 So we've seen there's a difference between the molar volume as we define for a single component solution or a single component system and the partial molar volume, the derivative of the volume with respect to added number of moles as we need to use for a multi-component system. Those differences turn out to be pretty important and allow us to define several other interesting properties. So let's back up and remind ourselves that for a single component system, we can think of the molar volume as just being a function of two thermodynamic properties, the temperature and the pressure, for example. So what that means, well, we'll explore what that means in a second. For a multi-component system, we've seen that it's a little more complicated than that. In a multi-component system, certainly if I change the temperature or the pressure, that will change the partial molar volume, but just like adding the second mole of sodium chloride to a solution changes the volume by a different amount than the first mole, that partial molar volume is going to depend on how many moles of substance one, how many moles of substance two, and so on. So it's going to depend on the composition and the concentration of the different components of the solution. So the partial molar volume depends on more than just temperature and pressure, depends on composition. What that means is when we take this partial derivative to obtain the partial molar volume, if I take the derivative with respect to moles of component I, moles of sodium chloride, for example, I need to do that not only while holding the temperature and pressure constant, but also holding number of moles of water, number of moles of all the other components constant. So we say take the derivative with respect to the number of moles of component I, add more moles of sodium chloride, for example, but don't change the number of moles of water. If I also had other solutes in there, I wouldn't change the number of moles of them as well. So a more careful definition of the partial molar volume would be derivative of the volume with respect to moles of I while holding all the other thermodynamic properties constant and the number of moles of each other component constant as well. So continuing this analogy or discussion of the differences between single component and multi-component volumes, in the single component case, we would write, for example, a differential expression saying that the change in volume might be due to some change in the temperature plus some change in the pressure. And as long as I know how quickly the volume changes when I change the temperature and how quickly the volume changes when I change the pressure at constant P in this case, at constant T in this case, then I can use those to compute the change in the volume. The same type of expression is going to be true and is going to be useful for multi-component systems, but in this case, what we'd have to write would be to understand how the volume changes. Certainly if I change the temperature, that'll change the volume. If I change the pressure, that'll change the volume. But now I also need to worry about the fact that if I change moles of some component, that will also change the volume as well. So the volume depends on not just T and P, but also moles. So not just moles of component one, moles of component two, but I have to do this for all the components of the solution. So I need to sum terms that look like this for every component, component one and two and three and so on. But this dv dni, and if I'm careful, it'll cause my equation to get a lot more complicated. But if I'm careful and add, sorry, take this derivative with respect to T, holding P and all the moles constant, take this one while holding T and all the moles constant, take this one while holding T and P and all the other moles other than I constant. That's an expression for the differential change in volume if I were to change the temperature or pressure or composition. I could rewrite that since we know that this derivative dv dni, that's the molar volume. This last term I can write if I want as sum of the partial molar volumes for each of my components. So that's, in fact, the way we would usually write this expression is a term relating to the change in temperature, a term due to the change in pressure, and another term due to the change in composition of the system. Any one of those could cause a change in the volume. So that's just an extension of some equations that we knew about for single component systems looking at the slightly more complicated way that they would look for a multi-component system as it relates to volumes, which we now call not molar volume, but partial molar volume. All those statements hold true, not just for volume, but for other thermodynamic properties as well. For example, I can think about the enthalpy or the molar enthalpy, whichever I want, for a single component system as a function of T and P. So again, this would be for a single component. We've talked about how the molar enthalpy will be a function of just T and P. For a multi-component system, on the other hand, the N1, N2, and so on. So the enthalpy, in fact, let's not talk about molar enthalpy until we define what that means, but the total enthalpy of the system will depend on the temperature, pressure, and number of moles of each component. If I change the composition of the system, that will change the enthalpy, just like if I change the temperature of the pressure. So instead of being able to define just a molar enthalpy as total enthalpy over a number of moles, as I would for the single component system, in this case, I can define the rate at which the enthalpy changes as I add substance I, that we would call a partial molar enthalpy. Just like we have a partial molar volume, we can define a partial molar enthalpy in exactly the same way. And again, that would be while holding T constant and P constant and moles of everything else constant, all the moles of all the J's that are not equal to I. Likewise, if I want to talk about how the enthalpy changes in response to thermodynamic changes, I could write that as a piece due to the temperature change, a piece due to the pressure change, and new now, a piece due to the rate at which the entropy changes when I change the number of moles, multiplied by the change in that number of moles, sum that up over all different substances. So this would be the more complicated version of the simpler expression, D H being D H D T D T plus D H D P D P that we would have used in a single component system. So just to add an extra term for the changes in composition, and again, the single component expression gets a little more complicated as we turn it into a multi-component expression. We can do this, of course, not just for the enthalpy. I won't write it out in detail for any more properties. But if we want to know partial molar internal energy, now we can just say the partial molar anything is equal to the rate of change of that property as I change number of moles by holding temperature pressure and other moles constant. So we can define partial molar volume, partial molar enthalpy, partial molar internal energy, partial molar any property we want using that same sort of expression. As you might expect, the most interesting property for which we can define a partial molar quantity will be the Gibbs free energy for a variety of reasons. And so we'll devote a whole lecture next to exploring the partial molar Gibbs free energy.