 Alright, so when we talk about thermodynamics, we're very often talking about macroscopic properties like let's say the pressure or the energy or the volume of the temperature, some other thermodynamic value. Sometimes we talk about the value of the property itself. Sometimes we talk about changes in that property. We might be interested in the difference between the pressure inside a container and outside a container or the change in the pressure as some process occurs. Sometimes what we're interested in is not the finite change in the property, but in the infinitesimally small change in the property, the differential change in the property. So this quantity dp, we call it differential, not a derivative. It's only half of the derivative, not dy dx, but it's only either the numerator, the denominator, infinitesimally small change in the pressure like you're used to from calculus class. And it turns out as we begin talking about the relationship between different properties, between pressure and volume or pressure and temperature, these differential quantities are going to become more and more important. Because when we have equations of state, like we've seen with the ideal gas law or the Van der Waals equation of state or other equations of state, those equations of state are like functions. They tell us how to calculate the pressure given values of other thermodynamic variables that specify the state of the system. And we need to know if we know that equation for the pressure, what does that allow us to say about the change in the pressure as I change the volume or as I change the temperature. So we'll talk about pressure, volume, and temperature in just a minute. But to keep things a little closer to what you may have learned in a math class previously, let's think about these functions more generically as just some function of two variables X and Y. If I have some function of X and Y, then the differential change in that quantity, if the function value is changing, it must be either because X is changing or Y is changing. And the amount that the function changes, I can write that as df dx times the change in X plus df dy times the change in Y. And as always with partial derivatives, df dx is done while holding Y constant, df dy is done while holding X constant. So in one sense, that's just the chain rule like you might remember from calculus class. Change in F comes either from change in X times the rate of change of F with X, multiplied by the change in X, and the rate of change of F with Y as I change Y. We can also think about what that means more graphically, more visually, which might make it help make it make a little more sense. So let's say that I draw a sketch of some function F of X and Y. So I've got some three-dimensional or two-dimensional function. So here's here's my function F of X and Y that I'm attempting to draw in 3D up above the XY plane. So this value, let's say, let's use a couple of different colors for this. This value corresponds to some particular value X comma Y in the XY plane. And at that point X comma Y, the function has some value F of X and Y. Now what does this quantity mean? What this means is if I move this point in the XY plane, if I move it, let's say, to the right in X, changing X, but without changing Y. I've moved it by some small distance DX. Then the function by moving in X but not moving in Y is going to slide down in that direction. So the amount by which the function changes from here to here, this is how much the function is changing. The amount the function changes is the slope of the function in the X direction, the slope of this curve in the purely X direction, multiplied by however much I shifted the point in the XY plane. The second term is similar. If I shift the point only in Y, then that's like moving along this contour. And the amount by which the function changes is whatever rate of change there is in the Y direction, the slope of the curve along one of these contours, multiplied by how far I've moved in that Y direction. And if I move from this initial point to some other point where I'm changing both X and Y, then the change in the function is due to some change in X and some change in Y. And I have to think about these as infinitesimally small differential sized quantities because then I can add, if I'm making very small steps, I can add the change in X and the change in Y together to get the change in Z. I'm sorry, the change in NF. So that's just a visual representation if that helps you make a little more sense out of this differential form, which is just change in F comes from two places, comes from any amount that I've changed X and the effect that that has on the function and some amount that I've changed in Y multiplied by the effect that that has on the function itself. So if we return now to a function like a thermodynamic equation of state where we know how to calculate pressure as a function of V bar and T, so that's our new F of X and Y. Let's start simple with the ideal gas equation of state. So I know how to write down pressure as a function of temperature and molar volume for an ideal gas. And now we have enough information to say how much the pressure would change in response to some change in the molar volume or some change in the temperature. That change in P is going to be dP dV bar multiplied by the change in V bar and dP dt multiplied by a change in T. And again, this is dP dV bar at constant T, dP dt at constant V bar. But since I know the equation of state, I know an equation that describes how pressure depends on T and on V bar, I know how to calculate those derivatives. The derivative of pressure with respect to V bar, the derivative of 1 over V is minus 1 over V squared. So I've got RT times 1 over V squared with a negative sign. That's what I have for the derivative of pressure with respect to volume. The derivative with respect to temperature is easier. Derivative of RT over V with respect to this T. We just lose the temperature and we have R over V bar. So what we can say is for a gas that obeys the ideal gas equation, then in response to a change in the pressure and the change in the temperature, what's going to happen to the pressure is the pressure is going to drop a little bit whenever I increase the volume. And it's going to increase a little bit with a positive sign every time I increase the temperature. And those two factors, the reason the amount by which the pressure drops when I increase the volume is this quantity RT over V bar squared because that's just the slope. That's how much P changes as I change V bar. And likewise R over V bar tells us the slope, how much P changes as I change the temperature. So this tells us how to evaluate a differential quantity both for an ordinary function and for a thermodynamic function like the pressure. Let me point out one more thing before we conclude, which is that we'll run into cases where we have more than just two variables. If I have some function that depends not just on x and y, but let's say lots of variables, x1, x2, x3, and so on, then the differential quantity for a change in that function f is going to have a term that looks like any one of these terms or any one of these terms, but for each individual variable of this function. So I'm going to have a df dx1 times the change in x1, df dx2 times the change in x2, and so on. So if I can write all those all together, it's just the sum of the rate of change of the function with respect to each variable multiplied by the change in that variable summed over all the different variables that we have. So we can do this not only for two-dimensional functions, but also if we want for three or four or five-dimensional functions if we need to. So this quantity of differentials that we've talked about, the title of this video is exact differential. So anytime we have something like this that is the differential of some function that we can write down an equation for, then we call that an exact differential. We wouldn't bother calling them exact differentials unless there was another case. So that's what we'll talk about in the next video lecture is some cases where the differentials are not exact.