 So the home holds free energy is a useful quantity. We've discovered that it can tell us something about the spontaneity of a process, which makes it more convenient than talking about the energy or the enthalpy. But its natural variables are temperature and volume. And in the lab, typically, we're more interested in working at constant temperature and constant pressure rather than constant volume. So it turns out there's a different flavor of the free energy that we'll define, that has natural variables of temperature and pressure. To see how the Gibbs energy comes about, let's recap what we know about the flavors of energy we've seen so far. Notably, we have the internal energy, U. And if we define the enthalpy as the energy plus PV, we've defined a new quantity called the enthalpy. That's just a definition that we made to make things more convenient at constant pressure. We've also defined a different flavor of the energy. The home holds free energy. And we defined that by taking the internal energy and subtracting the product T times S from it. So things we know about these energies, the home holds free energy is a natural function of T and V. The internal energy we've seen is a natural function of S and V. Enthalpy is a natural function of S and P. So these are three different combinations of these natural variables. What we're really interested in is some functions who is a natural function of T and P. Notice also what these transformations have done when we added P times V to the energy. It swapped the natural variable V out for a P. Likewise, when we subtracted T times S, instead of having natural variable of S, we transformed the function and the new function as a natural variable of T. So these transformations, those are called Legendre transformations. If I add or subtract an appropriate product of thermodynamic variables, I can switch natural variable of V to P or S to T. So we're interested in doing the same sort of thing now to get a function which is, we're going to call this new function G for the Gibbs free energy. That is a natural function of T and P. So we can think about doing that in two different ways. We can take the Helmholtz free energy, which is a natural function of T and V. In order to convert it to a natural function of T and P, we can use this Legendre transform. We can define it as the Helmholtz energy plus PV. That will give us a function whose natural variables are T and P. Or we could consider doing it from the other side. We could take the Helmholtz, I'm sorry, the enthalpy, subtract T S from it in order to transform this S into a T. So G and this expression will look closer to familiar to those of you who have seen the Gibbs free energy before. In general chemistry, for example, enthalpy minus temperature times entropy is another definition of the Gibbs free energy. Both of those will lead to the same product because both the enthalpy and the Helmholtz energy were themselves defined using the internal energy. So the more fundamental definition of the Gibbs free energy, whether I think of it as A plus PV where A is U minus T S. So here's the Helmholtz energy A to which I can add PV. Or I could think of it as enthalpy, U plus PV. So here's a U plus PV. If I subtract a T S from it, I get the same thing. So both of these roots took us to the same place. This Gibbs free energy we're defining, our definition of the Gibbs free energy is that it's the internal energy minus T times S plus P times V. So essentially what we've done is said, let's find some free energy that is just like the enthalpy is more convenient at constant pressure than the internal energy. The Gibbs free energy is more convenient at constant pressure than the Helmholtz free energy. Or likewise, the Helmholtz free energy tells us about spontaneity unlike the internal free energy to turn the enthalpy into a free energy and let us use it to tell something about spontaneity. We make this transformation just as we did for the turning the energy into the Helmholtz free energy. We've done the same thing for the enthalpy turning into the Gibbs free energy. So this is only a definition. We can move on next and do the same sort of things we've done for these other flavors of the energy. Notably, define a fundamental equation, find out what the thermodynamic derivatives of G are, write down a thermodynamic connection formula, talk about spontaneity. So those are all the things we'll pursue next.