Generalized Gibbs-Duhem relation

Equation (15) is called the general Gibbs—Duhem relation. Because it tells us that there is a relation between the partial molar quantities of a solution, we will learn how to use it to determine a Xt when all other X/ il have been determined. (In a two-component system, knowing Asolvent determines Asolute.) This type of relationship is required by the phase rule because, at constant T, P, and c components, a single-phase system has only c — 1 degrees of freedom. 

Equation (4.6-9) is called the generalized Gibbs-Duhem relation. 

For example, we may choose to as the average volume velocity, to = (c, V,) v,-. In more general terms, we may define to by X Pi vi > with Y Pi = 1 The s are weighting factors. If we formulate Eqn. (4.72) for two different reference velocities, to and to", and take into account the partial molar volumes (V,) which are not independent of each other (Gibbs-Duhem relation), we obtain after some algebraic rearrangements [H. Schmalzried (1981)] the quite general expression... 

The generalization for multicomponent systems is quite straightforward. The condition on the equality of all interaction potentials is sufficient for the SI behavior of all the components in the system. In the multicomponent system, we have to require that the SI behavior of the type (5.27) holds for c— 1 components the Gibbs-Duhem relation ensures the validity of the SI behavior for the cth component. 

We can see that these relations (4.218), (4.219) are in accord with classical thermochemistry if the sum in the right hand side of (4.218) is zero (which is known as (generalized) Gibbs-Duhem equation) for all y (4.210). Exceptions are chemical potentials ga and specific Gibbs energy g as may be seen from (4.206) and (4.217)... 

The problem with this equation lies in the formulation of the force term. In general, a particle may move in response to gradients in the electrochemical potentials of other species, leading to cross-terms in the flux equation. In principle, the presence of cross-terms will occur whenever a component is present whose chemical potential may vary independently of that of species i. Thus, the motion of i may depend not only on Vrji but also on Vrjj if rjj is independent of f], (i.e. it is not coupled through a Gibbs-Duhem relation). The flux equation may therefore be generalized... 

If we choose P as the thermodynamic pressure P, then oSf(E) is referred to as the generalized PF. It is clear from (4.5.81) that the integral in this case diverges. The reason is that E(T, L, X) is a function of the single extensive variable L. Transforming L into the thermodynamic intensive variable P gives a partition function which is a function of the intensive variables T, P, X only. However, the Gibbs-Duhem relation states that... 

This last relation is known as the generalized Gibbs-Duhem lelatioa If we keep constant all the intensive variables 7, this relation can be simplified... 

Suppose that, for all compositions between a known state (generally the reference state) and the composition under examination, we know the activities (or activity coefficients) of all the components of a solution in the same frame of reference, except for one of them, and we are going to calculate the unknown activity of that component for the chosen composition. This method is founded on the Gibbs-Duhem relation, which is valid regardless of the convention adopted. Thus, we obtain the activity (or the activity coefficient) in the chosen convention for the known values. 

The Gibbs-Duhem relation (1.14.13) may be used in conjunction with the foregoing to determine the molar activity of the ionic species in a binary solution. This is necessary because a direct determination is difficult. The subsequent relations are generally couched in terms of molalities. We proceed as follows since din = din a we use the form (T and P constant for the solvent, mi = 1000/Mi)... 

This equation is very similar to the Gibbs-Duhem equation under the condition that the temperature and pressure are constant. A more general relation can be obtained by differentiating Equation (6.10) and comparing the result with Equation (6.1). The differentiation of Equation (6.10) gives... 

Two methods may be used, in general, to obtain the thermodynamic relations that yield the values of the excess chemical potentials or the values of the derivative of one intensive variable. One method, which may be called an integral method, is based on the condition that the chemical potential of a component is the same in any phase in which the component is present. The second method, which may be called a differential method, is based on the solution of the set of Gibbs-Duhem equations applicable to the particular system under study. The results obtained by the integral method must yield... 

In the last chapter we established two powerful general theorems, those of Gibbs and Duhem, relating to heterogeneous systems. We shall now consider in more detail the quantitative behaviour of some simple systems, beginning with a study of the phase changes of a pure substance. A study of more complex heterogeneous systems will follow in later chapters after we have discussed the thermodynamic conditions of stability. 

These and all previous results of thermodynamic mixture which also fulfil Gibbs-Duhem equations (4.263) show the complete agreement with the classical thermodynamic of mixtures but moreover all these relations are valid much more generally. Namely, they are valid in this material model—linear fluid mixture—in all processes whether equilibrium or not. Linear irreversible thermodynamics [1-4], which studies the same model, postulates this agreement as the principle of local equilibrium. Here in rational thermodynamics, this property is proved in this special model and it cannot be expected to be valid in a more general model. We stress the difference in the cases when (4.184) is not valid—e.g. in a chemically reacting mixture out of equilibrium—the thermodynamic pressures P, Pa need not be the same as the measured pressure (as e.g. X =i Pa) therefore applications of these thermodynamic... 

Besides (3.4.4), another attribute of partial molar properties, also derived in Appendix A, is that they obey a set of relations known as Gibbs-Duhem equations. For the generic extensive property F(T, P, N ), the general form of the Gibbs-Duhem equation is... 

In the thermodynamic description of multicomponent systems, a principal relation is the Gibbs-Duhem equation. Astarita [1] has shown that the Gibbs-Duhem equation is not merely a thermodynamic relation it is a general repercussion of the properties of homogeneous functions. Consider a multivariant function, such as... 

Gibbs-Duhem restriction on the chemical potential (eq. 8.5-3). Eq.(8.5-5) is the generalized Maxwell-Stefan constitutive relation. However, such form is not useful to engineers for analysis purposes. To achieve this, we need to express the chemical potential in terms of mole fractions. This is done by using eq. (8.5-2) into the constitutive flux equation (8.5-5). 

The general principles established for ideal solutions, such as Raoult s law in its various forms, are of course applicable to solutions of any number of components. Similarly, the Gibbs-Duhem equation is applicable to nonideal solutions of any number of components, and as in the case of binary mixtures various relationships can be worked out relating the activity coefficients for ternary mixtures. This problem has now been attacked from several points of view, a most excellent summary of which is presented by Wolil (35). His most important results pertinent to the problem at hand are summarized here. 

One of the important general relations is the Gibbs-Duhem equation. This relation shows that the intensive variables T, p and Pjt independent. It... 

The Gibbs-Duhem equation provides a general relation for the partial molar properties of different species in a mixture that must always be true. For example, we just saw how the activity coefficient of different species can be related to one another. In this section, we explore one way to use this interrelation to judge the quality of experimental data. The basic idea is to develop a way to see whether a set of data conform to the constraints posed by the Gibbs-Duhem equation. If the data reasonably match, we say they are thermodynamically consistent. On the other hand, data that do not conform to the Gibbs-Duhem equation are thermodynamically inconsistent and should be considered unreliable. The development that follows is based on the relation between activity coefficients in a binary mixture of species a and b. It serves as an example to this methodology there are several other ways that have been developed to apply this same type of idea. 

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