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Considering the fundamental importance which we now attribute to entropy in physics and chemistry and natural philosophy one may find it odd that Clausius discovered the notion in the context of heat engines. We must realize, however, that thermodynamics started out as the science of the “motive power of heat” and the very name of the science — thermodynamics — recalls that origin.

Clausius’ proof of the existence of entropy and his investigation of its properties are very much “a physicist’s argument” starting from a loosely worded axiom. As such, the proof is looked upon with disdain by analysts. Even so, in this chapter we retrace the steps of Clausius. After all, he was first, and physicists at least find his arguments convincing; they love them!

Carathéodory felt uncomfortable with the notions of heat and heating and therefore Clausius’ axiom, discussed in Sect. 1.3, meant nothing to him. He had to find an alternative axiom and an alternative route to entropy. Being an analyst he found that route in his research on differential forms and their integrating factors. Altogether Carathéeodory’s arguments are more formally mathematical than those of Clausius, and therefore his analysis highlights some concepts of thermodynamics which Clausius took for granted. Therein lies the lasting value of Carathéeodory’s work. In this chapter we review his ideas and — for definiteness — we illustrate them for visco-elastic bodies.

Pupils in Hong Kong schools find writing in Chinese very boring. With regard to this phenomenon, the theoretically framework of ‘Hong Kong Writing Project’ aims to create an environment for language usage, and help pupils to learn how to write efficiently in a pleasurable environment. The project has been implemented in Hong Kong for five years, and the number of schools participating in this project is increasing. Theories of linguistics, psychology and language teaching serve as the project’s foundation and surveys were conducted after the end of the project. Results indicated that most pupils liked writing and did not consider it difficult. Compared with the traditional teaching method, pupils were considered to have performed better when taught by new teaching methods, especially the peer review strategy. The teachers also indicated that the Project helped them to have a better understanding of their pupils.

It is all very well to advertise a concept like entropy as important for natural philosophy, but it is much more efficient to introduce it as a means for making money, or saving money. And, indeed, that is what entropy does! It saves time and money, because it helps to determine the caloric equation of state of gases and fluids. We need to know the form of that equation when we build refrigerators, or when we refine natural gas or oil, or when we produce fertilizers and explosives.

It is often useful and very common to represent the caloric equation of state as an equation for enthalpy rather than internal energy. Enthalpy is a form of energy — appropriate for reversible processes — which combines the internal energy of the body with the potential energy of the loading device, e.g. a heavy piston, that creates and maintains the pressure of the body.

We demonstrate that the entropy is “driven” toward a maximum by the random character of the thermal motion. That drive is capable of exerting forces. Thus the expansive pressure of a gas and the elastic force of a stretched rubber strap are both forces. The study of these cases — gases and rubber — can provide a thorough appreciation of the “mechanism” by which entropy grows. At the same time it becomes clear that the growth is merely probable, albeit , but not strictly deterministic.

Entropy is a measure of disorder; this aspect of entropy is best understood by considering a polymeric rubber molecule. The knowledge of entropy of a rubber molecule implies knowledge of the entropy of a rubber strap which, in turn, allows us to calculate the thermal equation of state of rubber. Altogether this chain of arguments is known as the . Different as gases and rubber may be in appearance, thermodynamically those materials are essentially identical. A joker with an original turn of mind has once commented on this similarity by saying that “rubbers are the ideal gases among the solids”. Both exhibit entropic elasticity.

While Boltzmann did discover the statistical interpretation of entropy, his arguments were pertinent and useful only for systems of independent elements like monatomic ideal gases, or rubber molecules, cf. Chap. 5. In real gases, or liquids, or solids the atoms interact and their statistical treatment requires the statistical thermodynamics of ensembles to which Boltzmann’s ideas were extrapolated.

Statistical thermodynamics succeeds in expressing the thermodynamic equilibrium properties of arbitrary bodies in terms of a single function, . Most often the partition function cannot be calculated analytically, but it may sometimes be approximated. A case where it can be determined is the case of a hydrogen atom at rest in a heat bath.

Equivalent to the growth of entropy in an adiabatic body is the decrease of the available free energy in a body whose boundary is kept at a constant temperature. The form of the available free energy depends on the nature of the working on the body. The working of conservative body forces may be represented by a potential energy but the working on the boundary of the body may assume quite a variety of forms. Special cases are

Generally the available free energy contains an energetic part and an entropic part. And when the free energy assumes a minimum, we may consider this as a compromise between the energy which tends to a minimum and the entropy which tends to a maximum. Those two tendencies — the energetic and the entropic one — often compete, since generally the energy favours a different distribution of matter than the entropy.

The atmosphere of a planet is a case in point for the competition of energy and entropy. Energy tends to assemble all atmospheric molecules on the surface of the planet while entropy tends to spread the molecules evenly throughout space. In the end the entropy wins and planets will be bare of atmosphere. This will happen earlier for a hot planet than for a cold one. Earth, which is a temperate planet in the solar system, has already lost all light gases but it holds on — for the time being — to the heavier ones like nitrogen, oxygen and argon.

Mixing of different constituents is a process associated with a considerable increase of entropy. In fact, entropy opposes de-mixing strongly and complete separation absolutely. The entropic tendency for mixing may be interfered with by the tendency of potential energy to decrease. The Pfeffer tube furnishes an instructive example for such a competition between the gravitational potential energy and entropy. At the same time the Pfeffer tube may be considered as a model for the “mechanism” by which the sap in the capillary ducts of a tree reaches the tree tops.

If energy and entropy compete in arranging the mass of a body, the energy need not be the potential energy of the gravitational field. It may be the potential energy of the intermolecular van der Waals forces, cf. Sect.4.4. The non-convex character of the potential energy creates a qualitative difference between the effects of gravitational energy and of molecular interaction: The transition between the energy-dominated low temperature situation and the entropy-dominated high temperature case is no longer smooth for the particle interaction; rather it occurs abruptly at one temperature and one pressure in a phase transition.