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PROFESSOR: I want to remind and
clarify as needed the

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equilibrium constant Kp for gas
phase reaction was the ratios

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of the partial pressures
referenced to some reference

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pressure, which we usually
take as one, one bar.

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To the stoichiometry , pD
divided by p naught, to

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the mu D where species
C and D are products.

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And the reactants
are on the bottom.

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And usually we don't write p
naught, but it's important to

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remember that it's there.
 

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And then we can also write this
in terms of the Gibbs free

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energy for the reaction.
 

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The standard Gibbs free energy,
minus delta G naught of the

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reaction, divided by RT.
 

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And what this tells us is
that this is a number.

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This is a number, there's no
pre-factor here that has units.

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That's a unitless number.
 

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And it doesn't depend
on the total pressure.

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And the other thing to remember
is that delta G naught for the

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reaction is the process of
taking the reactants separated

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in separate boxes, separate
containers, and the final

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product, the final step, is the
products separated in

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individual containers.
 

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That's what we write when
we write delta G naught

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for the reaction.
 

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We also looked at K in
terms of mole fractions.

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So if you replace all the
partial pressures with the mole

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fraction times the total
pressure, you get an expression

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for K sub x, which we define as
the mole fraction of

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the products to the
stoichiometric powers.

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Which is also unitless.
 

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This is true, it's unitless.
 

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But, if you write it in terms
of K sub p, the total pressure

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comes in here. p total, divided
by the reference pressure to

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the minus delta nu, where this
is the change in the number

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of moles, in going from
reactants to products.

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And there's K sub
p sitting here.

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So unlike K sub p, which
doesn't depend on the total

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pressure, K sub x does depend
on the total pressure through

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this term right here.
 

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So when we look at problems
where we change the pressure,

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the total pressure of the
system, this is going

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to stay the same.
 

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Because it only cares
about delta G naught

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for the reaction.
 

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But this K sub x will depend
on the total pressure.

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And that's often a source of
confusion in doing problems.

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OK, any questions?
 

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We're going to do an example
where we change the

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pressure first.
 

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So there are examples in
the notes, and I'm going

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to skip the first one.
 

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I'm going to go to the second
one, which is the effect of the

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total pressure on the reaction.
 

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And Le Chatelier's
principle, for pressure.

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And the example we're
going to take is a fairly

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standard example, also.
 

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Which is the reaction of N2O4,
which is a gas, to 2 NO2, which

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is a gas, the kind of reaction
that happens when you have smog

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and, fairly common
in big cities.

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The question we're going to ask
is, what happens when we change

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the total pressure in
this reaction here.

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Which way does the
equilibrium go?

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Does it go to the right,
does it go to the left?

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Does it go to the products
or the reactants.

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And so, in order to answer that
question, we're going to ask a

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slightly different question.
 

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We're going to ask what is the
molar ratio, what is the

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fraction, what is fraction
of the reactant, the

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N2O4 that's reacted.
 

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That has reacted.
 

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And we're going to
call that alpha.

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It's the number of moles that
have reacted divided by

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number of moles initially.
 

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So we're going to need to find
at equilibrium what is the

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number of moles of N2O4
that has reacted.

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So we have to set
up the problem.

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And so the way that, the
standard way of setting up

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the problem is to write the
equilibrium, 2 NO2 gas.

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And then on this line here, we
write the initial conditions

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before we set up
the equilibrium.

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And let's say that we
have n moles of N2O4

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initially in the box.
 

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And zero moles of the NO2.
 

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So we have n moles here.
 

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And zero moles here.
 

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At equilibrium, let's write
the number of moles.

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A certain number of moles
of N2O4 will have reacted,

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let's call that x.
 

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So n minus x moles left.
 

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For every x moles of N2O4
that's reacted, we create

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two moles of NO2.
 

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So we have 2x here.
 

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And then we're going to need
the total number of moles,

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because we're going to be doing
mole ratios, mole fractions.

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So the total number of moles at
any time is the sum of these

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two, n minus x plus 2x.
 

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It's n plus x.
 

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00:07:18 --> 00:07:21
So if we're going to be writing
our equilibrium constant in

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terms of mole fractions, we're
going to need mole fractions.

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So the mole fraction at any
time is n minus x divided by

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the total number of moles,
which we just calculated

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as n plus x.
 

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And this is 2x divided
by the total number

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of moles, n plus x.
 

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And what we want is
this ratio here.

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We want the ratio of the number
of moles reacted, which is x,

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that's the number of
moles that's gone.

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That have reacted.
 

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Divided by the number of
moles initially, which is n.

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That's what we want.
 

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We want to see how that is
going to change with pressure.

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So we're going to deal first
with Kp, because Kp doesn't

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depend on total pressure.
 

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We're going to write that down.
 

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Then we're going to
go to Kx, somehow.

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And that's going to
depend on pressure.

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So let's see what Kp is here.
 

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So K sub p, you've got
the products on top.

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So it's the partial pressure of
NO2 to the second power,

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divided by the partial pressure
of N2O4 to the one power.

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And everything is referenced to
one bar, everything's in bar.

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And in terms of the molar
fractions, it's the total

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pressure squared, times the
mole fraction of NO2 squared,

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divided by the total pressure
to the first power.

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So the square root on
top gets divided by one

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00:09:10 --> 00:09:11
factor of pressure.
 

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So we have total
pressure in front.

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00:09:12 --> 00:09:25
Divided by x to the N2O4,
and that's p times Kx.

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00:09:25 --> 00:09:27
So let's plug in what
these mole fractions are

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from our table here.
 

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The mole fraction of NO2 is
2x divided by n plus x.

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00:09:35 --> 00:09:39
2x divided by n plus x
to the square power.

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Mole fraction of N2O4, n
minus x over n plus x. n

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00:09:42 --> 00:09:46
minus x over n plus x.
 

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Multiply, square the
top, 4x squared.

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Divided by n plus x squared.
 

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Things sort of cancel out here.
 

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Rearrange.
 

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4x squared divided by n
squared minus x squared.

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00:10:05 --> 00:10:09
What we're really interested in
is, we're not interested in x.

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00:10:09 --> 00:10:12
We're interested in
x divided by n.

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So let's divide both the top
and the bottom by n squared.

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00:10:17 --> 00:10:20
And we're going to
get alpha come up.

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00:10:20 --> 00:10:27
So this is then p times 4
alpha squared divided by

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00:10:27 --> 00:10:32
one minus alpha squared.
 

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00:10:32 --> 00:10:35
This is not, there's no
total pressure here.

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The only way, the only place,
where the total pressure

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comes in, is right here.
 

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This is just a number that
doesn't care what the

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00:10:44 --> 00:10:45
total pressure is.
 

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00:10:45 --> 00:10:47
Which is why we're using it.
 

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And not K sub x, which cares
what the total pressure is.

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So now we can solve for alpha.
 

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We can solve for alpha by
rearranging this equation.

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This is just a number.
 

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And this is where the
total pressure comes in.

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So you rearrange that, and you
get alpha is equal to 1 plus

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4p divided by Kp, to
the minus 1/2 power.

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And if I rewrite that slightly
to make it a little bit easier

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to see what's going to happen,
when I change the pressure, 1

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00:11:28 --> 00:11:37
plus 4p divided by Kp,
to the 1/2 power.

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And that's what I'm after.
 

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This tells me what happens at
equilibrium to the amount of

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NO2 as I change the
total pressure.

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This is the only place
where it comes in.

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So now I can see that if I
raise the pressure in my

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container, raise the pressure,
this is in the denominator, so

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this fraction gets smaller.
 

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Alpha gets smaller.
 

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I raise the pressure, the
fraction of material that

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reacts gets smaller.
 

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Therefore, the reaction goes
towards the reactants.

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00:12:30 --> 00:12:38
If I decrease the pressure,
this is a smaller number here.

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The fraction gets bigger.
 

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00:12:40 --> 00:12:40
Alpha goes up.
 

