Gibbs-Duhem relationship

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. 

In equation (5.27), we used the Gibbs-Duhem equation to relate changes in the chemical potentials of the two components in a binary system as the composition is changed at constant temperature and pressure. The relationship is... 

The osmotic coefficient 4> and activity coefficient are related in a simple manner through the Gibbs-Duhem equation. We can find the relationship by writing this equation in a form that relates a and 2-... 

Chapter 4 presents the Third Law, demonstrates its usefulness in generating absolute entropies, and describes its implications and limitations in real systems. Chapter 5 develops the concept of the chemical potential and its importance as a criterion for equilibrium. Partial molar properties are defined and described, and their relationship through the Gibbs-Duhem equation is presented. 

If we differentiate Equation (16.63) with respect to Xi, and simplify using the Gibbs-Duhem relationship [Equation (9.34)], the result is... 

The fundamental relationship between the chemical potentials of the two components of a solution at a fixed temperature and pressure is the Gibbs-Duhem Equation (9.34) ... 

A definite relationship exists between y+ and (p (the Gibbs-Duhem relationship), which may be expressed by the excess Gibbs energy of the solution of an electrolyte ... 

In this section, we wish to derive the Gibbs-Duhem equation, the fundamental relationship between the allowed variations dRt of the intensive properties of a homogeneous (singlephase) system. Paradoxically, this relationship (which underlies the entire theory of phase equilibria to be developed in Chapter 7) is discovered by considering the fundamental nature of extensive properties Xu as well as the intrinsic scaling property of the fundamental equation U = U(S, V, n, n2,. .., nc) that derives from the extensive nature of U and its Gibbs-space arguments. 

The latter fact may be emphasized by considering small samples of molar content n and n3 drawn from the bulk phases in regions far from the interface. The size and shape of these samples need have no relationship to the geometry of the interface—any irregularly shaped specimen of bulk phase will do. For each of these bulk phase samples a and p we have Gibbs-Duhem equations... 

If one or more chemical reactions are at equilibrium within the system, we can still set up the set of Gibbs-Duhem equations in terms of the components. On the other hand, we can write them in terms of the species present in each phase. In this case the mole numbers of the species are not all independent, but are subject to the condition of mass balance and to the condition that , vtpt must be equal to zero for each independent chemical reaction. When these conditions are substituted into the Gibbs-Duhem equations in terms of species, the resultant equations are the Gibbs-Duhem equations in terms of components. Again, from a study of such sets of equations we can easily determine the number of degrees of freedom and can determine the mathematical relationships between these degrees of freedom. 

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. 

Gibbs-Duhem equation — The relationship between partial molar quantities of different components when... 

Tpo obtain vapor-liquid equilibrium data for binary systems, it is now well established that under certain circumstances it can be more accurate and less time consuming to measure the boiling point, the total pressure, and the liquid composition and then use the Gibbs-Duhem relationship to predict vapor composition (I) rather than to measure it. The disadvantage is that there is no way of checking the thermodynamic consistency of the experimental data. 

Gibbs-Duhem Relationship The partial molar properties of a multicomponent phase cannot be varied independently (the mole fractions, jc, = ,/E of the components total unity). For example, for the chemical potentials, /i, the Gibbs-Duhem relationship is En, dni = 0 (for details, see e.g., Atkirs, 1990 Blandamer, 1992 Denbigh, 1971). Similar constraints apply to the partial molar volumes, enthalpies, entropies, and heat capacities. For pure substances, the partial molar property is equal to the molar property. For example, the chemical potential of a pure solid or liquid is its energy per mole. For gaseous, liquid, or solid solutions, X, = X,(ny), that is, the chemical potentials and partial molar volumes of the species depend on the mole fractions. 

The relationship between the activities of the components of the solution is expressed by the Gibbs-Duhem equation... 

The experimental values of the activity coefficients must satisfy the Gibbs-Duhem relationship [55]. If the data are collected at constant temperature and spreading pressure, then... 

From the Gibbs-Duhem relationship (see equation (1.4.8) and associated discussion), the last two terms in equations (1.10.11) and (1.10.12) are equal to zero. It follows that the derivatives of / RT) with respect to a and b give... 

In order to determine the activity of a component in solution, one must measure its vapor pressure. In the case of volatile liquids such as those discussed in most of this chapter, vapor pressure measurement is not a problem so that very accurate determination of activity is possible over the whole composition range for which a solution is formed. However, many solutes, for example, most solids, have negligible vapor pressures. Under these circumstances, one makes use of the Gibbs-Duhem relationship between the activities of the two-components in solution. Since the vapor pressure of the solvent can be measured, its activity can be determined, and then used to estimate the activity of the solute. 

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. 

Having obtained the solvent vapor pressure, one may immediately relate it to the activity of the electrolyte using the Gibbs-Duhem relationship. Thus,... 

The above results illustrate the importance of non-ideality for electrolyte solutions and also of the use of the Gibbs-Duhem relationship in obtaining electrolyte... 

Since it is impossible to change the chemical potential of A without also changing that of W, one cannot measure either Fa or F independently. The practical significance of this can be seen using the Gibbs-Duhem relationship, according to which... 

This is the Gibbs-Duhem equation, which relates the variation in temperature, pressure, and chemical potentials of the C components in the solution. Of these C + 2 variables, only C + 1 can vary independently. The Gibbs-Duhem equation has many applications, one of which is providing the basis for developing phase equilibrium relationships. 

In order to calculate the distribution coefficient by Equation 1.29, the activity coefficient Y must be evaluated. The activity coefficients are generally determined from the experimental data and correlated on the basis of thermodynamic phase equilibrium principles. The relationship most often used for this purpose is the Gibbs-Duhem equation (Equation 1.7). At constant temperature and pressure, this equation becomes... 

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