 Okay, so we have begun to talk about the difference between an exact differential that we denote as df and an inexact differential that gets a different symbol. And so the next thing we'll talk about is some connections between exact and inexact differentials. It turns out both exact differentials and inexact differentials turn out pretty frequently in physical chemistry. So, and being able to think about how they're related to each other is often fairly useful. So, as the first of these relationships, let's point out that if we have an exact differential, let's say, let's give a simple exact differential like xdy plus ydx. We know that's an exact differential even if we've not seen the original function that it came from because we can integrate backwards and find out that the function xy, if I take, so I've written these opposite order than I normally do, but if I take the x derivative of this function I get a y, if I take the y derivative of the function I get an x, so that's a perfectly legitimate exact differential. Both these terms come from the same original function. So, even though, actually let's modify this. Let's say my function is 3xy. So the exact differential would be 3xdy and 3ydx. I can write down some inexact differentials. For example, let's say I wrote down 2xdy and only 1ydx. That's an inexact differential if I attempt to figure out what function would give rise to this as an exact differential. Taking the y integral of this term would give me 2xy. Taking the x integral of this term would just give me 1xy. So there is no function which has the exact differential of 2xdy plus ydx. So that's an inexact differential. Likewise, for exactly the same reason, I can have an inexact differential that looks like only 1x times dy and 2y times dx. So both of those are inexact differentials, but notice that if I add those two together, 2x and x give me 3x, 2y and y give me 3y. So if I combine those two inexact differentials, the ones I've called dg and dh, those two inexact differentials happen to sum to the same as this exact differential df. So it's sometimes the case that we're interested in inexact differentials that by themselves are kind of inconvenient because they're not the differential of anything, but when we combine them together, they might well represent an exact differential when we combine them in a certain way. That's one interesting thing to keep in mind about inexact differentials. Another one is let's again start with an inexact differential. Let's use the same one that we've seen previously when we introduced the idea of an inexact differential. So that's an inexact differential again because there's no single function whose x derivative is 3y squared and whose y derivative is 12xy. You can't find a function for which this is the exact differential. But sometimes we can tweak that inexact differential and turn it into an exact differential. So let's say I take that inexact differential d bar f. Let's say I just multiply it by y squared. So what I've got left is y squared, what I want to multiply by. So I've got 3y to the fourth dx plus 12xy cubed dy. Yes, that's what I want to do. That quantity itself, this whole thing y squared times df, we can ask is that an exact or an exact differential? It has the form of the differential, some quantity multiplied by dx, another quantity multiplied by dy. And in fact, this one's okay if we ask what function this might have arisen from as a differential. If I take the x derivative of some function and I get 3y to the fourth, that function would have been 3xy to the fourth. And that same function, 3xy to the fourth, if I take the y derivative, 4 times 3 is 12, reducing the power of y by 1 gives me 12xy cubed. So this quantity, there is some function, let's call it g, whose exact differential is 3y to the fourth dx and 12xy cubed dy. So after multiplying my inexact differential by y squared, I can turn it into an exact differential. So that's a useful trick. In this particular case, what we say is that we've used an integrating factor. If I multiply my inexact differential by y squared by an integrating factor of y squared, basically what that does is I've used a factor that allows me to integrate it back up to a function g. So y squared is the integrating factor in this particular case. So that's another useful trick when you have an inexact differential. You can either combine it with another inexact differential to make it exact, or sometimes you can multiply it by something to turn it into an exact differential. And as a final example of being able to recognize when something is an exact or an inexact differential, let's take a case like, let's say, 3x squared dx and 4y cubed dy. And I'm asking, I've labeled it df, but I'm asking, do you know whether this would be an exact or an inexact differential? And we can, of course, do the same trick we've done every time I write these upwards arrows, and I can ask, is there some original function which gives rise to this differential? And in fact, there is, if I write, if I ask what is the integral of 3x squared, or what function do I need to take the derivative of to get 3x squared, that function is x cubed. Likewise here, is there some function whose derivative, y derivative looks like 4y cubed? Or if I integrate 4y cubed with respect to y, what do I get? The answer there is y to the fourth. And your first instinct may be to say, those are different functions. Every time I get a different function when I integrate this piece, and this piece, that means there is no parent function that gives rise to this differential. But notice in this case, this one only involves x's and this one only involves y's. So if my function is, in fact, x cubed plus y to the fourth, then when I take of the whole function, if I take df dx, the partial derivative with respect to x, we treat the y's as constants so they disappear. So the derivative of the 3x cubed is 3x squared. Likewise, if I take the partial derivative with respect to y, treating the x's as if they're constants, this part is irrelevant and the derivative of y to the fourth becomes 4y cubed. So what we've seen in this case is any time I have a function that looks like something that's purely a function of x times a dx and something that's purely a function of y times a dy, that's always going to be an exact differential. I can always find the integral of some function, pure function of x with respect to x, I can always integrate a pure function of y with respect to y and the parent function is going to be just the sum of those two integrals. So that's essentially the same as separation of variables if you want to think about it that way. Any time you have a differential that's purely x and another piece that's purely y, then the variables are completely separated into two terms that are added together and that's guaranteed to be an exact differential. So these are a few different tricks to keep in mind as we talk about the difference between exact and inexact differentials and the next thing we'll do is use what we've learned about exact and inexact differentials to start talking about thermodynamic properties.