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00:12:40 --> 00:12:47
If I decrease the pressure and
I compare what happens to the

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00:12:47 --> 00:12:51
number of moles of reactants
that react, more of it reacts.

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Equilibrium shifts
towards the product.

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And this is Le Chatelier
that you already know.

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00:13:02 --> 00:13:10
Le Chatelier's principle,
for pressure.

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00:13:10 --> 00:13:14
The way it works is that Le
Chatelier's principle states

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00:13:14 --> 00:13:22
that, this chemical system
wants to stay as close

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00:13:22 --> 00:13:23
to what it was before.
 

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00:13:23 --> 00:13:24
It doesn't like change.
 

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It doesn't want to have
any change happen.

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So if you increase the
pressure, the chemical system

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00:13:34 --> 00:13:37
says, hey, you know I'm not so
happy that you're increasing

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00:13:37 --> 00:13:38
the pressure on me.
 

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00:13:38 --> 00:13:43
I'd like to go back to
a smaller pressure.

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00:13:43 --> 00:13:44
It doesn't like change.
 

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00:13:44 --> 00:13:46
Very conservative.
 

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And the way to decrease the
pressure is to decrease the

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number of moles in
the container.

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00:13:53 --> 00:13:55
How does it decrease
the number of moles?

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00:13:55 --> 00:13:56
Goes back to where
there are fewer moles.

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00:13:56 --> 00:14:03
And that's on the
reactant side.

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00:14:03 --> 00:14:06
How many of you
know Lenz's law?

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00:14:06 --> 00:14:07
In magnetism.
 

207
00:14:07 --> 00:14:11
The diamagnetic materials.
 

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00:14:11 --> 00:14:11
Right.
 

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00:14:11 --> 00:14:13
At least one person knows it.
 

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00:14:13 --> 00:14:17
It's the same idea.
 

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00:14:17 --> 00:14:21
You take a diamagnetic material
in the absence of a magnetic

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00:14:21 --> 00:14:25
field, and you slowly move it
into a place where there's high

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00:14:25 --> 00:14:30
magnetic field, what does the
diamagnetic material do?

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00:14:30 --> 00:14:33
It orients its magnetic
moment to reverse the field.

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00:14:33 --> 00:14:35
So that there's
no field inside.

216
00:14:35 --> 00:14:36
It starts out with no field.
 

217
00:14:36 --> 00:14:38
Doesn't like change.
 

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00:14:38 --> 00:14:42
So Lenz's law says it's going
to do whatever it can so that

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00:14:42 --> 00:14:44
it retains no field inside.
 

220
00:14:44 --> 00:14:47
Le Chatelier's principle is
basically the same thing.

221
00:14:47 --> 00:14:50
Equilibrium systems are
very unhappy if you

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00:14:50 --> 00:14:52
try to change them.
 

223
00:14:52 --> 00:14:56
And that's what happens for Le
Chatelier here with pressure.

224
00:14:56 --> 00:15:01
Any questions?
 

225
00:15:01 --> 00:15:04
So before we go to Le
Chatelier's with temperature,

226
00:15:04 --> 00:15:06
and the van 't Hoff equation.
 

227
00:15:06 --> 00:15:11
Let's do a little detour here
and talk about equilibrium in

228
00:15:11 --> 00:15:14
solution, which is really as
important, or if not important

229
00:15:14 --> 00:15:20
for a lot of you, then
gas phase equilibrium.

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00:15:20 --> 00:15:23
Although gas phase equilibrium
was where everything started.

231
00:15:23 --> 00:15:26
And still a huge deal.
 

232
00:15:26 --> 00:15:32
OK, so in equilibrium now, when
we talk about equilibrium in

233
00:15:32 --> 00:15:35
solution, we still have to,
still going to be the

234
00:15:35 --> 00:15:36
chemical potential.
 

235
00:15:36 --> 00:15:39
It's still going to be looking
at how chemical potential

236
00:15:39 --> 00:15:41
likes to go downhill.
 

237
00:15:41 --> 00:15:46
And we're going to have to
write chemical potential for

238
00:15:46 --> 00:15:51
a species, A, let's say,
which is in solution.

239
00:15:51 --> 00:15:54
At some concentration
c sub A in solution.

240
00:15:54 --> 00:16:03
And the concentration could
be given in moles per liter.

241
00:16:03 --> 00:16:08
Or it could be in
grams per liter.

242
00:16:08 --> 00:16:13
Or it could be in
grams per 1000 grams.

243
00:16:13 --> 00:16:16
Whatever your favorite
unit of concentration is.

244
00:16:16 --> 00:16:17
Use it.
 

245
00:16:17 --> 00:16:20
Stick to it.
 

246
00:16:20 --> 00:16:22
And in order to do equilibrium,
we're going to have to

247
00:16:22 --> 00:16:28
reference it to, so this would
be the species at some

248
00:16:28 --> 00:16:30
arbitrary concentration.
 

249
00:16:30 --> 00:16:32
We're going to have to
reference it to some

250
00:16:32 --> 00:16:33
reference concentration.
 

251
00:16:33 --> 00:16:37
Just like we referenced
everything to one bar before,

252
00:16:37 --> 00:16:39
as our standard pressure.
 

253
00:16:39 --> 00:16:41
And we're going to take,
usually you take one mole

254
00:16:41 --> 00:16:45
per liter, or one gram per
liter, or one whatever.

255
00:16:45 --> 00:16:47
One as your reference
concentration.

256
00:16:47 --> 00:16:49
And the reference concentration
is going to disappear

257
00:16:49 --> 00:16:50
from the equation.
 

258
00:16:50 --> 00:16:52
It's just like the
reference pressure.

259
00:16:52 --> 00:16:54
Disappear from the equation.
 

260
00:16:54 --> 00:16:58
So we're going to reference
this to some standard

261
00:16:58 --> 00:17:01
state chemical potential.
 

262
00:17:01 --> 00:17:06
Where the naught refers now to
the standard concentration.

263
00:17:06 --> 00:17:12
And instead of having RT
log p, now we're going

264
00:17:12 --> 00:17:16
to have RT log cA.
 

265
00:17:16 --> 00:17:19
It's kind of like considering
the molecules in the solution

266
00:17:19 --> 00:17:26
to act like an ideal gas.
 

267
00:17:26 --> 00:17:31
Knowing fully well that behind,
that underneath the cA, is this

268
00:17:31 --> 00:17:33
reference concentration of one.
 

269
00:17:33 --> 00:17:36
One whatever is your
favorite units.

270
00:17:36 --> 00:17:39
Now, it's a little bit
more complicated than

271
00:17:39 --> 00:17:41
for the ideal gas.
 

272
00:17:41 --> 00:17:48
Because your solution may
contain other things than your

273
00:17:48 --> 00:17:50
reactants and your products.
 

274
00:17:50 --> 00:17:52
Especially if you're
doing biology.

275
00:17:52 --> 00:17:54
It could be a buffer.
 

276
00:17:54 --> 00:17:56
It could be a buffer,
it could have salt.

277
00:17:56 --> 00:18:00
It could have a pH that's not
equal to seven, whatever.

278
00:18:00 --> 00:18:04
And so this reference, chemical
potential, now needs to be

279
00:18:04 --> 00:18:13
referenced to a particular
pH or salt concentration.

280
00:18:13 --> 00:18:20
Or whatever the properties
of your solvent are.

281
00:18:20 --> 00:18:23
Or your solution are.
 

282
00:18:23 --> 00:18:25
And that's the big difference.
 

283
00:18:25 --> 00:18:28
In an ideal gas, it's reference
to vacuum, basically.

284
00:18:28 --> 00:18:30
There's nothing there.
 

285
00:18:30 --> 00:18:32
In here, in solution you have
all these molecules of solvent,

286
00:18:32 --> 00:18:39
molecules of salt, molecules of
acid, or whatever, that are

287
00:18:39 --> 00:18:42
going to be around
to buffer the pH.

