 So we can predict whether a process is spontaneous or not, at least if it's happening at constant volume, by looking at the change in internal energy minus t times ds. So this combination of the energy and the entropy together tells us whether the process is going to be spontaneous or not, spontaneous if it comes out negative, equilibrium if it comes out equal to zero, non-spontaneous if that quantity is positive. So that quantity is important enough. We're going to want to give it a name. So to do that, we will define a new quantity. So since this looks like some energy minus t times some entropy, changes in the energy and changes in the entropy, we'll start out by defining this quantity, this new quantity, energy minus temperature times entropy. And this turns out not to be exactly the same as this criterion we've got up here, but it's going to be very close, as we'll see in just a second. This new quantity that we've just defined is called the Helmholtz energy. So we can go with the Helmholtz energy, or we can go with the Helmholtz free energy, either one. And for now, we've just defined that quantity. I've just made up a new variable, energy minus t times entropy. We're interested in knowing the differential of u and minus t times the differential of s to predict whether a process is spontaneous or not. So let's take the differential of the Helmholtz energy on the left. A becomes dA on the right. The differential of u is du. And product rule tells us the differential of ts is tds and sdt. And those each have a negative sign in front of them, because the ts had a negative sign in front of it. So here's what I meant. This quantity, dA, is not exactly equal to du minus tds, the quantity we're interested in. It has this extra term in it, sdt. So in order to make this quantity look exactly like the property, the combination of things that predicts whether a process is spontaneous or not at constant volume, we have to do one more thing. In order to get rid of this last term, I have to say, let's also do this at constant temperature. If I do a process at constant temperature, then dt will be 0. If I do it at constant temperature. So then I have dA is equal to du minus tds, and the last term has gone away. du minus tds is exactly the quantity we know that must be negative or equal to 0 for a process that's actually going to happen. For a process that's spontaneous or equilibrium. So we've learned that dA must be less than or equal to 0 if we're doing the process at constant temperature and volume. Constant volume, so that this expression is correct. Remember that was because that we needed the PV work to be equal to 0, and constant temperature so that this term, the sdt term, disappears. So under these fairly restricted set of conditions, if I do something at constant temperature and constant volume, this new quantity that I've defined, the home holds free energy. If that change in home holds free energy is negative, the process will be spontaneous. In fact, I'll write these conditions over here. If I do something at constant temperature and constant volume, and I measure the change in the home holds free energy, this new thing that I've just defined. If that's negative, the process will be spontaneous. If the change in the home holds free energy is equal to 0, the process will be in equilibrium. And if the change in the home holds energy is positive, that's a sign of a non-spontaneous process. So with this definition and this extra restriction on constant temperature, then we've got a very useful way of predicting whether a process is spontaneous or non-spontaneous under this set of conditions under constant temperature and volume. So soon enough, we'll find a way of doing this under other sets of conditions also, but for now, we'll focus on the home holds free energy.