Gibbs-Duhem

It is strictly for convenience that certain conventions have been adopted in the choice of a standard-state fugacity. These conventions, in turn, result from two important considerations (a) the necessity for an unambiguous thermodynamic treatment of noncondensable components in liquid solutions, and (b) the relation between activity coefficients given by the Gibbs-Duhem equation. The first of these considerations leads to a normalization for activity coefficients for nonoondensable components which is different from that used for condensable components, and the second leads to the definition and use of adjusted or pressure-independent activity coefficients. These considerations and their consequences are discussed in the following paragraphs. 

The pressure at which standard-state fugacities are most conveniently evaluated is suggested by considerations based on the Gibbs-Duhem equation which says that at constant temperature and pressure... 

Equation (16) is a differential equation and applies equally to activity coefficients normalized by the symmetric or unsymme-tric convention. It is only in the integrated form of the Gibbs-Duhem equation that the type of normalization enters as a boundary condition. 

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

In this case, there is no superscript on y because, by assumption, Y is independent of pressure. The disadvantage of this procedure is that the reference pressure p" is now different for each component, thereby introducing an inconsistency in the iso-baric Gibbs-Duhem equation [Equation (16)]. In many, but not all, cases, this inconsistency is of no practical importance. 

The special case of equation (A2.1.27) when T and p are constant (dJ= 0, dp = 0) is called the Gibbs-Duhem equation, so equation (A2.1.27) is sometimes called the generalized Gibbs-Duhem equation . 

Moreover, using the generalized Gibbs-Duhem equations (A2.1.27) for each of the two one-component phases,... 

Kofke D A 1993 Gibbs-Duhem integration a new method for direot evaluation of phase ooexistenoe by moleoular simulation Mol. Phys. 78 1 331-6... 

Each of the coexisting phases will be governed by a Gibbs-Duhem equation so that... 

The Gibbs-Duhem equation also follows from the definition of partial molar quantities nid/Hi + r 2d 2 0. With the Gibbs-Duhem equation, d G/dc2 becomes... 

This result, known as the Gibbs-Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. In particular, at constant T and P it represents a simple relation among the Af/ to which measured values of partial properties must conform. 

If temperature and pressure are constant, equation 130 reduces to equation 131 (constant T,P) which is a common form of the Gibbs-Duhem equation. 

A great deal of study and research has gone into the development of working equations that can represent the curves of Figure 3. These equations are based on solutions of the Gibbs-Duhem equation ... 

Mathematical Consistency. Consistency requirements based on the property of exact differentials can be apphed to smooth and extrapolate experimental data (2,3). An example is the use of the Gibbs-Duhem coexistence equation to estimate vapor mole fractions from total pressure versus Hquid mole fraction data for a binary mixture. 

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). 

Pertinent examples on partial molar properties are presented in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed.. Sec. 10.3, McGraw-Hill, NewYonc, 1996). Gibbs/Duhem Equation Differentiation of Eq. (4-50) yields... 

Moreover, the Gibbs/Duhem equation for a solution at given T and P, Eq. (4-52), becomes... 

Moreover, Eq. (4-122), the Gibbs/Duhem equation, may be written for experimental values in a binaiy system as... 

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. 

The right-hand side of this equation is exactly the quantity that Eq. (4-289), the Gibbs/Duhem equation, requires to be zero for consistent data. The residual on the left is therefore a direct measure of deviations from the Gibbs/Diihem equation. The extent to which values of this residual fail to scatter about zero measures the departure of the data from consistency with respect to this equation. 

If the experimental values P and w are closely reproduced by the correlating equation for g, then these residues, evaluated at the experimental values of X, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P and y . Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-X data this is possible because the P-x -y data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters Ot, b, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals 6P. Known as Barkers method Austral. ]. Chem., 6, pp. 207-210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y values however, it provides a better fit than does the procedure that minimizes the sum of the squares of the 6g residuals. 

Worth noting is the fact that Barkers method does not require experimental yf values. Thus the correlating parameters Ot, b, and so on, can be ev uated from a P-X data subset. Common practice now is, in fact, to measure just such data. They are, of course, not subject to a test for consistency by the Gibbs/Duhem equation. The worlds store of X T.E data has been compiled by Gmehling et al. (Vapor-Liquid Lquilibiium Data Collection, Chemistiy Data Series, vol. I, parts 1-8, DECHEMA, Frankfurt am Main, 1979-1990). 

The sulphur pressure of tire sulphides is a sensitive function of die tnetal/sulphur ratio around the stoichiometric composition, rising sharply when the sulphur content exceeds this limit. It follows from die Gibbs-Duhem equation... 

Onee again, integrating as in the Gibbs-Duhem equation, yields... 

While a Gibbs-Duhem equation does not exist for general transformations dSo, ds a, a specialized (i.e., coarse-grained ) Gibbs-Duhem equation... 

Equating the expressions for dVt given in Eqs. (37) and (40) and rearranging terms yields the coarse-grained Gibbs-Duhem equation... 

The Gibbs-Duhem equation allows the determination of activity coefficients for one component from data for those of other components. 

Martinez-Ortiz, J. A., and D. B. Manley, Direct Solution of the Isothermal Gibbs-Duhem Equation for Multicomponent Systems, Ind. Eng. Chem. Process Des. Dev., 17, 3, (1978) p. 346. 

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 Gibbs-Duhem equation is extremely important in solution chemistry and it can be seen from equation 20.171 that it provides a means of determining the activity of one component in a binary solution providing the activity of the other is known. 

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