Lecture 14. The Gibbs Energy.
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Penner, Reginald
 In Chemistry 131C, students will study how to calculate macroscopic chemical properties of systems. This course will build on the microscopic understanding (Chemical Physics) to reinforce and expand your understanding of the basic thermo-chemistry concepts from General Chemistry (Physical Chemistry.) We then go on to study how chemical reaction rates are measured and calculated from molecular properties. Topics covered include: Energy, entropy, and the thermodynamic potentials; Chemical equilibrium; and Chemical kinetics. 
 So at UC Irvine, we don't boast about our athletic prowess very often. That's partly because we don't have a football team. What we do have is probably the best men's volleyball program in the country. Everybody know that? We sort of flies under the radar. We've won two national championships in the last five years on this campus. National NCAA championships where we beat every other team in the country. Pretty much means we're the best men's volleyball team, collegiate men's volleyball team in the world because let's face it, the U.S. dominates in this sport. So these guys beat Penn State last night and tomorrow they're going to beat USC. And then we're going to have our third national championship on this campus. At least that's what I'm hoping happens. So we can be very proud. We should talk more about these guys. So sadly we're going to start talking about the Gibbs energy today. Now this is really a Chapter 16 topic and I'm not sure we'll come back to anything else in Chapter 15. I'm going to think about it over the weekend but I'm guessing we're probably done with Chapter 15. We did it in sort of two lectures. That material is important but it's sort of abstract to us as chemists because where does the Carnot cycle fit in in chemistry? How does it affect our daily lives as chemists? I can't answer that question. It's a tenuous connection. The Gibbs energy for chemists that's where the rubber meets the road in thermodynamics is the Gibbs energy. It's going to allow us to tell whether a chemical reaction happens or not, whether we're at equilibrium or not. This is what thermodynamics is going to do for us as chemists. So we've really been building up to this thing called the Gibbs energy. In today's lecture it's extremely boring but important. We're going to have some of those. Lots of symbols today. But what we want to do is we want to figure out where this Gibbs energy comes from. It was really never spelled out for me in the physical chemistry class I had back in 1981. It was just here's the Gibbs energy, here's what it does, here's why it's important but no one ever really derived it but there is a nice derivation of the Gibbs energy. So that's what we're going to try and do today. Now we've only got 25 minutes so we may not get all the way through this lecture. If we don't we'll just continue it on Monday so don't worry about that. All of these guys, by the way, Clausius, not an Austrian. I saw that on a few forums. German, that's what that flag is right there. Only one of these guys won the Nobel Prize. This sounds like a quiz question for next week. K, B, C, D, which one of these guys won the Nobel Prize? Anybody know? They all did enough to win it but only one of them won it. So there's really a mystery here. Why did only one guy win the Nobel Prize? First of all, which guy was it? No, Gibbs and Lewis are good guesses because those guys should have won the darn thing. It was like the least famous guy on this slide. Nernst. Nernst won the Nobel Prize in 1920. Anybody know why? Not why Nernst won but why these other guys didn't win. I mean, objectively, they did more than Nernst and they didn't win it. They didn't survive until 1944. The answer is, you know, the way the Nobel Prize works, you do something that's really important and then it takes a while. There's an induction period while the rest of the world tries to understand what you've done and then once they understand it, they have to appreciate what you've done and then once they appreciate it, they've got to get organized and give you the award. And that process typically takes 20 years. When did the Nobel Prizes start? Anybody know? How do you know that? You just memorized that? That's an amazing statistic to have on your fingertips. That's the answer. He died in 1896 and he, in his will and final will and testament, he endowed the Nobel Prize and the first one was given 1901 and they don't give it to dead guys and most of these guys were dead. 1901 is here, in principle, Boltzmann and Lewis could have won it but by the time everyone understood what they had done, they were dead. Nernst hung on long enough to win the darn thing. What's that? Oh, yeah, this bar is not quite right. Yeah, you're right. There's no good reason why he didn't win the Nobel Prize. It's one of those things where everybody who wins this award deserves it and then there's another 50 guys who should have won it who never did. It's sort of the way this works, I think. That's the reason. The award started here and then it takes a while and they only got their act together for a minute. You're right. Lewis should have won the darn thing. So this is Gibbs. He's our hero in this whole thing because he's the only American we're going to be talking about pretty much. C.N. Lewis was an American too. G.N. Lewis, rather. But