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I've hopefully given you a bit of a gut feeling behind where the formula of Gibbs free energy comes from in this video I want to do something a little bit more rigorous and actually I guess you could say derive the formula so to do that let's just study two systems that have the exact same change in entropy and to depict that I'll I'll get out the our handy PV diagram PV diagram we're going to do is when compare two systems one that's this perfect reversible system and one that's irreversible or spontaneous system so and they're both going to start here they're going to go from this point on the PV diagram to this point right here and before we go into exactly what they're doing I think it's a good review of just kind of talking about what an irreversible process and what a reversible process really is and they're reversible process is this theoretical thing where you have no friction where you're always so close to equilibrium that you can always go backwards you're you're you're always you can kind of view that the reactions never really proceeding forward or backwards although obviously it is moving so it's a very it actually doesn't really exist in nature but it's a useful theoretical construct so let me draw a little so this is a I'll get the good old Pistons out so let's say that this isn't I'll do it twice so that's my irreversible piston and this is my I let that'll be my reversible it's my irreversible and let me draw them okay let me label them so this right here is going to be reversible and this right here is going to be irreversible irreversible or it could also be considered spontaneous spontaneous using the key word that's going to matter for Gibbs free energy but all spontaneous reactions are irreversible so they're going to go from this state to this state so in our reversible world we have this little list little the cap to our piston we have our gas in here exerting some pressure and we'd have as a bunch of pebbles we have a bunch of pebbles out here and we slowly remove the pebbles one by one and as we remove the pebbles our piston or this remove this movable ceiling up here will move upwards it'll move upwards so as we have the pressure will push up but as we move up we'll have lower and lower pressure because the gases will bounce into this surface less and will have a higher higher volume so as we move each infinitesimally small grain of sand and we're going to do it super slow that we're always infinitely close to equilibrium we're going to move from this state to this state and even better let's view that this is kind of the first stage of our Carnot cycle let's say in either case we're on top of a reservoir so in either case both of these systems are on top of this kind of infinite reservoir that has a temperature t1 and what that does is it kept keeps the temperature constant so we're going to travel along an isotherm because normally if we would remove these things in the DD if we just allowed to occur adiabatic Li we would actually lose temperature we would actually lose average kinetic energy as work is done but in this case we have this reservoir so heat is going to be transferred to my system heat is going to be transferred so let me call that Q R right if this didn't if the reservoir wasn't here if the reservoir wasn't here our temperature would go down but since the temp since we do have the reservoir where it will constantly be transferring heat so when we've seen this this is just the first stage of the Carnot cycle will move along an isotherm like this this is there is a reversible case and the only reason why we can even draw the state at every point to here is because reversible processes they're quasi static they're always infinitely close to equilibrium and when we say reversible we're also saying there's no friction between this little piston and the cylinder that if we put a grain of sand back it will go exactly to where it was before no energy was lost because there was no heat of friction there so this is what the graph of the reversible process would look like now what does the graph of the irreversible well actually know I won't draw the graph but let's talk about what the irreversible process is going to be like so it's going to look similar it's going to look like that it's going to have its gases there but for the sake of an argument to get from that state to that see instead of moving the pebbles one by one let's say I have these big blocks I have these big blocks and when I remove one of these big blocks I go from that state to that state but it obviously throws all hell breaks loose so I'm not really defined in this in-between state but I definitely go from that state to that state once I go back to equilibrium now the other key thing and the irreversible process is that and every real process in our world is irreversible is that you're going to have friction here you're going to have friction here as this moves up it's going to rub against the side of the container and generate heat of friction heat of friction so let me call that heat of friction so let me ask you a question if in this case Q sub R had to be added to the system to maintain its temperature does what's going to be the Q what's going to be the Q sub irreversible here how much has to be added to this system in order to keep it at the constant temperature t-1 will have to be more or less than what was added to the reversible system well this guy as this piston as this piston moves up he's generating some of his own heat so if this was an adiabatic process he wouldn't lose as much temperature as this guy would so he's been good he's going to need less heat from the reservoir in order to maintain his temperature at t1 in order to get to this point on the isotherm remember this irreversible process we don't know what happens over here you might be traveling on some crazy path in fact we can't even define the path because it goes out of equilibrium so it's going to be some crazy thing but we know it pops back on the PV diagram right there but because it's generating some of its own heat from the friction it's going to need less heat from the reservoir so let me write that down the heat absorbed by the reversible process is going to be greater than the heat absorbed by the irreversible process and that's because the irreversible process is generating friction fair enough now what is the change in entropy for both systems well they both started here and they both ended here and entropy is a state variable so the change in entropy the change in entropy for the reversal all process is going to be equal to the change in entropy is going to be equal to the change