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PROFESSOR NELSON: So, over the last few lectures we've worked
and struggled so formulate the second and third laws of
thermodynamics in addition to the first.
Last time we reach the third law which is telling us that
we can't quite get to zero degrees Kelvin, but that as
the temperature approaches zero degrees Kelvin, the
absolute entropy of a pure substance in perfect
crystalline form is zero.
And what that corresponds to, if you recall, is the idea
that in a perfect crystal at zero degrees Kelvin then you
have no disorder at all.
You have a perfectly ordered system.
And in that case, the entropy is absolutely zero.
And the bigger lesson from that is that entropy, unlike
energy u or enthalpy H, we could define an absolutely
number for it.
The zero of it wasn't arbitrary.
Unlike the case for energy like you've seen in lots and
lots of disciplines, where you can arbitrarily set the zero
in a way that makes it convenient for you.
Entropy really is not like that.
There is an absolute zero of entropy, and that's really
what we learn.
So now, now that we've got all three of the major laws of
thermodynamics, what I want to start in on is a discussion of
what happens spontaneously.
We've seen that in just one specialized case so far, but
we should in a more general way be able to tell when is a
system at equilibrium, where there's no net change taking
place, and when is it still undergoing spontaneous change
towards some other state, presumably toward an
equilibrium state.
How do we tell?
So that's the topic that I'd like to address today.
So, that's the big question, right?
So let's say that we've got our stuff in some state.
A, whatever it is.
And there's some other possible state,
B, whatever it is.
And maybe if it some well-defined
temperature and pressure.
What do we do to tell whether that change will happen
spontaneously?
Do we calculate, you know, delta S, delta u, delta H?
What tells us whether or not the change happens?
Certainly in principle we know how to calculate this and
other stuff for a change in state of this sort, for lots
of changes of state.
But calculating it alone doesn't necessarily tell us
whether or not it will just happened of its own accord.
And that's the issue that we'd like to be able to address.
Now, we have addressed this in some cases.
So, for example, we know that if we take, you know, gas A
and gas B with a barrier between them, and we remove
the barrier, they're going to mix.
And we saw that and went through it somewhat carefully
and saw that if the system is isolated, for that case we do
have a criterion that tells us whether change happens
spontaneously.
Namely, it's delta S is greater than zero.
That tells us whether the change is spontaneous.
And we saw that in fact in this case delta S of mixing,
we calculated it, saw that it is positive.
So, clearly, if we remove this barrier mixing takes place,
and obviously you know that that happens from lots of
experience.
In general, the second law gave us the Clausius
inequality for spontaneous change.
Namely dS is greater than dq over T. I suppose we could
specify the surroundings temperature.
We saw that in general dS is greater than or equal to dq
over T. if it's a reversible process then the equality
holds, but if it's irreversible, which means it
happens spontaneously, then dS is greater than this.
Just to illustrate the kind of issues that we're up against
here, let me just consider a few different chemical
reactions, all of which happen spontaneously.
So, I just want to write a few examples down with a few
values for delta u or delta H or delta S, and see whether we
can get any clues from what we see.
So here are some spontaneous chemical changes.
Here's one.
If we take hydrogen peroxide in the liquid state, it can
break down to form water and oxygen.
If we look at the thermodynamic quantities, the
enthalpy and the entropy of the reaction, what we find is
delta H is minus 209 kiloJoules and delta S is plus
132 joules per Kelvin.
So it seems like there's a favorable change in entropy
going this way.
That is, you've got lower energy on the right and also
higher entropy.
Higher entropy basically because you're forming
molecules of gas where there weren't any before, and
there's more disorder in the gas phase than in the liquid.
That is, the gas phase molecules have
more freedom to roam.
Okay, that's one example.
Here's another.
Let's just take hydrogen and nitrogen in the gas phase and
form ammonia.
Well, here we get, we find that delta H is negative 92
kiloJoules.
Delta S is negative 198 joules per Kelvin.
This one also happens spontaneously.
One thing that makes it pretty clear is that certainly delta
S or the sign of it alone is not
dictating the outcome here.
All right, here's a third example.
Let's take salt, solid, and dissolve it in a bunch of
liquid water.
Hopefully you've got experience saying that this
happens, of course it does.
