Figure 17.1: Cramped versus Uncramped : Without Cycles versus With Cycles
We have a choice. We can select at most one of the following two options, not both:
| (17.1) |
where
| (17.2) |
and where Q is a well-defined, well-behaved a function of state, just as E is a function of state.
There are numerous ways of demonstrating that it’s physically impossible to choose both options at the same time. The basic problem is that the physics that allows you to build a heat engine requires cycles, and if there are cycles there must be multiple inequivalent paths from one point to another in thermodynamic state-space. To say the same thing the other way, if the system is so constrained – so cramped – that there are no cycles, or if the integral of T dS is zero around any cycle, then you can’t build a heat engine. The situation is shown in figure 17.1 (which is a copy of figure 0.2 as seen in section 0.3).
The existence of the Carnot cycle A→X→Y→Z→A implies that there are (at least!) two inequivalent paths from A to Z, including the simple path A→Z along a contour of constant entropy, and the more complex path A→X→Y→Z
Not all cycles are Carnot cycles. The path A→Y→Z→A is a another perfectly legitimate thermodynamic cycle. Compared to a Carnot-cycle engine, a reversible heat engine that uses the A→Y→Z→A cycle is more complex and harder to analyze, but only slightly so.
Within uncramped thermodynamics, you are allowed to build things that aren’t heat engines. That is, you can have cycles that don’t convert any heat-bath energy into useful work. The various possibilities are summarized in the Venn diagram in figure 17.2.
We now focus attention on the immediate neighborhood of point A in figure 17.1. It must be emphasized that paths can depart from point A in innumerably many directions. The Carnot cycle uses only two of these directions (namely the contour of constant T during one part of the cycle, and the contour of constant S during another part of the cycle). However, there are infinitely many non-Carnot cycles, and infinitely many ways in which a reversible path can depart from point A such that neither T nor S is constant. The blue line in figure 17.1 is just one of many such paths.
In the immediate neighborhood of point A, we can distinguish these paths by their direction. The red line in figure 17.1 represents a change in T in the direction of constant S, while the blue line represents a change in T along some other direction.
Therefore, uncramped thermodynamics requires us to treat dT as a vector. If you think of dT as representing some kind of “change in T” you need to specify the direction of the change (as well as the magnitude). Whenever something has a direction and a magnitude, you should suspect that it is a vector.
For large excursions, we would need to specify the entire path, but in the immediate neighborhood of a given point, it suffices to know the magnitude and direction. Therefore a vector such as dT can be considered a function of state. It depends on the local state, not on the entire path. It is a vector-valued function of state.
| The existence of reversible heat engines is sufficient to guarantee the existence of innumerably many inequivalent paths from A to Z, and also to guarantee the existence of innumerably many directions for the vectors located at point A. |
In this chapter, for simplicity, we have not mentioned any
irreversible processes. Irreversibility is not the cause of the the multiple paths and multiple directions. Conversely, multiple paths and multiple directions are not the cause of irreversibility. |
As an even simpler example of the distinction between cramped and uncramped thermodynamics, consider the elementary example of so-called “heat content” or “thermal energy content” that might arise in connection with a measurement of the heat capacity of a cylinder of compressed gas. We have a problem already, because there are two heat capacities: the heat capacity at constant pressure, and the heat capacity at constant volume. So it is unclear whether the heat content should be CP T or CV T. Now we get to play whack-a-mole: You can remove the ambiguity by rigorously restricting attention to either constant volume or constant pressure … but that restriction makes it impossible to analyze a Carnot-type heat engine.