288
00:18:42 --> 00:18:44
And that's going to
change what the chemical

289
00:18:44 --> 00:18:46
potential of a species is.
 

290
00:18:46 --> 00:18:50
And if I change the pH and I've
got a molecule that I'm

291
00:18:50 --> 00:18:54
interested in, it may not have
an acidic moiety on it, but it

292
00:18:54 --> 00:18:59
could still care what
the pH is, slightly.

293
00:18:59 --> 00:19:01
And that would change what
the reference potential

294
00:19:01 --> 00:19:03
is, chemical potential is.
 

295
00:19:03 --> 00:19:05
So this is really
important to remember.

296
00:19:05 --> 00:19:07
And there are textbooks
that are written on how

297
00:19:07 --> 00:19:09
to do this the right way.
 

298
00:19:09 --> 00:19:10
We're not going
to do that here.

299
00:19:10 --> 00:19:13
We're just going to remember
this is, we're going to assume

300
00:19:13 --> 00:19:15
that this is done correctly.
 

301
00:19:15 --> 00:19:20
Once you take that as a given,
that you have a way to have a

302
00:19:20 --> 00:19:23
reference chemical potential at
a properly referenced pH and

303
00:19:23 --> 00:19:25
salt concentration, then you
can go through the same

304
00:19:25 --> 00:19:29
analysis that we went to
for partial pressures.

305
00:19:29 --> 00:19:34
This looks just like the ideal
gas, where the concentration

306
00:19:34 --> 00:19:38
replaces the partial pressure.
 

307
00:19:38 --> 00:19:41
Or the pressure of chemical A.
 

308
00:19:41 --> 00:19:44
And you can go through, then
the same argument, where

309
00:19:44 --> 00:19:52
you take your reaction.
 

310
00:19:52 --> 00:20:00
And you initially have some
delta G for the reactants.

311
00:20:00 --> 00:20:05
And you have some delta
G for the products.

312
00:20:05 --> 00:20:08
And the difference is the delta
G for the reaction, delta G

313
00:20:08 --> 00:20:17
naught for the reaction, and
then on this side here, you

314
00:20:17 --> 00:20:28
have a solution of A, so the
reaction would be nu A times

315
00:20:28 --> 00:20:30
A, which is in a solution.
 

316
00:20:30 --> 00:20:31
Temperature and pressure.
 

317
00:20:31 --> 00:20:37
Plus nu B of reactant B, in
a solution, temperature

318
00:20:37 --> 00:20:38
and pressure.
 

319
00:20:38 --> 00:20:44
Going to a nu C, C solution,
temperature and pressure plus

320
00:20:44 --> 00:20:50
nu D, D in a solution, constant
temperature and pressure.

321
00:20:50 --> 00:20:54
So this is taking a solution of
A, in one container, a solution

322
00:20:54 --> 00:20:56
of B in another the container.
 

323
00:20:56 --> 00:20:57
That's the initial point.
 

324
00:20:57 --> 00:20:57
Mix them together.
 

325
00:20:57 --> 00:20:59
Let them react.
 

326
00:20:59 --> 00:21:01
Then you take the
product, you put them

327
00:21:01 --> 00:21:02
in separate containers.
 

328
00:21:02 --> 00:21:06
And that gets you the stuff
that you have the reaction.

329
00:21:06 --> 00:21:08
So when you mix A and B,
you're going to have the

330
00:21:08 --> 00:21:10
same entropy of mixing.
 

331
00:21:10 --> 00:21:12
You're going to
lower the delta G.

332
00:21:12 --> 00:21:14
Of the solution.
 

333
00:21:14 --> 00:21:16
And you're going to have
the same curve that

334
00:21:16 --> 00:21:18
goes down like this.
 

335
00:21:18 --> 00:21:21
To the mixture of products.
 

336
00:21:21 --> 00:21:24
And just like for the gases,
where we wanted to know what is

337
00:21:24 --> 00:21:28
the bottom of this curve which
gives us the equilibrium,

338
00:21:28 --> 00:21:29
we can do the same thing.
 

339
00:21:29 --> 00:21:34
Exactly the same
thing, for solutions.

340
00:21:34 --> 00:21:37
And so we start out with a
mixture of the A and B in

341
00:21:37 --> 00:21:41
solution and C and D, reactants
and products together.

342
00:21:41 --> 00:21:45
And we let the reaction
proceed a little bit.

343
00:21:45 --> 00:21:53
And we look at the change in
delta G, going from, say this

344
00:21:53 --> 00:21:56
point here through
that point here.

345
00:21:56 --> 00:21:58
This will be delta
G of epsilon.

346
00:21:58 --> 00:22:01
We ask, is this positive,
negative, or zero.

347
00:22:01 --> 00:22:04
And if it's zero, that means
that we're in equilibrium,

348
00:22:04 --> 00:22:07
that we're actually
sitting down here.

349
00:22:07 --> 00:22:09
And that gives us the
equilibrium constant.

350
00:22:09 --> 00:22:13
So, just for the sake of
completeness, let me just

351
00:22:13 --> 00:22:15
write down what we would do.
 

352
00:22:15 --> 00:22:20
We would react it
for a small amount.

353
00:22:20 --> 00:22:24
And then we'd end up with nu C.
 

354
00:22:24 --> 00:22:28
So you would have the chemical
potentials of the products

355
00:22:28 --> 00:22:33
minus the chemical potentials
of the reactants, nu C mu C,

356
00:22:33 --> 00:22:43
plus nu D mu D, minus nu
A mu A, minus nu B mu B.

357
00:22:43 --> 00:22:46
And instead of these chemical
potentials, you would write

358
00:22:46 --> 00:22:51
them in terms of the pure
chemical potentials times

359
00:22:51 --> 00:22:55
their concentrations.
 

360
00:22:55 --> 00:23:01
And then you'd end up with
epsilon times delta G naught

361
00:23:01 --> 00:23:10
of the reaction, plus RT log,
and then the concentrations.

362
00:23:10 --> 00:23:12
And then you write them
in a different way.

363
00:23:12 --> 00:23:14
So if it's moles per liter,
you usually write that

364
00:23:14 --> 00:23:18
with these brackets here.
 

365
00:23:18 --> 00:23:21
That means concentration of A
in moles per liter, I'd say.

366
00:23:21 --> 00:23:29
So C to the nu C power, D to
the nu D power, A to the nu A

367
00:23:29 --> 00:23:37
power, and B to the nu B power,
in these concentrations.

368
00:23:37 --> 00:23:41
Where this ratio of logs
comes from expanding out the

369
00:23:41 --> 00:23:44
chemical potential here.
 

370
00:23:44 --> 00:23:46
And there's the log term here.
 

371
00:23:46 --> 00:23:48
Just like an ideal gas.
 

372
00:23:48 --> 00:23:50
Then at equilibrium, this is
equal to zero, you're at

373
00:23:50 --> 00:23:52
the bottom of that curve.
 

374
00:23:52 --> 00:23:57
And you set these two things
equal to each other.

375
00:23:57 --> 00:24:04
And you get the chemical,
you get your equation

376
00:24:04 --> 00:24:06
that you know well.
 

377
00:24:06 --> 00:24:11
For the equilibrium constant.
 

378
00:24:11 --> 00:24:15
And this time it's not
K sub p, it's just K.

379
00:24:15 --> 00:24:19
And that's what you know from
doing solution equilibrium.

380
00:24:19 --> 00:24:23
And it's just like V.
 

381
00:24:23 --> 00:24:29
So the thing to remember, which
is the slightly more advanced

382
00:24:29 --> 00:24:32
part, which you'll have to
worry about at some point if

383
00:24:32 --> 00:24:39
you stay in some sort of
biochemistry oriented field, is

384
00:24:39 --> 00:24:43
that you've got to reference
your initial solution properly.

385
00:24:43 --> 00:24:48
To get to the right
equilibrium constant.

386
00:24:48 --> 00:24:53
OK, any questions?
 

