Duhem equation generalization

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,... 

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). 

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

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... 

The five conclusions regarding zeolite synthesis given in this and the previous section are derived largely via the Gibbs-Duhem equation and are in general accord with practical experience. The physico-chemical interpretation of so much observed behaviour is of considerable interest. 

Any initial impression that there is something unreasonable about the Gibbs-Duhem equation should be instantly quelled. It merely tells one that (for a two-component system) when an increase in n a[uioccurs, it causes a decrease in nj dni of equal magnitude, a typically powerful and general result of thermodynamic reasoning. 

Equation (169) must hold for arbitrary 5 / satisfying E8rij = 0, and clearly the only possibility is that/ (X/) = 1/X/, that is,fiX,) = InX/. The logarithmic form is the only one that satisfies the stated conditions, and hence the discrete equivalent of Eq. (21) simply follows from the definitions. Clearly, the argument can be generalized to a continuous description [one only needs to apply the Gibbs-Duhem equation in its continuous formulation to a 6 (a ) satisfying <8n(y)> = 0], and hence Eq. (21) is identified as simply the definition of a continuous ideal mixture. 

The equations of the upper left quadrant of Table 4-6 reduce to those of the upper right quadrant for n = 1 and drit = 0. Each equation in the upper left quadrant has a partial-property analog, as shown in the lower left quadrant. Each equation of the upper left quadrant is a special case of Eq. (4-172) and therefore has associated with it a Gibbs-Duhem equation of the form of Eq. (4-173). These are shown in the lower right quadrant. The equations of Table 4-6 store an enormous amount of information, but they are so general that their direct... 

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