 Okay. So this is our definition of the chemical potential. Chemical potential is the partial molar Gibbs free energy. It's the rate of change of the Gibbs energy as we change the number of moles of some compound holding the temperature pressure and other moles constant. To understand a little better what the chemical potential means, what it's telling us about a particular system, we can explore that in a little more detail. And it turns out chemical potential is not just related to the Gibbs free energy, but it also has a connection to the other forms of energy as well. So to see why that's true, let me go back to one of our definitions or important statements about the Gibbs free energy. For example, I can say the Gibbs free energy is the enthalpy minus t times s. And since we're talking about differentials, if I take the differential of both of those sides of the equation, dg is equal to dh. And product rule on ts gives me tds and sdt. So now what I want to do if I bring the sdt term over to the other side, so I can say dg plus sdt is equal to dh minus tds. That allows me to say, since I often think about Gibbs free energy when the temperature is held constant, if the temperature is constant, the left side of this equation says dg at constant t under which conditions the dt term goes to zero. On the right side, if I want to make this term go to zero, I have to hold the entropy constant. So if I'm holding entropy constant on the right, this tds term goes away. So what this statement says is for an isothermal process, the change in the free energy is equal to the change in the enthalpy if that process were done at constant entropy instead. So that just looks like an interesting little fact to observe. But now let's suppose that the process we're doing that changes the free energy isothermally and is equal to the enthalpy change when I do it under a different set of conditions, let's suppose that the change I'm making is to change the number of moles of compound I. So changing the number of moles at constant temperature, dg dn at constant t is equal to dhd change in the same number of moles at constant s. That's now beginning to look, at least the left side is now beginning to look a lot like this chemical potential. If I in addition do it at constant pressure and do it without changing any other types of composition, any other moles of any other compounds on the left, I also don't change the pressure, don't change the moles of any other compounds on the right. Then I've manipulated the left side until it looks exactly like our definition of the chemical potential. Chemical potential is the partial molar free energy. That happens to be the same thing as the change in the enthalpy as I change number of moles, not while holding t and p constant, but while holding the entropy and the pressure constant. So it turns out the chemical potential is the partial molar Gibbs free energy, but it's also related to the change in the enthalpy as I change number of moles just under different conditions. So let me write those two statements down separately. So I can go back to our original definition. Very often the most useful one chemical potential is the partial molar Gibbs free energy. We've also just discovered that the chemical potential can be thought of as the rate of change of the enthalpy as I change the number of moles, but at constant s and p. And now it may not surprise you to learn that we have similar equations for each one of the four forms of the energy, Gibbs free energy, enthalpy, Helmholtz free energy, internal energy, and the only difference is what's being held constant. t and p, the natural variables of the Gibbs free energy are held constant in this expression. s and p, the natural variables of the enthalpy are what's held constant in this expression. And I won't bother to derive the other ones if you want to check me and do that for yourself. If we want to derive the one for the Helmholtz free energy, for example, we just have to go back and find a relationship, recall the relationship between Gibbs free energy and the Helmholtz free energy or for this last relationship for the internal energy, we'd use the relationship between Gibbs free energy and the internal energy. And it probably will not surprise you to learn that in each of these expressions, the variables that need to be held constant are the natural variables of that particular form of the energy. So we can treat these as definitions. They're worth putting in a box because there's occasions when we'll need each one of them. But the important thing to remember is the chemical potential, what it means, what it means to say the chemical potential of a particular substance is 500 kilojoules or 500 joules per mole or some particular value. That is the rate at which the free energy is changing when I add more moles of the substance, when I do that at constant temperature and pressure, or if I happen to be doing something at constant temperature and volume, then it's the Helmholtz free energy that's increasing by that number every time I add a mole of the substance and likewise for whatever other conditions there are. So we can use whichever one of these we want. They're all true statements. Depending on what conditions we find ourselves using, one of these statements may be more useful than the others. The other useful thing that this now lets us say is we've seen in the previous video that what we've called the fundamental equation for Dg that can be extended for a multi-component system as long as we remember to write down the chemical potential times the number of moles or the change in the number of moles. That tells us Dg. Again, if we use these expressions, if we go back and look at this one, for example, Dg differs by Dh only with these terms of Tds and an Sdt. So if I were to write down what does Dh look like, if I add a Tds and add an Sdt, one of them cancels the Sdt that's here with a negative sign. So that's the fundamental equation that we already know for the enthalpy. Nothing has happened to this expression. I still have chemical potential times the change in the number of moles. There's no, the only differences between Gibbs free energy and enthalpy involve the temperature and the entropy. Modifying this term doesn't modify the VDP term, doesn't modify the chemical potential term. So these two equations are related to one another. We can see looking at this expanded fundamental equation, the chemical potential, just like the entropy is one of the fundamental derivatives, the volume is one of the fundamental derivatives from this fundamental equation, the chemical potential is one of the fundamental derivatives. It tells us how the free energy is changing as I change the number of moles if I do it at constant temperature and pressure. This expression tells us that the chemical potential is the rate of change of the enthalpy as I change the number of moles as long as I do it at constant entropy and pressure. So these definitions of the chemical potential are related to these expanded versions of the fundamental equation and I'll just write down the last two for the Helmholtz free energy and the internal energy. We have Helmholtz energy being minus SDT minus PDV, again plus some of the chemical potentials times the number change in number of moles. Internal energy is TDS minus PDV plus chemical potential times change in number of moles. So those are our new versions of the fundamental equations for a multi-component system where we have more than one type of molecule in the system. So again, each one of these equations is true. We're able to use any one or all of them that we wish. Often which one is most convenient to use depends on whether the temperature and pressure are being held constant or the temperature and the volume and so on. So these are not the only examples where a single component expression like the fundamental equation we've had for the single component system gets expanded or needs to be written a little bit differently when we have a multi-component system. Many of those multi-component expressions do turn out to involve the chemical potential. So the next step is to see how some of our familiar single component thermodynamic expressions get changed when we're thinking about a multi-component system.