 All right, so armed with this new idea of a partial molar quantity, so here the x could stand for volume, partial molar volume, but it could stand for any other thermodynamic property as well. The partial molar volume or partial molar whichever property is defined as the rate at which that property changes as we change the moles of a particular component of the mixture that we're talking about. So now that we understand this, it turns out one of the most useful partial molar quantities we can define is the partial molar Gibbs free energy. I'll remind us that these derivatives, partial derivatives as always are taken holding other quantities constant, so we're holding the temperature pressure constant, we're holding the number of moles of all the other components of the mixture constant, only changing the number of moles of component I that we're interested in, so we're not changing any of the other moles. So this, what I've just written down is the definition of the partial molar Gibbs free energy. That turns out to be important enough that we give it another name as well. So we'll use the partial molar Gibbs free energy often enough that instead of calling it partial molar Gibbs free energy, we call that, by a different name we call it the chemical potential, and we give it a different symbol. Mu sub i is the partial molar Gibbs free energy of component i in a solution. The reason this turns out to be useful, you can probably predict some of these reasons. First of all, g, the Gibbs free energy, is among the more useful chemical energies, and so the partial molar Gibbs free energy is more useful often than partial molar other types of energies. And part of the reason that's true is because the partial molar quantities are defined at constant T and P, and of course the energy that's most useful when we are at constant temperature and pressure, as we often are at experimental conditions, is the Gibbs free energy. So knowing which substance has the higher or lower Gibbs free energy tells us a lot about what is the most stable state of a system, likewise the chemical potential is going to tell us a lot about what's the most stable phase or component in a solution. So I'll remind you something we already know about the Gibbs free energy, and extend that to what we can say about the chemical potential. So we know the fundamental equation for the Gibbs free energy, dG is equal to minus s dt plus v dp, that is true for a single component system, as we've talked about until now. That tells us, for example, that the natural variables of G are temperature and pressure. We can think of G as a function of T and P. So in a single component system, G depends on temperature and pressure. In a multi-component system, things get a little bit more complicated. If we have more than one component in a system, moles of compound one, compound two, compound three, the free energy is going to depend on how much of each of those quantity, how much of each of those compounds we have. So it turns out that the equivalent statement in a multi-component system for the change in a free energy, it's certainly going to depend on the temperature and pressure. When we change the temperature, the free energy changes proportionally to negative entropy. When we change the pressure, the free energy changes proportionally to the volume. But as this expression tells us, when we change the number of moles of any one of these components, the free energy changes proportionally to the chemical potential. So I could write down mu1 times dn1 plus mu2 times dn2 plus mu3 times dn3 and so on. Or I can just write down all of those at the same time and say that the change in the free energy is proportional to the change in the number of moles of any one of these components and that proportionality constant is the dgdn, or in this case the chemical potential. It's just the different name that we give to the partial molar Gibbs free energy. So this would be the fundamental equation for a multi-component system. If we have more than one component in the system, we need to remember that the free energy is going to depend on the chemical potentials times the changes in the number of moles of each one of those components. So that's an introduction to this idea of chemical potential. Chemical potential is going to turn out to be one of the most important thermodynamic variables we talk about. It's often a little bit uncomfortable and unfamiliar. If this is the first time you're seeing it, it probably seems a little bit abstract. The things to keep in mind for right now, number one, this definition. If you find yourself confused about what the chemical potential is or what it means, just go back to the definition. Chemical potential is the partial molar Gibbs free energy. It's the rate at which the free energy changes as we add or subtract moles of that particular component. As we've talked about briefly and we'll see a lot more coming up, the reason is useful. It will turn out to be very useful to us in predicting which phase of a compound is more stable, solid, liquid, or gas, for example, because it's related to the free energy. It tells us something about the stability of the different compounds. But since it's a fairly abstract idea, it's going to make a lot more sense as we do some examples and consider it in a little more detail. So what we'll move on with next is seeing how the chemical potential relates not only to the Gibbs free energy, but it also turns out to relate to other types of thermodynamic energies as well.