387
00:24:53 --> 00:24:57
Alright, then now we can do the
temperature dependence of the

388
00:24:57 --> 00:25:02
equilibrium constant
in a general way.

389
00:25:02 --> 00:25:05
Whether it be a gas
or a solution.

390
00:25:05 --> 00:25:06
It doesn't matter.
 

391
00:25:06 --> 00:25:13
It's going to be
the same thing.

392
00:25:13 --> 00:25:18
So the question that we ask now
is, suppose that I change the

393
00:25:18 --> 00:25:22
temperature of my equilibrium,
which way is the equilibrium

394
00:25:22 --> 00:25:23
going to shift?
 

395
00:25:23 --> 00:25:28
And you all know the
answer already, probably.

396
00:25:28 --> 00:25:31
But let's derive it out.
 

397
00:25:31 --> 00:25:33
So we're going to want to
know basically, we want

398
00:25:33 --> 00:25:34
to know what is dKp/dT.
 

399
00:25:34 --> 00:25:38
 


400
00:25:38 --> 00:25:41
How does equilibrium constant,
or dK/dT, if you're doing

401
00:25:41 --> 00:25:45
solution, how does the
equilibrium change

402
00:25:45 --> 00:25:46
with temperature?
 

403
00:25:46 --> 00:25:46
What's the slope?
 

404
00:25:46 --> 00:25:49
Is it positive, negative?
 

405
00:25:49 --> 00:25:51
If we have this, we
can integrate it out.

406
00:25:51 --> 00:25:53
We can do an integral
over temperature.

407
00:25:53 --> 00:25:56
And get an actual change.
 

408
00:25:56 --> 00:25:58
So that's our goal.
 

409
00:25:58 --> 00:26:00
To find how the equilibrium
constant changes

410
00:26:00 --> 00:26:03
with temperature.
 

411
00:26:03 --> 00:26:04
What do we know?
 

412
00:26:04 --> 00:26:11
Well, we know how Kp depends on
temperature, through the Gibbs

413
00:26:11 --> 00:26:14
free energy of the reaction.
 

414
00:26:14 --> 00:26:16
The Gibbs free energy,
delta G naught, has a

415
00:26:16 --> 00:26:18
temperature dependence.
 

416
00:26:18 --> 00:26:19
And then there's an RT
sitting on the bottom.

417
00:26:19 --> 00:26:23
There's another temperature
dependence here.

418
00:26:23 --> 00:26:27
Well, dKp/dT is sort of
like, we could also ask

419
00:26:27 --> 00:26:28
what's d log Kp dT.
 

420
00:26:28 --> 00:26:30
That might be an
easier question.

421
00:26:30 --> 00:26:32
It's basically the
same question.

422
00:26:32 --> 00:26:34
Especially since we have
something which is log K,

423
00:26:34 --> 00:26:35
is equal to something.
 

424
00:26:35 --> 00:26:38
So let's ask this
question instead.

425
00:26:38 --> 00:26:42
Let's ask, what is d log Kp dT?
 

426
00:26:42 --> 00:26:54
Alright, so let's differentiate
both sides. d/dT, d/dT here.

427
00:26:54 --> 00:26:56
Got to use the chain rule now.
 

428
00:26:56 --> 00:26:58
Because we've got temperature
as part of delta

429
00:26:58 --> 00:27:04
G Write it out.
 

430
00:27:04 --> 00:27:06
So let's take the derivative
with respect to the temperature

431
00:27:06 --> 00:27:07
on the bottom first.
 

432
00:27:07 --> 00:27:10
We have delta G naught, which
is a function of temperature,

433
00:27:10 --> 00:27:14
divided by RT squared.
 

434
00:27:14 --> 00:27:16
The minus sign here
disappears when you take the

435
00:27:16 --> 00:27:18
derivative on the bottom.
 

436
00:27:18 --> 00:27:30
Minus one over RT, d/dT
of delta G naught.

437
00:27:30 --> 00:27:36
So this is a derivative
of delta G, where zero

438
00:27:36 --> 00:27:38
means here one bar.
 

439
00:27:38 --> 00:27:39
Fixed at one bar.
 

440
00:27:39 --> 00:27:46
So really, d/dT, with delta G
naught fixed on one bar, is the

441
00:27:46 --> 00:27:50
same thing as the partial
derivative of delta G with

442
00:27:50 --> 00:27:53
respect to temperature,
keeping p is equal to

443
00:27:53 --> 00:27:55
constant at one bar.
 

444
00:27:55 --> 00:28:01
It's the same thing, just
different notation.

445
00:28:01 --> 00:28:05
And we know what this is.
 

446
00:28:05 --> 00:28:11
In terms of other things
that we can find in books,

447
00:28:11 --> 00:28:16
like delta H, or delta S.
 

448
00:28:16 --> 00:28:18
Because we can go to the
fundamental equations.

449
00:28:18 --> 00:28:27
To find out how delta G
depends on temperature.

450
00:28:27 --> 00:28:31
And our goal is to get rid of
delta G, which clearly has a

451
00:28:31 --> 00:28:33
nice temperature dependence
through the entropy term.

452
00:28:33 --> 00:28:36
And to replace delta G
with delta H, if we can.

453
00:28:36 --> 00:28:38
Because delta h is going to
be much less sensitive to

454
00:28:38 --> 00:28:43
temperature and it's, delta H
is going to be over small

455
00:28:43 --> 00:28:46
temperature ranges, is going to
be independent of temperature.

456
00:28:46 --> 00:28:49
And we know the temperature
dependence of delta H,

457
00:28:49 --> 00:28:51
because it's through
the heat capacities.

458
00:28:51 --> 00:28:55
So our goal is to get rid
of this delta G here.

459
00:28:55 --> 00:28:59
And to try to replace it with
delta H, if at all possible.

460
00:28:59 --> 00:29:04
So we go to the fundamental
equation for G, dG is equal

461
00:29:04 --> 00:29:10
to minus S dT plus V dp.
 

462
00:29:10 --> 00:29:17
And sitting right here
is dG/dT at constant p.

463
00:29:17 --> 00:29:19
Which is what we have here.
 

464
00:29:19 --> 00:29:24
So we get rid of this
derivative of G.

465
00:29:24 --> 00:29:27
And replace it with S.
 

466
00:29:27 --> 00:29:36
So now, now we have d log Kp
dT, and I mentioned already

467
00:29:36 --> 00:29:37
that I want to get rid of G.
 

468
00:29:37 --> 00:29:39
Because it has a strong
temperature dependence.

469
00:29:39 --> 00:29:42
And I want to somehow get
H in there, which is not

470
00:29:42 --> 00:29:44
going to have a strong
temperature dependence.

471
00:29:44 --> 00:29:48
And delta G naught, I can write
in terms of H and S, and T.

472
00:29:48 --> 00:29:59
Delta H naught minus T delta S
naught divided by RT squared.

473
00:29:59 --> 00:30:04
And then my derivative here,
I have minus one over RT.

474
00:30:04 --> 00:30:08
Partial of G with respect to
T. p is equal to one bar.

475
00:30:08 --> 00:30:13
Well, that's just delta S. p
is equal to one bar, well,

476
00:30:13 --> 00:30:14
that's just delta S naught.
 

477
00:30:14 --> 00:30:20
Times delta S naught.
 

478
00:30:20 --> 00:30:23
And that's great, because
now there's minus T

479
00:30:23 --> 00:30:24
divided by RT squared.
 

480
00:30:24 --> 00:30:27
That's one over RT.
 

481
00:30:27 --> 00:30:31
And somewhere I've lost a sign.
 

482
00:30:31 --> 00:30:33
And there's my sign that
I lost, right there.

483
00:30:33 --> 00:30:40
This minus sign here. d/dT of
delta G naught is minus S.

484
00:30:40 --> 00:30:43
It actually includes this
minus sign right here.