If we measure the thermodynamics, we discover
that delta H is 4 kiloJoules, plus 4 kiloJoules.
Delta S is 45 joules per Kelvin.
So now we have a different sign for delta H and it still
happens spontaneously.
So clearly, we've got signs and magnitudes of delta H and
delta S, and if we wanted to put delta u there, similar
things would happen.
They're all over the map.
And yet, these things are all spontaneous processes.
And I didn't specify the conditions, but if we were to
do this under ordinary chemical conditions of some,
you'd say room temperature and pressure, right, they all
happen spontaneously.
OK, clearly we'd be much better off if we had some
systematic quantitative way to tell whether something would
happen spontaneously.
In other words, we need criteria for equilibrium under
more general conditions than the ones that we've dealt with
so far, than the one set of conditions that we've dealt
with so far, which is isolated system.
Most chemical changes, most physical changes don't happen
in isolated systems.
So let's start by writing down our definition of equilibrium.
It's very simple.
The equilibrium state is the one, and it's just one, in
which there are no spontaneous changes that can take place to
any other state.
Now that's under whatever constraints there are.
There's a box around it.
The temperature or the pressure are
fixed, what have you.
But the point is, no spontaneous changes can occur
to any other state.
So for example, when we remove the barrier and the gases mix,
you know it's over.
Once the gases are mixed, there's not going to be any
further net change in the system.
It's at equilibrium, under the new condition, that is with
the barrier removed.
OK, so now let's try to formulate how to describe the
equilibrium state and what dictates spontaneity.
So what we're going to do is consider the
first and second laws.
So our first law, du is dq plus dw.
And our second law, dS is greater than dq over the
temperature of surroundings for a change that happens
spontaneously.
And now, I want to combine these two, which I
of course can do.
I can substitute dq from this expression in here.
I also want to assume for our present purposes that there's
only pressure volume work going on, which is to say I
want to put p dV in here minus p dV for dw.
So for combining for p V work, what we see then is du has to
be less than T surroundings dS minus p external dV.
Or we can rewrite this as du plus external pressure dV
minus T surroundings dS is less than zero.
Now this is fundamentally important, and as you know
that means that it warrants the exalted distinction of
being put up in colored chalk.
And, in fact, since we're going to reuse this again and
again during today's lecture, I'm going to put it over here
and leave it sacrosanct for our further use. du plus p
external dV minus T surroundings dS
is less than zero.
This Is our condition for spontaneous change.
Now this is a really quite useful expression.
For one thing what we have here are all functions of
state and parameters that we can control like temperature
and pressure.
So that's a big help.
And equilibrium happens when there isn't any possible
change of state that would satisfy this.
In other words, you've got your system in some state.
You know it's in some state A, there are some
other states around.
Here is what we calculate to tell whether it happens
spontaneously.
So we now have a real usable criterion to help guide our
understanding of whether things happen by themselves of
their own accord or not.
Now, this is still a little bit cumbersome, in part
because of the variables involved, including S. That
is, most processes that we're concerned with, they'll happen
with something held constant like pressure or temperature
or maybe volume.
So this isn't the most useful form that we can have, but
what we'll see shortly is that from this, we can then derive
further criteria for essentially any set of
variables or any set of external constraints, like
constant temperature or pressure or volume and so
forth that we might set.
And so that's what I now want to do.
So I just want to use that again and again, starting from
that, for various different sorts of conditions and derive
the criterion for equilibrium in each set of conditions.
So first, let's start with the one that we already know, and
make sure that it still works, starting from here, mainly our
isolated system.
So remember what that means?
It means no heat, no work.
Delta V is zero.
Delta u is zero.
So looking at this, du is zero. dV is zero.
So all that's left is negative T dS is less than zero.
In other words, T surrounding dS has to be greater than
zero, and of course
temperature is always positive.
So dS for u and V fixed is greater than zero.
All right, so that's sounds right.
That's what we saw before.
When we have an isolated system, the criterion that
determines whether something happens spontaneously is the
entropy has to increase.
Now, what this means too is if we imagine a bunch of
different states, and this is the entropy of them, so this
could be any sort of variable but the point is there are a
bunch of possible states around, whichever one has the
maximum entropy, that's the equilibrium state.
In other words, you know we've got the two gases on either
side of that partition.
