 So among the collection of many different thermodynamic relationships, there's a handful of them that are so commonly used and so useful that they get names of their own. So as one example of this one that we've seen so far is the heat capacity. So our definition of the heat capacity, heat capacity is defined to be how much the internal energy changes as we change the temperature while doing so at constant volume. That gives us the constant volume heat capacity. That's just a definition. We've defined that quantity because it's useful enough to know how much the energy is changing when we change the temperature. And it's also something that we can measure relatively easily in a calorimetry experiment that we just give that a name and say whatever that property is, however quickly the energy changes as I raise the temperature of an object. That's something we can measure, put in a table to be looked up later, and we call that the heat capacity. That's different in some way from other thermodynamic relationships like this one, DDT at constant P we've seen is negative entropy. That actually tells us something deep and fundamental about the connection between these different variables. The fact that this derivative is equal to negative the entropy, that fact that came from the fundamental equation for G. That's a useful thermodynamic relationship between variables that existed before we wrote this equation down. This equation, the heat capacity is just the thing we've defined to be equal to this ratio of constant. So we've just given a name to this particular thermodynamic relationship. Some others that fall in that category, if we didn't ask for DUDT but DHDT and if we do it at constant pressure, that of course is the constant pressure heat capacity. The other named derivative we've seen so far is the Joule-Thompson coefficient, which tells us how quickly the temperature increases as we increase the pressure on an object isenthalpically as we do it at constant enthalpy. So that coefficient is useful enough in reverberation that that particular ratio gets a name. There's a couple of others that are important that we'll use going forward that we haven't run across yet. So we can talk about those, define them now for the first time. The first of them is related to this quantity. How quickly does the volume of something change as I heat it up and when I do that at constant pressure? So that quantity is called the thermal expansion coefficient. And in particular, when we define that, the definition of the thermal expansion coefficient alpha includes that extra factor of 1 over V in front of it. So that thermal expansion coefficient is physically very useful because that's a useful thing to want to know. Essentially, as the temperature changes, how much is an object going to swell? How much is its volume going to change? That's useful both in chemistry for measuring the, for example, if I change the temperature of a solution, how much will the volume change? It's very useful to engineers, for example, if you build a bridge you'd like to know in the swing between cold winter days and warm summer days how much the volume of the iron that you're using or the steel that you're using to build that bridge is going to increase. So that's in a practical sense a very important quantity to know for a material is its thermal expansion coefficient. How much does its volume change when you heat it up? The reason for this 1 over V in front of the dV dt at constant P is to keep this quantity intensive. So volume is an extensive property and I don't really care whether the volume of a 1 cubic meter object is going to increase by a few cubic centimeters. What I want to know is the fractional change in volume, the intensive change in volume. I divide this extensive volume by the volume and that turns it into an intensive quantity. I have to do it that way for this definition a little bit differently than we do for these quantities. For extensive versus intensive heat capacities, for example, we're used to just, if I want to make this an intensive quantity, I just use the intensive internal energy and throw a bar over top of both the energy and the heat capacity. In this case the thermal expansion coefficient is always an intensive quantity as defined as 1 over V dV dt at constant P. A second equally important name thermodynamic quantity is the isothermal compressibility. That one looks similar but now instead of asking how much the volume changes as I heat something up, now we're asking how much the volume changes as I increase the pressure on something. So if I squeeze on an object, increase the pressure, its volume is going to go down. That's an important property of the material to know how much it will decrease in volume as its pressure increases. In particular when we do that isothermally, this defines the isothermal compressibility or isothermal compressibility coefficient. This variable, that's a Greek letter kappa, it's not a capital K or lower case K, that's Greek letter kappa that's used for the isothermal compressibility coefficient. It also includes this factor of 1 over V so that it's an intensive property. It includes the negative sign as you may have guessed because when I increase the pressure, when the pressure goes up, the volume of an object is going to go down. So volume is going to decrease as the pressure increases. That means this derivative dV dP at constant T is going to be a negative quantity. Normally for properties that we measure, list, tabulate, we like those to be positive numbers so we throw the negative sign in here so that kappa itself is a positive number because this negative multiplies the negative change in the volume when the pressure is increased. So that's also a practically important quantity. If the pressure is changing, this will tell us again how much the volume of the object will change in response to that change in pressure. So with this collection of five different named quantities, the heat capacities, the Joule-Thompson coefficient, thermal expansion and isothermal compressibility coefficients, that completes the list of things that we simply give names to because they're practically important and because they can be measured relatively easy in the lab. So what do we do with these named thermodynamic quantities? We do much the same with them as we've done in the past for heat capacities, which is to say determine how much the numerator changes in response to the denominator so we can use these quantities to work numerical problems and we'll work an example of that next.