 So we have a lot of thermodynamic results we've developed, primarily so far for single component systems, and now we need to think a little bit carefully about how those thermodynamic results apply for multi-component systems. And in general, the equations are still true, so if I just write down any thermodynamic result we have, it's still true whether it's a single component system or a multi-component system that the enthalpy is internal energy plus p times v, where we have to think carefully is when we use the intensive versions of those expressions. So this is an extensive equation, total enthalpy is total energy plus pressure times total volume. If I want to write down the intensive version of that equation, I'll do this in a little more detail than we've done since we first defined intensive and extensive properties. So this extensive property, if I want to think about the intensive version of that equation, if I just divide by number of moles on each side, h over n, u over n, v over n, I can then write this as molar enthalpy is equal to molar internal energy plus pressure times molar volume. So I've just converted all the extensive properties into intensive properties and everything is fine. That works fine for a single component system, but we can't think about it in quite the same way for a multi-component system. If I have two components a and b or one and two, I can't just divide by total number of moles. I need to think about, am I talking about moles of compound one, moles of compound two, and as we've seen for a multi-component system, the right way to think about that would be instead of dividing by number of moles. If I ask how quickly is the enthalpy changing as I change the number of moles of a particular substance, that's going to be just take d, d, n on all sides. Those same derivatives apply for the internal energy and for the volume and now I can write that expression as the partial molar enthalpy of compound i is equal to the partial molar enthalpy, I'm sorry, partial molar internal energy of compound i plus pressure times partial molar volume of compound i. If we want to take a single component thermodynamic result and change it into a multi-component thermodynamic result, very often all we have to do is change the molar quantities into partial molar quantities, although it can help to step back and think about how we got to that expression we got here by dividing, we get to this expression by differentiating. Just to give another example or two to make sure that's clear, let's take a result like one of the fundamental derivatives that we obtain from the fundamental equation for the Gibbs free energy, dGDP at constant T is equal to the volume. Gibbs free energy increases with pressure proportionally to the volume. As an intensive property for a single component system, I can just put bars over those two properties, the molar Gibbs free energy changes with pressure proportionally to the molar volume. If I want to think about that as a multi-component system, then I'd have to say the partial molar, the change in the partial molar Gibbs free energy with respect to pressure is proportional to the partial molar volume. Derivatives of these quantities with respect to moles of compound i rather than just the molar quantity. Of course, once I've said partial molar Gibbs free energy, I can also choose to write partial molar Gibbs free energy as the chemical potential. This expression looks somewhat new, if I were to ask how much does the chemical potential of something change as I change the pressure, well that's just the partial molar volume for the same reason as we've already known that the Gibbs free energy, the molar Gibbs free energy changes proportionally to the molar volume in a single component system. These expressions can end up looking a little bit different until you remember that the chemical potential is just the partial molar Gibbs free energy. Let's work one more example with a chemical potential in it. So we've seen for the particular case of an ideal gas that the Gibbs free energy, the molar Gibbs free energy of an ideal gas is its standard molar Gibbs free energy plus the RT log of the pressure relative to the standard pressure. If I want to convert that to a multi-component expression, again I can just turn these G's into partial molar G's or in this case go directly to recalling that partial molar Gibbs free energy is equal to the chemical potential. So the chemical potential of an ideal gas at a pressure different than standard pressure is equal to its standard chemical potential, turning that molar G into a partial molar Gibbs free energy which is the chemical potential and again plus RT log of P over P naught gives us an expression similar to this one in fact for how the chemical potential changes with pressure proportionally to the molar volume for an ideal gas that RT over P when we integrate it ended up looking like RT P over P naught. One more example, the Gibbs Helmholtz expression that we've seen before in extensive terms. We've seen that the derivative of, turned out recall that the derivative of the Gibbs free energy with respect to temperature was a little bit complicated. It's a little bit simpler says the Gibbs Helmholtz expression. If I think about the derivative of G over T with respect to temperature and that's just negative enthalpy over T squared, intensive version of that equation for a single component expression would have been this. The two extensive properties have been named as molar Gibbs free energy and molar enthalpy in a multi-component system. Now I would write that as again write all the molar properties as partial molar properties. So partial molar Gibbs free energy and partial molar enthalpy. But partial molar Gibbs free energy again is chemical potential. So this would be the version of the Gibbs Helmholtz equation in a multi-component system. Likewise we have these versions of equations that we've seen before and we'll see again that have slightly different appearance in a multi-component system. Number one, because we need to think about partial molar quantities instead of just molar quantities. Number two, because we call the partial molar Gibbs free energy the chemical potential. So switching from a single component thermodynamic result to a multi-component thermodynamic result is usually just as straightforward as making those two changes.