Gibbs-Duhem equation general form

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). 

But Langmuir s isotherm for the solute entails the generalized form of Raoult s law (Eq. 13) as a necessary thermodynamic consequence. This can best be seen from the Gibbs-Duhem equation,... 

The general form of the Gibbs-Duhem equation for an -component system can be expressed as... 

Equation (169) must hold for arbitrary 5 / satisfying E8rij = 0, and clearly the only possibility is that/ (X/) = 1/X/, that is,fiX,) = InX/. The logarithmic form is the only one that satisfies the stated conditions, and hence the discrete equivalent of Eq. (21) simply follows from the definitions. Clearly, the argument can be generalized to a continuous description [one only needs to apply the Gibbs-Duhem equation in its continuous formulation to a 6 (a ) satisfying <8n(y)> = 0], and hence Eq. (21) is identified as simply the definition of a continuous ideal mixture. 

The equations of the upper left quadrant of Table 4-6 reduce to those of the upper right quadrant for n = 1 and drit = 0. Each equation in the upper left quadrant has a partial-property analog, as shown in the lower left quadrant. Each equation of the upper left quadrant is a special case of Eq. (4-172) and therefore has associated with it a Gibbs-Duhem equation of the form of Eq. (4-173). These are shown in the lower right quadrant. The equations of Table 4-6 store an enormous amount of information, but they are so general that their direct... 

These results are forms of the generalized Gibbs-Duhem equation. 

The approach has been extended in two fundamental ways (1) one may consider variations in field variables [1] other than temperature and pressure and (2) one may consider additional phases, so that (for example) three-phase coexistence lines are produced. Both extensions start with a more general form of the Gibbs-Duhem equation, which we write as follows ... 

Besides (3.4.4), another attribute of partial molar properties, also derived in Appendix A, is that they obey a set of relations known as Gibbs-Duhem equations. For the generic extensive property F(T, P, N ), the general form of the Gibbs-Duhem equation is... 

The partial molar functions of the component B can be calculated using the Gibbs-Duhem equation. The Gibbs-Duhem equation for any partial molar function Z (Z equal to AG, AS, AH, etc.) has the general form... 

By means of Eqs. (13) and (14) a generalized form of the Gibbs-Duhem equation is obtained (at constant T and P), which is expressed by... 

Chemical thermodynamics was developed by Pierre Maurice Martin Duhem (Paris, lo June i86i-Cabrespine, 14 September 1916), professor of theoretical physics in Bordeaux, who published on the equations for heats of solution and dilution which had been deduced by Kirchhoff, on the liquefaction of gaseous mixtures, eutectic and transition points for binary mixtures which can form mixed crystals, and a long series of papers on false equilibrium of doubtful value. He published some books on thermodynamics and later on the history of science. An important general thermodynamic equation (Gibbs-Duhem equation) was deduced independently by Gibbs and Duhem. ... 

The Gibbs-Duhem equation is one of the most important relationships of mixture thermodynamics. A general form of the Gibbs-Duhem equation can be derived in combination with the general definition of the partial molar properties. The Gibbs-Duhem equation allows for the development of so-called consistency tests, which are a necessary but not sufficient condition for the correctness of experimental data. The total differential of the function... 

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. 

Equation 9.2.12 is the Gibbs-Duhem equation for a binary mixture, applied to partial molar volumes. (Section 9.2.4 will give a general version of this equation.) Dividing both sides of the equation by a + b gives the equivalent form... 

Different forms of Gibbs-Duhem equation are used and the choice of independent variables is determined by the vapour species being easily registered. The replacement of low intensities by the ratio of measured ones is a general approach in KCMS. Any type of equilibrium involving molecules j can be used for this purpose. Eor example, consider the reactions A + B AB... 

Here A(g) represents any gas-phase species, is the DFT electronic energy, is the zero-point correction due to vibrations, P° is a reference pressure often chosen as 1 bar, and AG°(r) = G T,P°) - G 0K,P°) and is calculated either from ideal gas statistical mechanics or from experimental data, often tabulated using the Shomate equation. This form of the Gibbs-Duhem equation is derived assuming ideal gas behavior, which is generally acceptable for the low partial pressures used in catalytic applications, and even for cases where the ideal gas assumption may not be valid it still serves as a first approximation. Applying the 0 K reference state of eqn (2.30), eqn (2.34) can be rewritten as ... 

Furthermore, we may express the change in the chemical potential with the spreading pressure in the form of a generalized Gibbs-Duhem type equation ... 

Gibbs-Duhem restriction on the chemical potential (eq. 8.5-3). Eq.(8.5-5) is the generalized Maxwell-Stefan constitutive relation. However, such form is not useful to engineers for analysis purposes. To achieve this, we need to express the chemical potential in terms of mole fractions. This is done by using eq. (8.5-2) into the constitutive flux equation (8.5-5). 

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