A look at a seductive but wrong Gibbs spontaneity proof

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.