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Ludwig Meier e. Bergisch Gladbach, Germany. It will enable us to establish CH forproducts of several systems, and thereby show, via Theorem 2, that entropy exists andis additive. They might look a bit abstract, so a few words of introduction mightbe helpful. In order to relate systems to each other, in the hope of establishing CH for compounds,and thereby an additive entropy function, some way must be found to put them into contactwith each other.
The total energy U will notchange but the individual energies, U1 and U2 will adjust to values that depend on Uand the work coordinates. This new system with the thread permanently connected then behaves like a simple system with one energy coordinate but with several workcoordinates the union of the two work coordinates. This holds for everychoice of U1 and U2 whose sum is U.
Moreover, after thermal equilibrium is reached, thetwo systems can be disconnected, if we wish, and once more form a compound system,whose component parts we say are in thermal equilibrium. That this is transitive is the 14 Thus, we can not only make compound systems consisting of independent subsystems which can interact, but separate again , we can also make a new simple system out of two simple systems.
To do this an energy coordinate has to disappear, and thermal contact does this for us. All of this is formalized in the following three axioms. Thermal contact. Thermal splitting. T T TA Zeroth law of thermodynamics. A11 and A12 together say that for each choice of the individual work coordinates there is a way to divide up the energy U between the two systems in a stable manner. A12 is the stability statement, for it says that joining is reversible, i. A13 is the famous zeroth law, which says that the thermal equilibrium is transitive, and hence an equivalence relation.
It will appear later. Two more axioms are needed. This assumption essentially prevents a state-space from breaking up into two pieces that do not communicate with each other. Without it, counterexamples to CH for compound systems can be constructed. A14 implies A8, but we listed A8 separately in order not to confuse the discussion of simple systems with thermal equilibrium.
Transversality, A14, requires that each X have points on each side of its adiabat that are in thermal equilibrium.
Entropy – the key concept of thermodynamics, clearly explained and carefully illustrated. This book presents an accurate definition of entropy in classical. The Entropy Principle book. Read reviews from world's largest community for readers. Entropy is the most important and the most difficult to understand t.
A15 is a technical and perhaps can be eliminated. When temperature is introduced later, A15 will have the meaning that all systems have the same temperature range. This postulate is needed if we want to be able to bring every system into thermal equilibrium with every other system. Universal temperature range. First, the thermal join establishes CH for the scaled productof a simple system with itself. The basic idea here is that the points in the product that lieon the thermal diagonal are comparable, since points in a simple system are comparable.
Thisis the key point needed for the construction of S, according to 9. The importance oftransversality is thus brought into focus. With some more work we can establish CH for multiple scaled copies of a simple system. Thus, we have established S within the context of one system and copies of the system,i. As long as we stay within such a group of systems thereis no way to determine the unknown multiplicative or additive entropy constants. This is based on the following. Withits aid we arrive at our chief goal, which is CH for compound systems.
The com-parison hypothesis CH is valid in arbitrary scaled products of simple systems. The entropy function is unique, up to an overall multiplicative constant andone additive constant for each simple system under consideration.
Concavity of S implied by A7 ,Lipschitz continuity of the pressure and the transversality condition, together with some 17 It follows that T is not always capable of specifying a state, and thisfact can cause some pain in traditional discussions of the second law—if it is recognized,which usually it is not. Mixing and Chemical Reactions. Nevertheless, a nagging doubt will occur to some, because thereare important adiabatic processes in which systems are not conserved, and these processesare not yet covered in the theory. A critical study of the usual textbook treatments shouldconvince the reader that this subject is not easy, but in view of the manifold applicationsof thermodynamics to chemistry and biology it is important to tell the whole story andnot ignore such processes.
As long as we consider only adiabatic processes that preserve the amountof each simple system i. It thenbecomes a nontrivial question whether the additive constants can be chosen in such away that the entropy principle holds. Oddly, this determination turns out to be far morecomplex, mathematically and physically than the determination of the multiplicative con-stants Theorem 2.
We present here a general and rigorous approach which avoids all this. The additive entropy constants do not matterhere since each function Si appears on both sides of this inequality. It is important tonote that this applies even to processes that, in intermediate steps, take one system intoanother, provided the total compound system is the same at the beginning and at the endof the process. This is excluded by Axiom Hence we arrive at 20 The other is that there might be a true gap, i. In nature only states containing the same amount of the chemical elements can be transformed into each other.
The other possible source of non-uniqueness, a nontrivial gap 32 for systems with the same composition in terms of the chemical elements is, as far as we know, not realized in nature. Note that this assertion can be tested experimentally without invoking semiper- meable membranes. London A, Physics Reports in press. Austin Math. To include mixing processes and chemical reactions as well, the entropy constants for different mixtures of given ingredients, and also of compounds of the chemical elements, have to be chosen in a consistent way. The converse, i. There exist many variants of non-equilibrium thermodynamics.
A concise overview is given in the monograph by Lebon et al. Most of these formalisms focus on states close to equilibrium. A further point to note is that the role of entropy in non-equilibrium thermodynamics is considerably less prominent than in equilibrium situations.
Equilibrium entropy is a thermodynamic potential when given as a function of its natural variables U and V , i. For a description of non-equilibrium phenomena, on the other hand, more input than the entropy alone is needed. It is a meaningful question, nevertheless, to ask to what extent an entropy can be defined for non-equilibrium states, preserving as much as possible of the useful properties of equilibrium entropy. We emphasize that need not contain all non-equilibrium states that the system might, in principle, possess, but only a part that is relevant for the problems under consideration.
