Gibbs-Duhem integration applications

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... 

Once a state point of coexistence is established, additional state points can be determined expeditiously through application of the Gibbs-Duhem integration method [48,85,86]. In this approach a differential equation for the coexistence line is used to guide the establishment of state points away from the known coexistence point. The most well known such formula is the Clapeyron equation [41]... 

Summary of Applications of Gibbs-Duhem Integration Method"... 

A similar situation exists for alloys where a component pressure is not measurable under the temperatures for measurable vapor pressures of other components. Examples are Hf in Ni-Al-Hf [85]), Cr in Mn-Cr [104], and rare earth (RE) in Mg-RE alloys [105]. The latter study was based on a Knudsen cell without the use of a mass spectrometer. Nonetheless the approach is applicable to a mass spectrometric study. Depending on the alloy system, several approaches can be taken. In some cases the effect of controlled additions of the low-pressure component on a fixed ratio of the measured elements is the only required information [100,102]. Albers et al. [85] and Zaitsev et al. [104] did a Gibbs-Duhem integration to obtain the activities of the low vapor pressure component. Pahlman and Smith [105] assumed Raoultian behavior in the terminal RE-Mg solution and moved across the phase diagram to derive the activity of the RE component in each two-phase region. 

The Gibbs-Duhem integration method excels in calculations of solid-fluid coexistence [48,49], for which other methods described in this chapter are not applicable. An extension of the method that assumes that the initial free energy difference between the two phases is known in advance, rather than requiring it to be zero, has been proposed by Meijer and El Azhar [51]. The procedure has been used in [51] to determine the coexistence lines of a hard-core Yukawa model for charge-stabilized colloids. 

Moderate temperature changes result in such minor changes in activity coefficient that constant-pressure data are ordinarily satisfactory for application of the various integrated forms of the Gibbs-Duhem equation. 

Applications of the Integrated Equations. The usefulness of the Gibbs-Duhem equation for establishing the thermodynamic consistency of, and for smoothing, data has been pointed out. The various integrated forms are probably most useful for extending limited data, sometimes from even single measurements, and it is these applications that are most important for present purposes. 

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