485
00:30:43 --> 00:30:45
Which is great, because now
things work out, because

486
00:30:45 --> 00:30:49
this becomes a plus sign.
 

487
00:30:49 --> 00:30:54
And this and this cancel out.
 

488
00:30:54 --> 00:31:03
And this becomes delta H
naught over RT squared.

489
00:31:03 --> 00:31:04
Great, so we have what we want.
 

490
00:31:04 --> 00:31:08
We have how the equilibrium
constant depends on temperature

491
00:31:08 --> 00:31:10
in a way which is very clear.
 

492
00:31:10 --> 00:31:13
Where the top part is only
very weakly dependent on

493
00:31:13 --> 00:31:14
temperature, usually.
 

494
00:31:14 --> 00:31:23
And this is called the
van 't Hoff equation.

495
00:31:23 --> 00:31:26
And this will tell us what
happens to equilibrium when

496
00:31:26 --> 00:31:30
we change the temperature.
 

497
00:31:30 --> 00:31:34
So if you want to do a finite
temperature change, now what

498
00:31:34 --> 00:31:52
you need to do is, you
need to integrate.

499
00:31:52 --> 00:31:59
You integrate both sides here.
 

500
00:31:59 --> 00:32:02
From some T1 to T2.
 

501
00:32:02 --> 00:32:09
From T1 to T2, and that tells
you, then, that the log of the

502
00:32:09 --> 00:32:13
equilibrium constant at the new
temperature is equal to the log

503
00:32:13 --> 00:32:19
of the equilibrium constant of
the old temperature, T1, plus

504
00:32:19 --> 00:32:26
the integral from T1 to T2
of delta H over RT delta H

505
00:32:26 --> 00:32:28
naught, over RT squared.
 

506
00:32:28 --> 00:32:31
And this could be slightly
temperature dependent. dT.

507
00:32:31 --> 00:32:34
And this is the integrated
van 't Hoff equation.

508
00:32:34 --> 00:32:39
And if you're going to be
designing a chemical plant

509
00:32:39 --> 00:32:44
where you have high
temperatures and high pressures

510
00:32:44 --> 00:32:48
around, you better use that.
 

511
00:32:48 --> 00:32:50
Because there is some
temperature dependence in

512
00:32:50 --> 00:32:53
delta H, through the heat
capacities of the reactants

513
00:32:53 --> 00:32:54
and the products.
 

514
00:32:54 --> 00:32:57
And that could make the
difference between your plant

515
00:32:57 --> 00:33:00
running nice and smoothly
or your plant exploding.

516
00:33:00 --> 00:33:04
And you don't want to have
exploding plants around.

517
00:33:04 --> 00:33:09
So for heavy-duty uses of
this equation, you've got to

518
00:33:09 --> 00:33:11
do the integral properly.
 

519
00:33:11 --> 00:33:14
But for most normal
applications, like if you're

520
00:33:14 --> 00:33:17
doing biology, where the
temperature changes by a few

521
00:33:17 --> 00:33:19
degrees, like today I have a
little bit of a cold.

522
00:33:19 --> 00:33:22
I don't have a fever, but
I could have a fever.

523
00:33:22 --> 00:33:25
So my biochemistry would
change if I had a fever.

524
00:33:25 --> 00:33:27
The equilibrium constant
of all my reactions would

525
00:33:27 --> 00:33:28
change a little bit.
 

526
00:33:28 --> 00:33:31
It's a small change
in temperature.

527
00:33:31 --> 00:33:34
I'm not going to explode.
 

528
00:33:34 --> 00:33:37
And so you can then take
the approximation.

529
00:33:37 --> 00:33:47
In that case, the delta H
naught, is independent of T.

530
00:33:47 --> 00:33:53
And this is fine over
small temperature ranges.

531
00:33:53 --> 00:33:55
And that's the one that
you're most used to.

532
00:33:55 --> 00:33:58
Is this approximation here,
this approximate van

533
00:33:58 --> 00:34:00
't Hoff equation.
 

534
00:34:00 --> 00:34:02
Which is really fine for most
cases that you're going

535
00:34:02 --> 00:34:05
to be dealing with.
 

536
00:34:05 --> 00:34:10
So then, if that's the case
then you can take your delta H.

537
00:34:10 --> 00:34:12
Ignore the temperature
dependence and take it

538
00:34:12 --> 00:34:14
outside of the integral.
 

539
00:34:14 --> 00:34:17
And now you can do
the integral fine.

540
00:34:17 --> 00:34:21
And then you have an analytic
expression for the change in

541
00:34:21 --> 00:34:23
the equilibrium constant
with temperature.

542
00:34:23 --> 00:34:29
Log Kp at a new temperature,
T2, is log Kp temperature T1.

543
00:34:29 --> 00:34:31
And I'm carrying this little
p around everywhere.

544
00:34:31 --> 00:34:33
But really, it doesn't
have to be there.

545
00:34:33 --> 00:34:35
This could be solution.
 

546
00:34:35 --> 00:34:38
I shouldn't really have written
this for the specific case

547
00:34:38 --> 00:34:40
of partial pressures.
 

548
00:34:40 --> 00:34:43
But it's equally
valid for solutions.

549
00:34:43 --> 00:34:46
Then we have delta
H naught over R.

550
00:34:46 --> 00:34:49
And then we have the integral
from T1 to T2, over

551
00:34:49 --> 00:34:51
one of RT squared.
 

552
00:34:51 --> 00:34:53
And if you do that the
right way, you get T2

553
00:34:53 --> 00:34:57
minus T1 over T1 T2.
 

554
00:34:57 --> 00:35:02
And that gives you
the approximate van

555
00:35:02 --> 00:35:04
't Hoff equation.
 

556
00:35:04 --> 00:35:06
Which is fine.
 

557
00:35:06 --> 00:35:09
And you'll know that it's fine
in problem sets or exam,

558
00:35:09 --> 00:35:11
because we'll say assume that
delta H is temperature - yes.

559
00:35:11 --> 00:35:18
STUDENT: You said
that the [INAUDIBLE]

560
00:35:18 --> 00:35:20
PROFESSOR: For K, if
K is solution K.

561
00:35:20 --> 00:35:23
STUDENT: Right.
 

562
00:35:23 --> 00:35:23
PROFESSOR: Yeah.
 

563
00:35:23 --> 00:35:26
STUDENT: So then, can
you also use Kx?

564
00:35:26 --> 00:35:28
PROFESSOR: Can you use Kx?
 

565
00:35:28 --> 00:35:33
Well, as long as you keep the
pressure, the total pressure,

566
00:35:33 --> 00:35:38
constant, then you should
be able to use Kx.

567
00:35:38 --> 00:35:43
Let me think about this.
 

568
00:35:43 --> 00:35:46
Yeah, here you would have
p, you have a log c so,

569
00:35:46 --> 00:35:47
you can use Kx, fine.
 

570
00:35:47 --> 00:35:50
Because then you would have
log p to the minus delta

571
00:35:50 --> 00:35:53
nu times Kx, log p to
the delta nu minus Kx.

572
00:35:53 --> 00:35:56
And the log of the
multiplication is the

573
00:35:56 --> 00:35:58
sum of the logs.
 

574
00:35:58 --> 00:35:59
And the logs will
just fall out.

575
00:35:59 --> 00:36:01
So it could be any
K that you want.

576
00:36:01 --> 00:36:04
Doesn't matter.
 

577
00:36:04 --> 00:36:06
As long as you keep the
pressure constant.

578
00:36:06 --> 00:36:10
If you change the pressure,
then you're in trouble.

579
00:36:10 --> 00:36:14
So now we can see what
happens when you do

580
00:36:14 --> 00:36:18
change the temperature.
 

581
00:36:18 --> 00:36:25
If I have some equilibrium,
and it's all going to depend

582
00:36:25 --> 00:36:26
on the sign of delta H.
 

583
00:36:26 --> 00:36:33
Whether the reaction is
exothermic or endothermic.