we're not going to, you know, we talk about him a lot when you talk about bonding. Lewis dot structures, that's the guy, right? But we mentioned him at the beginning of the class. Probably you might not remember this but he got his, he got the first PhD in engineering in the whole country. The very first one that was given at Yale was in 1863, right before, or actually right during the Civil War, right? And then, you know, one of the sad realizations you come to if you're an experimentalist is all the guys whose names are attached to equations and chemistry are all theoreticians pretty much. All right? So Lewis was a theoretician. It's not the people in the lab mixing the chemicals together and plotting the data. It's the theoreticians who sit at their desks and with their pencils and figure stuff out. He was a theoretician. Like Einstein, Dirac, Schrodinger, all right, these guys never did an experiment outside of school. All right? They just looked at other people's experiments and figured them out. So he had an appointment at Yale. He graduated from Yale. He got appointed there without salary. All right? And this was common at that time. If you hadn't proven yourself as a scholar, it was not uncommon to get hired by a university. You teach, you do all this work, but they wouldn't pay you anything until you started to publish, which was hard to do in those days. Now, you write a paper. It's in your computer. In 10 minutes, you can submit it to a journal electronically. All right? Boom. It's a link on your browser opens. There's a way to upload it to the website. It's gone. You're submitted. In those days, everything had to be handwritten manuscripts in triplicate delivered by horseback to the journal office, which was somewhere on the east coast. This process could take years to publish a paper. All right? It's a lot different than it was now. So it wasn't until he got an offer from Hopkins in Baltimore that Yale, which is up in New Haven, Connecticut, said, okay, well, if Hopkins is going to offer you $3,000 a year, we'll give you $2,000, and that was enough for him to stay. And then right after that, he wrote his treaties on the equilibrium of heterogeneous substance. This basically spelled out everything that we think about as thermodynamics these days. He spelled it out in one. Now these days, one of the standard strategies that we all have as academics is you take some body of data that you've got, and you try to carve it up into the smallest publishable unit. So whatever you've done, you can get the most papers out of. Right? Because that's how you get promoted in this business. He took everything that he had done and put it into one 300-page paper and published it. It's like the opposite of what we would do today. He's even got a Facebook page. If you write a 300-page paper, you too can have a Facebook page for your paper. So here's his grave. He's buried on the Yale campus because the Yale campus is one of the coolest cemeteries that you've ever seen on it. It's right on the campus. It's called the Grove Street Cemetery. Here it is. Here's Grove Street. This is the Yale campus in New Haven. New Haven is a dump. Except for Yale. You've got this beautiful university in this town that is really not so beautiful anymore. Getting better, I guess. You can, anybody who wants to go and look at the cemetery and the reason you might want to do that is because Gibbs is buried there. Next to his dad, here's a map. The cemetery has its own website. You can see the name of everybody who was buried in the cemetery and what they did. It's all people from like the 1800s. It's pretty cool. Noah Webster. Yes, that Webster. Here's the gate that you go into. It's really, really cool. Huge grave markers like that. If you're ever in New Haven, Grove Street Cemetery. What we've been doing is learning the underpinnings of thermodynamics. That's what statistical mechanics is. It's the underpinnings of thermodynamics. As chemists, what we really care about is equilibrium. We haven't said the word yet until today. We haven't used the word equilibrium, I don't think, or maybe once or twice. We need to get to this because we're ready to be done talking about this whole subject, aren't we? Let's get to the meat of the issue and then we can put it behind us. Now, oops, my ear, I'll slip. That's supposed to be up there. What we're going to try and do today is derive the Gibbs function. There's going to be a whole bunch of symbols, but I really think it's worth understanding this. Here we go. The system is surrounding. We talked about this already. There's open systems, closed systems, and isolated systems. Everybody remember that? Isolated systems, no matter or energy, can change. Close systems, energy only, open both energy and matter. Most of what we've said so far pertains to isolated systems because they're easier to understand. They're a lot simpler. For isolated systems, we've already derived that the entropy increases, has to increase. For any non-reversible process, if the process is reversible, I'm going to put an equal sign underneath this, greater than or equal to zero. Remember that? For an isolated system, entropy increases during a spontaneous process. That's what that means. We'll actually derive this again today. If you're in either one of these two categories, and we virtually always are, if we're actually doing an experiment, we're not in an isolated system. We're either in an open