in entropy for thee is going to be equal to the change in entropy for the irreversible process they're both going from there to there now what is the total and obviously the entropy has changed we're going from one state to another state and the entropy well I won't I won't go too much into it but let's ask another question what is the total change in entropy of the universe for the reversible process so the for the universe so that's going to be our universe here is the reservoir and our system so let me write here reversible and I want to run out of space let me see I'll do it in different colors so this is the reversible process so change in entropy of the universe is equal to the change in entropy of our reversible process plus the change in entropy of well I already use R for reversible so you know let's call it the reservoir force three letters are the same so let me call it of the of our environment right and in the reversible process the change in entropy of our versatile process is the heat added for a reversible process and we can use this definition because it is a reversible process it's that over t1 and then what's the change in entropy of our environment well it's giving away Q sub R right so it has its its heat absorbed as minus Q sub R and of course it's at a constant temperature it is a it is a heat reservoir so it's t1 so it equals zero equals zero interesting so that will agree I mean actually I should take a little side note here that the change in entropy of the universe for a reversible process is zero and actually that should make a lot of intuitive sense because the whole point of a reversible process is you could go in this direction or you could go in that direction it's always so close to equilibrium you can move in either direction and if the entropy was greater than zero in one direction it would have to be less than zero in the other direction so it wouldn't be able to go in the other direction by the second law of thermodynamics so it actually makes sense that the entropy the of the universe the change in entropy of the universe not just of the system when a reversible process occurs is zero let's see if we can relate that to the irreversible process so if I wanted to figure out the change in entropy of the irreversible process what's the change in entropy of the irreversible process and then let me say let me subtract from that the heat the heat that was taken away from the reservoir from the irreversible process Q IR and of course all of this over t1 what is this going to be equal to as I guess in relation to zero remember this is an irreversible spontaneous process well this value we've it's going the irreversible process is starting over here and going there so it's change in Delta s is the exact same thing as that it's a change in Delta s it's the same as the reversible process so these two things are equivalent now I just told you that since there's some heat of friction this guy has to take in less less heat from the reservoir than this guy so if this and why I wrote it here let me clean this thing up a little bit I wrote it right here Q sub I are the heat that the irreversible process has to take from the reservoir because generating its own heat from friction is less than the heat from the reversible process so this number right here is less than this number here or this you could view it this way this number here was equal to this so this number here is going to be less than this so this has to be this has to be greater than zero for the reversible process of sorry for the irreversible irreversible spontaneous spontaneous process now let's just do a little bit of mathematics so this is the heat that was essentially given to the irreversible system and it's a minus here because this is the terms kind of taken away from the actual reservoir so let's just do a little bit of of let's multiply all sides of this equation by T 1 and we get T 1 times Delta s of the irreversible process - queue of the irreversible process is greater than zero now I well let's just so what what is this how can we say this if we if we wanted to but let's actually let's just multiply both sides of this by negative one and remember this is true for any irreversible spontaneous process if you multiply both sides of this by negative one you get this the heat added to the irreversible process minus T times Delta s of the irreversible process is going to be less than zero that this is true for any irreversible spontaneous process and at this point this should look reasonably familiar to you when we wrote our Gibbs free energy formula we said change in G change in G is equal to Delta H minus T times Delta s and we said that if this is less than zero then we have a spontaneous process but these two and this all makes sense because these two are equivalent to each other what's the only difference here here we wrote heat added to the system here we wrote change in enthalpy and I've done three or four videos right now where I say that change in enthalpy is equal to the heat added to a system as long as as long as we're dealing with a constant pressure system constant pressure so if we get this result just by comparing a reversible to reversible system and this is true for all irreversible spontaneous processes and then if we assume that we're dealing with a constant pressure system so you can kind of forget a little bit of what I just drew but because we know that this is true and we assume constant pressure then we get to this and then we know that if something is spontaneous then this right here must be less than zero so hopefully you found that a little bit interesting actually I'll do one more one more point here this kind of gels with the idea that any spontaneous with the second law of thermodynamics that tells us that for any spontaneous process Delta s is going to be greater than or equal or well for any spontaneous process is going to be greater than zero because although this right here it isn't the formal of entropy 4/4 because we're not dealing with a reversible process here you can kind of think of it that way and so it at least on an intuitive level it kind of gets you there that where we have Delta s greater than zero and I won't fixate on that too much because I want to what I did earlier in the video is more rigorous than what I'm heading to right now so hopefully that gives you a sense that gives you a sense of where you can get to the Gibbs free energy formula and how it drives spontaneous reactions from just our basic understanding of what reversible and irreversible processes are and how they relate to entropy and heat exchange and enthalpy

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