27
When I go from here to
here, before there's any

426
00:25:27 --> 00:25:30
reaction that happens.
 

427
00:25:30 --> 00:25:32
Shall we vote?
 

428
00:25:32 --> 00:25:36
How many say that it's
going to go down?

429
00:25:36 --> 00:25:39
How many say that
it's going to go up?

430
00:25:39 --> 00:25:42
How many say that it's
going to stay the same?

431
00:25:42 --> 00:25:45
We have about three people that
said it's going to go up.

432
00:25:45 --> 00:25:48
And a lot of people that
say it's going to go down.

433
00:25:48 --> 00:25:52
And a few people that
are abstaining.

434
00:25:52 --> 00:25:57
OK.
 

435
00:25:57 --> 00:25:59
We're mixing things.
 

436
00:25:59 --> 00:26:02
Anybody want to
change their votes?

437
00:26:02 --> 00:26:05
Something is happening here.
 

438
00:26:05 --> 00:26:08
And this entropy term here,
when A and B come together

439
00:26:08 --> 00:26:14
and start mixing up.
 

440
00:26:14 --> 00:26:15
You're changing your vote.
 

441
00:26:15 --> 00:26:17
Alright.
 

442
00:26:17 --> 00:26:18
It's going to go down.
 

443
00:26:18 --> 00:26:21
Now we have near-unanimity.
 

444
00:26:21 --> 00:26:22
So the first thing that
happens is, you're

445
00:26:22 --> 00:26:24
going to go down here.
 

446
00:26:24 --> 00:26:27
You're going to have delta
G naught of the reactants

447
00:26:27 --> 00:26:32
in the mixture.
 

448
00:26:32 --> 00:26:37
And if it were to go all the
way, if A and B were to

449
00:26:37 --> 00:26:40
disappear, it still
would be mixed.

450
00:26:40 --> 00:26:42
And so the end product here
would actually be down here.

451
00:26:42 --> 00:26:44
It wouldn't be up
in the products.

452
00:26:44 --> 00:26:48
It would be the mixture.
 

453
00:26:48 --> 00:26:50
So that's the first
thing we know.

454
00:26:50 --> 00:26:53
That this delta G naught
reaction is not the full story.

455
00:26:53 --> 00:26:57
And along the way, here I have
two species to begin with.

456
00:26:57 --> 00:26:59
I've got two species
to end with.

457
00:26:59 --> 00:27:04
But I've got three different
species here in the middle.

458
00:27:04 --> 00:27:12
So as soon as I form a little
bit of products, in this case

459
00:27:12 --> 00:27:14
here, or if I start from
products and go the other way

460
00:27:14 --> 00:27:17
in the reaction and form the
reactants, the first thing

461
00:27:17 --> 00:27:19
that's going to happen is I'm
going to decrease the delta G.

462
00:27:19 --> 00:27:22
Just from the
entropy of mixing.

463
00:27:22 --> 00:27:27
And so if I plot my delta G as
a function of reaction

464
00:27:27 --> 00:27:34
conditions, I'm going to get a
bowing curve like that.

465
00:27:34 --> 00:27:38
If the entropy wasn't there,
then it would just be a

466
00:27:38 --> 00:27:40
straight line from
one to the other.

467
00:27:40 --> 00:27:43
The entropy of mixing of
reactants and products

468
00:27:43 --> 00:27:43
wasn't there.
 

469
00:27:43 --> 00:27:45
Actually, let me
put it up here.

470
00:27:45 --> 00:27:48
If entropy of mixing
wasn't there, I would

471
00:27:48 --> 00:27:50
start from up here.
 

472
00:27:50 --> 00:27:54
And as my stoichiometry
changed, I would have a linear

473
00:27:54 --> 00:27:57
curve from here to there
as a function of the

474
00:27:57 --> 00:27:58
process of reaction.
 

475
00:27:58 --> 00:28:01
But the entropy of mixing is
causing my initial state and

476
00:28:01 --> 00:28:02
my final state to go down.
 

477
00:28:02 --> 00:28:05
And it's causing a
bow in this here.

