Gibbs-Duhem equation derivation

We have already derived the Gibbs-Duhem equation in Chapter 1.4. At constant p and T ... 

We derived Equation (4.30) from first principles, using pure mathematics. An alternative approach is to prepare a similar equation algebraically. The result of the algebraic derivation is the Gibbs-Duhem equation ... 

By examining the compositional dependence of the equilibrium constant, the provisional thermodynamic properties of the solid solutions can be determined. Activity coefficients for solid phase components may be derived from an application of the Gibbs-Duhem equation to the measured compositional dependence of the equilibrium constant in binary solid solutions (10). 

Most thermodynamic data for solid solutions derived from relatively low-temperature solubility (equilibration) studies have depended on the assumption that equilibrium was experimentally established. Thorstenson and Plummer (10) pointed out that if the experimental data are at equilibrium they are also at stoichiometric saturation. Therefore, through an application of the Gibbs-Duhem equation to the compositional dependence of the equilibrium constant, it is possible to determine independently if equilibrium has been established. No other compositional property of experimental solid solution-aqueous solution equilibria provides an independent test for equilibrium. If equilibrium is demonstrated, the thermodynamic properties of the solid solution are also... 

The activity of the water is derived from this expression by use of the Gibbs-Duhem equation. To utilize this equation, the interaction parameters fif ) and BH must be estimated for moleculemolecule, molecule-ion and ion-ion interactions. Again the method of Bromley was used for this purpose. Fugacity coefficienls for the vapor phase were determined by the method of Nakamura et al. (JO). 

The use of the Gibbs-Duhem equation to derive the limiting laws for coUigative properties is based on the work of W. Bloch. 

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. 

Let us now derive some simple consequences of the Gibbs-Duhem equation for special cases. 

Pitzer s solution to the problem was the development of a set of analytical equations that are thermodynamically consistent after transformations through the Gibbs-Duhem equation. These equations are known as the Pitzer equations, in recognition of the major role that he played in developing them and the major contributions he made in the understanding of electrolyte solutions through a lifetime of work. We will now summarize these equations and describe their usefulness. For details of the derivation we refer the reader to Pitzer s original paper.6... 

Starting with equation (18.85), equations can be derived for , L, Cp, and osmotic coefficient is obtained from a transformation using the Gibbs-Duhem equation. The result is... 

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 derivation of the phase rule is based upon an elementary theorem of algebra. This theorem states that the number of variables to which arbitrary values can be assigned for any set of variables related by a set of simultaneous, independent equations is equal to the difference between the number of variables and the number of equations. Consider a heterogenous system having P phases and composed of C components. We have one Gibbs-Duhem equation of each phase, so we have the set of equations... 

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. 

The derivatives (dP/dT)S3t and (dxt/dT)sat may be determined experimentally or by solution of the set of Gibbs-Duhem equations applicable to each phase, provided we have sufficient knowledge of the system. If the system is multivariant, a sufficient number of intensive variables—the pressure or mole fractions of the components in one or more phases—must be held constant to make the system univariant. Thus, for a divariant system either the pressure or one mole fraction of one of the phases must be held constant. When the pressure is constant, Equation (9.9) becomes... 

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... 

We discuss in the next few sections the applications of the Gibbs-Duhem equations to various phase equilibria. In so doing we obtain expressions for the derivatives of one intensive variable with respect to... 

The expressions for the derivatives become somewhat simpler when one of the components is not present in one of the phases. For the purposes of discussion we assume that the second component is not present in the double-primed phase. Then the two Gibbs-Duhem equations become... 

There are six variables and five equations, and the system is univariant. We wish to determine the change of the partial pressure of the species A2B with change of composition of the condensed phase that is, the derivative (din PAB/dx1)T- In the solution of the set of Gibbs-Duhem equations, we therefore must retain nAB and or expressions equivalent to these quantities. The solution can be obtained by first eliminating n2 and fiB from Equations (11.167) and (11.168) by use of Equations (11.169) and (11.171). The expressions... 

Thus the Gibbs-Duhem equation represents one of the 2D thermodynamic potentials that can be defined for a system, but this thermodynamic potential is equal to zero. It should be emphasized that there is a single Gibbs-Duhem equation for a one-phase system and that it can be derived from U, H, A, or G. 

The Gibbs-Duhem equations for the two phases at equilibrium can be derived from equations 8.2-3 and 8.2-4 ... 

Equation 8.5-3 indicates that the number of natural variables for the system is 6, D = 6. Thus the number D of natural variables is the same for G and G, as expected, since the Legendre transform interchanges conjugate variables. The criterion for equilibrium is dG 0 at constant T,P,ncAoi, ncA(3, /icC, and The Gibbs-Duhem equations are the same as equations 8.4-8 and 8.4-9, and so the number of independent intensive variables is not changed. Equation 8.5-3 yields the same membrane equations (8.4-13 and 8.4-14) derived in the preceding section. 

These clearly are not the same as the suggested expressions, which are therefore not correct. Note that application of the summability equation to the derived partial-property expressions reproduces the original equation for H. Note further that differentiation of these same expressions yields results that satisfy the Gibbs/Duhem equation, Eq. (11.14), written ... 

The Gibbs-Duhem equation (50.6), derived below, proves to be a useful starting point for an alternative derivation of the Clausius Claperyron equation to that offered in Frame 26) and offers an alternative proof of the Phase Rule to that given in Frame 30. 

Derivation of Clausius-Clapeyrron Equation using the Gibbs-Duhem Equation. [ ln the form of Equation (26.5), Frame 26]... 

Derivation of the Phase Rule using the Gibbs-Duhem Equation... 

Since the Margules expansions represent a convergent power series in the mole fractions,8 they can be summed selectively to yield closed-form model equations for the adsorbate species activity coefficients. A variety of two-parameter models can be constructed in this way by imposing a constraint on the empirical coefficients in addition to the Gibbs-Duhem equation. For example, a simple interpolation equation that connects the two limiting values of f (f°° at infinite dilution and f = 1.0 in the Reference State) can be derived after imposing the scaling constraint... 

Jones liquid. Circles denote molecular dynamics results dashed curves were derived from integration of the Gibbs-Duhem equation using molecular dynamics data. Convenient molecular dimensions were used. Reprinted with permission from Deitrick et al. [3],... 

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. 

The Gibbs-Duhem equation is used in several cases in electrochemistry, e.g., in the derivation of - Gibbs adsorption equation or -> Gibbs-Lippmann equation since Eq. (4) can be extended by surface work ... 

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