system or a closed system. We have to consider both the system and the surroundings in this non-inequality. In other words, it's the total entropy of the universe that's got to get bigger. The universe includes the system that you care about and everything else. There's two terms in this total entropy. There's the system that you care about and everything outside of it because your system is in communication with the outside world either by exchanging matter or by exchanging energy. If the system is in isolated entropy, does the system do anything to it? Yes. Now, this is the only thing that matters in terms of understanding where something is spontaneous or not. We're going to derive this Gibbs function, but the Gibbs function is going to be derived from this. It's all about entropy. If you remember nothing else about this lecture, you need to remember that spontaneity of a chemical process is about entropy. The Gibbs energy is just a way to parameterize in a convenient way the entropy for us. That's all we're doing with the Gibbs function. We're parameterizing the entropy in a convenient way. If this is true, then I can move that guy over to the right-hand side and put a minus sign on him, right? We've already said we've got an expression for the entropy in terms of the heat that's transferred and the temperature. I can make a substitution for the surroundings. Minus dS here is going to be dQ. It's plus dQ because there's normally a minus sign that connects this minus dQ over T with dS. If that's minus dS, that's going to be plus dQ over T surroundings. Now, Q is a conserved quantity. In other words, if plus Q enters the system, minus Q is removed from the surroundings. If Q goes into the system, it had to come from the surroundings. If it's a plus for the system, it's a minus for the surroundings. What we've already beaten into the ground is that dU, where you use the internal energy, that equals dW plus dQ. That means dQ is dU minus dW. I can just plug this in for dQ and this equation right here. dU minus dW. I've got this equation right here. We haven't done anything fancy yet. Everything's very simple. All I've done here is assume that the only work that's being done is pressure volume work. I've got PDV here. Now, let's multiply both sides by T surroundings. When we do that, we're going to get T surroundings on that side. We're going to lose it from the right side. Here we go. All I did is multiply by D surroundings. Now it's on the left-hand side. It's gone from over here. This doesn't look like our conservation of entropy equations anymore, does it? Starting to morph. We're still talking exclusively about entropy here. We've just made substitutions from symbols. This equation is going to keep coming up. We're going to call it the pink equation. What we're going to do is we're going to find convenient expressions for dU, plug them into the pink equation, and we're going to find the Gibbs equation. We're going to find the Gibbs energy rather, the Helmholtz energy. That's what we're going to do in the next 10 minutes. We're going to keep coming back to the pink equation. At equilibrium, for an isolated system, dV equals dU equals zero. In other words, the internal energy of the system doesn't change and the volume doesn't change. If I prevent the volume from changing, and I prevent dU from changing, this term is zero and this term is zero, in other words, dS of the system is going to have to be greater than or equal to zero for an isolated system. If I just take the pink equation and I say, if my system is isolated, this term is zero and this term is zero, if I divide by T surroundings, dS system is going to be greater than or equal to zero. We've derived that now for an isolated system from this pink equation, which came from this entropy argument. The total entropy of the system and the surroundings has to be greater than or equal to zero for any chemical process if it's spontaneous. Now I'm going to stop writing cis every time I'm talking about the system. If you see a big letter with no subscript, assume it means the system. I'm talking about the surroundings, I'll keep using that. Now this is a general expression for any system moving towards a new equilibrium. That's what the non-equality means. It means that the system is in flux towards equilibrium. I'm going to explain in detail exactly how we think about that. At equilibrium, the external pressure and the system pressure and the external temperature and the system temperature are equal to one another. In other words, the temperature is the same outside and inside the system. The pressure is the same outside and inside the system. It's intuitively obvious that this would have to be true if we're talking about equilibrium. All this means is the pressure is in the same direction. At equilibrium, the pink equation becomes the yellow equation. Only thing that's different is the equal sign. That's what the inequality means that we're evolving towards equilibrium. When we get there, we get an equal sign. We'll come back to this equation. This is the so-called master equation of thermodynamics, but the pink equation is actually more useful. We'll keep coming back to the pink equation. We won't continue to mention the yellow equation. The pink equation again addresses many processes of interest. For example, the volume and entropy are held constant. If the volume and entropy are held constant, volume and entropy, if I hold