Pfaff equation

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. 

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... 

As we have seen earlier, the thermodynamic variables p, V, T, U, S, H, A, and G (that we will represent in the following discussion as W, X, T, and Z) are state functions. If one holds the number of moles and hence composition constant, the thermodynamic variables are related through two-dimensional Pfaffian equations. The differential for these functions in the Pfaff expression is an exact differential, since state functions form exact differentials. Thus, the relationships that we now give (and derive where necessary) apply to our thermodynamic variables. 

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. 

We have previously shown that the Pfaff differential <5pressure-volume work equation (2.43) is an inexact differential. It is easy to show that division of equation (2.43) by the absolute temperature T yields an exact differential expression. The division gives... 

One must now examine the integrability of the differentials in equation (A2.1.12) and equation (A2.1.13), which are examples of what mathematicians call Pfaff differential equations. If the equation is integrable, one can find an integrating denominator 7, a function of the variables of state, such that = d( ) where d( ) is... 

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