We remove the partition, and they mix.
Well, the equilibrium state is the one with the gases
completely mixed.
Of course there are lots of states that would have maybe
local pockets of one substance in excess and another
substance in excess somewhere else.
In other words, there would be lots of states nearby to the
equilibrium state.
That is, the chain, they could they could be approached with
very little change from the equilibrium state.
They aren't the equilibrium state.
The entropy in all of those states will be lower than the
entropy of the fully mixed state.
So the point is, once you're at equilibrium none of the
other states, they're accessible, the system could
rearrange itself to form them, but there is no accessible
state that has higher entropy than the equilibrium state.
OK, that's our familiar isolated system.
Now let's try moving to unfamiliar territory and
extending what we know.
So, let's try constant entropy and volume.
And the motivation for choosing a pair like that is
easy to see, if we look at our condition for spontaneous
change or general condition.
Well, entropy and volume constant means dV and dS are
equal to zero.
What does that say?
It means du is less than zero.
That's our condition.
So we immediately get du at constant S and V
is less than zero.
That's our condition for spontaneous change.
In other words, if we don't have to worry about entropy or
volume equilibrium is achieved when energy is at a minimum.
Now this is what you learn in elementary physics and in
mechanics, right.
You're not worried about entropy.
So, you know, if you've got a hill or valley and there's a
cart on wheels, it's going to go down to the bottom.
The spontaneous change lowers the potential
energy in that case.
So this is a simple condition that's very familiar.
Now, the reason this condition always holds in ordinary
mechanics is because you're never, in that case, concerned
with a huge statistical population of particles where
the disorder among them is an issue.
We're not worrying then about the fact that, well like in
the case of gas molecules mixing, the macroscopic state
of the whole thing, all those molecules, how many different
microscopic configurations are there?
Remember, I mentioned then we'll go further later on into
this, that entropy can be related to
the extent of disorder.
That is, how many different possible configurations of all
those molecules there would be for a particular state.
The reason the entropy of the mixed gases is the highest is
because that has the most possible configurations.
If you start segregating the gases, there are fewer
possible configurations that the whole system can be in
because you're forcing a particular set of
circumstances.
When you don't have to worry about criteria like that,
ordinary, mechanical energy rules supreme, and that's
dictating where equilibrium lies.
But of course, most chemical and biological systems aren't
that simple precisely because you have to worry about many
particles and their statistics and the way they might order
or disorder.
So, and of course, you know, keeping entropy as a fixed
variable for a system like that is extremely cumbersome.
As soon as you allow anything to mix, like you might if you
want to do any chemistry, entropy changes.
If you change the temperature entropy changes and so on.
So let's go on.
Let's consider a few other examples.
Let's hang on for a little while longer to a set of
conditions where we will maintain constant entropy,
namely constant entropy and pressure.
So, the dS term is zero, but the other two are not.
So, du plus p dV is less than zero.
I can write this as d(u + pV) less than zero.
Normally I couldn't do that because this term would have p
dV plus V dp, but we've specified the pressure is
constant, so the dp part is zero.
And this is a quantity that you know, right?
What's u plus pV?
STUDENT: dH.
PROFESSOR NELSON: dH less than zero, criterion for
spontaneous change.
In the case where S and p are held constant.
Now, once again, like I illustrated for entropy, and I
could have done the same for energy here, you know, if we
again look at a bunch of different states, and look at
their enthalpy, well, like before, they'll be invariably
lots of possible states.
And now, what is this saying, the equilibrium state is the
one with the lowest possible enthalpy.
In the case here, that I just illustrated with the little
cart going down the valley, would be exactly the same with
regular energy, the equilibrium state is one of
lowest energy, right.
And of course there are lots of nearby states.
The cart could be a little ways up the hill, and in this
case, it's enthalpy, but again, there would be lots of
accessible states.
But if the system is in equilibrium, none of those
states has lower enthalpy.
It's already in the lowest enthalpy state.
That is the equilibrium state.
OK, well, now, let's get to the big ones.
That is, in real life, the variables that you'd normally
control aren't some combination of entropy and
these variables, but really their temperature, volume and
pressure, any couple of those, might be what you'd really
have under experimental control.
So now let's go to them.
Let's control T and V. So, all right, so now
we're getting serious.
