Entropy and the Second Law in Open Systems


In our two preceding chapters, we have seen The Definitions of Entropy, and The Second Law of Thermodynamics. In this 3rd chapter, we will enlarge the discussion to open, or nonequilibrium, systems. You should read both of the prior chapters, before trying to cover this one.

As we saw in the previous chapter, the 2nd law of thermodynamics applies only to isolated systems in thermodynamic equilibrium. There are ways to use the 2nd law, in systems that don't meet these fundamental criteria, and we will look at those here. But it must be emphasized that you cannot take the 2nd law off the shelf, and apply it "as is", without regard to the isolated or equilibrium state of the system.

Temperature and Equilibrium

You all have a pretty good idea of what temperature is. You read it from a thermometer, and it tells you whether things are "cold" or "hot". You recognize what the common number mean, and you know that you would be much happier running a marathon of the temperature were 65F, as opposed to 105F.

But the temperature you are used to is an average quantity, not in the sense of being "mediocre", but in the sense of "average" from simple arithmetic. You stick a thermometer into something, and a zillion atoms or molecules run into it, some really fast, some really slow, and most at more or less the same speed. Each one has some kinetic energy ("energy of motion"), and leaves some of that energy behind when it hits the thermometer. The average of all those kinetic energies in collision with the thermometer is the temperature that you measure with a thermometer, and that you can feel on your skin.

If the temperature is the same, everywhere in a system, then we say that the system is in an equilibrium state. Most real systems are always just a little bit out of equilibrium, but we don't worry about it, and we pretend that they are true equilibrium systems. But if the temperature is remarkably different, from one part of a system to another, then we can't ignore it, and we have to admit that the system is in a nonequilibrium state.

Entropy is rigorously defined only for systems that are in equilibrium. Just look at the defining equation, from classical thermodynamics, S = Q/T. There can't be a T unless there is equilibrium.

Open, Closed and Isolated

If anything can pass into, or out of, a system, we say it is an open system. If only matter can pass into, or out of, a system, but not energy, then we call it a closed system. If neither matter nor energy can pass into, or out of, a system, then we call it an isolated system.

We have a definition of the 2nd law from our previous chapter, a standard definition from standard thermodynamics.

Processes in which the entropy of an isolated system would decrease do not occur, or, in every process taking place in an isolated system, the entropy of the system either increases or remains constant
The definition explicitly requires the system in question to be isolated. This is a non trivial observation. If the system were not isolated, then entropy could pour out over the boundary, and the entropy decrease instead of increase.

The 2nd Law in Nonequilibrium Systems

So, with all the stress on equilibrium and isolated, how does one use the 2nd law in systems that don't measure up? There's really only one answer: Fake it. In this case, the "fake" is to take your nonequilibrium system, and carve it up (Mathematically, not physically) into smaller subdomains, each of which has a fairly constant temperature throughout. They don't have to all have the same temperature, they only need to have their own temperature. You treat each subdomain like an "isolated" system, computing all the internal changes in entropy and energy, and then add in any energy and/or entropy that comes across the boundary from any other subdomain that the subdomain in question is in contact with. In practice, this requires one to solve all of the relevant equations, for all of the subdomains, simultaneously (so you don't lose track of anything important).

The only real trick is to notice that if your system is not isolated, then you have to keep track of all the entropy and energy that goes in or out, along with the strictly internal sources & sinks, for both entropy and energy. Of course, it's not just the subdomains that count, you also have to handle the outer boundary of the whole system as well. If you can create curcumstances where the outer boundary is impassable, and the system as a whole is isolated, so much the better, but you don't really need to.

If the outer boundary is impassable, and the system isolated, then you know that the aggregate change in entropy must be 0. If not, just replace 0 with the net entropy change across the system outer boundary, and you know thet system as a whole can't go beyond those limits.

In this way, you can apply the essential spirit of the 2nd law, even in the case of a system that is neither in equilibrium, nor isolated.

Go back to The Second Law of Thermodynamics
Go back to The Definitions of Entropy
Go back to The Collected Writings of Tim Thompson
Go all the way to Tim Thompson's Home Page
Page dated 19 February 2002