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# A look at a seductive but wrong Gibbs spontaneity proof

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In the video that I just did,
where I try to more rigorously prove the Gibbs free energy
relation, and that if this relation is less than zero
then this is spontaneous. I took great pains to make sure
that we use the proper definition of entropy. That every time that we said,
OK, a change in entropy from here to here is the heat
absorbed by a reversible process divided by the
temperature at which it was absorbed. And the change in entropy of
the environment is the opposite of that and, of course,
that is equal to zero. And I was very careful to
use this definition. And so you might have been
asking, hey Sal, there's a much simpler definition or
proof in my textbook. And I don't if it's in your
textbook, but it's in some of the ones that I've seen
and in some of the web pages I've looked at. Where they use a much simpler
argument that gets us eventually to this Gibbs
free energy relation. And I thought I would go over
it because as far as I can tell, it's incorrect. And what the argument tends to
go, is it says, look, the second law of thermodynamics
tells us, that for any spontaneous process, that delta
S is greater than zero. I agree with that completely
right now. And in order for delta S, and
that's delta S of the universe, is greater
than zero. And that means that delta S of
the system plus dealt S of the environment is going to
be greater than zero. and then this is the step that
you'll often see in a lot of textbooks and a lot of websites
that I disagree with. They'll say delta S of the
environment is equal to the heat or, let me say, the heat
absorbed by the environment, divided by the temperature
of the environment. And let's just say for
simplicity that everything here it's in some type of
temperature equilibrium. And it tends to be when we're
dealing with stuff in our chemistry sets in our
labs, whatever else. But the the reason why I
disagree with this step right here, that you see in a lot of
textbooks, is that this is not saying anything about the
reversibility of the reaction. You can only use this
thermodynamic definition of entropy if you know this heat
transfer is reversible. And when we're doing it in
general terms, we don't know whether it's reversal. In fact, if we're saying to
begin with that the reaction is spontaneous, that means
by definition that it's irreversible. So this is actually an
irreversible transfer of heat, which is not the definition
of entropy. The thermodynamic definition
of entropy is a very delicate one. You have to make sure that it
is a reversible reaction. Obviously, in a lot of first
year chemistry classes this doesn't matter. You're going to get the
question right. In fact, the question might be
dependent upon you making this incorrect assumption. So I don't want to confuse
you too much. But I want to show you that
this is not a right assumption. Because if you're assuming
something is spontaneous, and you're saying, OK, the change in
entropy in the environment is equal to the amount of heat
the environment absorbs, divided by T-- this is wrong
because this is not an irreversible reaction. But let's just see how this
argument tends to proceed. So they'll say, OK, this is
equal to delta S of our system plus the change in heat of our
environment, divided by the temperature of our
environment. They'll call this for the
environment, and that of course has to be
equal to zero. And they'll say, look, the
heat absorbed by the environment is equal to the
minus of the heat absorbed by the system. Right? It's either the system is
giving energy to the environment or the environment
is giving energy or heat to the system. So they are just going to be
the minus of each other. So the argument would go, well
this thing I can rewrite. This equation is the change in
entropy of the system, instead of writing a plus Q of the
environment here, I could write a minus Q of the system
over T is greater than zero. And then they multiply both
sides of this equation by T and you get T delta S of the
system minus the heat absorbed by the system is greater
than zero. You multiply both sides of this
by negative one and you get the heat absorbed by the
system minus the temperature times the change in entropy of
the system is greater than zero-- I'm sorry, is less than
zero when you multiply both sides by a negative, you
switch the signs. And then if you assume constant
pressure, this is the change in enthalpy
of the system. So you get the change in
enthalpy minus the temperature times delta S of the system
is less than zero. And they say, see this shows
that if you have a negative Gibbs free energy or change in
Gibbs free energy, then you're spontaneous. But all of that was predicated
on the idea that this could be rewritten like this. But it can't be rewritten like
that, because this is not a reversible process. We're starting from the
assumption that this is a spontaneous irreversible
process. And so you can't make this
substitution here. And that's why in the earlier
video I was very careful not to make that substitution. I was very careful to say, oh,
you know, the change in entropy of an irreversible
system that goes from here to here is the same as the change
in entropy as an irreversible system that goes from
there to there. Or let me say this
differently. The change in entropy of a
reversible system from there to there is the same as
an irreversible system from there to there. Although you don't know what
goes on in between for the irreversible. And so that's why I made
this comparison. This thing, and this thing are
the same, but then we compared the heat absorbed by an
irreversible system, and we showed that it's less than the
heat absorbed by a reversible system because it's generating
its own friction. And from that, we got this
relation, which we were able to then go and get the Gibbs
free energy relation. So, anyway, I don't want to make
a video that's too geeky or too particular, or
kind of trying to really pick at the details. But I think it's an important
point to make because so much of what we talk about,
especially in thermodynamics, is our definition of entropy. And it's very important we use
the correct one, and we don't take what I would argue are
incorrect shortcuts, because this is not the definition
of entropy right there.