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 this is spontaneous I took great pains to make sure that we use the proper definition of entropy that every time that we said okay a change in entropy from here to here is a change 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 this definition and so you might have been asking hey Sal you know there's a much there's a much simpler definition or proof in my textbook and I know if it's in your textbook but since some of the ones that I've seen some of the web pages I've looked at where they kind of 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 looks the second law of thermodynamics second law tells us that for any spontaneous process 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 this Delta s of the universe is greater than zero and that means that Delta s of the system plus Delta 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 they'll write Delta s of the environment is equal to the write Delta s of the environment is equal to the heat the heat or let me say the heat absorbed by the environment the heat absorbed by the environment divided by the temperature of the environment and let's just say for the 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 in our 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 is if you know this heat transfer is reversible is reversible and we're using when we're doing it in general terms we don't know whether it's reversible in fact if we're saying to begin with that the reaction is spontaneous it that means by definition that it's irreversible it's irreversible so this is actually an irreversible transfer of heat which is not the definition of entropy there the thermodynamic definition of entropy is a very delicate one you have to make sure that it's at a reversible reaction and 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 then you're saying okay the change the change in entropy of the environment is equal to oh the amount of heat the entropy 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 okay look this is 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 then let's say look the heat of the environment 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 there's going to be the minus of each other so the argument will go well you know this thing I can rewrite I can rewrite this equation is Delta 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 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 Sarris is less than zero when you multiply both sides by 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 it's a see this this shows that if you have a negative Gibbs free energy this is or change in Gibbs free energy then you're spontaneous but all of that was predicated on the idea that this could be re-written like this but it can't be rewritten like that because this is not a reversible process we're starting from the assumption that 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 a irreversible system that goes from here to here is the same as the change in entropy of an irreversible system of a of as an irreversible system that goes from there to there or let me say that it's it the change in entropy of a reversible system from there 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 reversible system because it's generating its own 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 to you know too particular kind of you know trying to really pick up the details but I think it's an important point to make because so much of what we talk about especially you know in thermodynamics is our definition of entropy it's very important that we use the correct one and we don't take what would I would argue are incorrect shortcuts because this is not the definition of entropy right there