Integrating denominator

One must now examine the integrability of the differentials hi equation (A2.1.121 and equation (A2.1.13), which are examples of what matliematicians call Pfoff differential equations. If the equation is integrable, one can find an integrating denominator k, a fiiiictioii of the variables of state, such that = d( ) where d( ) is... 

If a Pfaff differential expression DF = Xdx + Tdy+Zdz has the property that every arbitrary neighbourhood of a point P(x, y, z) contains points that are inaccessible along a path corresponding to a solution of the equation DF = 0, then an integrating denominator exists. Physically this means that there are two mutually exclusive possibilities either a) a hierarchy of non-intersecting surfaces (x,y, z) = C, each with a different value of the constant C, represents the solutions DF = 0, in which case a point on one surface is inaccessible... 

So far, the themiodynamic temperature T has appeared only as an integrating denominator, a fiinction of the empirical temperature 0. One now can show that T is, except for an arbitrary proportionality factor, the same as the empirical ideal-gas temperature 0jg introduced earlier. Equation (A2.1.15) can be rewritten in the fomi... 

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. 

Pfaffian Differential Expressions with Three or More Variables and the Conditions for the Existence of an Integrating Denominator We extend the expression for the Pfaffian differentials in three or more variables, by writing it as... 

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. 

Application of the condition for exactness shows that both derivatives equal zero, so equation (2.45) must be exact. Thus, we have determined that when an ideal gas is involved, T is an integrating denominator for Sqrev, and the right hand side of equation (2.45) is the total differential for a state function that we will represent as dS.cc... 

Our analysis above has shown that an integrating denominator must exist for the composite system and for the two subsystems. Let us designate these integrating denominators by 0, 0], and 2 for the composite, subsystem 1 and subsystem 2, respectively. In general, we would expect the integrating denominator to be a function of the variables associated with each entity. Thus,... 

Given these relationships, we can show that neither 0, 0], nor 02 can depend upon the z variables.ff Thus, the integrating denominators, 0, 0i, and 02, assumed originally for generality to include a dependence on the z variables, are in fact, independent of them. This leaves a dependency only on 9 and the respective E s. 

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. 

We have already shown that the absolute temperature is an integrating denominator for an ideal gas. Given the universality of T 9) that we have just established, we argue that this temperature scale can serve as the thermodynamic temperature scale for all systems, regardless of their microscopic condition. Therefore, we define T, the ideal gas temperature scale that we express in degrees absolute, to be equal to T 9), the thermodynamic temperature scale that we express in Kelvins. That this temperature scale, defined on the basis of the simplest of systems, should function equally well as an integrating denominator for the most complex of systems is a most remarkable occurrence. 

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... 

In summary, the Carnot cycle can be used to define the thermodynamic temperature (see Section 2.2b), show that this thermodynamic temperature is an integrating denominator that converts the inexact differential bq into an exact differential of the entropy dS, and show that this thermodynamic temperature is the same as the absolute temperature obtained from the ideal gas. This hypothetical engine is indeed a useful one to consider. 

In the Pfaffian form (1.11.13) let us first hold Xi.-.-.x, constant Eq. (1.11.13) then reduces to the form Xu.jdx,- + XjjdXjt this is certainly integrable as was demonstrated in part (a). Thus, one must be able to find a function H with a total differential of the form (1.11.7), as well as an integrating denominator , such that... 

Inasmuch as the same function T of an empirical temperature scale serves as an integrating denominator of specific properties of the system under study, it is called the absolute temperature (function ) in thermodynamics and may be used as a universal function for measuring temperature. 

A reversible incremental transfer of heat, dr Q, between a given system and its surroundings is related to a new incremental function of state, termed the empirical entropy, ds, through an integrating denominator, k, whose physical significance is to be established later. 

If a function L does not admit of an exact differential of the form (9.1.2) it may nevertheless be possible, under conditions established below, to set up functions q x, ..., Xi,..., Xn) such that the ratio dL/q = dR does constitute an exact differential. Pfaffian forms of this genre are of special interest they are said to be holonomic or integrable. For obvious reasons q is said to be an integrating denominator and 1/, an integrating factor. 

The manner in which y is transformed to y(x) is arbitrary, but is commonly done by applying an integrating denominator , or an integrating factor. Thus, if the differential expression... 

As we shall see. S is a function of state, hence dS is exact. Hence, T is an integrating denominator for the Pfaff differential Dq. That is, dividing the inexact differential Dq by T produces the exact differential dS. 

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. 

It is clear that, if a Pfaffian expression admits one integrating denominator, it must admit an infinite number of them. Any function of a, S a), can be used to describe the family of surfaces, by the relation... 

---