478
00:28:05 --> 00:28:08
Because if I have equal amounts
of A, B, C, and D, that's a lot

479
00:28:08 --> 00:28:12
of entropy of mixing there.
 

480
00:28:12 --> 00:28:14
So equilibrium is actually
somewhere down here.

481
00:28:14 --> 00:28:20
It's where delta G of the
mixture is at its lowest.

482
00:28:20 --> 00:28:25
Delta G of the mixture
is at its lowest.

483
00:28:25 --> 00:28:31
Any questions?
 

484
00:28:31 --> 00:28:33
Now we going to do the math.
 

485
00:28:33 --> 00:28:42
We're going to see how
that comes about.

486
00:28:42 --> 00:28:52
Let me do it here.
 

487
00:28:52 --> 00:28:55
So the question we're going
to ask is, suppose that

488
00:28:55 --> 00:28:56
I've got my mixture here.
 

489
00:28:56 --> 00:29:02
And I'm sitting somewhere
on this curve here.

490
00:29:02 --> 00:29:07
So I've got pA for partial
pressure of A, partial pressure

491
00:29:07 --> 00:29:10
of B, partial pressure of C,
and partial pressure

492
00:29:10 --> 00:29:12
of D in my mixture.
 

493
00:29:12 --> 00:29:14
And I want to know, I've
got this mixture of

494
00:29:14 --> 00:29:16
reactants and products.
 

495
00:29:16 --> 00:29:21
Which way is the
reaction going to go?

496
00:29:21 --> 00:29:22
Is it going to go
towards the products?

497
00:29:22 --> 00:29:24
Is it going to go
towards the reactants?

498
00:29:24 --> 00:29:26
Or is it at equilibrium?
 

499
00:29:26 --> 00:29:31
And to answer that question,
I'm going to let the reaction

500
00:29:31 --> 00:29:34
react a little bit more to
create a little bit

501
00:29:34 --> 00:29:36
more products.
 

502
00:29:36 --> 00:29:38
Remove a little
bit of reactants.

503
00:29:38 --> 00:29:42
And see what the sign of
delta G is for that process.

504
00:29:42 --> 00:29:51
So I'm going to go from a moles
of A, b moles of B, c moles of

505
00:29:51 --> 00:29:55
C and d moles of D in my
mixture, which is that

506
00:29:55 --> 00:29:57
point on my graph here.
 

507
00:29:57 --> 00:29:59
Partial pressure A, partial
pressure of B, partial pressure

508
00:29:59 --> 00:30:02
of C, and partial pressure D,
and I'm going to react it a

509
00:30:02 --> 00:30:13
little bit more. a plus, a
minus epsilon times nu A, where

510
00:30:13 --> 00:30:17
epsilon is a very small number.
 

511
00:30:17 --> 00:30:24
Of A, b minus epsilon
times nu B times B.

512
00:30:24 --> 00:30:28
And I'm going to have, now,
products being formed. c moles

513
00:30:28 --> 00:30:31
plus epsilon times nu C,
I've going to have the

514
00:30:31 --> 00:30:33
stoichiometry in there.
 

515
00:30:33 --> 00:30:37
For every nu A moles of
A that I lose, I create

516
00:30:37 --> 00:30:38
mu C moles of C.
 

517
00:30:38 --> 00:30:42
Times epsilon as the
scaling factor.

518
00:30:42 --> 00:30:47
And d plus epsilon
times nu D moles of D.

519
00:30:47 --> 00:30:50
That's going to be
my new mixture.

520
00:30:50 --> 00:30:53
And I'm going to ask, as I go
from this initial state to that

521
00:30:53 --> 00:30:56
final state, I'm sitting on
that curve, which

522
00:30:56 --> 00:30:58
sign is delta G.
 

523
00:30:58 --> 00:31:01
How do I know what the
sign of delta G is?

524
00:31:01 --> 00:31:04
What is delta G
for this process?

525
00:31:04 --> 00:31:07
Not the reaction with the
isolated reactants and

526
00:31:07 --> 00:31:10
products, but the reaction
where everything

527
00:31:10 --> 00:31:11
is mixed together.
 

