Differentials inexact

The work depends on the detailed path, so Dn is an inexact differential as symbolized by the capitalization. (There is no established convention about tliis symbolism some books—and all mathematicians—use the same symbol for all differentials some use 6 for an inexact differential others use a bar tln-ough the d still others—as in this article—use D.) The difference between an exact and an inexact differential is crucial in thennodynamics. In general, the integral of a differential depends on the path taken from the initial to the final state. Flowever, for some special but important cases, the integral is independent of the path then and only then can one write... 

The coefficients Y and Z are, of course, fiiiictioiis of V and 9 and therefore state fiiiictioiis. However, since in general dpiddy) is not zero, dYIdd is not equal to dZIdV, so is not die differential of a state fiiiictioii but rather an inexact differential. 

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... 

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... 

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. 

As we have seen above, a path must be specified to integrate an inexact differential expression between two states, because different paths give different integration results. Since the value of the integral... 

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. 

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... 

In classical thermodynamics the (inexact) differential change in heat Bq, is related to the... 

Many thermodynamic relationships can be derived easily by using the properties of the exact differential. As an introduction to the characteristics of exact differentials, we shall consider the properties of certain simple functions used in connection with a gravitational field. We will use a capital D to indicate an inexact differential, as in DW, and a small d to indicate an exact differential, as in dU. 

The thermodynamic changes for reversible, free, and intermediate expansions are compared in the first column of Table 5.1. This table emphasizes the difference between an exact differential and an inexact differential. Thus, U and H, which are state functions whose differentials are exact, undergo the same change in each of the three different paths used for the transformation. They are thermodynamic properties. However, the work and heat quantities depend on the particular path chosen, even though the initial and final values of the temperature, pressure, and volume, respectively, are the same in all these cases. Thus, heat and work are not thermodynamic properties rather, they are energies in transfer between system and surroundings. 

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,... 

It was initially appreciated by R. Clausius that Carnot s theorem (4.25) allows the second law to be reformulated in a profoundly improved form. Clausius recognized that (4.25) is nothing more than the exactness condition (1.16a) for the differential dqmY/Ti.e., that L = 1 /T is an integrating factor for the inexact differential state property, a conserved quantity that... 

We must recall that the process or processes that we have been discussing have not been completely defined that is, we have not stated whether the process is adiabatic or isothermal, or whether any specific quantity of heat has been added to or removed from the system during the process. Although we have essentially defined the initial state of the system, we have not defined its final state, neither have we defined the path that we choose to connect the two states. When we do so, we find by experience that the quantity of work done by the system depends upon the path and, therefore, the differential quantity of work, dlT, is an inexact differential quantity. 

When this equation is substituted for dV in Equation (2.16), the inexact differential expression... 

CP heat capacity at constant pressure Cv heat capacity at constant volume d inexact differential D electric displacement E energy... 

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