Inexact

Analogous methods are used to calculated the measuring error related to inexact measuring of voltage Ur and/or Up. It is counted that the value of U is determined with measuring error 5%. The measuring error 5hu-, Shut at too low and high value of U are separately analyzed. 

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... 

Note that, since T>w is inexact, so also must be Dq. 

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. 

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. 

Thermodynamically, the activity of a single ionic species is an inexact quantity, and a conventional pH scale has been adopted that is defined by reference to specific solutions with assigned pH(5) values. These reference solutions, in conjunction with equation 3, define the pH( of the sample solution. 

U.S. producers of benzene from petroleum and their approximate production capacities are shown in Table 5. These figures are inexact because the size of the market and instabiUty of benzene prices causes frequent changes in capacity. Dow Chemical, with total armual benzene capacity of 8.3 x 10 t (250 million gallons) is the largest producer in the United States. Other companies with total domestic capacity of over 3.3 x 10 t (100 million gallons) per year are Amoco Corp., Lyondell, British Petroleum America, Chevron, Exxon Chemical, Occidental Petroleum, Shell Oil, and Mobil. These companies account for approximately 60% of total U.S. benzene capacity (65). 

Numerical techniques therefore do not yield exact results in the sense of the mathematician. Since most numerical calculations are inexact, the concept of error is an important feature. The error associated with an approximate value is defined as... 

This inexact performance leads to the recommendation that measurement sets should be discarded in their entirety when gross errors are detected. Therefore, actual isolation of which measurements contain error is not necessary when entire sets are discarded. 

The // -free potential is thus always the average of the single potentials U . The method of determination (i.e., the contribution of I ) theoretically has no influence. In practice, however, with a small difference I2- Iy, a. related inexact potential... 

Because of the inexactly defined conditions causing metal dusting, mitigation methods will not be the same for each occurrence. Each problem must be carefully studied to determine the most effective and economic measures that will be compatible with the process stream. 

A typical example of this class of polymer may be obtained by reaeting ethylenediamine and dimer fatty aeid , a material of inexact structure obtained by fractionating heat-polymerised unsaturated fatty oils and esters. An idealised strueture for this acid is shown in Figure 18.21. These materials are dark coloured, ranging from viscous liquids to brittle resins and with varying solubility. 

The theory of molecular interactions can become extremely involved and the mathematical manipulations very unwieldy. To facilitate the discussion, certain simplifying assumptions will be made. These assumptions will be inexact and the expressions given for both dispersive and polar forces will not be precise. However, they will be reasonably accurate and sufficiently so, to reveal those variables that control the different types of interaction. At a first approximation, the interaction energy, (Ud), involved with dispersive forces has been calculated to be... 

Equations (1.47) and (2.42) differ in that dV is an exact differential while 8qrcv is inexact. We again use the designations d and 8 to distinguish the two types of differentials. 

Because the two derivatives are not equal, the differential expression is inexact. 

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... 

---