528
00:31:11 --> 00:31:13
Where I'm going to have to
worry about the chemical

529
00:31:13 --> 00:31:15
potentials of mixtures.
 

530
00:31:15 --> 00:31:18
And I'm going to go from one
mixture to another mixture, and

531
00:31:18 --> 00:31:20
that's going to be the key to
telling me if I'm in

532
00:31:20 --> 00:31:24
equilibrium or not.
 

533
00:31:24 --> 00:31:25
But we know how to do this.
 

534
00:31:25 --> 00:31:28
We know how to do this.
 

535
00:31:28 --> 00:31:35
So I want to calculate delta G
for the process is delta G

536
00:31:35 --> 00:31:39
after minus delta G before.
 

537
00:31:39 --> 00:31:46
That's delta G after
minus delta G before.

538
00:31:46 --> 00:31:53
And I know that delta G,
rather, let's call it G after

539
00:31:53 --> 00:31:59
minus G before, the Gibbs
free energy after minus the

540
00:31:59 --> 00:32:01
Gibbs free energy before.
 

541
00:32:01 --> 00:32:05
And I know the Gibbs free
energy is just the sum of the

542
00:32:05 --> 00:32:06
chemical potentials, right?
 

543
00:32:06 --> 00:32:09
You take all the species,
and you take all the

544
00:32:09 --> 00:32:10
chemical potentials.
 

545
00:32:10 --> 00:32:12
You add it up together
times the stoichiometry.

546
00:32:12 --> 00:32:14
And that gives you the
Gibbs free energy.

547
00:32:14 --> 00:32:15
That's what we
learned last time.

548
00:32:15 --> 00:32:18
So if I want to find what the
Gibbs free energy at the end

549
00:32:18 --> 00:32:23
here is, I look at the chemical
potentials of A, B, and C and D

550
00:32:23 --> 00:32:26
times their number of moles.
 

551
00:32:26 --> 00:32:30
So I look at a minus epsilon
times nu A times the chemical

552
00:32:30 --> 00:32:36
potential of A, plus b minus
epsilon times nu B times the

553
00:32:36 --> 00:32:41
chemical potential of B plus c,
plus epsilon times nu C, the

554
00:32:41 --> 00:32:46
chemical potential at C, plus
d, plus epsilon times nu D,

555
00:32:46 --> 00:32:53
the chemical potential of D.
 

556
00:32:53 --> 00:32:59
And then I subtract
what G was before.

557
00:32:59 --> 00:33:03
This infinitesimally
small process.

558
00:33:03 --> 00:33:13
Was a times mu A plus b
times mu B plus c times

559
00:33:13 --> 00:33:20
mu C, plus d times mu D.
 

560
00:33:20 --> 00:33:25
And we're assuming that my
small change, the small amount

561
00:33:25 --> 00:33:29
of A and B that get destroyed
to form C and D is small enough

562
00:33:29 --> 00:33:31
that the chemical potential
basically stays the same during

563
00:33:31 --> 00:33:34
this infinitesimally
small change.

564
00:33:34 --> 00:33:36
That's why I can use the
same chemical potentials

565
00:33:36 --> 00:33:41
before and after.
 

566
00:33:41 --> 00:33:44
OK, a lot of things drop out.
 

567
00:33:44 --> 00:33:46
This term drops out
from that term.

568
00:33:46 --> 00:33:48
This term drops out
from this term.

569
00:33:48 --> 00:33:50
This term drops out
from this term.

570
00:33:50 --> 00:33:54
This is a minus, no,
this is all plus.

571
00:33:54 --> 00:33:55
That's fine.
 

572
00:33:55 --> 00:33:57
There's the minus
sign right here.

573
00:33:57 --> 00:34:00
So then this term drops
out from that term.

574
00:34:00 --> 00:34:09
And I am left with epsilon
times nu C mu C, the chemical

575
00:34:09 --> 00:34:16
potentials of the products
minus the sum of the chemical

576
00:34:16 --> 00:34:29
potentials of the reactants,
nu A, mu A plus nu B mu B.

577
00:34:29 --> 00:34:37
That's the delta G for
this small change.