that constant, then dv is zero. If I hold the entropy constant, then ds is zero, and that means that du has to be less than zero. What does that mean? It means that it's u that is minimized as we evolve from a non-equilibrium state to an equilibrium state. It's u that is minimized if and only if entropy and volume are held constant. Do we normally hold entropy and volume constant when we do chemistry? I don't know how you would. Volume, yes, you can do it, but entropy, difficult. It's not super useful for us as chemists to use the internal energy as a marker for whether we're at equilibrium or not because the internal energy is only going to be a minimum. When volume and entropy are held constant, we can't hold entropy constant very easily. We make note of this and we move on. In other words, the spontaneous process occurring at constant volume and entropy will minimize the internal energy. This is true, but not terribly useful. Last week we talked about the enthalpy. Maybe that's more useful. The enthalpy is u plus pv, that's just a definition. If I take the total derivative of h, that's going to be du plus vdp plus pdv, and I can just solve for u and plug it into the pink equation. That's going to be our standard strategy for the next six minutes. We'll plug it into the pink equation, rearrange these variables, and what do we do? Plug it into the pink equation. What the pink equation tells us, now what I do, I've got this whole thing now that I'm going to plug into there. There it is. The first thing that we notice is that these two terms will cancel if the pressures are equal. Just imagine that we're approaching the equilibrium state, and they are equal. They won't necessarily be equal until that's true, but when we're close to equilibrium, this term and this term will cancel. I hope you'll agree. Then, if pressure and entropy are now held constant, we held volume and entropy constant earlier, now we're going to hold pressure and entropy constant. That is not more useful to us, by the way. If we do that, entropy goes to zero, pressure goes to zero, these guys cancel with one another, so we're left with d. The enthalpy is the variable that we're going to care about if we can control and maintain constant the entropy and the pressure. Is that a useful thing for us to know as chemists? It's a useful thing to know, but as a practical matter, we're not going to be constantly doing this. Enthalpy is not going to tell us when we're at equilibrium most of the time. We can't satisfy this requirement of maintaining the entropy constant. Now temperature actually is a variable that we frequently hold constant as chemists. If we consider dt equal to zero and dv equal to zero, those are requirements that we are sometimes going to be able to satisfy in the laboratory. We can do an experiment at constant temperature and we can do an experiment at constant volume. In principle, that's actually going to be quasi useful for us to understand whether we're at equilibrium under those conditions. The thermodynamic variable we're going to care about here is something called the Helmholtz energy. All we're going to do here is write a definition for it. Helmholtz energy is called A, A is equal to u minus Ts. That's just how it's defined. That equation doesn't really mean anything to us. We didn't derive it from any place. We just defined the Helmholtz energy. That's what it is. Then what we do is write the derivative of that equation. D A is du minus d Ts, which is minus Tds minus Sdt. Solve for du. What are we going to do? Plug it into the pink equation is the right answer. Here's the pink equation again. We put this guy in for du and there it is. Right there. Now what we're talking about, that's going to cancel that without thinking too hard. If we assume we're close to equilibrium and these two temperatures are going to be equal to one another, that's the system temperature, that's the surroundings. If we further assume that dt is zero and dv is zero, that goes away and dv. That guy goes away. So dA, the Helmholtz energy ends up being the thing that we're going to care about. If the Helmholtz energy is getting lower, that should be spontaneous. If the Helmholtz energy is getting higher, that should be non-spontaneous. For any system that we care about. So if we're able to maintain temperature and volume constant, A is going to be our go-to variable. Now the way that you do that is by using one of these guys. In case you're wondering. This is a par bomb. It's universally called a par bomb even though there's like five companies that make these things but par makes the best ones. All this is a stainless steel container into which there's a glass jacket that goes inside here and these are gas inlets and outlets and you see these two tubes right there? Those are pressure, those are over pressure valves. They're thin aluminum membranes and if the pressure of this thing goes above 3000 ATM, those blow. It sounds like a gunshot when they go off. You're heating something in here, this needle goes all the way up to here and when this thing is about to blow up, those two valves prevent that from happening but when they go off by golly, it's spectacular. But I think you'll agree this thing can maintain constant volume no matter what the pressure is doing as long as the pressure stays below 3000 ATM, boom. That's how you do constant volume and constant temperature. Now most of the time we're not going to use this guy. So we have to still talk about the Gibbs energy but we'll do that on Monday.