584
00:36:33 --> 00:36:35
And it's the same thing
as Le Chatelier's for

585
00:36:35 --> 00:36:37
pressure, or Lenz's law.
 

586
00:36:37 --> 00:36:40
The system doesn't want to
have change happening.

587
00:36:40 --> 00:36:43
So if you have something
that's, delta H is less than

588
00:36:43 --> 00:36:46
zero, it's exothermic.
 

589
00:36:46 --> 00:36:51
Exothermic, that means that
it's putting out heat. it wants

590
00:36:51 --> 00:36:55
to heat up its environment.
 

591
00:36:55 --> 00:36:59
And if I take temperature and
I raise the temperature, the

592
00:36:59 --> 00:37:02
system's not going to
like that very much.

593
00:37:02 --> 00:37:06
It doesn't want to get hotter,
and going from reactants to

594
00:37:06 --> 00:37:08
products makes things hotter.
 

595
00:37:08 --> 00:37:10
And if you go from products to
reactants, that's the opposite.

596
00:37:10 --> 00:37:13
Go from reactants to products,
that becomes endothermic.

597
00:37:13 --> 00:37:15
It sucks in heat.
 

598
00:37:15 --> 00:37:16
You raise the temperature.
 

599
00:37:16 --> 00:37:19
The system said no, no, no,
I'm happy where I am at

600
00:37:19 --> 00:37:20
my original temperature.
 

601
00:37:20 --> 00:37:22
I'm going to start sucking
in heat, to try to get

602
00:37:22 --> 00:37:24
the temperature down.
 

603
00:37:24 --> 00:37:26
And it's going to try
to make more product.

604
00:37:26 --> 00:37:29
More reactants.
 

605
00:37:29 --> 00:37:39
So, equilibrium K is going to
go down. and the reaction

606
00:37:39 --> 00:37:43
goes towards the reactants.
 

607
00:37:43 --> 00:37:45
And the opposite if you
have something that's

608
00:37:45 --> 00:37:49
endothermic to begin with.
 

609
00:37:49 --> 00:37:51
OK, delta H is positive here.
 

610
00:37:51 --> 00:37:54
Delta H naught is positive.
 

611
00:37:54 --> 00:37:57
You raise the temperature,
delta H naught is

612
00:37:57 --> 00:37:58
positive, T2's bigger.
 

613
00:37:58 --> 00:38:00
This is a positive number.
 

614
00:38:00 --> 00:38:06
K becomes larger at
higher temperature.

615
00:38:06 --> 00:38:08
K goes up.
 

616
00:38:08 --> 00:38:10
The reaction goes to products.
 

617
00:38:10 --> 00:38:15
So if you think of it in terms
of the system, the system

618
00:38:15 --> 00:38:16
is at some temperature.
 

619
00:38:16 --> 00:38:18
You raise the temperature.
 

620
00:38:18 --> 00:38:20
System doesn't like it.
 

621
00:38:20 --> 00:38:22
Says, I want to go back to
my original temperature,

622
00:38:22 --> 00:38:24
what can I do.
 

623
00:38:24 --> 00:38:26
I can try to suck in heat
that you're trying to

624
00:38:26 --> 00:38:28
put in the environment.
 

625
00:38:28 --> 00:38:31
That's great because if
I make more products,

626
00:38:31 --> 00:38:32
that's endothermic.
 

627
00:38:32 --> 00:38:34
And I'm just going to make
more products until I try

628
00:38:34 --> 00:38:35
to lower my temperature.
 

629
00:38:35 --> 00:38:41
So I move the equilibrium
to the products.

630
00:38:41 --> 00:38:56
OK, Le Chatelier
for temperature.

631
00:38:56 --> 00:38:59
Any questions?
 

632
00:38:59 --> 00:39:08
On equilibrium.
 

633
00:39:08 --> 00:39:09
OK, let's do a quick example.
 

634
00:39:09 --> 00:39:13
Because this was the example
that we started out with,

635
00:39:13 --> 00:39:14
talking about, the
Haber process.

636
00:39:14 --> 00:39:21
This important industrial
reaction that started the

637
00:39:21 --> 00:39:23
chemical industry, essentially.
 

638
00:39:23 --> 00:39:27
That uses up 1%, or close to
1%, of the world's energy.

639
00:39:27 --> 00:39:29
If you think about it,
that's an amazing number.

640
00:39:29 --> 00:39:35
1% of all energy produced in
the world goes to one reaction.

641
00:39:35 --> 00:39:37
One industrial reaction.
 

642
00:39:37 --> 00:39:45
Just shows how important it is.
 

643
00:39:45 --> 00:39:47
OK, why does it take
so much energy?

644
00:39:47 --> 00:39:48
We're going to find out.
 

645
00:39:48 --> 00:39:55
We're going to find out why
it takes so much energy to

646
00:39:55 --> 00:39:57
run this here reaction.
 

647
00:39:57 --> 00:40:00
Alright, let's look at
this Haber process.

648
00:40:00 --> 00:40:04
Take some nitrogen gas.
 

649
00:40:04 --> 00:40:11
Plus some hydrogen gas.
 

650
00:40:11 --> 00:40:14
And this is usually done over
catalysts, like an iron oxide

651
00:40:14 --> 00:40:15
catalyst or something.
 

652
00:40:15 --> 00:40:17
To try to speed it up.
 

653
00:40:17 --> 00:40:18
It doesn't change
the thermodynamics.

654
00:40:18 --> 00:40:22
As you know, and you'll hear
again in this class, catalysts

655
00:40:22 --> 00:40:23
just affect the kinetics.
 

656
00:40:23 --> 00:40:25
They don't change
the thermodynamics.

657
00:40:25 --> 00:40:27
So this is usually done
over some catalyst to

658
00:40:27 --> 00:40:29
try to speed things up.
 

659
00:40:29 --> 00:40:30
To make ammonia.
 

660
00:40:30 --> 00:40:35
And ammonia becomes the
feedstock for fertilizers, for

661
00:40:35 --> 00:40:37
almost anything that contains
an amine in it, or

662
00:40:37 --> 00:40:38
a nitrogen in it.
 

663
00:40:38 --> 00:40:42
If you're going to make
proteins or whatever.

664
00:40:42 --> 00:40:45
You've got to have ammonia
somewhere in the process.

665
00:40:45 --> 00:40:48
OK, delta H naught of the
reaction, we're given

666
00:40:48 --> 00:40:50
all these numbers.
 

667
00:40:50 --> 00:40:54
At 298 degrees Kelvin.
 

668
00:40:54 --> 00:40:56
And they're in your notes,
so I'm not going to go

669
00:40:56 --> 00:40:58
through them in detail.
 

670
00:40:58 --> 00:41:00
Delta G naught for
the reaction, we're

671
00:41:00 --> 00:41:01
given that number.
 

672
00:41:01 --> 00:41:06
At 298 degrees Kelvin, that's
minus 16, roughly minus

673
00:41:06 --> 00:41:12
16 kilojoules per mole.
 

674
00:41:12 --> 00:41:16
And we want to know, what is
the equilibrium constant.

675
00:41:16 --> 00:41:17
Room temperature.
 

676
00:41:17 --> 00:41:18
So you know how to
calculate that.

677
00:41:18 --> 00:41:22
Minus RT log Kp, log K
is equal to minus RT.

678
00:41:22 --> 00:41:25
 


679
00:41:25 --> 00:41:28
Minus delta G naught over RT.
 

680
00:41:28 --> 00:41:30
So you put that in there.
 

681
00:41:30 --> 00:41:33
You get Kp is equal to 860.
 

682
00:41:33 --> 00:41:35
A number, no units.
 

683
00:41:35 --> 00:41:36
It's a big number.
 

684
00:41:36 --> 00:41:39
It's a big number, you've
got the products divided

685
00:41:39 --> 00:41:40
by the reactants.
 

686
00:41:40 --> 00:41:42
It means that the
products are favored.