To repeat: It may at first be tempting to think that the gas cylinder has a so-called “thermal energy” related to T and S, plus a “nonthermal energy” related to P and V, but if you try to build a theory of thermodynamics on this basis you are guaranteed to fail. The sooner you give up, the happier you will be.
| Cramped thermodynamics is a legitimate option. This is option 1 as mentioned at the beginning of this chapter. It is only a small subset of thermodynamics, but it’s not crazy. Almost everyone learns about cramped thermodynamics before they learn about uncramped thermodynamics. Consider for example warming the milk in a baby-bottle. This is almost always carried out under conditions of constant pressure. You’re not trying to build a steam engine (or any other kind of engine) out of the thing. In this case, for this narrow purpose, there is a valid notion of the “heat content” of the system. | Since this document is mostly about uncramped thermodynamics, I have chosen option 2. Therefore you will find almost no mention of “heat content” or “thermal energy” (except in warnings and counterexamples). |
Within limits, the choice is yours: If you want to do cramped thermodynamics, you can do cramped thermodynamics. Just please don’t imagine your results apply to thermodynamics in general. Cramped thermodynamics by definition is restricted to situations where the state-space is so low-dimensional that there is no hope of building a heat engine or a refrigerator or anything like that. There are no Carnot cycles, nor indeed any other kind of nontrivial cycles.
Trying to divide the energy along the lines suggested by equation 17.1 is allowable within cramped thermodynamics, but is utterly incompatible with uncramped thermodynamics. The Q that appears in this equation could be called “heat content” or “thermal energy” or caloric. Long ago, there was a fairly elaborate theory of caloric. The elaborate parts are long dead, having been superseded by thermodynamics during the 19th century. The idea of caloric (aka heat content, aka thermal energy) remains valid only within very narrow limits.
By way of contrast, note that the Locrian versus non-Locrian distinction is compatible with thermodynamics, as discussed in section 8.2.
| It is a bad idea, incompatible with thermodynamics, to overlook the path-dependence of QΓ and pretend that Q is a state function. (See section 6.7.) | It is a fine idea to distinguish Locrian from non-Locrian. (See section 8.2.) |
To repeat, it is OK to talk about “heat content” in the context of warming up a baby bottle. It is OK to talk about “caloric” in connection with a swimming pool as it warms up in the spring and cools down in the fall. It is OK to talk about “thermal energy” in connection with the heat capacity of a chunk of copper in a high-school lab experiment.
However, just because it works in cramped situations doesn’t mean it works in uncramped situations.
It is not OK to talk about “heat content” or “thermal versus nonthermal energy” or “caloric” in the context of uncramped thermodynamics, i.e. in any situation where a thermodynamic cycle is possible. Energy is energy. Energy doesn’t recognize the distinction between thermal and nonthermal, and thermodynamics allows us to convert between the two (subject to important restrictions).
The problem is that the Q that appears in equation 17.1 simply cannot exist in the context of uncramped thermodynamics.
The problem still is that Q exists as a state function only within cramped thermodynamics, not more generally, not in any situation where a thermodynamic cycle is possible. In uncramped thermodynamics, Q may exist as a functional of some path, but not as a function of state.
For a list of constructive suggestions about things that actually do exist as functions of state, see section 7.2.
You can visualize the situation by reference to figure 17.1.
| On the LHS, if we restrict attention to the subspace define by the red line, there is only one path from A to Z. | On the RHS, there are many ways of getting from A to Z, including A→Z, or A→Y→Z, or even paths that include cycles, such as A→X→Y→A→X→Y→A→Z, and so forth. |
| Within the subspace defined by the red line in figure 17.1, you can represent Q as height, and this Q is well defined everywhere in this small, cramped subspace. | You cannot define a Q value as a function of position in a way that is consistent throughout the (T, S) space. The peculiar thing is that you can take almost any simple one-dimensional subspace in the plane and define a consistent Q function there, but you cannot extend this to cover the entire space. The problem is nowhere in particular, yet the problem is everywhere: you cannot assign a consistent height to points in this space. |
Pedagogical remarks: Virtually everyone begins the study of thermodynamics by considering cramped situations. This is traditional … but it is a pedagogical disaster for anyone trying to learn uncramped thermodynamics. Cramped thermodynamics is a not a good foundation for learning uncramped thermodynamics; it is aggressively deceptive.
Virtually every newcomer to thermodynamics tries to extend the ‘heat content” idea from cramped thermodynamics to uncramped thermodynamics. It always almost works … but it never really works.
The next time you feel the need for a measure of “heat content” in the context of uncramped thermodynamics, lie down until the feeling goes away.
