Gibbs-Duhem relation presentation

When both solutes are present in large amounts, i.e. greater than about 1 at. 7o of each, no simple theoretical treatment is available to predict their mutual effects on thermodynamic properties. In this case, recourse must be made to the various solutions of the ternary Gibbs-Duhem relation... 

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... 

In polymer solvent systems, it is usually the chemical potential of the solvent that is determined (i.e., osmotic pressure). Due to the Gibbs-Duhem relation, however, x l and are directly related to each other and knowledge of one also determines the other. Scattering experiments on polymer blends are used to determine yet another interaction parameter Xs,. In the mixed state, small concentration fluctuations will be present that increase the free energy. For small composition fluctuations around the average composition pa/ the... 

In the case of multi-components adsorption, the partial molar differential adsorption enthalpies and entropies of each component i present in the gas mixture cannot be directly measured by experiments. However, it is possible to estimate them by mean of the tangent method based on the well-known Gibbs-Duhem relation. 

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 present relations differ from the KM approximation since the factor 3 is replaced by the bridge function at zero separation. This feature does not seem to be unreasonable because, from diagrammatic expansions, B (r) = B r)/3 is supposed to be accurate only at very low densities. Eq. (112) presents two advantages at high density i) it provides a closed-form expression for Bother fluids than the HS model and ii) it allows to ensure a consistent calculation of the excess chemical potential by requiring only the use of the pressure consistency condition (the Gibbs-Duhem constraint, no longer required, is nevertheless implicitly satisfied within 1%). 

Let us consider a solution composed of hard-sphere droplets of a single size g present in a multicomponent solvent. The expression for the osmotic pressure due to the hard spheres allows us to calculate the chemical potentials of the components in the mixed solvent. Subsequently, the Gibbs—Duhem equation is used to calculate the chemical potentials of the hard-sphere droplets.26 The mole fraction X, of the components and their volume fraction d> are related via the expressions... 

A review of chemical thermodynamics, especially as it relates to the properties of liquid solutions, has also been presented. Partial molar quantities such as the chemical potential are an important feature of the treatment of this subject. It is often the case that the activity and chemical potential of one quantity is relatively easy to determine directly by experiment, whereas that of another component is not. Under these circumstances, the change in chemical potential of one component can be related to that of another through the Gibbs-Duhem equation. This relationship and its use in estimating thermodynamic properties are extremely important in solution chemistry. 

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. 

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