578
00:34:37 --> 00:34:40
Now, you remember way back,
maybe from even just last

579
00:34:40 --> 00:34:43
semester if you've taken 5.112
last semester, or from last

580
00:34:43 --> 00:34:46
year, or from high school, that
the partial pressure is going

581
00:34:46 --> 00:34:49
to the equilibrium constant.
 

582
00:34:49 --> 00:34:52
Somehow we're going to have to
get partial pressures in there.

583
00:34:52 --> 00:34:54
But we know now how to go
from chemical potentials

584
00:34:54 --> 00:34:57
to partial pressures.
 

585
00:34:57 --> 00:34:59
It's written right here.
 

586
00:34:59 --> 00:35:04
The chemical potentials of the,
and we also know how to go from

587
00:35:04 --> 00:35:08
the chemical potential in the
mixed species, in the,

588
00:35:08 --> 00:35:12
mixture to the chemical
potential in a pure.

589
00:35:12 --> 00:35:18
We saw that mu A in the
mixture, temperature, pressure

590
00:35:18 --> 00:35:25
was equal to mu A pure
temperature, pressure

591
00:35:25 --> 00:35:29
plus RT log xA.
 

592
00:35:29 --> 00:35:32
So those are things that we're
going to be using, to go from

593
00:35:32 --> 00:35:35
something that has chemical
potential to something where

594
00:35:35 --> 00:35:38
we'll be able to get back
pressures, partial pressures,

595
00:35:38 --> 00:35:41
and delta G of the reaction,
because we're going to need the

596
00:35:41 --> 00:35:45
chemical potential
of the pure stuff.

597
00:35:45 --> 00:35:51
So let me go forward
just a little bit.

598
00:35:51 --> 00:35:57
Remind you, if I look at the
delta G of the reaction, delta

599
00:35:57 --> 00:36:00
G of the reaction, in terms
of chemical potentials.

600
00:36:00 --> 00:36:04
Delta G naught of the reaction.
 

601
00:36:04 --> 00:36:09
This is the delta G of the
products minus the delta G of

602
00:36:09 --> 00:36:13
the reactants when they're
pure, not when they're mixed.

603
00:36:13 --> 00:36:24
So delta G naught of the
reaction is nu C mu C at one

604
00:36:24 --> 00:36:34
bar pure plus nu D mu D pure.
 

605
00:36:34 --> 00:36:45
Minus nu A mu A pure minus nu
B mu B in the pure state.

606
00:36:45 --> 00:36:48
That's what delta G naught of
the reaction is in terms of

607
00:36:48 --> 00:36:51
the chemical potentials
of all these species.

608
00:36:51 --> 00:36:54
Everything's at one bar,
and everything is pure.

609
00:36:54 --> 00:37:06
Somehow this is going to
have to come out of this.

610
00:37:06 --> 00:37:09
So let's keep going.
 

611
00:37:09 --> 00:37:17
And see how it falls out.
 

612
00:37:17 --> 00:37:20
So for each one of these
chemical potentials, I'm

613
00:37:20 --> 00:37:22
going to write it in
terms of the pressure.

614
00:37:22 --> 00:37:24
What I just covered up here.
 

615
00:37:24 --> 00:37:33
Mu(T, p) is mu naught
of T times RT log p.

616
00:37:33 --> 00:37:39
So now, delta G is going to
be equal to epsilon, and

617
00:37:39 --> 00:37:41
I'm going to do a little
massaging quickly.

618
00:37:41 --> 00:37:45
And I'll let you take this, go
home and see how I went from

619
00:37:45 --> 00:37:45
one state to the other.
 

620
00:37:45 --> 00:37:50
The secret is to put in here
the pressure dependence.

621
00:37:50 --> 00:38:05
Nu C mu C naught, plus nu D mu
D naught, minus nu A mu A

622
00:38:05 --> 00:38:19
naught, plus nu B mu B naught,
plus RT log pC to the nu C, pD

623
00:38:19 --> 00:38:28
to the nu D, divided by pA to
the nu A, pB to the nu B.

