Caratheodory thermodynamics

In the orientation given to it by Caratheodory, thermodynamics is made to depend upon the postulate that in the immediate vicinity of any state of a system of more than one body there exist other states which cannot be reached by reversible adiabatic transitions. 

A careful analysis of the fundamentals of classical thermodynamics, using the Born-Caratheodory approach. Emphasis on constraints, chemical potentials. Discussion of difficulties with the third law. Few applications. 

It can be shown mathematically that a two-dimensional Pfaffian equation (1.27) is either exact, or, if it is inexact, an integrating denominator can always be found to convert it into a new, exact, differential. (Such Pfaffians are said to be integrable.) When three or more independent variables are involved, however, a third possibility can occur the Pfaff differential can be inexact, but possesses no integrating denominator.x Caratheodory showed that expressions for SqKV appropriate to thermodynamic systems fall into the class of inexact but integrable differential expressions. That is, an integrating denominator exists that can convert the inexact differential into an exact differential. 

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

Like the engine-based statements, Caratheodory s statement invokes limitations. From a given thermodynamic state of the system, there are states that cannot be reached from the initial state by way of any adiabatic process. We will show that this statement is consistent with the Kelvin-Planck statement of the Second Law. 

Thus, we can conclude that, within the neighborhood of every state in this thermodynamic system, there are states that cannot be reached via adiabatic paths. Given the existence of these states, then, the existence of an integrating denominator for the differential element of reversible heat, Sqrev, is guaranteed from Caratheodory s theorem. Our next task is to identify this integrating denominator. 

To summarize, the Carnot cycle or the Caratheodory principle leads to an integrating denominator that converts the inexact differential 8qrev into an exact differential. This integrating denominator can assume an infinite number of forms, one of which is the thermodynamic (Kelvin) temperature T that is equal to the ideal gas (absolute) temperature. The result is... 

Figure 3.2 compares a series of reversible isothermal expansions for the ideal gas starting at different initial conditions. Note that the isotherms are parallel. They cannot intersect since this would give the gas the same pressure and volume at two different temperatures. Figure 3.3 shows a similar comparison for a series of reversible adiabatic expansions. Like the isotherms, the adiabats cannot intersect. To do so would violate the Caratheodory principle and the Second Law of Thermodynamics, since the gas would have two different entropies at the same temperature, pressure, and volume. 

Within each solution surface are numerous subsets of points that also satisfy the differential equation bQ = dF = 0. These subsets are referred to as solution curves of the Pfaffian. The curve z — 0, y + y2 = 25.00 is one of the solution curves for our particular solution surface with radius = 5.00. Others would include x = 0, y2 + z2 — 25.00, and r — 0,. v2 + r2 = 25.00. Solution curves on the same solution surface can intersect. For example, our first two solution curves intersect at two points (5, 0, 0) and (-5, 0. 0). However, solution curves on one surface cannot be solution curves for another surface since the surfaces do not intersect. That two solution surfaces to an exact Pfaffian differential equation cannot intersect and that solution curves for one surface cannot be solution curves for another have important consequences as we see in our discussion of the Caratheodory formulation of the Second Law of Thermodynamics. 

Schottky effect in solids 580-5 Second law of thermodynamics 56-90 absolute temperature, identification of as integrating factor 71-8 Caratheodory principles differentials 63-7 and inaccessible states 68-71... 

Most branches of theoretical science can be expounded at various levels of abstraction. The most elegant and formal approach to thermodynamics, that of Caratheodory [1], depends on a familiarity with a special type of differential equation (Pfaff equation) with which the usual student of chemistry is unacquainted. However, an introductory presentation of thermodynamics follows best along historical lines of development, for which only the elementary principles of calculus are necessary. We follow this approach here. Nevertheless, we also discuss exact differentials and Euler s theorem, because many concepts and derivations can be presented in a more satisfying and precise manner with their use. 

An essential step in the Caratheodory formulation of the second law of thermodynamics is a proof of the following statement Two adiabatics (such as a and b in Fig. 6.12) cannot intersect. F rove that a and b cannot intersect. (Suggestion Assume a and b do intersect at the temperature Ti, and show that this assumption permits you to violate the Kelvin-Planck statement of the second law.)... 

Second Law of Thermodynamics. There have been numerous statements of the second law. To paraphrase Clausius It is impossible to devise an engine or process which, working in a cycle, will produce no effect other than the transfer of heat from a colder to a warmer body. According to Caratheodory, the Second Law can be stated as follows Arbitrarily close to any given state of any closed system, there exists an unlimited number of other states which it is impossible to reach from a given state as a result of any adiabatic process, whether reversible or not . 

Caratheodory s9 principle derives the three laws of thermodynamics using differential geometry, from certain limits on the possible paths between adjacent differential surfaces. 

The above definitions reflect the Clausius view of the origin of entropy at the beginning of the twentieth century a reformulation of thermodynamics by -> Born and Caratheodory showed firstly that the formulation of the second law of - thermodynamics requires a consideration of the heat and work relationships of at least two bodies, as implicitly discussed above, and that entropy arises in this formulation from the search for an integrating factor for the overall change in heat, dq when the simultaneous changes in two bodies are considered. The Born-Caratheodory formulation then leads naturally to the restriction that only certain changes of state are possible under adiabatic conditions. 

The above immediately leads to the application of Caratheodory s theorem to the equation HQ - 0, which holds for adiabatic systems. Since the heat flow is related to changes in the thermodynamic coordinates of the system through the Pfaffian form HQ - this means there are states that... 

