 So as we've seen, the key feature of a phase transition, a transition between two different phases of matter, is that the free energy change between the two phases is equal to zero at the particular temperature of the phase transition. So at the temperature where those two phases coexist, the difference in their free energy is zero. So we know something about free energy. We know Gibbs free energy is equal to enthalpy minus t times entropy. And if the temperature is constant, then we can say the change in Gibbs free energy is the change in the enthalpy minus t times the change in the entropy. And that's true for molar quantities as well as it's true for the extensive quantities. So if we're doing that at the phase transition, delta G of the phase transition is enthalpy change of the phase transition minus the temperature of the phase transition multiplied by the change in entropy when the phase transition happens. The delta G of the phase transition is the thing that we know is equal to zero. That means that these two terms must be equal to each other. The enthalpy of a phase transition must be equal to the temperature times the entropy change of that phase transition. Or the form where that equation is most useful most of the time is to isolate the temperature. The temperature at which a phase transition occurs is enthalpy of that phase transition divided by entropy of that phase transition. So here's a relationship between three quantities. The temperature, let's say the melting point, the enthalpy change, when a substance melts and the entropy change when a substance melts. Or same thing for boiling or any other phase change. So to see how that equation works, let's say we have some data about water. We can look up the enthalpy and the entropy of melting or fusion. Those values, enthalpy of melting is about six kilojoules per mole. And the entropy is 22 joules per mole kelvin. And while we're at it, we can also look up data for boiling. The molar enthalpy of vaporization when water boils is equal to 40.68 kilojoules per mole and the molar entropy of vaporization when water boils. Its entropy increases by 109.02 joules per kelvin per mole of water. So those are all values that we can look up. Those have been measured and tabulated many times before. What we can do with that information is we can make predictions. The temperature of a phase change, in particular, let's say the temperature of fusion, the melting point for water, is just the ratio of the entropy of fusion, the enthalpy of fusion to the entropy of fusion. So if we take the ratio of these two values, 6.01 kilojoules per mole, that would be 6,010 joules per mole. If I divide that by 22 joules per mole kelvin, the units, joules cancel, moles cancel, 1 over 1 over kelvin gives me units of kelvin as the temperature should be. And that ratio, it won't surprise you to learn that if you divide 6,010 by 22, that will come out to be 273 kelvin. So that's a value you already knew. I don't need to tell you that water melts at 0 degrees Celsius or 273 kelvin or 32 Fahrenheit. But that value happens to be also the ratio between its enthalpy of fusion and its entropy of fusion. Likewise, we can predict something else. We already know the boiling point of water, the temperature at which it vaporizes, enthalpy of vaporization over the entropy of vaporization. The ratio of these two quantities, 40,680 joules per mole divided by 109.02 joules per mole kelvin. Units cancel in the same way. We have a few more sig figs here. So that ratio, if I take 40,680 divided by 109.02, you can confirm that that ratio comes out to be 373.1 kelvin is our prediction for the boiling point of water. So again, not a surprise that 100 degrees Celsius is the boiling point of water. Just reinforces the fact that this ratio of the entropy to the entropy tells us what the phase change temperature is for any sort of phase change, melting, boiling, whatever phase change you want. And of course, this is true not just for water, but other substances where you may be less familiar with the melting and boiling points. You can use it just as well.