 So the fundamental equation for the Gibbs free energy led us directly to these two statements. Just tell us how the Gibbs free energy changes as we change the pressure at constant temperature or the temperature at constant pressure. We've already made good use of this expression to tell us how the Gibbs free energy changes for an ideal gas, for example, if we change the pressure. We can make similar use of this expression to tell us how the free energy changes as we change the temperature. So if we focus on this expression a little more closely, if I break that expression apart and rewrite it as dg is equal to minus s dt. So break the derivative up into two differentials. Once I've written it in this form, we have to remember that expression is only good at constant pressure. So at constant pressure, the free energy is changing proportionally to the temperature and that proportionality constant being negative the entropy. So that already tells us something quite interesting. We know quite a bit about entropy at this point. We know entropy is always positive. The third law of thermodynamics tells us entropy starts out at zero at zero Kelvin and the hotter it gets, the larger the entropy gets. So the entropy is always a positive number. So what that means is that negative s is of course a negative number. So as the temperature increases, this expression tells us that the free energy is going to decrease because of this negative sign here. So when if I heat something at constant pressure, its free energy is automatically going to go down. It's guaranteed to go down. So that's an interesting statement that has some consequences that we'll run into when we start talking about phase changes, but that's a little bit in the future. What we can do with this expression for now is to say if I actually want to know what is the quantitative amount of a Gibbs free energy change when I change the temperature of an object, when I heat it up at constant pressure, if I integrate dg to get delta g, then on the right side I can just take the integral of s times dt, if I do that from some initial temperature to some final temperature, then doing this integral, the integral of s as a function of temperature from t1 to t2, throw a negative sign in front of that, that tells me how much the free energy is changing. But the entropy of course depends on the temperature. Unless we're only talking about a very small temperature range, we can't pull the entropy out of this integral. We would have to know how the entropy depends on temperature in order to perform this integral. We do know something about how the entropy depends on the temperature, but in general the entropy is not a terribly convenient quantity. It's not like the expression we used for pressure changes for the Gibbs free energy. It's very easy to go into the lab and measure the volume of something. Volume doesn't change necessarily that much for some substances as we change the pressure for solids and liquids. This was a very convenient expression that we can use to talk about free energy change as we change the pressure. This one is less convenient in that we don't have an entropy meter that we use to measure the entropy of something. It's a little more difficult to come up with a value of the entropy. What we'll typically do when we want to know how the free energy is just changing as we change the temperature is actually not to use this expression, but to use a slightly different expression and we'll explore that one coming up next.