 So at UC Irvine, we don't boast about our athletic prowess very often.  That's partly because we don't have a football team.  What we do have is probably the best men's volleyball program in the country.  Everybody know that?  We sort of flies under the radar.  We've won two national championships in the last five years on this campus.  National NCAA championships where we beat every other team in the country.  Pretty much means we're the best men's volleyball team, collegiate men's volleyball team in  the world because let's face it, the U.S. dominates in this sport.  So these guys beat Penn State last night and tomorrow they're going to beat USC.  And then we're going to have our third national championship on this campus.  At least that's what I'm hoping happens.  So we can be very proud.  We should talk more about these guys.  So sadly we're going to start talking about the Gibbs energy today.  Now this is really a Chapter 16 topic and I'm not sure we'll come back to anything  else in Chapter 15.  I'm going to think about it over the weekend but I'm guessing we're probably done with  Chapter 15.  We did it in sort of two lectures.  That material is important but it's sort of abstract to us as chemists because where  does the Carnot cycle fit in in chemistry?  How does it affect our daily lives as chemists?  I can't answer that question.  It's a tenuous connection.  The Gibbs energy for chemists that's where the rubber meets the road in thermodynamics  is the Gibbs energy.  It's going to allow us to tell whether a chemical reaction happens or not, whether we're at equilibrium  or not.  This is what thermodynamics is going to do for us as chemists.  So we've really been building up to this thing called the Gibbs energy.  In today's lecture it's extremely boring but important.  We're going to have some of those.  Lots of symbols today.  But what we want to do is we want to figure out where this Gibbs energy comes from.  It was really never spelled out for me in the physical chemistry class I had back in 1981.  It was just here's the Gibbs energy, here's what it does, here's why it's important but  no one ever really derived it but there is a nice derivation of the Gibbs energy.  So that's what we're going to try and do today.  Now we've only got 25 minutes so we may not get all the way through this lecture.  If we don't we'll just continue it on Monday so don't worry about that.  All of these guys, by the way, Clausius, not an Austrian.  I saw that on a few forums.  German, that's what that flag is right there.  Only one of these guys won the Nobel Prize.  This sounds like a quiz question for next week.  K, B, C, D, which one of these guys won the Nobel Prize?  Anybody know?  They all did enough to win it but only one of them won it.  So there's really a mystery here.  Why did only one guy win the Nobel Prize?  First of all, which guy was it?  No, Gibbs and Lewis are good guesses because those guys should have won the darn thing.  It was like the least famous guy on this slide.  Nernst.  Nernst won the Nobel Prize in 1920.  Anybody know why?  Not why Nernst won but why these other guys didn't win.  I mean, objectively, they did more than Nernst and they didn't win it.  They didn't survive until 1944.  The answer is, you know, the way the Nobel Prize works, you do something that's really  important and then it takes a while.  There's an induction period while the rest of the world tries to understand what you've  done and then once they understand it, they have to appreciate what you've done and then  once they appreciate it, they've got to get organized and give you the award.  And that process typically takes 20 years.  When did the Nobel Prizes start?  Anybody know?  How do you know that?  You just memorized that?  That's an amazing statistic to have on your fingertips.  That's the answer.  He died in 1896 and he, in his will and final will and testament, he endowed the Nobel Prize  and the first one was given 1901 and they don't give it to dead guys and most of these guys  were dead.  1901 is here, in principle, Boltzmann and Lewis could have won it but by the time everyone  understood what they had done, they were dead.  Nernst hung on long enough to win the darn thing.  What's that?  Oh, yeah, this bar is not quite right.  Yeah, you're right.  There's no good reason why he didn't win the Nobel Prize.  It's one of those things where everybody who wins this award deserves it and then there's  another 50 guys who should have won it who never did.  It's sort of the way this works, I think.  That's the reason.  The award started here and then it takes a while and they only got their act together  for a minute.  You're right.  Lewis should have won the darn thing.  So this is Gibbs.  He's our hero in this whole thing because he's the only American we're going to be talking  about pretty much.  C.N. Lewis was an American too.  G.N. Lewis, rather.  But we're not going to, you know, we talk about him a lot when you talk about bonding.  Lewis dot structures, that's the guy, right?  But we mentioned him at the beginning of the class.  Probably you might not remember this but he got his, he got the first PhD in engineering  in the whole country.  The very first one that was given at Yale was in 1863, right before, or actually right  during the Civil War, right?  