Hello, in this video, we're going to talk about the stability of thermodynamic systems. The Second Law implies that since S is a maximum at equilibrium, then dS = 0. But that also implies that the second derivative is less than 0, and we haven't explored that requirement yet. The problem of stability that we typically face is twofold. One is what's called mutual stability. That's a stability of two systems with respect to each other concerning the partition of energy, volume, or mole numbers. That's our fundamental problem of thermodynamics, where we have a part A and Part B, are they stable with respect to each other? In other words, if I remove the barrier, does something happen? The other problem is intrinsic stability. That's the stability of a simple closed system. And the classic example of that, that you may have been exposed to in your undergraduate chemistry class, is supersaturation, a supersaturated solution. And the classic example they show is sugar that has been dissolved in water at a higher temperature. And then it's very slowly cooled, and the sugar does not come out of solution until you tap the side of the vessel that the sugar is in. In intrinsic stability, you can think of it as some division within the material where for whatever reason you might ask, is this little portion in equilibrium with the rest of it? Intrinsic stability of sugar, for example, when you have the supersaturated solution, no, it isn't in equilibrium. And you have to perturb it in order to get it to go into equilibrium. Stability, consider two identical subsystems isolated from their surroundings and initially isolated from each other. If the fundamental relation for both is the same, then the initial entropy is going to be 2S(U, V, N) of the two systems. If we were to transfer energy from A to B in some small amount delta U, then the final entropy would be S of U minus delta U and V and N plus S of U plus delta U, V, and N. Now suppose the entropy is of the form shown in the plot here, in that sort of curvy shape, and we start out at with an energy of U0. If we transfer a little bit of U from one side to the other, then the two entropies of the two subsystems are different. And in this case, I mean, sorry, S-final, the final entropy would be greater than the initial entropy. And if that's the case, that means that the initial condition was unstable. And if the constraint were removed, there would be energy flow across the wall. So we just, that is the case, that is an example of the case of mutual stability. But if in a single system, a small, local perturbation produces a region with a higher entropy, then that system is intrinsically unstable. So in essence, the criterium is the same for both neutral and intrinsic stability. So the condition for the stability of the initial state is that the final entropy be less than the initial entropy. And the limit as delta U goes to 0, that means the second derivative of S with U at constant V and N must be less than or equal to 0. Likewise, if we allow changes in volume, then the second derivative of S with volume at constant U and N must be less than or equal to 0. And these require the fundamental relation to be concave in the region of interest. And these are a local criterion. So one might have a fundamental relationship that looks like this curve ABCDEFG. The region CDE is locally unstable. And the region BCDEF, we say is globally unstable. However, if we were to draw a tangent line BE'F, that tangent line is stable. And that, in fact, will represent the equilibrium fundamental relation in the region, as the states along BCDEF will not occur as equilibrium states. More generally, if both U and V vary, this type of stability criterion becomes as shown. This is just the problem of multi-variable maximization. And for each of the potentials, there's a different set of derivative criteria, and they're shown on this slide. You can get all this from your advanced calculus book. So Maxwell's relations can be used to explore the consequence of these stability requirements. And it turns out that local stability requires that the isothermal compressibility be greater than the adiabatic compressibility, that C sub P be greater than C sub V. And that the partial of P with v at constant temperature, which is equal to 1 over the isothermal compressibility, be less than or equal to 0. So that's it for this video. Thanks for listening. And in the next video, we'll explore this subject some more.