Differential equation Pfaffian

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. 

Thus, exact or integrable Pfaffians lead to non-intersecting solution surfaces, which requires that solution curves that lie on different solution surfaces cannot intersect. For a given point p. there will be numerous other points in very close proximity to p that cannot be connected to p by a solution curve to the Pfaffian differential equation. No such condition exists for non-integrable Pfaffians, and, in general, one can construct a solution curve from one point to any other point in space. (However, the process might not be a trivial exercise.)... 

A second Pfaffian differential of interest to us now is the one for the differential quantity of heat, 8c/KV, associated with a reversible process.11 We obtain it by combining equation (1.47) with the first law statement (equation (2.4) that relates U, vv, and q... 

In this discussion, we will limit our writing of the Pfaffian differential expression bq, for the differential element of heat flow in thermodynamic systems, to reversible processes. It is not possible, generally, to write an expression for bq for an irreversible process in terms of state variables. The irreversible process may involve passage through conditions that are not true states" of the system. For example, in an irreversible expansion of a gas, the values of p. V, and T may not correspond to those dictated by the equation of state of the gas. 

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. 

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

The quantities dX and d Y are called differentials, the coefficients in front of dX and dT are called partial derivatives,11 and dZ is referred to as a total differential because it gives the total change in Z arising from changes in both X and Y. If Z were to depend upon additional variables, additional terms would be included in equation (A 1.1) to represent the changes in Z arising from changes in those variables. For much of our discussion, two variables describe the processes of interest, and therefore, we will limit our discussion to two independent variables, with the exception of the description of Pfaffian differentials in... 

A Pfaffian form is also known as a total differential equation [10, pp. 326-330], This type of differential equation plays an important role in thermodynamics. Consider the vector function f(x) of the vector argument x. The scalar product f(x) dx is... 

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. 

The expression (3.3) is known as the Gibbs Jundamental form of the system (it represents a special Pfaffian form, as it is called in differential calculus). Every process that can be performed by the system must satisfy this differential form. On the left-hand side is the total differential of energy (because energy is a state function ) the energy forms located on the right-hand side of the equation generally do not constitute total differentials, although they can often be summed up into total differentials (see later). 

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