687
00:41:42 --> 00:41:45
This is great.
 

688
00:41:45 --> 00:41:48
What a wonderful reaction.
 

689
00:41:48 --> 00:41:51
Shouldn't take energy to
for us to do that, right?

690
00:41:51 --> 00:41:53
It's a room
temperature reaction.

691
00:41:53 --> 00:41:55
Thermodynamics is great.
 

692
00:41:55 --> 00:41:58
But even over a catalyst,
this is a really, really,

693
00:41:58 --> 00:42:01
really slow reaction.
 

694
00:42:01 --> 00:42:04
We'd still be waiting here for
Mr. Haber to produce his first

695
00:42:04 --> 00:42:08
mole of amine, if you were
doing it, or ammonia if we were

696
00:42:08 --> 00:42:10
doing it at room temperature.
 

697
00:42:10 --> 00:42:13
It's just so slow.
 

698
00:42:13 --> 00:42:16
Thus, not at all practical.
 

699
00:42:16 --> 00:42:18
We're not going to run the
world on room temperature

700
00:42:18 --> 00:42:23
Haber process.
 

701
00:42:23 --> 00:42:27
But it turns out, if you raise
the temperature, kinetics is

702
00:42:27 --> 00:42:29
wonderful in terms of the
temperature dependence.

703
00:42:29 --> 00:42:31
It's exponential.
 

704
00:42:31 --> 00:42:32
Arrhenius rate law.
 

705
00:42:32 --> 00:42:34
Great thing, you raise the
temperature by a little bit.

706
00:42:34 --> 00:42:37
Rates speed up,
things go faster.

707
00:42:37 --> 00:42:39
So if you were to raise the
temperature from 298 degrees

708
00:42:39 --> 00:42:49
Kelvin to 800 degrees
Kelvin, the rate speeds up.

709
00:42:49 --> 00:42:50
You're going to
need some energy.

710
00:42:50 --> 00:42:53
As input here, to feed that.
 

711
00:42:53 --> 00:42:57
Hence the 1% energy use.
 

712
00:42:57 --> 00:42:58
Rate speeds up, that's great.
 

713
00:42:58 --> 00:43:00
Things happen faster.
 

714
00:43:00 --> 00:43:00
It becomes practical.
 

715
00:43:00 --> 00:43:04
But, what happens if you
raise the temperature?

716
00:43:04 --> 00:43:04
Let's see.
 

717
00:43:04 --> 00:43:09
This is an endothermic, or
exothermic, negative sign.

718
00:43:09 --> 00:43:10
And negative sign's exothermic.
 

719
00:43:10 --> 00:43:12
I raise the temperature,
K goes down.

720
00:43:12 --> 00:43:14
I know how to
calculate it here.

721
00:43:14 --> 00:43:17
And if I want to be super
careful, because it's a fairly

722
00:43:17 --> 00:43:20
large temperature range, I can
even use the exact form of

723
00:43:20 --> 00:43:21
the van 't Hoff equation.
 

724
00:43:21 --> 00:43:27
And what I find, if I do that,
and putting the heat capacities

725
00:43:27 --> 00:43:32
for all these gases, I
find that Kp does go down.

726
00:43:32 --> 00:43:34
In fact, it goes
down quite a bit.

727
00:43:34 --> 00:43:39
It becomes 0.0007.
 

728
00:43:39 --> 00:43:40
Two zero's.
 

729
00:43:40 --> 00:43:43
Still really small.
 

730
00:43:43 --> 00:43:45
That's not practical.
 

731
00:43:45 --> 00:43:48
Not practical at all.
 

732
00:43:48 --> 00:43:49
No good.
 

733
00:43:49 --> 00:43:53
We can't run an industrial
plant with this kind of yield.

734
00:43:53 --> 00:43:56
There's just no way
it's going to work.

735
00:43:56 --> 00:43:59
So, you're a chemical engineer.
 

736
00:43:59 --> 00:44:01
Or a chemist, like
Haber and Bosch were.

737
00:44:01 --> 00:44:05
And you're trying, you know
by Le Chatelier, you know

738
00:44:05 --> 00:44:07
that it went in the wrong
direction for you here.

739
00:44:07 --> 00:44:10
And then you look at your
reaction and you say, how many

740
00:44:10 --> 00:44:12
moles of reactants do I have?
 

741
00:44:12 --> 00:44:15
3/2 plus 1/2, that's two
moles of reactants.

742
00:44:15 --> 00:44:18
And I've got one
mole of product.

743
00:44:18 --> 00:44:20
Two moles reactants,
one mole of product.

744
00:44:20 --> 00:44:21
Two moles reactant...
 

745
00:44:21 --> 00:44:25
What happens if I
change the pressure?

746
00:44:25 --> 00:44:27
If I change the pressure, if I
increase the pressure, the

747
00:44:27 --> 00:44:29
system is going to say,
no, I don't want the

748
00:44:29 --> 00:44:31
pressure increased.
 

749
00:44:31 --> 00:44:33
It's going to go to where
there's less moles.

750
00:44:33 --> 00:44:36
And the less moles in
the product area.

751
00:44:36 --> 00:44:37
It's going to go to my product.
 

752
00:44:37 --> 00:44:38
That's great.
 

753
00:44:38 --> 00:44:39
I've got to increase
the pressure.

754
00:44:39 --> 00:44:40
Wonderful.
 

755
00:44:40 --> 00:44:41
Let's start increasing
the pressure.

756
00:44:41 --> 00:44:43
Again, we need some
energy to do that.

757
00:44:43 --> 00:44:46
We're going to go from one
bar to some higher pressure.

758
00:44:46 --> 00:44:48
It's going to make our
lives more complicated.

759
00:44:48 --> 00:44:49
The plant might explode now.
 

760
00:44:49 --> 00:44:50
If the pressure's too high.
 

761
00:44:50 --> 00:44:53
All sorts of problems
going to come into play.

762
00:44:53 --> 00:44:56
But, let's do it.
 

763
00:44:56 --> 00:44:59
Let's increase the pressure.
 

764
00:44:59 --> 00:45:10
So, you increase the pressure
from one bar to 100 bar.

765
00:45:10 --> 00:45:18
And you calculate Kx.
 

766
00:45:18 --> 00:45:20
Which is really what you want.
 

767
00:45:20 --> 00:45:24
So Kx, in this case here,
is equal to p times Kp.

768
00:45:24 --> 00:45:25
Kp doesn't change.
 

769
00:45:25 --> 00:45:27
Kp doesn't care what
the total pressure is.

770
00:45:27 --> 00:45:29
It's Kx that cares.
 

771
00:45:29 --> 00:45:32
At one bar, Kx is equal Kp.
 

772
00:45:32 --> 00:45:35
Kx is p to the minus delta nu.
 

773
00:45:35 --> 00:45:40
Number of moles of products
minus the number of reactants.

774
00:45:40 --> 00:45:47
If I go from one bar to 100
bars, Kx goes from 0.007 to

775
00:45:47 --> 00:45:53
100 times 0.007, which
is equal to 0.7.

776
00:45:53 --> 00:45:55
That's a lot better.
 

777
00:45:55 --> 00:45:58
Kx is the mole fraction
or the, of products

778
00:45:58 --> 00:46:00
divided by reactants.
 

779
00:46:00 --> 00:46:07
And if I go to p is equal
to 300 bars, then Kx goes

780
00:46:07 --> 00:46:11
to 2.1. three times 0.7.
 

781
00:46:11 --> 00:46:12
This is great.
 

782
00:46:12 --> 00:46:14
Now I'm really starting
to make good products.

783
00:46:14 --> 00:46:16
But I've got to go to 300 bar.
 

784
00:46:16 --> 00:46:22
I've got to go to 300 bar,
and 800 degrees Kelvin.

785
00:46:22 --> 00:46:25
That is incredibly
energy-intensive.

786
00:46:25 --> 00:46:27
But it works.
 

787
00:46:27 --> 00:46:29
That's why Haber and
Bosch made this work.