624
00:38:28 --> 00:38:34
These log partial pressures all
come from expanding out the

625
00:38:34 --> 00:38:57
chemical potential as mu
naught plus RT log p.

626
00:38:57 --> 00:39:06
And then we recognize that this
part right here, this part

627
00:39:06 --> 00:39:16
right here is delta
G naught reaction.

628
00:39:16 --> 00:39:22
So I have delta G now, for this
process of taking reactants to

629
00:39:22 --> 00:39:25
products just a little bit, let
me take it as a function

630
00:39:25 --> 00:39:27
of epsilon here.
 

631
00:39:27 --> 00:39:35
Delta G naught of the reaction
plus RT log of this ratio

632
00:39:35 --> 00:39:37
of partial pressures.
 

633
00:39:37 --> 00:39:40
I'm going to call that Q.
 

634
00:39:40 --> 00:39:42
I'm going to call this
thing here Q, which

635
00:39:42 --> 00:39:44
you've seen before.
 

636
00:39:44 --> 00:39:51
The reaction quotient.
 

637
00:39:51 --> 00:39:56
And this tells me that for this
very small process, epsilon,

638
00:39:56 --> 00:40:05
very small, if delta G naught,
delta G, of this process, is

639
00:40:05 --> 00:40:09
less than zero, then the
reaction will keep

640
00:40:09 --> 00:40:09
going forward.
 

641
00:40:09 --> 00:40:12
It means that I'll be on the
side of the curve here.

642
00:40:12 --> 00:40:15
I'm going to go down a little
bit. delta G naught is going

643
00:40:15 --> 00:40:18
to be great, it's going
to be spontaneous.

644
00:40:18 --> 00:40:20
I'm not in equilibrium.
 

645
00:40:20 --> 00:40:25
I'm going to go
towards the products.

646
00:40:25 --> 00:40:30
If delta G is equal to zero,
then I'm at the bottom

647
00:40:30 --> 00:40:32
of this curve here.
 

648
00:40:32 --> 00:40:33
I'm here.
 

649
00:40:33 --> 00:40:36
If I go forward a little
bit, the slope is zero.

650
00:40:36 --> 00:40:42
Delta G is zero, I'm
in equilibrium.

651
00:40:42 --> 00:40:50
And if delta G is greater than
zero, then I'm going to back to

652
00:40:50 --> 00:40:54
reactants, and I'm sitting,
then, on that part

653
00:40:54 --> 00:40:56
of the curve here.
 

654
00:40:56 --> 00:40:58
And if I try to make more
products I'm just going

655
00:40:58 --> 00:41:02
uphill a little bit.
 

656
00:41:02 --> 00:41:07
So now, I've done this all for
epsilon is equal to very small.

657
00:41:07 --> 00:41:10
What you usually find written
in books is where the

658
00:41:10 --> 00:41:11
epsilon equals to one.
 

659
00:41:11 --> 00:41:14
Basically taking one
mole of this process.

660
00:41:14 --> 00:41:18
So you'll see it per-mole
of this reaction.

661
00:41:18 --> 00:41:19
But it's the same idea.
 

662
00:41:19 --> 00:41:22
It's the same thing.
 

663
00:41:22 --> 00:41:24
And in fact it doesn't really
matter, because the quantity

664
00:41:24 --> 00:41:27
that matters is this delta G
naught reactions plus RT log Q.

665
00:41:27 --> 00:41:30
It's the sign of
this quantity here.

666
00:41:30 --> 00:41:32
This epsilon is just
an arbitrary number.

667
00:41:32 --> 00:41:36
I can pick it, really
whatever I want.

668
00:41:36 --> 00:41:42
So it's the sign of this
thing that's important.

669
00:41:42 --> 00:41:46
Let me go back.
 

670
00:41:46 --> 00:41:50
So what you'll see, then, is
delta G is equal to delta G

671
00:41:50 --> 00:41:56
naught reaction plus RT log Q.
 

672
00:41:56 --> 00:41:59
Basically, taking epsilon
is going to zero.

673
00:41:59 --> 00:42:12
As determining where the
equilibrium is going to be.

674
00:42:12 --> 00:42:20
OK, any questions?
 

