 So we'd like to come up with a relationship between the Gibbs free energy and the temperature. So we know how much the Gibbs free energy is going to change when the temperature changes. We have such a relationship, but that Gibbs free energy changing with the temperature proportionally to the negative of the entropy, that's a true thermodynamic relationship but it's not the most convenient one. So we'd like to come up with a more convenient relationship between the Gibbs energy and the temperature and that's going to be this Gibbs Helmholtz equation. So we can start with our definition of the Gibbs free energy. That's certainly one thing that tells us how the Gibbs energy is related to the temperature, although there's also a temperature dependence hiding inside this enthalpy and hiding inside the entropy. What I'll do next is I'll go through and divide on the left and right side of this equation by temperature. So g divided by t is going to be equal to the enthalpy divided by t minus the entropy. I've divided away this temperature. So that's a relationship between not the Gibbs free energy and enthalpy and entropy but g over t and how that depends on the enthalpy and the entropy. So this expression now, if I take the temperature derivative of each side of this expression, turns out we'll get something very useful. So take the temperature derivative at constant pressure. When I do that on the left, I'm just going to write the derivative of g over t with respect to t at constant pressure. On the right, I'll do that slightly differently. The derivative of this ratio h over t, I can use the product rule and say the temperature derivative is first taking the derivative of the numerator, it's 1 over t times the derivative of h. And then product rule says also take h and then the derivative of the 1 over t is negative 1 over t squared. So these two terms together are the temperature derivative of h over t. I've got left this negative s, so the derivative of s. I'll go ahead and write ds dt at constant p as our derivative of s with respect to t at constant pressure. So I've evaluated the derivative on the right-hand side, except we actually know what two of these derivatives are. dhdt at constant p should look familiar. That's by definition, that's the heat capacity at constant pressure. The second term, we can't do too much with, minus h over t squared. The third term, we also know something about this. ds dt at constant p, that's the derivative we've seen before. That is, in fact, heat capacity divided by temperature. If the derivative's at constant pressure, it's cp, the constant pressure heat capacity divided by temperature. So notice now that the cp over t in the first term and the negative cp over t in the second term, those cancel each other. And what we're left with, I'll go ahead and write up here, is this result, which is called the Gibbs-Helmholtz equation. It's a true statement, but not terribly convenient that the free energy changes with the temperature proportionally to the negative of the entropy. It's also a true statement, more convenient to use that if we're not interested in g, if what we think about is g over t, this quantity Gibbs free energy divided by the temperature, that quantity changes as I change the temperature proportionally to minus h over t squared. It looks like a more complicated relationship, but it's easier to use in practice because it's fairly simple to go into lab and determine the enthalpy of an object, determine how much the enthalpy changes. When I heat something up or cool it down, that's just the heat of the process which we can determine with methods like calorimetry. So measuring the enthalpy, measuring the temperature, that's usually much easier than measuring the entropy. So in practice, this turns out to be a more useful way of measuring or predicting how much g over t changes as I change the temperature. It's also an important result in a theoretical sense because this will have occasion to use this several times when we talk about solutions, when we talk about colligative properties or phase changes. So this is also an important result that we'll use as a starting point to get other thermodynamic results as well.