We begin here the discussion of the Second Law of Thermodynamics. This law has been enunciated in many different forms, the most prominent being the formulations by Kelvin and by Planck. These will be presented later as consequences of the approach derived below. Undoubtedly, the most elegant statement of this Law was provided by Caratheodory in the following form ... 

We follow here the presentation provided by H.A. Buchdahl, The Concepts of Classical Thermodynamics, Cambridge University Press, 1966, Chapters 5, 6. Readers interested in a somewhat more detailed exposition of Carath6odory s approach and in the introduction of the thermodynamic temperature concept may also consult Chapter 9, where it was placed so as not to interrupt the current derivation. In retrospect it seems to be more direct to start with the formulation of the Second Law adopted here rather than working with the more elegant theory developed by Caratheodory. 

We now utilize the machinery of the preceding Sections to rationalize the Second Law of Thermodynamics as specified by Caratheodory ... 

The present volume involves several alterations in the presentation of thermodynamic topics covered in the previous editions. Obviously, it is not a trivial exercise to present in a novel fashion any material that covers a period of more than 160 years. However, as best as I can determine the treatment of irreversible phenomena in Sections 1.13, 1.14, and 1.20 appears not to be widely known. Following much indecision, and with encouragement by the editors, I have dropped the various exercises requiring numerical evaluation of formulae developed in the text. After much thought I have also relegated the Caratheodory formulation of the Second Law of Thermodynamics (and a derivation of the Debye-Hiickel equation) as a separate chapter to the end of the book. This permitted me to concentrate on a simpler exposition that directly links entropy to the reversible transfer of heat. It also provides a neat parallelism with the First Law that directly connects energy to work performance in an adiabatic process. A more careful discussion of the basic mechanism that forces electrochemical phenomena has been provided. I have also added material on the effects of curved interfaces and self assembly, and presented a more systematic formulation of the basics of irreversible processes. A discussion of critical phenomena is now included as a separate chapter. Lastly, the treatment of binary solutions has been expanded to deal with asymmetric properties of such systems. 

Redlich [3] has criticized the so-called zeroth law on the grounds that the argument applies equally well for the introduction of any generalized force, mechanical (pressure), electrical (voltage), or otherwise. The difference seems to be that the physical nature of these other forces has already been clearly defined or postulated (at least in the conventional development of physics) while in classical thermodynamics, especially in the Bom-Caratheodory approach, the existence of temperature has to be inferred from experiment. 

The second law of thermodynamics, like the first, represents a generalization of the results of a large number of experiments. In Sec. 4-1 we present two equivalent physical statements of the second law. In Sec. 4-2 we present the mathematical statement of the second law and determine how a criterion for equilibrium can be set up, making use of the mathematical statement. In Sec. 4-3 the mathematical statement of the second law is shown to be equivalent to the physical statements. The argument proceeds by demonstrating that Caratheodory s principle can be derived from the physical statements. 

The main purpose of this book is to present a rigorous and logical discussion of the fundamentals of thermodynamics and to develop in a coherent fashion the application of the basic principles to a number of systems of interest to chemists. The concept of temperature is carefully discussed, and special emphasis is placed on the appropriate method for the introduction of molecular weights into thermodynamics. A new treatment of the second law of thermodynamics is presented which demonstrates that Caratheodory s principle is a necessary and sufficient consequence of the physical statements of Clausius and Kelvin. 

There are thus three main avenues of approach to thermodynamics. The first is that which starts from the laws of probability, and in its origins is specially associated with the name of Boltzmann. The second is that of Carnot, of Clausius, and of Thomson, and sets out from experience of the flow of heat and of the convertibility of heat into work. The third is that of Caratheodory. 

For proof, Stokes theorem is used. Note that in Sect. 9.10 and Chap. 10 of Kestin (1979) we find detailed descriptions concerning the Born-Caratheodory theorem based on conventional thermodynamics. 

This lack of unification can be illustrated by the following example. Margenau and Murphy state that the most satisfactory formulation of the laws of thermodynamics is probably that of Caratheodory, based on the properties of PfaflBan differential equations, yet the Caratheodory formulation is dealt with in such a cursory manner (p. 98) in the revision of Lewis and Randall s Thermodynamics , by Pitzer and Brewer, that it is not listed in either the name or the subject index. Nevertheless, many practical workers in this country and in America will undoubtedly study and use this modern version of Lewis and Randall s book. 

The Molecular Theory of Solutions , by Prigogine, includes thermodynamic considerations, and Zemansky s book is a later edition of the book mentioned earlier. Wilson s volume is of a fairly advanced level and requires a fairly high standard of mathematical ability. It is mainly designed for the use of physicists. Chapters include accounts of partial differentiation, and the book approaches entropy through Carnot cycles but also describes Caratheodory s axiomatic approach. Superconductivity and solutions are considered thermodynamically. Caldin s introduction is designed for chemistry undergraduates. A student who has mastered the text should be well prepared to go on to more advanced work. Chisholm and de Borde s book develops the equations for Bose-Einstein, Fermi-Dirac, and classical statistics by unusual routes and then applies the... 

Kirkwood and Oppenheim s book is based on lectures by Kirkwood from notes taken by Oppenheim, Karplus, and Rich. The purpose of the book is to present a rigorous and logical discussion of fundamentals, but it does not treat statistical thermodynamics. The treatment of the Second Law discusses Caratheodory s principle. 

The year 1970 saw the publication of the second edition of Guggenheim s monograph and of an Annual Review with a section on the thermodynamics of solid surfaces. Halberstadt has made a translation of Munster s Classical Thermodynamics , a book which is not for the beginner but is suitable for the advanced student. The development of the laws is presented classically, after Joule, Clausius, and Carnot, and in an axiomatic manner after Caratheodory. The treatment is concentrated and formal,... 

---