Caratheodory’s theorem

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. 

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... 

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. 

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)). 

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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