Exact differential properties

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. 

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... 

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... 

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

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 ... 

The coefficient of expansion, a, can also be related to (dpfdVm)T. From the definition of a and the properties of the exact differential, we can write... 

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. 

In Chapter 3, we defined a new function, the internal energy U, and noted that it is a thermodynamic property that is, dU is an exact differential. As Q was defined in Equation (3.12) as equal to At/ when no work is done, the heat exchanged in a constant-volume process in which only PdV work is done is also independent of the path. For example, in a given chemical reaction carried out in a closed vessel of fixed volume, the heat absorbed (or evolved) depends only on the nature and condition of the initial reactants and of the final products it does not depend on the mechanism by which the reaction occurs. Therefore, if a catalyst speeds up the reaction by changing the mechanism, it does not affect the heat exchange accompanying the reaction. 

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. 

Because dU is an exact differential and (dU/dV)T is equal to zero, the crossderivative property of the exact differential leads to the conclusion that dCy/dV)r is equal to zero, so Cy is a function of T only. 

In the case of ternary or higher-order mixtures, solution of the Gibbs-Duhem equation is again based on application of the properties of the exact differentials (Lewis and Randall, 1970) ... 

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... 

The general criterion (1.16a) for a state property (conserved quantity, exact differential) can be expressed as... 

Thermodynamic properties or coordinates are derived from the statistical averaging of the observable microscopic coordinates of motion. If a thermodynamic property is a state function, its change is independent of the path between the initial and final states, and depends on only the properties of the initial and final states of the system. The infinitesimal change of a state function is an exact differential. 

Therefore, -S is a state property or an exact differential. Entropy cannot be easily defined but can be described in terms of entropy increase accompanying a particular process. 

To calculate the internal energy per unit mass 0) firom the variables that can be measured, we make use of a special property of internal energy, namely, that it is an exact differential (because it is a point or state property, a matter to be described shortly) and, for a pure component, can. be expressed in terms of just two intensive variables according to the phase rule for one phase ... 

To calculate the enthalpy per unit mass, we use the property that the enthalpy is also an exact differential. For a pure substance, the enthalpy for a single phase can be expressed in terms of the temperature and pressure (a more convenient variable than specific volume) alone. If we let... 

However, we also notice from Eq. (5.83) that in the absence of the switching function, the remaining fluid substrate potential would depend only on the x-coordinate of a fluid molecule. In this case, fluid properties would be translationally invariant in the x- and y-directions. Hence, in this case, the exact differential of the grand potential would be given by Eq. (1.63), where this higher symmetry of the confined fluid has alread been exploited. As a consequence we obtain a closed expression for the grand potential [see Eq. (1.65)1 in terms of the tran.sverse stress T as we show in Section 1.6.1. 

This quantity is the exact differential of some state property and it is later defined as the entropy of the system ... 

The state functions are properties of the present state of the system and do not depend on the path by which that state was reached. The differential of a state function is an exact differential its integral between the initial and the final state is independent of the path taken. 

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