788
00:46:29 --> 00:46:32
And why Germany stayed in the
war longer than after 1916.

789
00:46:32 --> 00:46:35
The first world war.
 

790
00:46:35 --> 00:46:39
Ended in 1918.
 

791
00:46:39 --> 00:46:41
Nobel Prizes.
 

792
00:46:41 --> 00:46:44
Merck, Bayer, all these
german companies.

793
00:46:44 --> 00:46:48
Because they figured how to
do this at high pressure

794
00:46:48 --> 00:46:49
and high temperature.
 

795
00:46:49 --> 00:46:54
Without blowing everything up.
 

796
00:46:54 --> 00:46:55
OK.
 

797
00:46:55 --> 00:47:01
Any questions?
 

798
00:47:01 --> 00:47:02
Alright.
 

799
00:47:02 --> 00:47:07
The last topic is, so far we've
seen equilibria where you had

800
00:47:07 --> 00:47:11
things that were well mixed.
equilibria of ideal gases.

801
00:47:11 --> 00:47:14
Or in solutions, where
your solutes, your solute

802
00:47:14 --> 00:47:17
molecules are mixing around.
 

803
00:47:17 --> 00:47:20
And the entropy of mixing
was really super important.

804
00:47:20 --> 00:47:23
Our curve, our going down for
delta G, was all because

805
00:47:23 --> 00:47:24
of the entropy of mixing.
 

806
00:47:24 --> 00:47:28
Now, suppose that I have
a heterogeneous mixture.

807
00:47:28 --> 00:47:32
I've got some solids or some
pure liquids that are refusing

808
00:47:32 --> 00:47:35
to share their environment.
 

809
00:47:35 --> 00:47:38
And staying as pure materials.
 

810
00:47:38 --> 00:47:43
So, for instance, if I have
a beaker with some solid

811
00:47:43 --> 00:47:45
reactant on the bottom here.
 

812
00:47:45 --> 00:47:55
And the products
are in solution.

813
00:47:55 --> 00:47:57
How do I deal with
that equilibrium?

814
00:47:57 --> 00:48:00
Well, you know the answer, but
let's just do it out again.

815
00:48:00 --> 00:48:04
So we're going to have nu
A moles of A, of solid.

816
00:48:04 --> 00:48:06
Not mixed in the solution.
 

817
00:48:06 --> 00:48:08
Let's say we have
multiple phases here.

818
00:48:08 --> 00:48:10
Nu B moles of B,
which is a gas.

819
00:48:10 --> 00:48:13
Instead of a solution, let's
do a gas phase reaction.

820
00:48:13 --> 00:48:18
Nu C moles of C, which
is a pure liquid.

821
00:48:18 --> 00:48:22
And nu D moles of
D, which is a gas.

822
00:48:22 --> 00:48:23
So these two gases can mix.
 

823
00:48:23 --> 00:48:26
But the pure solid and the
pure liquid can't mix.

824
00:48:26 --> 00:48:28
So let's think again, where
does the equilibrium

825
00:48:28 --> 00:48:29
constant come from?
 

826
00:48:29 --> 00:48:34
It comes from looking at this
delta G of the mixture and

827
00:48:34 --> 00:48:36
letting it react a
little bit more.

828
00:48:36 --> 00:48:39
And taking out these
chemical potentials for the

829
00:48:39 --> 00:48:40
species and the mixture.
 

830
00:48:40 --> 00:48:46
Expanding it out in terms of
log p or log concentration.

831
00:48:46 --> 00:48:48
So we need to have
this delta G in.

832
00:48:48 --> 00:48:51
Let's take epsilon equal
to one, to make our

833
00:48:51 --> 00:48:51
life simpler here.
 

834
00:48:51 --> 00:48:57
So now we have nu C
mu C of the pure.

835
00:48:57 --> 00:48:59
The pure solid.
 

836
00:48:59 --> 00:49:04
Plus nu D mu C of the gas.
 

837
00:49:04 --> 00:49:07
Which is in the mixture.
 

838
00:49:07 --> 00:49:14
Minus nu A mu A of the
pure liquid, minus nu

839
00:49:14 --> 00:49:17
B mu B of the gas.
 

840
00:49:17 --> 00:49:20
Which is in the mixture.
 

841
00:49:20 --> 00:49:24
That's what this delta G is,
when we allow the reaction to

842
00:49:24 --> 00:49:27
proceed for a little bit more.
 

843
00:49:27 --> 00:49:30
We add a little bit of chemical
potentials from the products.

844
00:49:30 --> 00:49:32
Subtract a little bit
of chemical potential

845
00:49:32 --> 00:49:34
from the reactants.
 

846
00:49:34 --> 00:49:36
And then we expand it out
in terms of the standard

847
00:49:36 --> 00:49:39
chemical potentials for
everything being pure.

848
00:49:39 --> 00:49:41
Entropy of mixing
comes in here.

849
00:49:41 --> 00:49:42
Comes in here.
 

850
00:49:42 --> 00:49:46
Delta G of mixing
comes in here.

851
00:49:46 --> 00:49:54
And we end up with something
that looks like nu C mu C

852
00:49:54 --> 00:50:02
naught, plus nu D mu D naught,
minus nu A mu A naught,

853
00:50:02 --> 00:50:06
minus nu B mu B naught.
 

854
00:50:06 --> 00:50:12
Plus RT log, and the only place
where we have these log p's, or

855
00:50:12 --> 00:50:15
log concentration coming in,
is for those species

856
00:50:15 --> 00:50:17
that were not pure.
 

857
00:50:17 --> 00:50:19
And those are only
these two guys here.

858
00:50:19 --> 00:50:22
The ones that are
in the gas phase.

859
00:50:22 --> 00:50:23
D and B.
 

860
00:50:23 --> 00:50:26
So we end up with partial
pressure of D to

861
00:50:26 --> 00:50:27
the nu D power.
 

862
00:50:27 --> 00:50:32
Partial pressure of B,
to the nu B power.

863
00:50:32 --> 00:50:34
The other two species
don't come in there.

864
00:50:34 --> 00:50:36
Because they started
out as pure.

865
00:50:36 --> 00:50:39
There's no mixing going on.
 

866
00:50:39 --> 00:50:44
And there's no expansion
of the log for them.

867
00:50:44 --> 00:50:49
And so now, when we look at
the Q for the reaction,

868
00:50:49 --> 00:50:52
the reaction quotient,
it doesn't contain any

869
00:50:52 --> 00:50:53
of the pure species.
 

870
00:50:53 --> 00:50:56
It only contains those species
that are allowed to mix.

871
00:50:56 --> 00:51:00
Those that are in the gas
phase or in solution.

872
00:51:00 --> 00:51:08
And so K, then, for this
reaction only takes in

873
00:51:08 --> 00:51:11
those products like D.
 

874
00:51:11 --> 00:51:16
Or reactants like B, which
are in the gas phase.

875
00:51:16 --> 00:51:18
The pure solids or pure
liquids don't come in.

876
00:51:18 --> 00:51:19
They come in for
delta G naught.

877
00:51:19 --> 00:51:23
There's delta G naught
sitting right here.

878
00:51:23 --> 00:51:28
Delta G naught for the reaction
is sitting right there.

879
00:51:28 --> 00:51:32
So when you write your log K is
equal to minus delta G naught

880
00:51:32 --> 00:51:36
over RT, the delta G naught
has everything in it.

881
00:51:36 --> 00:51:40
The pure stuff, the solution
stuff, the gas phase stuff.

882
00:51:40 --> 00:51:44
But the K only has the gas
phase and solution stuff.

883
00:51:44 --> 00:51:49
Alright, any questions?
 

884
00:51:49 --> 00:51:49
Good.
 

885
00:51:49 --> 00:51:51
Next time we'll do an example.
 

886
00:51:51 --> 00:51:54
And then we'll go on
phase transitions.

887
00:51:54 --> 00:51:54