675
00:42:20 --> 00:42:22
All driven by
entropy of mixing.

676
00:42:22 --> 00:42:26
Without entropy of mixing,
we would be sitting

677
00:42:26 --> 00:42:28
on this curve here.
 

678
00:42:28 --> 00:42:31
Delta G naught of the reaction
would tell us that everything

679
00:42:31 --> 00:42:33
should go to completion.
 

680
00:42:33 --> 00:42:36
Things don't go to completion,
and that's a good thing.

681
00:42:36 --> 00:42:38
Otherwise there would
not be life on Earth.

682
00:42:38 --> 00:42:43
We're basically a set of
equilibria in a big membrane.

683
00:42:43 --> 00:42:44
Which is our skin, right?
 

684
00:42:44 --> 00:42:46
All these biochemical
cycles are in equilibrium

685
00:42:46 --> 00:42:47
with each other.
 

686
00:42:47 --> 00:42:49
And it's a complicated process.
 

687
00:42:49 --> 00:42:52
And it's all driven by entropy.
 

688
00:42:52 --> 00:42:55
Ultimately.
 

689
00:42:55 --> 00:42:58
And other things, but
entropy is very important.

690
00:42:58 --> 00:42:59
Alright, so equilibrium now.
 

691
00:42:59 --> 00:43:02
Equilibrium is when we have
this delta G equal to zero.

692
00:43:02 --> 00:43:10
That's when delta G naught of
the reaction equals RT log Q.

693
00:43:10 --> 00:43:14
And at that point, we replace
Q with to equilibrium,

694
00:43:14 --> 00:43:19
and we call that the
equilibrium constant.

695
00:43:19 --> 00:43:22
And we're going to put a little
p here, because it's in terms

696
00:43:22 --> 00:43:25
of the partial pressures.
 

697
00:43:25 --> 00:43:32
And it's equal to the partial
pressures of the products

698
00:43:32 --> 00:43:36
raised to their stoichiometry.
 

699
00:43:36 --> 00:43:40
And divided by the partial
pressures of the reactants,

700
00:43:40 --> 00:43:45
raised to the power of
their stoichiometry.

701
00:43:45 --> 00:43:48
And this, you've seen
before, I'm sure.

702
00:43:48 --> 00:43:54
At equilibrium.
 

703
00:43:54 --> 00:43:58
And there's a minus
sign somewhere here.

704
00:43:58 --> 00:44:00
Because it's this plus that
that's equal to zero, and

705
00:44:00 --> 00:44:04
there's the minus sign that
I forgot to write down.

706
00:44:04 --> 00:44:08
So we define, then, equilibrium
constant this way.

707
00:44:08 --> 00:44:15
And one of the things to note
is that Kp as written here

708
00:44:15 --> 00:44:16
is actually unitless.
 

709
00:44:16 --> 00:44:18
It doesn't look like it.
 

710
00:44:18 --> 00:44:19
The way that I've written it.
 

711
00:44:19 --> 00:44:25
Because I did a shorthand way
of writing the pressures,

712
00:44:25 --> 00:44:33
when I wrote RT log p here.
 

713
00:44:33 --> 00:44:35
There's the assumption, when
you have the log of the

714
00:44:35 --> 00:44:39
pressure, that there's always
one bar sitting behind there.

715
00:44:39 --> 00:44:40
Underneath.
 

716
00:44:40 --> 00:44:43
It's always referenced to
a reference pressure.

717
00:44:43 --> 00:44:47
There's always a reference
pressure. p naught dividing it

718
00:44:47 --> 00:44:51
by, because you don't want to
have any units inside the log.

719
00:44:51 --> 00:44:53
And this reference pressure,
we took as one bar.

720
00:44:53 --> 00:44:57
And it's pretty common to just
ignore the fact that you've got

721
00:44:57 --> 00:44:59
one bar sitting in
the denominator.

722
00:44:59 --> 00:45:04
And so, actually, all these
pressures here are divided by

723
00:45:04 --> 00:45:10
p reference divided by p
reference, divided by p

724
00:45:10 --> 00:45:13
reference, divided by
p reference, which

725
00:45:13 --> 00:45:15
happens to be 1 bar.
 

