Principle Caratheodory

We have proved Caratheodory s principle. Caratheodory s principle states From an arbitrary point (state), there is a finite region (set of states of finite measure) which cannot be reached by an adiabatic process, reversible or irreversible. This region may be taken arbitrarily near to the initial point. 

We acknowledge S.M. Blinder s treatment in Advanced Physical Chemistry, A Survey of Modern Physical Principles, Macmillan, London, 1969, pp. 300-336, as a primary source for the Caratheodory development presented here. 

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... 

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. 

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. 

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. 

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

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. 

In this section, we present two derivations of Caratheodory s principle from the physical statements of the second law one is based on physical statement a and the other is based on physical statement b. Then we derive the mathematical statement of the second law from Caratheodory s principle. 

Thus, we have again proved Caratheodory s principle. 

The well-known procedure of Caratheodory applies a restricted Caratheodory s principle (there exist a set of states of finite measure neighboring a given one which are inaccessible along reversible adiabatics) to a treatment of the geometry of solutions of Pfaffian expressions... 

Equation (4-35) can be proved most directly by using the arguments of Proof I of Caratheodory s principle. In Fig. 4-3 we have drawn a representation of the various processes undergone by the system in the T-S. 3... 

Equation (4-35) can also be proved using the arguments of Proof II of Caratheodory s principle.f Again we consider various processes represented in the T-S plane. We consider the following cycle represented in Fig. 4-4 ... 

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. 

Titulaer, U.M., van Kampen, N.G. On the deduction of Caratheodory s axiom from Kelvin s principle. Physica31, 1029-1032 (1965)... 

States A and B can be arbitrarily close. We conclude that every equilibrium state of a closed system has other equilibrium states infinitesimally close to it that are inaccessible by a reversible adiabatic process. This is Caratheodory s principle of adiabatic inaccessibility. ... 

There is, however, the additional dimension of temperature in the A-dimensional space. Do the paths for possible reversible adiabatic processes, starting from a common initial point, lie in a volume in the A-dimensional space Or do they fall on a surface described by r as a function of the work coordinates If the paths he in a volume, then every point in a volume element surrounding the initial point must be accessible from the initial point by a reversible adiabatic path. This accessibility is precisely what Caratheodory s principle of adiabatic inaccessibility denies. Therefore, the paths for all possible reversible adiabatic processes with a common initial state must lie on a unique surface. This is an (A — 1)-dimensional hypersurface in the A-dimensional space, or a curve if N is 2. One of these surfaces or curves will be referred to as a reversible adiabatic surface. 

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. 

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