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.