All right, well, there's our equilibrium criterion.
We're still going back to it.
It still holds.
So we've now got the dV part equal to zero.
So what this says is that du minus T dS is less than zero,
and we can combine those to say d(u - TS)
is less than zero.
And again, just like before, we can do that although this
normally would say this has T dS and also minus S dT.
T is fixed.
So that part is zero.
So this is really the equivalent of this.
So when we did this here, we conveniently found that that
quantity u plus pV is something we know and love,
and we're familiar with it.
It's our enthalpy H. So we could write that criterion as
dH less than zero.
So here, let's combine these to define a new quantity.
It obviously has importance because what it's going to say
is that that's the quantity that defines equilibrium, that
tells us about equilibrium, under the very important
practical constraints of having fixed
temperature and volume.
Realistically, the more likely constraints
than either of those.
So let's -- we'll go to a brand new color, define
A as u minus TS.
it's called the Helmholtz free energy.
OK, and then our criterion for equilibrium under these
conditions is dA, V and T equal to the temperature of
the surroundings, is less than zero.
OK, and once again, you know if we wanted to look at a
bunch of states that could be accessed, well, we would find
lots of states near by, in character to
the equilibrium state.
The one that is at equilibrium, there is only one
macroscopic state at equilibrium.
It has the lowest A.
In some sense, that's one reason to associate this as a
kind of energy, just like mechanical energy u or
enthalpy H, it's the minimum free energy state that is the
equilibrium state under the relevant conditions.
Now, let's take the step to the biggest set of
conditions of all.
What is it when you run a chemical reaction under
ordinary circumstances, what's constant?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR NELSON: A little louder.
STUDENT: Pressure and temperature.
PROFESSOR NELSON: Pressure and temperature, right.
You're running, you're shaking a beaker up here at room
temperature.
So let's look at that set of conditions.
All right, there it is.
This is the condition for really the lion's share of
chemistry, biology, and other kinds of changes we'll be
concerned with.
So, there's our condition for equilibrium.
We don't get to set any of them to zero, right?
So, okay and we can handle that. du plus p dV minus T dS
is less than zero, but we do get to simplify in writing
this as d(u + pV - TS) is less than zero, and just like we've
seen before, yes, this has p dV and V dp, but the dp is
zero because we're at constant pressure.
This has minus T dS minus S dT, but the dT part is zero
because we're at constant temperature.
So the result is we can combine all of these as a
single differential, and just like we've seen before, what
that suggests is that we define another new quantity
given by this expression.
And that is the last one we're going to describe.
And that is G, u plus pV minus TS.
The Gibbs free energy.
Notice, we could also write, let's rewrite that.
G is u plus the pV minus TS, but u plus pV is H. So we also
can write this as H minus TS and u minus TS is what we just
defined a minute ago as A. So we can also write
this as A plus pV.
And the main thing of crucial importance is what, by
defining this in the way we have, what that's saying is
that dG at constant p and T is less than zero.
There's our condition for equilibrium at constant
temperature and pressure.
Boy, is that going to be important for the whole rest
of the course.
So, and of course, I hardly need to emphasize further, but
we could do the exact same consideration that we have for
H and A, there's G. There's our equilibrium state.
It's the state that has the lowest Gibbs free energy.
All these things though are incredibly
practical, useful criteria.
This is only defined in terms of state functions.
And just like we saw before for the case of entropy in an
isolated system, now we have something we can calculate.
It's a state function, so we're at constant temperature
and pressure, and now we want to consider some chemical
change or a phase transition or you name it.
Does it happen of its own accord?
Well now we know what needs to be calculated in order to
determine that.
So this one is so uniquely pervasive, let's just really
explicitly write it all out.
For constant pressure and temperature delta G is less
than zero, means A going to B is, all right, let's consider
some process state A and state B. If delta G is less than
zero, it happens spontaneously.
If delta G equals zero, then we're already in equilibrium.
And if delta G is greater than zero, then it goes
spontaneously in the other direction.
Any questions about any of this?
Let me just give a couple of examples.
If we go back to any of those chemical reactions that I
wrote on the board before, right, well certainly we can
calculate what delta G would be for each one of them.
Because we know how to calculate all the parts of it.
