Gibbs-Duhem equation chemical potential

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). 

In summary, in the limit as x2 —> 0 and xi — 1, /i —>.V /f and f2 —> x2A h..x-It can be shown from the Gibbs-Duhem equation that when the solute obeys Henry s law, the solvent obeys Raoult s law, To prove this, we start with the Gibbs-Duhem equation relating the chemical potentials... 

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... 

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. 

The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

The physical significance of the Gibbs-Duhem equation is that the chemical potential of one component in a solution cannot be varied independently of the chemical potentials of the other components of the solution. This relation will be further discussed and used in Chapter 3. 

We can show that if the solute obeys Henry s law in very dilute solutions, the solvent follows Raoult s law in the same solutions. Let us start from the Gibbs-Duhem Equation (9.34), which relates changes in the chemical potential of the solute to changes in the chemical potential of the solvent that is, for a two-component system... 

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) ... 

Once values of g as a function of solution composition have been obtained, the Gibbs-Duhem equation can be used to relate the osmotic coefficient of the solvent to the activity coefficient of the solute. For this purpose, the chemical potential of the solvent is expressed as in Equation (19.42), with the approximation given in Equation (19.53), so that... 

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). 

There are ( -l) equations of type (8.3). Along with the Gibbs-Duhem equation, they can be solved for the unknown chemical potential gradients V. In combination with the (n-1) mass conservation equations... 

Since in addition to the chemical potentials also the electrical potential 99, affects the charged species, electrochemical potentials //, must be used. We use the symbol 99 instead of -ip because this is the Galvani potential (see Section 5.5). The Gibbs-Duhem equation for changes of state functions at constant temperature is... 

This is the Gibbs equation, which is particularly important for understanding phase equilibria. A related expression, called the Gibbs-Duhem equation, states that at equilibrium the change of chemical potential of one component results in the change of the chemical potentials of all other components... 

This equation is extremely important (see Section 5.12 for some applications). It is known as the Gibbs-Duhem equation, and such equations as the Duhem-Margules equation may be derived from it. Since no limitation has been put on the type of system considered in the derivation, this equation must be satisfied for every phase in a heterogenous system. We recognize that the convenient independent variables for this equation are the intensive variables the temperature, the pressure, and the chemical potentials. 

The Gibbs-Duhem equation is applicable to each phase in any heterogenous system. Thus, if the system has P phases, the P equations of Gibbs-Duhem form a set of simultaneous, independent equations in terms of the temperature, the pressure, and the chemical potentials. The number of degrees of freedom available for the particular systems, no matter how complicated, can be determined by the same methods used to derive the phase rule. However, in addition, a large amount of information can be obtained by the solution of the set of simultaneous equations. 

In the previous examples we have assumed that all comonents are present in all of the phases, and we have not introduced any chemical reactions or restrictions. When a component does not exist in a phase, the mole number of that component is zero in that phase and its chemical potential does not appear in the corresponding Gibbs-Duhem equation. As an example, consider a two-component, three-phase system in which two of the phases are pure and the third is a solution. The Gibbs-Duhem equations are then... 

Finally, consider a two-phase, two-component system in which the two phases are separated by an adiabatic membrane that is permeable only to the first component. In this case we know that the temperatures of the two phases are not necessarily the same, and that the chemical potential of the second component is not the same in the two phases. The two Gibbs-Duhem equations for this system are... 

This equation then gives the differential of the chemical potential of a component in terms of the experimentally determined variables the temperature, pressure, and mole fractions. It is this equation that is used to introduce the mole fraction into the Gibbs-Duhem equation as independent variables, rather than the chemical potentials. The problem of expressing the chemical potentials as functions of the composition variables, and consequently the determination of (dpjdx j P x, is discussed in Chapters 7 and 8. 

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 common characteristics of phase transitions are that the Gibbs energy is continuous. Although the conditions of equilibrium and the continuity of the Gibbs energy demand that the chemical potential must be the same in the two phases at a transition point, the molar entropies and the molar volumes are not. If, then, we have two such phases in equilibrium, we have a set of two Gibbs-Duhem equations, the solution of which gives the Clapeyron equation (Eq. (5.73))... 

Three different uses of the Gibbs-Duhem equation associated with the integral method are discussed in this section (A) the calculation of the excess chemical potential of one component when that of the other component is known (B) the determination of the minimum number of intensive variables that must be measured in a study of isothermal vapor-liquid equilibria and the calculation of the values of other variables and (C) the study of the thermodynamic consistency of the data when the data are redundant. 

When the excess chemical potential of the solute in the liquid phase is required as a function of the mole fraction at the constant temperature T0 and pressure P, an integration of the Gibbs-Duhem equation must be used. For this the infinitely dilute solution of the solute in the solvent must be... 

This equation gives the change of pressure with a change of temperature at the maximum or minimum. The system becomes univariant. However, because of the cancellation of the terms containing the chemical potentials, the determination of the change of the composition of the phases at the maximum or minimum with change of temperature or pressure cannot be determined from the Gibbs-Duhem equations alone. In order to do so we introduce the equality of the differentials of the chemical potentials in the two phases for one of the components, so... 

The experimental studies of three-component systems based on phase equilibria follow the same principles and methods discussed for two-component systems. The integral form of the equations remains the same. The added complexity is the additional composition variable the excess chemical potentials become functions of two composition variables, rather than one. Because of the similarity, only those topics that are pertinent to ternary systems are discussed in this section of the chapter. We introduce pseudobinary systems, discuss methods of determining the excess chemical potentials of two of the components from the experimental determination of the excess chemical potential of the third component, apply the set of Gibbs-Duhem equations to only one type of phase equilibria in order to illustrate additional problems that occur in the use of these equations, and finally discuss one additional type of phase equilibria. 

The quantity ps can be considered as the chemical potential of the mixed solvent. The Gibbs-Duhem equation may be written as... 

On eliminating the chemical potential of the first component between the two Gibbs-Duhem equations applicable to the two phases, we obtain... 

These four Legendre transforms introduce the chemical potential as a natural variable. The last thermodynamic potential U T, P, /<] defined in equation 2.6-6 is equal to zero because it is the complete Legendre transform for the system, and this Legendre transform leads to the Gibbs-Duhem equation for the system. 

This last equation is the Gibbs-Duhem equation for the system, and it shows that only two of the three intensive properties (T, P, and fi) are independent for a system containing one substance. Because of the Gibbs-Duhem equation, we can say that the chemical potential of a pure substance substance is a function of temperature and pressure. The number F of independent intensive variables is T=l — 1+2 = 2, and so D = T + p = 2 + l = 3. Each of these fundamental equations yields D(D — l)/2 = 3 Maxwell equations, and there are 24 Maxwell equations for the system. The integrated forms of the eight fundamental equations for this system are ... 

When the pH is specified, we enter into a whole new world of thermodynamics because there is a complete set of new thermodynamic properties, called transformed properties, new fundamental equations, new Maxwell equations, new Gibbs-Helmholtz equations, and a new Gibbs-Duhem equation. These new equations are similar to those in chemical thermodynamics, which were discussed in the preceding chapter, but they deal with properties of reactants (sums of species) rather than species. The fundamental equations for transformed thermodynamic potentials include additional terms for hydrogen ions, and perhaps metal ions. The transformed thermodynamic properties of reactants in biochemical reactions are connected with the thermodynamic properties of species in chemical reactions by equations given here. 

Following our development of the Gibbs-Duhem equation in Chapter 8, we now apply Eq. (45) to a process at constant T and P, where components are added to the surface in an arbitrary ratio. In this process, the chemical potentials will vary ... 

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