And then, you know, one of the sad realizations you come to if you're an experimentalist is  all the guys whose names are attached to equations and chemistry are all theoreticians pretty  much.  All right?  So Lewis was a theoretician.  It's not the people in the lab mixing the chemicals together and plotting the data.  It's the theoreticians who sit at their desks and with their pencils and figure stuff out.  He was a theoretician.  Like Einstein, Dirac, Schrodinger, all right, these guys never did an experiment outside  of school.  All right?  They just looked at other people's experiments and figured them out.  So he had an appointment at Yale.  He graduated from Yale.  He got appointed there without salary.  All right?  And this was common at that time.  If you hadn't proven yourself as a scholar, it was not uncommon to get hired by a university.  You teach, you do all this work, but they wouldn't pay you anything until you started  to publish, which was hard to do in those days.  Now, you write a paper.  It's in your computer.  In 10 minutes, you can submit it to a journal electronically.  All right?  Boom.  It's a link on your browser opens.  There's a way to upload it to the website.  It's gone.  You're submitted.  In those days, everything had to be handwritten manuscripts in triplicate delivered by horseback  to the journal office, which was somewhere on the east coast.  This process could take years to publish a paper.  All right?  It's a lot different than it was now.  So it wasn't until he got an offer from Hopkins in Baltimore that Yale, which is up in New  Haven, Connecticut, said, okay, well, if Hopkins is going to offer you $3,000 a year, we'll  give you $2,000, and that was enough for him to stay.  And then right after that, he wrote his treaties on the equilibrium of heterogeneous substance.  This basically spelled out everything that we think about as thermodynamics these days.  He spelled it out in one.  Now these days, one of the standard strategies that we all have as academics is you take some  body of data that you've got, and you try to carve it up into the smallest publishable  unit.  So whatever you've done, you can get the most papers out of.  Right?  Because that's how you get promoted in this business.  He took everything that he had done and put it into one 300-page paper and published it.  It's like the opposite of what we would do today.  He's even got a Facebook page.  If you write a 300-page paper, you too can have a Facebook page for your paper.  So here's his grave.  He's buried on the Yale campus because the Yale campus is one of the coolest cemeteries  that you've ever seen on it.  It's right on the campus.  It's called the Grove Street Cemetery.  Here it is.  Here's Grove Street.  This is the Yale campus in New Haven.  New Haven is a dump.  Except for Yale.  You've got this beautiful university in this town that is really not so beautiful anymore.  Getting better, I guess.  You can, anybody who wants to go and look at the cemetery and the reason you might want  to do that is because Gibbs is buried there.  Next to his dad, here's a map.  The cemetery has its own website.  You can see the name of everybody who was buried in the cemetery and what they did.  It's all people from like the 1800s.  It's pretty cool.  Noah Webster.  Yes, that Webster.  Here's the gate that you go into.  It's really, really cool.  Huge grave markers like that.  If you're ever in New Haven, Grove Street Cemetery.  What we've been doing is learning the underpinnings of thermodynamics.  That's what statistical mechanics is.  It's the underpinnings of thermodynamics.  As chemists, what we really care about is equilibrium.  We haven't said the word yet until today.  We haven't used the word equilibrium, I don't think, or maybe once or twice.  We need to get to this because we're ready to be done talking about this whole subject,  aren't we?  Let's get to the meat of the issue and then we can put it behind us.  Now, oops, my ear, I'll slip.  That's supposed to be up there.  What we're going to try and do today is derive the Gibbs function.  There's going to be a whole bunch of symbols, but I really think it's worth understanding  this.  Here we go.  The system is surrounding.  We talked about this already.  There's open systems, closed systems, and isolated systems.  Everybody remember that?  Isolated systems, no matter or energy, can change.  Close systems, energy only, open both energy and matter.  Most of what we've said so far pertains to isolated systems because they're easier to  understand.  They're a lot simpler.  For isolated systems, we've already derived that the entropy increases, has to increase.  For any non-reversible process, if the process is reversible, I'm going to put an equal sign  underneath this, greater than or equal to zero.  Remember that?  For an isolated system, entropy increases during a spontaneous process.  That's what that means.  We'll actually derive this again today.  If you're in either one of these two categories, and we virtually always are, if we're actually  doing an experiment, we're not in an isolated system.  We're either in an open system or a closed system.  We have to consider both the system and the surroundings in this non-inequality.  In other words, it's the total entropy of the universe that's got to get bigger.  The universe includes the system that you care about and everything else.  There's two terms in this total entropy.  