Caratheodory

The approach that will be outlined here is due to Caratheodory [1] and Bom [2] and should present fresh insights to those familiar only with the usual development in many chemistry, physics or engineering textbooks. However, while the fonnulations differ somewhat, the equations that finally result are, of course, identical. 

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 thennodynamics, especially in the Bom-Caratheodory approach, the existence of temperature has to be inferred from experiment. 

Caratheodory C 1909 Untersuchungen tiber die Grundlagen der Thermodynamik Math. Ann. 67 355-86... 

Basic iaws, using the Caratheodory approach. Appiications to gases, mixtures and soiutions, chemicai and phase equiiibria, eiectrochemicai systems, surfaces. 

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. 

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. 

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. 

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... 

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. 

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. 

The Caratheodory treatment is grounded in the mathematical behavior of Pfaffian differential expressions (equation (2.44), and the observation that a... 

The Caratheodory analysis has shown that a fundamental aspect of the Second Law is that the allowed entropy changes in irreversible adiabatic processes can occur in only one direction. Whether the allowed direction is increasing or decreasing turns out to be inherent in the conventions we adopt for heat and temperature as we will now show. 

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. 

The sign is consistent with our convention for work if the system does work on the environment, the energy of the system must decrease, and vice versa. This formulation of the definitions of adiabatic systems and of energy and the subsequent discussion of the first law originated with Caratheodory [6]. 

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 . 

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). 

Caratheodory, C. (1967). Calculus of Variations and Partial Differential Equations of the First Order, Part II, tr. R.B. Dean and J.J. Brandstatter (Holden-Day,... 

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

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