 So, we've seen in the second law of thermodynamics that entropy is a pretty crucial quantity in being able to predict whether a process is spontaneous or not, whether it will happen or not. So, entropy is very important. It's also very common that we want to know whether the entropy is going up or going down as the temperature changes for a particular process. So, that raises the question of how quickly does the entropy change as we change the temperature in some chemical process. So, it turns out we can make quite a bit of progress determining what this rate of change of the entropy with the temperature is, and we can start with the Clausius theorem. That is a relationship between the entropy and the temperature. Change in entropy is the heat for reversible process divided by the temperature. So if we additionally assume that we're doing a process at constant volume so that the heat, so at constant volume dq is the same thing as du, then we can write this as, let's see, I'll write it as 1 over temperature times du. If we're doing the process at constant volume, there's this relationship between the entropy and the energy and the temperature. As one more step, we know something else about the internal energy. We know that since we're at constant volume already, the heat capacity at constant volume is du dt at constant volume. So if I just rearrange this expression, if we're at constant volume, the change in the energy is heat capacity times change in temperature. So du is cv dt, we can use that expression to replace the du in this expression. So I can rewrite that the change in the entropy is writing du as cv dt and keeping the 1 over t. We found that change in entropy, if we're at constant volume, is heat capacity divided by temperature, multiplied by the change in temperature. So now, if I just bring the dt over to the left-hand side, this equation can be written ds dt is equal to constant volume heat capacity over temperature. And in particular, since we've done all this at constant volume, both this ds and this dt are happening at constant volume. So this derivative that we're taking is ds dt at constant volume. So that expression, and actually the form in which we'll usually use that, is not the extensive version of this expression with the entropy and the heat capacity, but if I stick a bar over both of these, those are now intensive quantities. The change in the molar energy, I'm sorry, the change in the molar entropy with temperature is the molar constant volume heat capacity divided by the temperature. So that's an expression we'll use fairly often, so I'll put that in a box. And that tells us what we're interested in. At least if we're interested in a process at constant volume, it tells us how quickly the entropy is changing as we heat up or cool down the system as we change its temperature. That rate of change of the entropy with temperature is heat capacity divided by temperature. What if we're not interested in constant volume? There's other things that we could hold constant or other processes we could imagine. Everything is going to proceed very similarly if we're at constant pressure instead of constant volume, so just to illustrate that, it's still true. The Clausius theorem is still true. DS is dq reversible over T. At constant pressure, dq is equal to dh. So then entropy would be dh over T. In the next step, we would say the constant pressure heat capacity, dcp, is dh dt. So I can write dh as cp times dt all divided by temperature. So that would be the result DS is cp over T dt. Or if I write that as a derivative, ds divided by dt, if it's done at constant pressure, will be heat capacity over temperature. Again, I could use the extensive formula. I'll usually prefer to use the intensive version of that equation. So that's the equivalent of the statement we obtained over here. If we're at constant volume, entropy changes with temperature as heat capacity over temperature, but it's the constant volume heat capacity. If we're doing it at constant pressure, it's the constant pressure heat capacity divided by temperature. So at least under these two conditions, constant volume and constant pressure, we have an expression for how quickly the entropy is changing with temperature. We can use those expressions to actually evaluate the numerical amount that the entropy increases when a system is heated up or cooled down when it's temperature changes. So we'll do that next.