A natural requirement is that states in are reproducible. It is not clear to us that the entropy of an exploding bomb, for instance, is a meaningful concept although the energy might be. Another point to keep in mind is that a non-equilibrium state is, typically, either time dependent or it is not isolated from its environment, as in the case of a non-equilibrium steady state that has to be connected to reservoirs that cause fluxes of heat or electric current to flow through it.
The physical meaning of on is supposed to be the same as before, i.
X Y means that Y can be reached from X by a process that in the end leaves no traces in the surroundings except that a weight may have been raised or lowered. For the non-equilibrium states in , it is not natural to require A4 scaling and A5 splitting , but we shall assume the following:. We consider this to be very natural, physically. The basic question we now ask is: What can be said about possible extensions of S to functions on that are monotone with respect to , i.
For define. The essential properties of these functions are collected in the following proposition. The arrows indicate adiabatic state changes. Online version in colour. But then X Y by transitivity. From the additivity of the equilibrium entropy S and. The following theorem clarifies the connection between adiabatic comparability and uniqueness of an extension of the equilibrium entropy to the non-equilibrium states. Recall that two states X and Y are called comparable w.
Particularly noteworthy is the equivalence of i , iii and vi below, which may be summarized as follows: a non-equilibrium entropy, characterizing the relation , exists if and only if every non-equilibrium state is adiabatically equivalent to some equilibrium state. That i is equivalent to ii follows from d and g in proposition 3. The implications are obvious.
If v holds, then either Z X or X Z. Such a reservoir can be regarded as an idealization of a simple system without work coordinates that is so large that an energy change has no appreciable effect on its temperature defined, as usual, to be the inverse of the derivative of the entropy with respect to the energy. X is an equilibrium state, then. This follows as usual by considering the total entropy change of the system plus reservoir, i.
Equality is reached if the process is reversible. This can be seen as follows. The GB entropy is therefore one possible choice of a function, which is monotone w. According to our analysis, it characterizes the relation if and only if the CP is valid on the whole state space as GB assume as part of their second law; see also assumption 2 in [ 15 ] , in which case all entropies on extending S coincide. As we have already stated, however, and shall discuss further below, we consider it implausible to assume adiabatic comparability for general non-equilibrium states.
If CP does not hold on , the entropy S GB may depend in a non-trivial way on the choice of the thermal reservoir and the final state X 0. The inequalities 3.
According to theorem 3. Although there are idealized situations when such comparability can be conceived, it seems to be a highly implausible property in general. The problem can already be expected to arise close to equilibrium as we now discuss. The states in are here described by local values of equilibrium parameters such as temperature, pressure and matter density.
In particular, one can define a local entropy density by using the equilibrium equation of state, and, by subsequently integrating this entropy density over the volume of the system, one obtains a global entropy. In this situation, also the local temperature has to be replaced by a different variable cf.
These six conditions are all highly plausible if is interpreted as the relation of adiabatic accessibility in the sense of the operational definition. To do this an energy coordinate has to disappear, and thermal contact does this for us. They hold protests, make new laws, create new forms of technology, work to alleviate poverty, and pursue other noble goals. This book focuses on parameter estimation using entropy for a number of distributions frequently used in hydrology. The cyclist is going uphill and so she will be doing work pushing the pedals around, i. This article may be too technical for most readers to understand.
Also, the argument above for establishing adiabatic equivalence with equilibrium states no longer applies. Although we do not have a rigorous proof, we consider it highly implausible that a state that is significantly influenced by the flux can be adiabatically equivalent to an equilibrium state, where no flux is present, for this would mean that turning the flux on or off could be done reversibly. Unless this can be done, however, CP does not hold on and there is no unique entropy. If one moves further away from equilibrium, not even EIT may apply and CP becomes even less plausible. In extreme cases like an exploding bomb, one may even question whether it is meaningful to talk about entropy as a state function at all, because the highly complex situation just after the explosion cannot be described by reproducible macroscopic variables.
They provide bounds on the possible adiabatic state changes in the system and the maximum work that can be extracted from the system in a given state and a given environment. The difference. To elucidate the concepts and issues discussed above, we may consider a simple toy example. The system consists of two identical pieces of copper that are glued together by a thin layer of finite heat conductivity. We regard the state of the system as uniquely specified by the energies or, equivalently, the temperatures T 1 and T 2 of the two copper pieces that are assumed to have constant heat capacity.
The layer between them is considered to be so thin that its energy can be ignored. As equilibrium entropy, we take. If we extend the relation defined above by allowing the copper pieces to be temporarily taken apart and using them as thermal reservoirs between which a Carnot machine can run to equilibrate the temperatures reversibly, then the previous forward sector will be extended and is now characterized by the unique extension of S to. If the parts are unbreakably linked together, however, the situation is different. An irreversible heat flux between the two parts during the adiabatic state change is then unavoidable.
If the heat conduction is governed by Cattaneo's rather than Fourier's law, it is necessary to introduce the heat flux as a new independent variable and apply EIT as discussed in the last section. The general objections against the CP and hence the existence of a unique entropy mentioned then apply. But even if we stay with Fourier's law and the two-dimensional state space of the toy model, it is clear that the extended forward sector, obtained by applying Carnot machines in addition to rubbing and equilibration, will depend on the relation between the heat conductivity of the separating layer between the parts, and the heat conductivity between the Carnot machine and the copper pieces.
PMID: Elliott H.