Differential Exact

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. 

The factor in wavy brackets is obviously an exact differential because the coefficient of d9 is a fiinction only of 9 and the coefficient of dVis a fiinction only of V. (The cross-derivatives vanish.) Manifestly then... 

Equation (A2.1.26) is equivalent to equation (A2.1.25) and serves to identify T, p, and p. as appropriate partial derivatives of tire energy U, a result that also follows directly from equation (A2.1.23) and the fact that dt/ is an exact differential. 

Since G is a state variable and forms exact differentials, an alternative expression for dG is... 

Because these are exact differential expressions. Maxwell equations can be written by inspection. The two most useful ones are derived from equations 67 and 68 ... 

Mathematical Consistency. Consistency requirements based on the property of exact differentials can be apphed to smooth and extrapolate experimental data (2,3). An example is the use of the Gibbs-Duhem coexistence equation to estimate vapor mole fractions from total pressure versus Hquid mole fraction data for a binary mixture. 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

In addition, the common Maxwell equations result from application of the reciprocity relation for exact differentials ... 

Heat-Capacity Relations In Eqs. (4-34) and (4-41) bothdH and dU are exact differentials, and application of the reciprocity relation leads to... 

The reciprocity relation for an exact differential applied to Eq. (4-16) produces not only the Maxwell relation, Eq. (4-28), but also two other usebil equations ... 

The statistical expression for the internal energy is simply (U) = PjNs-Ej, from which its exact differential follows as... 

In equation (1.4), the infinitesimal change dZ is an exact differential. Later, we will describe the mathematical test and condition to determine if a differential is exact. 

Of special interest are the properties of the exact differential. We have seen that our thermodynamic variables are state functions. That is, for a thermodynamic variable Z... 

If one causes an infinitismal change dZ to occur, the quantity dZ is an exact differential (whenever Z is a state function). We will now describe the test that determines if a differential is exact and summarize the relationship between exact differentials. 

Example 1.1 Test the following differential expressions to determine which is an exact differential... 

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. 

When the state of a variable Z that forms an exact differential is fixed by specifying two others, X and Y, one can write... 

Other relationships between the partial derivatives of the variables W, X, Y, and Z that form exact differentials are as follows ... 

Equation (1.34) states that the order of differentiation is immaterial for the exact differential. The Maxwell relation follows directly from this property,... 

Derivation of Thermodynamic Equations Using the Properties of the Exact Differential... 

By using relationships for an exact differential, equations that relate thermodynamic variables in useful ways can be derived. The following are examples. 

Example 1.4 Start with equation (1.18) relating H and U and the properties of the exact differential and prove that... 

El.l Use the Maxwell s relation to determine if dr is an exact differential in the following examples ... 

Pl.l Use the properties of the exact differential and the defining equations for the derived thermodynamic variables as needed to prove the following relationships ... 

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... 

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. 

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... 

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... 

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