 So when we defined the Gibbs free energy, we did so because we wanted a function whose natural variables were temperature and pressure. So let's investigate that and see what in fact is true about the Gibbs free energy when we control the temperature and pressure. In particular, let's see what the Gibbs free energy does when we keep the temperature and the pressure constant. So let's start by writing down the differential of this equation, dG. So we could just write down the fundamental equation, but I want to do that slightly differently this time for reasonable c in just a second. dG, take the differential of the terms on the right and I've got du minus Tds minus ss dt plus pdv and vdp. So having taken the differential of each of these terms, we get this long expression. I would like to know what is the change in the Gibbs free energy if we hold the temperature and pressure constant. So if we keep the temperature constant, then of course dt is equal to zero, so this term goes away. And if we keep the pressure constant, this term will go away. So under conditions of constant temperature and pressure, I could rewrite this as du minus Tds plus pdv. And now this is the reason I didn't want to start directly from the fundamental equation for dG. Instead of using the fundamental equation for du, I'm going to use the first law and rewrite du as heat plus work. And I've still got minus Tds and a plus pdv. But we can make some connections between dq and Tds and between dw and pv. In particular, let me rearrange this equation so those terms are nearer to each other. dq, I'll move the minus Tds over here. So we can do a couple things. The easier one, I can think of work as minus pdv so that this term is now going to cancel this term. Work is going to cancel the pdv term, so those two terms go away. The relationship between heat and T times ds is not quite as simple. We know from the Clausius theorem that depending on whether the process is reversible or irreversible, the ds, the change in the entropy will be larger than or perhaps equal to the change in heat divided by the temperature. So rearranging that slightly. We know that Tds is larger than dq. So in this expression, if I have dq minus Tds, dq minus a term that's larger than dq, that number will be some number that's negative. So what we've determined is that the change in the Gibbs free energy, if we're keeping the temperature and pressure constant, will always be less than or equal to zero. If it's an irreversible or spontaneous process, that change will be negative. If it's a reversible process, the change will be equal to zero. So summarizing those statements, if we have the change in the Gibbs free energy negative, that means the same thing as when the change in the entropy was larger than dq over T, that is for a spontaneous process. If the equality sign holds, if the change in the Gibbs free energy is exactly equal to zero, that must be true because ds is equal to dq over T, and that's true for a reversible process or an equilibrium process. Likewise, we could say irreversible or spontaneous process. And the last possibility, if the Gibbs free energy change for a process would be positive, that is a non-spontaneous process. That's a process that won't actually happen. So we won't observe a process, measure its Gibbs free energy change at constant temperature and pressure, because that process wouldn't have happened. But hypothetically, if a process would have a Gibbs free energy change greater than zero at constant temperature and pressure, we know that process won't happen and it's not spontaneous. So this is going to turn out to be a very useful criterion to use to predict spontaneity of reactions at constant temperature and pressure. We'll be able to use it for a lot of different things, be able to predict which phase will spontaneously transform into what other phase. So we can understand phase changes, we'll understand what salts will or won't dissolve in different solvents. We'll understand what reactions will or won't happen at constant temperature and pressure.