726
00:45:15 --> 00:45:17
But it's there.
 

727
00:45:17 --> 00:45:20
And that means that the bars
on top and the bars on

728
00:45:20 --> 00:45:22
the bottom cancel out.
 

729
00:45:22 --> 00:45:25
Which means that K sub p
doesn't have any units.

730
00:45:25 --> 00:45:27
It is unitless.
 

731
00:45:27 --> 00:45:28
It's a number.
 

732
00:45:28 --> 00:45:30
Straight number.
 

733
00:45:30 --> 00:45:35
Very common mistake to make,
to write it and forget that

734
00:45:35 --> 00:45:37
there's one bar sitting
on the bottom here.

735
00:45:37 --> 00:45:40
And you take all the bars
to the new powers on top,

736
00:45:40 --> 00:45:42
and the bars to the new
powers on the bottom.

737
00:45:42 --> 00:45:45
And make units there.
 

738
00:45:45 --> 00:45:48
And get Kp is equal to some
number, to the, times

739
00:45:48 --> 00:45:49
bar to some power.
 

740
00:45:49 --> 00:45:54
That would be wrong.
 

741
00:45:54 --> 00:45:56
I know I've made that
mistake before.

742
00:45:56 --> 00:45:59
But you are not going to
make that mistake, right?

743
00:45:59 --> 00:46:01
Because you've been warned.
 

744
00:46:01 --> 00:46:03
That this is a common mistake.
 

745
00:46:03 --> 00:46:07
This is unitless.
 

746
00:46:07 --> 00:46:13
Now, you can invert this to get
Kp as a function of delta G. e

747
00:46:13 --> 00:46:20
to the minus delta G naught of
the reaction divided by RT.

748
00:46:20 --> 00:46:22
And those are the things that
you use to go back and forth

749
00:46:22 --> 00:46:27
between the thermodynamic
quantities, like G that you

750
00:46:27 --> 00:46:31
calculate, to equilibrium
quantities like the K, to

751
00:46:31 --> 00:46:34
finding equilibria between
something like the

752
00:46:34 --> 00:46:36
Haber process.
 

753
00:46:36 --> 00:46:39
Now, there's other equilibrium
constant that is used a lot.

754
00:46:39 --> 00:46:42
Which is the form of the
equilibrium constant not

755
00:46:42 --> 00:46:44
in terms of the partial
pressures, but in terms

756
00:46:44 --> 00:46:45
of the mole fraction.
 

757
00:46:45 --> 00:46:48
That's also an important
one when you're looking

758
00:46:48 --> 00:46:54
at solution cases.
 

759
00:46:54 --> 00:46:56
Because we can rewrite the
partial pressures using

760
00:46:56 --> 00:47:02
Dalton's law. pC is equal
to the mole fraction of

761
00:47:02 --> 00:47:06
species C times the total
pressure, I'll call it p.

762
00:47:06 --> 00:47:09
So if I replace every one of
these partial pressures using

763
00:47:09 --> 00:47:19
Dalton's law, I get K sub p is
equal to xC pC times p to the

764
00:47:19 --> 00:47:29
nu C, times xD pD to the nu
D, divided by xA pA - no,

765
00:47:29 --> 00:47:32
not pA, total pressure.
 

766
00:47:32 --> 00:47:33
Dalton's law.
 

767
00:47:33 --> 00:47:39
To the nu A, xB p to the nu B.
 

768
00:47:39 --> 00:47:42
OK, so I've got p, p, p, p.
 

769
00:47:42 --> 00:47:44
They all come out.
 

770
00:47:44 --> 00:47:48
And that's p to the
minus delta nu.

771
00:47:48 --> 00:47:51
Where delta nu is the
difference in the number of

772
00:47:51 --> 00:47:55
moles of reactants minus the
number of moles of products.

773
00:47:55 --> 00:48:05
So delta nu is nu C plus nu
D minus nu A minus nu B.

774
00:48:05 --> 00:48:10
It's the moles of products
minus moles of reactants.

