Gibbs-Duhem equation generalized

The special case of equation (A2.1.27) when T and p are constant (dJ= 0, dp = 0) is called the Gibbs-Duhem equation, so equation (A2.1.27) is sometimes called the generalized Gibbs-Duhem equation . 

Moreover, using the generalized Gibbs-Duhem equations (A2.1.27) for each of the two one-component phases,... 

However, from the general Gibbs-Duhem equation (6.34) for each phase, we can write... 

THE PARTIAL MOLAR GIBBS ENERGY AND THE GENERALIZED GIBBS-DUHEM EQUATION... 

The Partial Molar Gibbs Energy and the Generalized Gibbs-Duhem Equation 347 However, from H = G -r TS and Eq. 8.2-1. we have ... 

These results are forms of the generalized Gibbs-Duhem equation. 

The Partial Molar Gibbs Energy and the Generalized Gibbs-Duhem Equation 349... 

Equation (A2.1.39) is the generalized Gibbs-Duhem equation previously presented (equation (A2.1.27)). Note that the Gibbs free energy is just the sum over the chemical potentials. 

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)... 

Moreover, from (4.265), (4.206), (4.211) we obtain the classical relationships (for them the general Gibbs-Duhem equation (4.263) are necessary)... 

In this molar formulation, all the variables are intensive but they are all generalized densities in the sense that, like the density itself, the values differ in two coexisting phases. Unlike these density variables, a potential or field variable e.g. T, p, J) must have the same value in every phase at equilibrium. Equation (2) can be transformed into the generalized Gibbs-Duhem equation in which the independent variables are fields ... 

This is known as the generalized Gibbs-Duhem equation. The most common and useful of these expressions involves the Gibbs free energy (A = G) for which one... 

From eqs 2.98 and 2.99 the generalized Gibbs-Duhem equation is readily obtained ... 

The well-known Gibbs-Duhem equation (2,3,18) is a special mathematical redundance test which is expressed in terms of the chemical potential (3,18). The general Duhem test procedure can be appHed to any set of partial molar quantities. It is also possible to perform an overall consistency test over a composition range with the integrated form of the Duhem equation (2). 

This general result, the Gibbs/Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. For the special case where T and P are constant ... 

Because experimental measurements are subject to systematic error, sets of values of In y and In yg determined by experiment may not satisfy, that is, may not be consistent with, the Gibbs/Duhem equation. Thus, Eq. (4-289) applied to sets of experimental values becomes a test of the thermodynamic consistency of the data, rather than a valid general relationship. 

While a Gibbs-Duhem equation does not exist for general transformations dSo, ds a, a specialized (i.e., coarse-grained ) Gibbs-Duhem equation... 

But Langmuir s isotherm for the solute entails the generalized form of Raoult s law (Eq. 13) as a necessary thermodynamic consequence. This can best be seen from the Gibbs-Duhem equation,... 

We cannot generalize from this example. In some systems Pj < P 1 while P > P . In other examples Vf- > P . 1 while V < Pm. . The thing that must be true from the Gibbs-Duhem equation is that... 

In general, for an arbitrary system with i components, the Gibbs-Duhem equation is obtained by combining eq. (1.78) and eq. (1.90) ... 

The general form of the Gibbs-Duhem equation for an -component system can be expressed as... 

Because variations in solvent chemical potential are generally much easier to determine experimentally (e.g., by osmotic pressure measurements, as described in Section 7.3.6), (6.37) gives the recipe for determining the more difficult solute from its Gibbs-Duhem dependence on other easily measured thermodynamic intensities. Equations such as (6.35)-(6.37) are sometimes referred to as Gibbs-Duhem equation(s), but they are really only special cases of (and thus less general than) the Gibbs-Duhem equation (6.34). 

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... 

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. 

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