 So in a beautiful world things would be fine now. We would have our free energy and we could use the free energy to study absolutely any system. Unfortunately the real world is not necessarily as beautiful as we might hope. If I put my physicist hat on myself for a second, as a physicist I love to study isolated systems. I already mentioned that. So here's a system and by isolated I mean that it's not really it's not exchanging anything with the outside. It's not exchanging particles and I'm not changing the shape of it or anything. So the only thing that will go out or in of this system is heat energy in terms of heat. That means in the system I have my free energy f equals e minus ts. Now if that was the real world things would be simple. The only problem in the real world things aren't that simple. Rather it is that simple but there are very few systems we could study that way. So let's take my chemistry hat instead. Now I'm a chemist and a lab rat here. Here is the chemist system and I call this system but imagine me sitting here and holding a beaker or test tube. Sure I too am exchanging energy with the rest of the world that happens all the time if I put this on a Bunsen burner or if there is a reaction happening that's heating the room. But what if I'm solvating a salt or something the volume might actually change because I have a test tube right it's open. So if the pressure is changes it can adjust the volume up and down. So here I'm going to need to account for the fact that my system might become smaller or larger because I'm performing either I am performing work on the outside world if I'm expanding or I let the outside world perform work on me if I'm contracting the system. To the physicists this is a royal pain in the A. To a chemist that's come on you don't have a real world unless you can describe this. So we know exactly how much work it takes to change how much work it takes to change the volume of the system V. That's the pressure multiplied by the volume so the chemist is quite happy. We have a way to describe this leave some space here. So here I'm going to say that this is the work I do here is E plus PV minus TS. If we were astronomically stupid I would call this F because then I would have an F but my F is not the same as that F and then you would always have to check is the author and the chemist or a physicist. Let's call this G and this factor its energy and volume occasionally we call that H and the word for that would be enthalpy. So note S is entropy H is enthalpy. So enthalpy describes the internal energy of this system plus the pressure multiplied by the system's current volume and that including that will account for the fact what happens if I change the volume of my system. Now if the world was good we would all be very careful and say are we using F or G. They actually have names too. The physicist would say that this is the Helmholtz free energy and the chemist would rather say that they're working with the Gibbs free energy. Occasionally we use A as a letter for the Helmholtz free energy too. F and A there on that side G on that side it's a simple and beautiful world right? Well if it wasn't for the fact that occasionally chemists are sloppy and we use F for this one too sorry why on earth are we that stupid? Well it turns out that for most reactions let's say that I'm salivating one I'm having one protein molecule fold. The volume change of that system is going to be one part in Avogadro's number. It's completely negligible so that short technically formally it is correct that I have to include the PV term here and everything. In practice I very rarely calculate the absolute values of things and if I'm only calculating a delta a change the delta in the PV term is going to be so small that I can't always say that it's zero. Again not for any chemical reaction but for reactions if I'm salivating a protein or breaking a hydrogen bond or something and that means that you will occasionally see even me write F equals H minus Ts or F equals E minus Ts sorry just deal with it you need to understand the difference between these you need to understand that you're occasionally having A if you're just seeing a value and you believe that PV might important that's when you should ask your colleague so wait a second you mean the Gibbs or the Helmholtz free energy here sorry the world would be better if we were perfectly standardized but in practice we're scientists only humans