Exact differentials integration

An exact differential integrates to a finite difference, Jf dU = U2 — U, which is independent of the path of integration. An inexact differential integrates to a total quantity, li 6 = 6> which depends on the path of integration. The cyclic integral of an exact differential is zero for any cycle, Eq. (7.7). The cyclic integral of an inexact differential is usually not zero. 

Inspection reveals that dynamic pressure P(r, 0) is an exact differential. Integration of (8-144Z ) with respect to 0 at constant r yields... 

If du x, y) is an exact differential, integration along a closed curve will give the value 0 ... 

If a process is reversible and adiabatic (i.e., dq = 0), then the change in entropy is zero. Also notice that whereas the product of two state functions is a state function, the product of a state function and a differential of a state function (i.e., TdS) do not have to be an exact differential. Integration of Equation 3.20 yields a value for AS in some overall process, but since S is a state function, AS is independent of the path. This means we will be able to find AS for irreversible processes by finding reversible paths with the same end points. 

If the adiabatic work is independent of the path, it is the integral of an exact differential and suffices to define a change in a function of the state of the system, the energy U. (Some themiodynamicists call this the internal energy , so as to exclude any kinetic energy of the motion of the system as a whole.)... 

In the example of the previous section, the release of the stop always leads to the motion of the piston in one direction, to a final state in which the pressures are equal, never in the other direction. This obvious experimental observation turns out to be related to a mathematical problem, the integrability of differentials in themiodynamics. The differential Dq, even is inexact, but in mathematics many such expressions can be converted into exact differentials with the aid of an integrating factor. 

Equation (2.18) is another example of a line integral, demonstrating that 6q is not an exact differential. To calculate q, one must know the heat capacity as a function of temperature. If one graphs C against T as shown in Figure 2.8, the area under the curve is q. The dependence of C upon T is determined by the path followed. The calculation of q thus requires that we specify the path. Heat is often calculated for an isobaric or an isochoric process in which the heat capacity is represented as Cp or Cy, respectively. If molar quantities are involved, the heat capacities are C/)m or CY.m. Isobaric heat capacities are more... 

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... 

Since dS is an exact differential, equations for dS = 0 can be integrated. The integration yields a family of solution surfaces, S = S(.vi,... x ) = constant. Each solution surface contains a set of thermodynamic states for which the entropy is constant.hh... 

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. 

A1.4 State Functions and Exact Differentials Inexact Differentials and Line Integrals... 

By similar reasoning, one can show that differential expressions for which equation (Al.18) is true must yield integrals between two fixed states whose values depend upon the path. Such differential expressions cannot be associated with state functions because of the dependence upon path. Therefore, equations (Al.17) and (Al.18) distinguish between differentials that can ultimately be associated with state functions and that cannot. Expressions for which equation (Al.17) is true are called exact differentials while those for which equation (Al.18) is true are called inexact differentials. 

When the Pfaffian expression is inexact but integrable, then an integrating factor A exists such that AbQ = d5, where dS is an exact differential and the solution surfaces are S = constant. While solution surfaces do not exist for the inexact differential 8Q, solution curves do exist. The solution curves to dS = 0 will also be solution curves to bQ = 0. Since solution curves for dS on one surface do not intersect those on another surface, a solution curve for 8Q — 0 that lies on one surface cannot intersect another solution curve for bQ = 0 that lies on a different surface. 

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.)... 

Euler s theorem 612 exact differentials 604-5 extensive variables 598 graphical integrations 613-15 Simpson s rule 614-15 trapezoidal rule 613-14 inexact differentials 604-5 intensive variables 598 line integrals 605-8... 

Modern mathematical software, such as Mathematica, allows us to compute symbolically the mean square deviation of this approximation from the exact acceleration, integrated over the feasible region, differentiate the resulting expression symbolically with respect to the parameters a and b, set the results to zero and solve the equations symbolically, and simplify the whole lot to find the following remarkably simple expressions... 

While Eq. 5.56 cannot be solved generally as a single equation, it may be solved for a given r. That is, there is a local similarity. Since both terms are exact differentials, the equation may be integrated once with respect to z to yield... 

The important point is that the final value of the integral depends only on the two endpoints, i.e., the value of the function z at (x, yi) and (x2, y2), but not the chosen path of integration (as illustrated in Sidebar 1.4). Moreover, in the special case of a cyclic integral (denoted ), where initial and final limits coincide, the integral (1.15) necessarily vanishes for an exact differential, independent of how the cyclic path is chosen. We can therefore state the following integral criterion for exactness ... 

Finally, we briefly mention the concept of an integrating factor, a multiplicative factor (L) that converts an inexact differential (ctf) to an exact differential (dg), namely,... 

The essential content of the first law is that dU =dq +dw is the (exact) differential of a state property, and hence independent of the path from A to B. The path integral over dU from A to B therefore depends only on the values of internal energy (UA, UB) at the two endpoints... 

From mathematics we recognize that the quantity (dQ + dW) is an exact differential, because its cyclic integral is zero for all paths. Then, some function of the variables that describe the state of the system exists. This function is called the energy function, or more loosely the energy. We therefore have the definition... 

However, the cyclic integral of an exact differential is zero and therefore QJT is an exact differential of some function. The notation dQ, is used to emphasize that the process is reversible. The new function is called the entropy function and is defined in terms of its differential, so... 

In applying Eq. (13), the variables that determine q and w will often be continuously changing. Such change can be handled by adding up very small changes of U or, more exactly, by integrating over infinitesimal changes in U The differential form of Eq. (13) is... 

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