 So at this point we've seen several different flavors of energy. We've got a couple different flavors of free energy, the Helmholtz free energy and the Gibbs free energy. We've also seen the internal energy and the enthalpy. So we've got four different flavors of the energy, two of which are free energies. So it's useful at this point to stop and recap and summarize what we know about these different flavors of the energy, in particular as affected by the natural variables of each of these. So we know, for example, most recently we've seen that the natural variables of the Gibbs free energy, in fact the reason we constructed the Gibbs free energy was to find a function whose natural variables were T and P. Natural variables mean two related but separate things. First of all, if we write the fundamental equations for these energies, the fundamental equation for DG is minus SDT plus VDP and we've talked about how when written as a function of T and P, the fundamental equation is quite simple. It just has a thermodynamic variable multiplying each of these two variables that's being changed. Likewise for the Helmholtz energy, when written as a function of T and V, we get this fundamental equation which is quite simple. We also have fundamental equations for the internal energy and the enthalpy that take on different forms, but again they're simple when expressed in terms of their natural variables. So we can see from the fundamental equation that the enthalpy has natural variables of S and P while the internal energy has natural variables S and V. So that's one thing we mean by saying the natural variables of each of these energies are our temperature and pressure or entropy and pressure or whatever. The fundamental equation is simple when expressed in those variables in particular, but also the natural variables have a relationship to the second law of thermodynamics. So the second law when we ran across it first, we said the second law meant that the entropy of the universe is increasing or the entropy of any isolated system increases if a process is spontaneous. We've also seen that an alternate way of thinking about the second law in terms of the Gibbs free energy is that the Gibbs free energy decreases, that should be a less than, the Gibbs free energy decreases for any spontaneous process that's performed at constant temperature and pressure. So that's one way of thinking about the second law. There's a related statement about the Helmholtz free energy. The Helmholtz free energy is also negative for a spontaneous process, but a spontaneous process that takes place at constant T and V. So again the natural variables, because the natural variables are T and P for the Gibbs free energy, those are the variables that are held constant in order for the Gibbs free energy to be negative for spontaneous process. We haven't seen the corresponding equations for the energy and the enthalpy, but it won't surprise you to see that equivalent statements of the second law say that the energy is negative and the enthalpy is negative for spontaneous processes, but those are first processes that take place at constant S and V for the internal energy and constant S and P for the enthalpy because those are the natural variables of those functions. So like I said, we haven't seen those before, but if we want to we can confirm or derive those statements. This one's probably the easiest one to do, just very quickly. First law tells us that the energy is heat plus work. If we're doing a process at constant volume, then there's no PV work. So the work term disappears at constant volume du and dq are the same thing as each other. So du is equal to dq, but we know for a spontaneous process, dq is less than or equal to Tds, and then if we're also doing that process at constant entropy, ds is equal to zero. So the net result of that is that du is less than or equal to zero. To get to that result, I had to assume constant volume so there's no PV work and constant entropy so the ds term goes away, so that would be how we arrive at this result, for example. The point is, number one, this connection between natural variables and what must be held constant for a particular flavor of the energy to be a signal for whether a process is spontaneous or not. These are all equivalent statements of the second law, along with saying the entropy is positive for any isolated system for a spontaneous process. The key difference is in what variables are being held constant. Probably this fourth statement, delta G being less than or equal to zero, that's the one we'll use most often going forward because it's convenient to work at constant temperature and pressure. Be careful, though, to always remember which variables are being held constant. It's not true that the delta G is negative for any spontaneous process, period. We can come up with processes at constant volume or at constant entropy or other conditions that might have a negative delta G and be non-spontaneous or a positive delta G and be spontaneous, but if we're at constant temperature and pressure in particular, delta G gives us a sign for whether the process is going to be spontaneous or equilibrium or non-spontaneous, likewise for the other variables. So be sure and keep in mind what variables are being held constant. If you're doing a process at constant T and V, you don't want to be looking at the Gibbs free energy, you want to be looking at the Helmholtz free energy to decide whether the process is spontaneous.