It's state functions, it's composed of state functions
that we know how to calculate.
So we could tell.
When delta G is zero, you know, it doesn't mean that
you've got all of one side, all reactants and zero
products or all products and zero reactants.
There is some mixture of them.
What this will tell us is what mixture.
You know the stuff is in there in equilibrium, you know the
hydrogen and nitrogen that will form ammonia.
And in the end, when it's at equilibrium, and you look and
you'd make a measurement, right, you could do
spectroscopy.
You could easily see how much of each thing is there.
It doesn't go all the way to absolutely 100 percent
ammonia, zero hydrogen zero nitrogen if they were mixed
together with the right ratios.
Doesn't happen.
There would be some of the reactants
and some of the products.
In the biochemical reactions that are taking place in your
body, there is equilibrium between a whole myriad of
reactants and products, and thank heavens that gets
maintained.
So that's what this will guide us through, and of course
that's incredibly, incredibly important.
Here's another thing that's worth thinking about.
There's a balance here between ordinary energy or enthalpy
and entropy.
Energy means, you know, chemical reactions happen, and
you end up with something that might be exothermic, that is,
the products are more stable then the reactants.
You burn methane, and it combines with oxygen to form
water, to form CO2.
And if you work out the energetics as we've gone with
thermochemistry, you discover there's a huge negative delta
H. In other words, the bonds are much stronger.
CO2 is really a stable molecule.
Methane, there are certainly some solid bonds there, but
breaking those to form CO2 and water, well it's worth it,
right, energetically.
Still, the actual equilibrium depends on entropy also, not
only on the energy.
And that's why, when I put up those three different
reactions, and we saw the signs could vary.
It's because there's a balance between the two.
Energy may be favoring reaction in one direction,
toward let's say products that have lower energy.
But at the same time, entropy is going to be favoring
whichever side has higher entropy, has more disorder,
and there's a balance that's achieved.
And that's why all those reactions, first of all, in
some sense what I put up was kind of a trivial statement in
actual fact saying they all happen spontaneously, because
I didn't specify what we were starting with exactly, what
concentrations we were starting with.
Even something quite unfavorable might happen at
least a little bit spontaneously.
You'll have equilibrium.
In those cases, though, you'd have quite a reasonable
equilibrium, spontaneously, that is there would be a lot
of reaction that went if you simply started under practical
conditions and let it go.
Even though the signs of the enthalpy changed, and the
signs of the entropy changed because it's a combination of
the two that matters.
Here's a really simple example.
Mixing of oil and water.
You know from experience if you've ever mixed them to make
salad dressing, they don't mix too well.
And you may know that if you heat them up,
they mix much better.
Why?
You know, we've done a bunch of thermochemistry, and we've
kind of seen that the energy of mixing, your energetics
don't change too much as a function of temperature.
What's changing?
Why does it mix better when you warm it up?
But, you know, looking at our definition of Gibbs free
energy, here it is, right, or here.
Let me just say, actually if you calculated delta S for the
mixing as a function of temperature, it doesn't change
all that much.
You know, the amount of disorder upon mixing is not
really sensitive to temperature.
What does change though?
T, and the entropy is weighted by the temperature, so the
entropy matters more and more the hotter it gets.
And that's consistent with other things
that we've seen, right?
Remember the whole thing about the perfect crystal at zero
degrees Kelvin has zero entropy.
It's completely ordered.
Entropy doesn't matter anymore.
It'll go to the lowest energy state.
Raise the temperature, and now entropy plays a bigger role.
So the point is, this balance between energy that you could
think of as say bond energies in chemical reactions, and
entropy that you can think of in terms of disorder, how many
different possible combinations or configurations
of something wrong, will dictate where
the equilibrium lies.
And knowing now how to calculate these free energies
especially the Helmholtz and the Gibbs free energies,
that's what's going to guide us in really calculating
quantitatively, OK, where will equilibrium lie.
And before long, we'll start in on discussing chemical
equilibrium, does deriving where they lie phase
equilibrium?
Does stuff change phase to go from liquid to solid and so
forth, right?
And where does that happen, at what temperature and pressure
and so forth.
And it's always going to come down to calculating the
appropriate free energy, and how it changes in the process.
So this is going to be a guide for us for essentially all
that we're going to do in the rest of the term.