21  Thermodynamics, Restricted or Not

There are various ways of restricting the applicability of thermodynamics, including

• microcanonical only (i.e. constant energy)
• equilibrium only
• reversible only
• ideal gases only
• vertical size small compared to kT/mg.
• et cetera.

Indeed, there are some people who seem to think that thermodynamics applies only to microcanonical reversible processes in a fully-equilibrated ideal gas.

To make progress, we need to carefully distinguish two ideas:

  a) Simplifying assumptions made in the context of a particular scenario. Depending on details, these may be entirely appropriate. Sometimes the gases involved are ideal, to an excellent approximation … but not always. Sometimes a process is reversible, to an excellent approximation … but not always.

  b) Restrictions applied to the foundations of thermodynamics. We must be very careful with this. There must not be too many restrictions, nor too few. Some restrictions are necessary, while other restrictions are worse than useless.

Some thermodynamic concepts and/or formulas necessarily have restricted validity.

• As discussed in section 11.4, there are situations where it is impossible to define a temperature.
• The Boltzmann distribution law (equation 9.1 and figure 9.1) is valid only in equilibrium.
• The notion of equiprobable states (equation 9.8) applies exactly only in microcanonical equilibrium, although it may be a worthwhile approximation in other situations.
• Deciding how many macroscopic variables are needed to describe the macrostate requires some judgment, and depends on knowing the context. For example, equation 7.8 and similarly equation 15.14 are restricted to cases where advection of energy is insignificant, changes in the number of particles are insignificant, changes in magnetic fields or other applied fields have no significant effect, et cetera. If you want to lift these restrictions, you have to add additional terms to the equations.

In contrast, very importantly, the law of conservation of energy applies without restriction. Similarly, the law of paraconservation of entropy applies without restriction. You must not think of E and/or S as being undefined in regions where “non-ideal” processes are occurring. Otherwise, it would be possible for some energy and/or entropy to flow into the “non-ideal” region, become undefined, and never come out again, thereby undermining the entire notion of conservation.

The ideas in the previous paragraph should not be overstated, because an approximate conservation law is not necessarily useless. For example, ordinary chemistry is based on the assumption that each of the chemical elements is separately conserved. But we know that’s only approximately true; if we wait long enough uranium will decay into thorium. Still, on the timescale of ordinary chemical reactions, we can say that uranium is conserved, to an excellent approximation.

When a law has small exceptions, you shouldn’t give up on the law entirely. You shouldn’t think that just because a process is slightly non-ideal, it becomes a free-for-all, where all the important quantities are undefined and none of the laws apply.

If you want to make simplifying assumptions in the context of a specific scenario, go ahead … but don’t confuse that with restrictions on the fundamental laws.

Also, in an elementary course, it might be necessary, for pedagogical reasons, to use simplified versions of the fundamental laws … but you need to be careful with this, lest it create misconceptions.

• As an example: an imperfect notion of entropy in terms of multiplicity (equation 9.8) is better than no notion of entropy at all. However sooner or later (preferably sooner) you need to understand that entropy is really defined in terms of statistics (equation 2.2 or equation 27.6), not multiplicity.
• As another example: In an elementary course, it might be appropriate to start by applying thermo to ideal gases. However, sooner or later (preferably sooner) it is very important to consider other systems; otherwise you risk horrific misconceptions, as discussed in section 9.3.3.

Finally, it must be emphasized that one should not ask whether thermodynamics “is” or “is not” applicable to a particular situation, as if it were an all-or-nothing proposition. Some concepts (such as energy and entropy) are always valid, while other concepts (such as equilibrium and temperature) might or might not be valid, depending on the situation.