 What we want to do now is we will take a look at ways of evaluating the properties of gas mixtures. And then what we'll do is we will work an example problem that demonstrates how to calculate the properties and applies it to the first law in entropy generation and things like that. So we'll begin however looking at properties of gas mixtures and in doing this what we'll do will begin by looking at internal energy. So essentially what we're doing is we want to add the individual components for each of the components of our gas mixture. So we want to add the contributions. So taking a look at the internal energy, we wanted to find the internal energy of a mixture, a gas mixture. What we would need to do is take the internal energy of each individual component and then sum those up. And we can rewrite that in terms of the internal energy per unit mass and the mass of each component. So notice I'm using little u there, that would be kilojoules per kilogram. And we can also express that in terms of the number of moles or per molar bases. So if we know the number of moles of each component and then we know internal energy and we'll use the over bar there, whenever you see the over bar that's usually denoting kilojoules per kilo mole. And the final units of this expression would be in kilojoules. Now quite often we're not wanting to determine internal energy at a given state. Quite often we're looking for changes of internal energy between two different states. So if we're looking for a change during a process, we could write the change in internal energy for the mixture. Again it's going to be the sum of the change of each individual component, going through the same process, introducing the mass of each component. Now that's a little u and again if we're looking at properties kilojoules per kilo mole and again the units here would be in kilojoules. So another way that we can express this, we can use our mass fractions if we're looking at kilojoules per kilogram to express the internal energy of the mixture. And if we're looking on a per mole basis, we would then be using our mole fraction. So that is the approach that we typically take. And the above approach holds for internal energy as well as enthalpy and entropy to a certain extent. We have to be a little careful with entropy and I'll talk about that in a moment. And I put the disclaimer here, use care with entropy. And the reason for that, let's take a look at what happens with entropy. So if we're trying to evaluate the change of entropy of one of our components in the mixture, we've seen equations for computing the change in entropy and I'm going to be using the one for an exact calculation here. So we have to address the fact that we have the term that looks at the change in pressure between state two and one. And we can approximate this as well if we want to look at the idea of variable specific heats. And that would be the expression that we would obtain. Now what we're looking at here, the subscript i refers to the individual component that we're looking at and when we have comma one that would be pertaining to state one and when we have comma two that would be pertaining to state two. So this pertains to state one and then when you see the comma two that pertains to state two. So that's what we're talking about when we use that nomenclature. And the other thing we can say here is that the pi comma one or comma two, these are the partial pressures. These are partial pressures that we talked about last lecture. And if you recall, they were defined in terms of the mole fractions multiplied by the mixture pressure. And given we have comma two, we have to talk about the mixture pressure at state two and then again mole fraction component i at state one multiplied by the mixture pressure at state one. So that's what that is referring to. The other thing in this equation, these terms here, those represent determining entropy with respect to absolute zero Kelvin. So there are no approximations there. So these would be tabulated in the back of your book. And this is where you're taking into account the fact that you have variable specific heats. And with this integral, what we do, we use the convention that at zero Kelvin, the entropy is zero. And that enables us to determine entropy exactly. So the exact formulation is in the first part here. And the approximation is in the second part there. So that is how we can determine property change when we have a gas mixture. What we're going to do next is we're going to take a look at an example problem where we have to apply all these things. And we'll be looking at using the first law as well. I think, yeah, they want us to calculate entropy generation. So we'll move ahead and look at that in the next segment.