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Since ancient times smelting of ore has been a primary task of the foundrymen, and the modern chemical engineer faces a thermodynamically similar task when he refines oil and natural gas. Nature offers her resources mostly as mixtures and man mostly wants them in the pure form, or at least in enriched form. It has long since been learned that the recipe for enrichment is smelting or destillation, but the thermodynamic knowledge behind the tricks governing these processes are more recent. They consist of outwitting the entropy of mixing, and sometimes the heat of mixing can promote that purpose.

In chemical reactions the additive constants in the internal energies and entropies of the constituents play a significant role. To be sure, it is not the constants themselves that matter but certain combinations, which we may call the heat of reaction and the entropy of reaction. These values can be determined experimentally and have been tabulated.

The balance of the heat of reaction and the entropy of reaction goes a long way in determining whether a reaction can proceed and in which direction it proceeds. We give some examples: Dissociation of oxygen, ammonia synthesis and photosynthesis of glucose. The latter appears to be impossible, if we did not know better.

The rich thermomechanical properties of shape memory alloys result from an austenitic-martensitic phase transition of the metallic lattice and from a twinning deformation in the low temperature phase, the martensite.

The thermodynamicist is interested in the phenomena, because — once again like in all phase transitions — they provide a non-trivial example for the competition between energy and entropy, with entropy gaining the upper hand at high temperatures. Recognizing this we are able to produce a model for shape memory behaviour which is capable of simulating all the observed features, at least qualitatively. Actually we describe several models, a thermodynamic one, a kinetic one and a numerical one. The latter employs molecular dynamics; it is particularly instructive and the accompanying CD permits the reader to view the crystalline rearrangement by observing the atoms shifting between their potential wells.

Shape memory alloys have found numerous small technical applications, and applications in the medical field. We start the chapter with a selective review.

It is obvious that as temperature decreases, the entropy plays a smaller and smaller role in minimizing the available free energy . This is so, because entropy enters into as —. However, the 3 law implies more: The entropy itself tends to zero with , irrespective of . This is so, because even tiny potential barriers between different states become unsurmountable for the body when the thermal motion becomes ever weaker.

The zeroth law of thermodynamics defines the as quantity that is continuous at an interface between two bodies. This definition is essentially the same as saying that T is the factor of proportionality between heat flux and entropy flux.

On the other hand we often define the temperature as a measure for the mean kinetic energy of the atoms of a body. We might call that quantity the . In equilibrium both definitions are equivalent, but there is a difference in non-equilibrium. In this chapter we demonstrate this observation.

We also show that, according to the kinetic theory of gases, there is a tensorial relation between heat flux and entropy flux. Indeed, for a strong degree of rarefaction of the gas, the entropy flux is not related to the heat flux alone. Thermodynamics has not, however, progressed far enough to evaluate such cases.

The easiest way to take care of a paradox is to firmly close the eyes and claim that it does not exist — or does not exist anymore. This usually happens to the Gibbs paradox. To be sure the situation is complex. To begin with, there are two Gibbs paradoxes, one in thermodynamics of mixtures and the other one in statistical mechanics. The latter one is due to an overinterpretation of Boltzmann’s formula for the entropy, cf. Sects. 3.6 and 3.7.

Here we review the situation and show how the Gibbs paradox of statistical mechanics can be resolved. The argument provides the opportunity to speak about degenerate gases and the reason for degeneracy.

Non-equilibrium thermodynamics is an extensive and successful field which, however, is not the subject of this book. Here we merely review — in the briefest possible manner — the Eckart theory which provides us with an explicit expression for the dissipative entropy source in a viscous, heatconducting fluid. We need this in Chap. 18 for an estimate of dissipation in the context of the entropy increase of radiation. And once we have that expression, we use it to disprove the “principle of minimum entropy production”, a popular misconception in irreversible thermodynamics.

The thermodynamic densities and fluxes of a radiation field are best calculated by considering the field as a photon gas and using the methods of the kinetic theory of gases, cf. Chap. 3. Even without specifying the collision term — due to the interaction of matter and radiation — one may obtain important results about thermal and viscous dissipation in matter, if the radiative fluxes are known, and when stationary conditions prevail. This is so because in the stationary state the dissipative entropy source is balanced by the in- and outgoing fluxes of entropy.

This chapter prepares the reader for the subsequent one in which the dissipation of the earth’s atmosphere is considered.

A bare planet or satellite, like mercury or the moon, have a fairly trivial balance of incoming and outgoing radiation. That balance determines the surface temperature and the entropy sources. It is always true — as it was for the concrete block of Chap. 18 — that the entropy source of radiation due to matter is between one and two orders of magnitude bigger than the dissipative entropy source due to heat conduction and viscous friction.

The earth’s climate is determined by the incoming solar radiation and by the efflux of radiation emanating from the clouds and from the earth’s surface. Meteorologists have determined, how these fluxes are partitioned and what the temperatures of the atmospheric layers are. We use these data to determine the dissipative entropy source in the atmosphere and compare it to the dissipation of entropy by mankind. The latter is numerically negligible.

A population of metaphorical hawks and doves, competing for the same resource is a well-known model of game theory characterized by a certain contest strategy. This model is modified here by making rewards and penalties dependent on the prevailing price level. Also a second strategy is invented which the hawks and doves may choose or not depending on which strategy provides a higher gain.

It turns out that at intermediate and high price levels the population will not achieve maximum gain, if hawks and doves remain homogeneously mixed. Rather they will partly or even fully segregate to maximize their gain. The ideas presented here are extrapolations of thermodynamics of binary mixtures whose components may mix at high temperature while at low temperature they exhibit miscibility gaps.