775
00:48:10 --> 00:48:18
And then I have xA to the nu A,
xC to the nu C, xD to the nu D,

776
00:48:18 --> 00:48:24
xA to the nu A, xB to the nu B.
 

777
00:48:24 --> 00:48:28
And that ratio, which is in
terms of mole fractions,

778
00:48:28 --> 00:48:31
we call K sub x.
 

779
00:48:31 --> 00:48:32
The mole fractions.
 

780
00:48:32 --> 00:48:36
Because p, to the minus
delta nu, times Kx.

781
00:48:36 --> 00:48:40
 


782
00:48:40 --> 00:48:44
Now, this was unitless.
 

783
00:48:44 --> 00:48:47
This pressure to the minus
delta nu sitting here, this has

784
00:48:47 --> 00:48:50
bars to the minus delta nu.
 

785
00:48:50 --> 00:48:53
Kx has units.
 

786
00:48:53 --> 00:48:56
It may not look like it because
it's a bunch of mole fractions,

787
00:48:56 --> 00:48:59
and it certainly doesn't
look like it has any units.

788
00:48:59 --> 00:49:11
Mole fractions or ratio,
but it's got units.

789
00:49:11 --> 00:49:20
So let's rewrite Kx
in terms of Kp.

790
00:49:20 --> 00:49:28
Rewrite Kx as equal to p to
the minus delta nu, K sub p.

791
00:49:28 --> 00:49:34
And the units are in terms
of bars, let's say, bar

792
00:49:34 --> 00:49:37
to the minus delta nu.
 

793
00:49:37 --> 00:49:39
Always want to check your
units at the end of the day.

794
00:49:39 --> 00:49:41
If your units don't work
out after you've done a

795
00:49:41 --> 00:49:48
calculation, you're in trouble.
 

796
00:49:48 --> 00:49:49
Any questions?
 

797
00:49:49 --> 00:49:51
Because we're going to
end today, at here.

798
00:49:51 --> 00:49:51
Yes.
 

799
00:49:51 --> 00:49:57
STUDENT: [INAUDIBLE]
 

800
00:49:57 --> 00:50:00
PROFESSOR: In the notes it says
Kp and Kx are both unitless.

801
00:50:00 --> 00:50:01
That is a mistake.
 

802
00:50:01 --> 00:50:05
I'm going to fix that,
and put it on the Web.

803
00:50:05 --> 00:50:08
So it may be that there's
a special case where the

804
00:50:08 --> 00:50:12
number of moles before
and after are the same.

805
00:50:12 --> 00:50:15
But generally that is not true.
 

806
00:50:15 --> 00:50:19
Where did I say that?
 

807
00:50:19 --> 00:50:21
Oh, yeah.
 

808
00:50:21 --> 00:50:24
Both unitless.
 

809
00:50:24 --> 00:50:37
Well, that is actually true.
 

810
00:50:37 --> 00:50:38
Let me think through this.
 

811
00:50:38 --> 00:50:42
STUDENT: [INAUDIBLE]
 

812
00:50:42 --> 00:50:43
PROFESSOR: About one
bar, one bar, one bar.

813
00:50:43 --> 00:50:44
That is actually true.
 

814
00:50:44 --> 00:50:49
They're both, that is true.
 

815
00:50:49 --> 00:50:53
They are both unitless.
 

816
00:50:53 --> 00:50:56
Well, this is a big
boob on my part here.

817
00:50:56 --> 00:50:57
Because you are
absolutely right.

818
00:50:57 --> 00:51:01
Because there's one bar sitting
here. one bar, one bar, one

819
00:51:01 --> 00:51:05
bar, one bar, and the
bars cancel out.

820
00:51:05 --> 00:51:08
Good catch.
 

821
00:51:08 --> 00:51:10
OK, the notes are right.
 

822
00:51:10 --> 00:51:13
My notes are right.
 

823
00:51:13 --> 00:51:14
Alright.
 

824
00:51:14 --> 00:51:16
Next time we'll talk about the
temperature dependence and the

825
00:51:16 --> 00:51:19
pressure dependence of
equilibrium constants.

826
00:51:19 --> 00:51:20