 So, having our free energy equation, I'm going to write it down again because you can never comment on this too many times. It allows us to do a number of things. Free energy is actually surprisingly difficult to measure in one specific state. We can measure it through the available work if I'm performing work on a system to see how it changes. And in most cases, it's going to turn out that I can measure E in the lab. And if I then do a change and see how E changes, I might also be able to see how F changes. And that means that I can then solve for the entropy. Measuring the entropy directly is impossible for all, but the very simplest systems, or if you have a very simple model and run it in a computer. But let's try to reason a little bit about the entropy. This is, I wish this was my screen, it's not. If we think of this, the organization of the icons in the screen as a number of states or something, is this a well-ordered system or a disordered system and how many states are there? And then we compare this to something that is, sadly, a much better representation of my desk. Is this a ordered or well-ordered system and how many states are there? The answer might surprise you. And sadly, the disadvantage of not doing this interactively is that I can't have you provide feedback. The number of states here is exactly one. How many states do we have here? If you just guess, I think you would say many, and that's wrong. There is exactly one state here. And the trick is there is exactly one state that have all these icons placed exactly the way they are. And yet I bet all of you are going to think that that's a tidy desk and that's a ugly desk even if it's mine. The reason for that is that we, on the microscopic level, it's exactly the same. Each of this is one specific state. But on the macroscopic state, what we can see in a room, the reason why you think different about this is that there are very few states like this one and there are very, very many states like this one. If you just start throwing things out randomly on the floor, the likelihood that they will up in a well-ordered state is fairly low, right? But each specific organization they are in is one microscopic state. And this starts to be the first connection between these so-called macro states that we think of as the collections of states and all the microscopic states inside of them. You can think of that as let's put some small balls in holes. And let's say that I have on this side, I might have two holes. And I'm going to put the letters A and B in them. And here, let's say that I have three holes. It's a much. So sure, I can say A, B, or I can say B, A. And if we say, well, whether you like alphabetical order or whether things are neatly packed, if you want to just compare the alphabetical order, there's just one perfect state here that they're next to each other. Or if they're neatly packed, you can say that both those two states are neatly packed. If I'm going to place the same two letters here in this space that just have one more available position, I can do A, B, A, B, A, B. And the A, B, A, B, A, B, A. No, sorry, I got that wrong. A, B there. And then I can do A, B, and A, B. One, two, three, four, five, six. Yes, that's right. So here we already have six states instead of two. The number of states available will go up very quickly as the volume of your space goes up. And in this case, well, if they're going to be in alphabetical order and neatly packed, there's only one of the states available that has A, B, and then a space. So as the number of available space in your volume goes up, the disorder will go up astronomically quickly. And if you now were to randomly throw out letters in these states, here you would have 50% of them in alphabetical order and neatly packed. Here, depending on how you count, either it should just be that state or we might allow that state to. But in this case, when the available volume is larger, the particles here will spread over many more states. And you're much more likely to end up with what we call what we think of as disordered states. But the point is that particular state is one state. And that particular state is not really more likely than that state. It is just that on the macroscopic states, we start to think of order in a different way. Now, starting to calculate this, how exactly what is the exact number of macroscopic states all the particles in this room can adopt that would be insanity. We're not going to do that. The other problem is that I say that the number of combinations there goes up very quickly. And that it turns out that the entropy has a very nice properties. So first, I get a number that I can use to count it. Second, I take the logarithm of that number, meaning that I don't have to worry so much as the volume here of states goes up exponentially. Because the logarithm will mean that that will increase, but it will increase linearly. And then I also have the Boltzmann constant in front of it, which also takes this down to much more reasonable numbers. And second, they will have units of energy. So the entropy that you previously feared, that is actually a very nice way for me not to have to deal with all these complications internally, but have a way that I'm basically describing the volumes in the number of states. Sure, if you like to think of this as disorder, that is fine. But remember that there is no hand waving here. So exactly what S is, because we defined it, S equals Boltzmann's constant multiplied by the number of, well, in this case I'm not going to say volume. Occasionally we use the letter omega, because it's the number of specific states. We can count these states. It's not in terms of square cubic meters or something. It's numbers, one, two, three. And that gives a number of states that we can put the particles in. So this is simple math. Don't let reality confuse you and make you believe that math is hard. It's not.