There's the system that you care about and everything outside of it because your system  is in communication with the outside world either by exchanging matter or by exchanging  energy.  If the system is in isolated entropy, does the system do anything to it?  Yes.  Now, this is the only thing that matters in terms of understanding where something is  spontaneous or not.  We're going to derive this Gibbs function, but the Gibbs function is going to be derived  from this.  It's all about entropy.  If you remember nothing else about this lecture, you need to remember that spontaneity of a  chemical process is about entropy.  The Gibbs energy is just a way to parameterize in a convenient way the entropy for us.  That's all we're doing with the Gibbs function.  We're parameterizing the entropy in a convenient way.  If this is true, then I can move that guy over to the right-hand side and put a minus  sign on him, right?  We've already said we've got an expression for the entropy in terms of the heat that's  transferred and the temperature.  I can make a substitution for the surroundings.  Minus dS here is going to be dQ.  It's plus dQ because there's normally a minus sign that connects this minus dQ over T with  dS.  If that's minus dS, that's going to be plus dQ over T surroundings.  Now, Q is a conserved quantity.  In other words, if plus Q enters the system, minus Q is removed from the surroundings.  If Q goes into the system, it had to come from the surroundings.  If it's a plus for the system, it's a minus for the surroundings.  What we've already beaten into the ground is that dU, where you use the internal energy,  that equals dW plus dQ.  That means dQ is dU minus dW.  I can just plug this in for dQ and this equation right here.  dU minus dW.  I've got this equation right here.  We haven't done anything fancy yet.  Everything's very simple.  All I've done here is assume that the only work that's being done is pressure volume  work.  I've got PDV here.  Now, let's multiply both sides by T surroundings.  When we do that, we're going to get T surroundings on that side.  We're going to lose it from the right side.  Here we go.  All I did is multiply by D surroundings.  Now it's on the left-hand side.  It's gone from over here.  This doesn't look like our conservation of entropy equations anymore, does it?  Starting to morph.  We're still talking exclusively about entropy here.  We've just made substitutions from symbols.  This equation is going to keep coming up.  We're going to call it the pink equation.  What we're going to do is we're going to find convenient expressions for dU, plug them  into the pink equation, and we're going to find the Gibbs equation.  We're going to find the Gibbs energy rather, the Helmholtz energy.  That's what we're going to do in the next 10 minutes.  We're going to keep coming back to the pink equation.  At equilibrium, for an isolated system, dV equals dU equals zero.  In other words, the internal energy of the system doesn't change and the volume doesn't  change.  If I prevent the volume from changing, and I prevent dU from changing, this term is zero  and this term is zero, in other words, dS of the system is going to have to be greater  than or equal to zero for an isolated system.  If I just take the pink equation and I say, if my system is isolated, this term is zero  and this term is zero, if I divide by T surroundings, dS system is going to be greater than or equal  to zero.  We've derived that now for an isolated system from this pink equation, which came from  this entropy argument.  The total entropy of the system and the surroundings has to be greater than or equal to zero for  any chemical process if it's spontaneous.  Now I'm going to stop writing cis every time I'm talking about the system.  If you see a big letter with no subscript, assume it means the system.  I'm talking about the surroundings, I'll keep using that.  Now this is a general expression for any system moving towards a new equilibrium.  That's what the non-equality means.  It means that the system is in flux towards equilibrium.  I'm going to explain in detail exactly how we think about that.  At equilibrium, the external pressure and the system pressure and the external temperature  and the system temperature are equal to one another.  In other words, the temperature is the same outside and inside the system.  The pressure is the same outside and inside the system.  It's intuitively obvious that this would have to be true if we're talking about equilibrium.  All this means is the pressure is in the same direction.  At equilibrium, the pink equation becomes the yellow equation.  Only thing that's different is the equal sign.  That's what the inequality means that we're evolving towards equilibrium.  When we get there, we get an equal sign.  We'll come back to this equation.  This is the so-called master equation of thermodynamics, but the pink equation is actually more useful.  We'll keep coming back to the pink equation.  We won't continue to mention the yellow equation.  The pink equation again addresses many processes of interest.  For example, the volume and entropy are held constant.  If the volume and entropy are held constant, volume and entropy, if I hold that constant,  then dv is zero.  If I hold the entropy constant, then ds is zero, and that means that du has to be less than zero.  What does that mean?  It means that it's u that is minimized as we evolve from a non-equilibrium state to  an equilibrium state.  