At this point in the course, we've gone through the first, second, and third laws of thermodynamics. This is the last week, and we're going to wrap up with the introduction and discussion of two last state functions, let's start with the Helmholtz free energy. So I'll remind you that. We discover the condition for a spontaneous process, is that the entropy change is greater than or equal to zero, in isolated system. That is a system at constant internal energy and constant volume. Now, if the system is not isolated, then the entropy change that's evaluated has to consider both the system and the surroundings, and evaluating the total entropy change is not very convenient. That might be the total entropy of the universe we'd be interested in. So instead, let's consider a system at constant temperature and volume, instead of internal energy and volume. So the system's no longer isolated, and that means that heat can flow in order to maintain constant temperature, heat from the outside or to the outside exchanged with the system. And as a result, the change in entropy being greater than or equal to zero for the system alone, is no longer the criterion for a spontaneous process. And so we can ask ourselves, what is the criterion for a spontaneous process? A constant T and V. So before we do that, it's a new week, let's get the intellectual juices flowing. Ill let you take a quick self assessment that's a little bit of a review and then we'll move on. Alright, so with that review out of the way, you remember that heat and work are path functions, but that their sum is a state function and it's the differential of the internal energy. That's the first law of thermodynamics. We also have learned the second law says that DS is greater than or equal to the del q over t or I can rearrange that and have something I can insert in for del q, namely TdS. Moreover, I know at constant volume that the work is going to be minus external pressure times dV, and that's equal to 0 at constant volume. And so, I established that I have a relationship of dU is less than or equal to TdS at constant volume. Where the less than symbol applies for an irreversible process, because dS will be greater than del q over T, and the equal symbol applies for a reversible process where dS is equal to del q over T. Of course, if the system is isolated. Then dU is equal to 0, right, that's constant internal energy. And we would recover that dS is greater than or equal to 0. That was actually what we determined to be the condition for spontaneity when the system is isolated. But when it's not, that suggests that we can rearrange this inequality, dU being less than or equal to tDS. And say that, at constant volume, dU minus TdS is less than or equal to 0. And again, the less than symbol holds for irreversible processes. The equal symbol holds for reversible processes. And this refers to constant T and V. Given that, we can define a new state function. Namely, U minus TS. And that defines the Helmholtz free energy which, is abbreviated capital A. And it was first suggested by Hermann von Helmholtz shown here on the left hand side of the slide. And notice that the total differential, then, d of U minus TS. That would be dU minus. And then, with the chain rule, it would be dTS. But we're at constant temperature. So dT is 0. And what's left is TdS. And that's this inequality up here. So this total differential, these are state functions. And hence, the Helmholtz Free Energy must be a state function. Thus, at constant T and V, we have that dA, Ud of us minus TS, is less than or equal to 0. Less than 0 for spontaneous processes. Equal to 0 at equilibrium. So, if we imagine a system that starts out of equilibrium, it will have some value of the Helmholtz free energy. And as spontaneous processes occur over time, they will lower that free energy. DA always has to be less than 0. So the Helmholtz free energy is going down, down, down, down, down until finally, the system achieves equilibrium. At which point, the Helmholtz free energy is minimized and it stays at that value until the system is perturbed from equilibrium again, indefinitely. So spontaneous processes will be reversible equilibrium processes. So before we explore that state function a little more, I can't resist doing a little bit of history of chemistry and physics. So von Helmholtz was a fascinating individual in the history of thermodynamics. He has a wonderful quote that most physical scientists keep close to heart. It says, whoever in the pursuit of science, seeks after immediate practical utility, may generally rest assured that he will seek in vain. And this quote is often trotted out as a basis for a fundamental scientific research, curiosity driven research, you just never know what might come of it. What's ironic to some extent is that Helmholtz himself was remarkably successful in pulling out practical utility. So for instance, Helmholtz invented the opthalmoscope. So that device that an optometrist or a doctor uses to look at the inside of your eye. That was invented by von Helmholtz in the nineteenth century and he figured out the way to structure the optics and the light beams so that people could see the inside of the eye, and it revolutionized the medicine of the eye. In fact he was a physician, he was also a professor of physiology and psychology, he did fundamental work on the transmission of nerve signals. Optimology, auditory perception, oh and thermodynamics just on the side. He ended up with academic appointments beginning in Konigsberg, then Bonn, then Heidelberg, and ultimately he moved to Berlin where he was a professor of physics. And a senior colleague of Max Planck, who appeared early in this course, as one of the founders of quantum mechanics. And so, one of the wonderful things about Germany, is that it tends to honor its scientists a great deal. And in this case, on the 100th anniversary of his death, the German Bundespost issued a stamp honoring von Helmholtz. You see here a picture of an eye to recognize what he did in, in the area of physiology. As well as other features that honored his scientific career. So a real towering figure in early thermodynamics and in early German science in the nineteenth century. All right, well enough digression on history. Let's return to our state function, Helmholtz free energy equals U minus TS, that's the definition. And so if I were to consider an isothermal change, that is one where temperature is going to be held constant. I have delta A is equal to delta U minus T delta S. Recalling that our condition for spontaneity is dA less than or equal to 0. I could write that for a non infinitesimal change as delta U minus T delta S less than or equal to 0. And just emphasizing, again, this is under constant temperature. And volume conditions. And so what it says is that if a process is going to take place spontaneously, there needs to be a compromise. So there's an energy term, and an entropy term. So if I increase the entropy because it's preceded by a negative sign, that will make delta A more negative. Delta U, on the other hand. Rather than becoming more positive, would need to become more negative to make delta A more negative. So these two can balance one another. And entropy becomes more important at high temperature, because it's multiplied by temperature. A process where delta A is positive will not take place spontaneously at constant T and V. Instead, you'll have to add something to the system, perhaps work, in order to drive that process. And so why was Helmholtz so interested in this quantity, the Helmholtz free energy? And it's because it gives you insight into the work you can extract from a system. So given that we have delta A is equal to delta U minus T delta S. And given that A is a state function, we can evaluate a change in the Helmholtz free energy. Simply by knowing the beginning point and the end point, and indeed we can follow a reversible path which tends to allow us to compute more readily what the change is. And so, on a reversible path I would have that delta S is equal to the reversible heat divided by temperature. So T delta S is just the reversible heat. And so if I plug that in, I will get that delta A is equal to delta U minus the reversible heat. But we know from the first law of thermodynamics, that delta U is equal to the reversible heat plus the reversible work. So if I take away the reversible heat I am left with. Delta A is equal to the reversible work. So isothermal reversible work that is. Given a process for which delta A is less than 0, that is a spontaneous process, the reversible work will be the maximum work. Arbeit is work in German, capital A, and that's why Helmholtz used A as the symbol for the Helmholtz free energy. That can be extracted from the system if you do anything irreversibly. Then the entropy inequality will come into play. You'll have wasted some of the heat energy, and you won't be able to do as much work. If delta A is greater than 0, well then the process is not spontaneous. And W reversible, the reversible work is the minimum work that must be done on the system. In order to make the process occur. And again if there are irreversible processes, you'll need to increase the amount of work in order to accomplish those. All right. Well, that is the Helmholtz free energy in a nutshell. Next, I want to take a look at another kind of free energy, and in particular the Gibbs free energy.