Theorem Caratheodory

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. 

According to the Caratheodory theorem, the existence of an integrating denominator that creates an exact differential (state function) out of any inexact differential is tied to the existence of points (specified by the values of their x, s) that cannot be reached from a given point by an adiabatic path (a solution curve), Caratheodory showed that, based upon the earlier statements of the Second Law, such states exist for the flow of heat in a reversible process, so that the theorem becomes applicable to this physical process. This conclusion, which is still another way of stating the Second Law, is known as the Caratheodory principle. It can be stated as... 

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. 

The Caratheodory theorem establishes the existence of an integrating denominator for systems in which the Caratheodory principle identifies appropriate conditions — the existence of states inaccessible from one another by way of adiabatic paths. The uniqueness of such an integrating denominator is not established, however. In fact, one can show (but we will not) that an infinite number of such denominators exist, each leading to the existence of a different state function, and that these denominators differ by arbitrary factors of . Thus, we can make the assignment that A F (E ) = = KF(E) = 1. 

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. 

In this chapter, the basic elements of convex analysis are introduced. Section 2.1 presents the definitions and properties of convex sets, the definitions of convex combination and convex hull along with the important theorem of Caratheodory, and key results on the separation and support of convex sets. Further reading on the subject of convex sets is in the excellent books of Avriel (1976), Bazaraa et al. (1993), Mangasarian (1969), and Rockefellar (1970). 

APPENDIX MATHEMATICAL PROOF FOR THE NECESSARY CONDITION OF CARATHEODORY S THEOREM... 

We establish in this Section that Caratheodory s Theorem also formulates a necessary statement, by proving the following ... 

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... 

It is of interest to note that the First Law may be set up on a more mathematical basis by adapting the so-called Caratheodory theorem to the differential Pfaffian form for work dW = Yidyi, Eq. (1.5.11a). As explicitly developed in Chapter 9, when adiabatic conditions are imposed the theorem, under the constraint dW = 0, necessitates the existence of an associated function of state that is constant under these conditions. 

We call attention to the mathematical construction in Chapter 9 when the transfer of heat assumes the linear form dQ — dxi Caratheodory s theorem necessitates the existence of a function of state that is fixed under adiabatic conditions, and whose change is tied to the transfer of heat under reversible conditions. 

Mathematical Proof of the Caratheodory Theorem and Resulting Interpretations Derivation of the Debye-Hiickel Equation... 

According to (9.3.17) we then require either that all f be zero or that all Fijn vanish. The first alternative cannot be correct since all the functions except fk may be chosen arbitrarily and fk is absent from the summation over i. This leaves only the alternative that all Fijk = 0. From the earlier discussion involving Eq. (9.2.20) it follows that the Pfaffian dL = dxi is integrable. We have thereby established the necessary condition for the Caratheodory theorem of Section 9.2 to hold. Given the fact that in the neighborhood of a point in phase space other points are inaccessible via solution curves of the form X, dxi — 0, the Pfaffian form is integrable. 

Use is made of Caratheodory s theorem If a Pfaffian expression has the property that, in every neighborhood of a point P, there are points which cannot be connected to P along curves which satisfy the Pfaffian equation, dQ = 0, then the Pfaffian expression must admit an integrating denominator. 

From Section 6.2.1.3, the dimension of the AR is equal to the dimension of S, it follows that the maximum number of independent reactor structures is directly related to the number of independent reactions taking part in the system. Moreover, this analysis may be determined in the absence of reaction kinetics and a feed point— the results are a consequence of the system reaction stoichiometry and Caratheodory s theorem only. 

A useful consequence of the dimension of the AR may be used to relate the maximum number of parallel structures needed to generate the AR, which is achieved by use of Caratheodory s theorem (Carathdodory, 1911 Eckhoff, 1993). Feinberg (2000a) shows that for an AR constructed in IR, the following limits, in terms of parallel reactor structures, may be enforced ... 

As a consequence of Caratheodory s theorem, the maximum number of parallel structures needed to generate the AR is equal to the dimension of the AR (which is computed from rank(A)). 

EckhofF. Hetty, Radon, and Caratheodory type theorems, tn P. Gruber and j. Wilts, editors. Handbook of Convex Geometry, pages 389-448. North-Holland, Amsterdam, 1993. 

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. 

Reprise to the Second Law. Mathematical Proof of the Caratheodory s Theorem and Resulting Interpretations... 

Is the converse also true That is to say, from the assmnption of nonaccessibility can one deduce that J, X,dr, is holonomic The answer is in the affirmative and is furnished through Caratheodory s theorem ... 

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