It's u that is minimized if and only if entropy and volume are held constant.  Do we normally hold entropy and volume constant when we do chemistry?  I don't know how you would.  Volume, yes, you can do it, but entropy, difficult.  It's not super useful for us as chemists to use the internal energy as a marker for whether  we're at equilibrium or not because the internal energy is only going to be a minimum.  When volume and entropy are held constant, we can't hold entropy constant very easily.  We make note of this and we move on.  In other words, the spontaneous process occurring at constant volume and entropy will minimize  the internal energy.  This is true, but not terribly useful.  Last week we talked about the enthalpy.  Maybe that's more useful.  The enthalpy is u plus pv, that's just a definition.  If I take the total derivative of h, that's going to be du plus vdp plus pdv, and I can  just solve for u and plug it into the pink equation.  That's going to be our standard strategy for the next six minutes.  We'll plug it into the pink equation, rearrange these variables, and what do we do?  Plug it into the pink equation.  What the pink equation tells us, now what I do, I've got this whole thing now that I'm  going to plug into there.  There it is.  The first thing that we notice is that these two terms will cancel if the pressures are  equal.  Just imagine that we're approaching the equilibrium state, and they are equal.  They won't necessarily be equal until that's true, but when we're close to equilibrium,  this term and this term will cancel.  I hope you'll agree.  Then, if pressure and entropy are now held constant, we held volume and entropy constant  earlier, now we're going to hold pressure and entropy constant.  That is not more useful to us, by the way.  If we do that, entropy goes to zero, pressure goes to zero, these guys cancel with one another,  so we're left with d.  The enthalpy is the variable that we're going to care about if we can control and maintain  constant the entropy and the pressure.  Is that a useful thing for us to know as chemists?  It's a useful thing to know, but as a practical matter, we're not going to be constantly doing  this.  Enthalpy is not going to tell us when we're at equilibrium most of the time.  We can't satisfy this requirement of maintaining the entropy constant.  Now temperature actually is a variable that we frequently hold constant as chemists.  If we consider dt equal to zero and dv equal to zero, those are requirements that we are  sometimes going to be able to satisfy in the laboratory.  We can do an experiment at constant temperature and we can do an experiment at constant volume.  In principle, that's actually going to be quasi useful for us to understand whether we're  at equilibrium under those conditions.  The thermodynamic variable we're going to care about here is something called the Helmholtz  energy.  All we're going to do here is write a definition for it.  Helmholtz energy is called A, A is equal to u minus Ts.  That's just how it's defined.  That equation doesn't really mean anything to us.  We didn't derive it from any place.  We just defined the Helmholtz energy.  That's what it is.  Then what we do is write the derivative of that equation.  D A is du minus d Ts, which is minus Tds minus Sdt.  Solve for du.  What are we going to do?  Plug it into the pink equation is the right answer.  Here's the pink equation again.  We put this guy in for du and there it is.  Right there.  Now what we're talking about, that's going to cancel that without thinking too hard.  If we assume we're close to equilibrium and these two temperatures are going to be equal  to one another, that's the system temperature, that's the surroundings.  If we further assume that dt is zero and dv is zero, that goes away and dv.  That guy goes away.  So dA, the Helmholtz energy ends up being the thing that we're going to care about.  If the Helmholtz energy is getting lower, that should be spontaneous.  If the Helmholtz energy is getting higher, that should be non-spontaneous.  For any system that we care about.  So if we're able to maintain temperature and volume constant, A is going to be our go-to  variable.  Now the way that you do that is by using one of these guys.  In case you're wondering.  This is a par bomb.  It's universally called a par bomb even though there's like five companies that make these  things but par makes the best ones.  All this is a stainless steel container into which there's a glass jacket that goes inside  here and these are gas inlets and outlets and you see these two tubes right there?  Those are pressure, those are over pressure valves.  They're thin aluminum membranes and if the pressure of this thing goes above 3000 ATM,  those blow.  It sounds like a gunshot when they go off.  You're heating something in here, this needle goes all the way up to here and when this  thing is about to blow up, those two valves prevent that from happening but when they  go off by golly, it's spectacular.  But I think you'll agree this thing can maintain constant volume no matter what the pressure  is doing as long as the pressure stays below 3000 ATM, boom.  That's how you do constant volume and constant temperature.  Now most of the time we're not going to use this guy.  So we have to still talk about the Gibbs energy but we'll do that on Monday.
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