Gibbs-Duhem equation integrations

Here we have chosen AO, B203, and oxygen O as components of the ternary compound AB204 (or rather (A,B)304+l5). Since nAO+Pb,o3 — U°ab2o, (which is the Gibbs-Duhem equation integrated under the assumption that the spinel is strictly stoichiometric and stress effects can be neglected), we obtain from the cation fluxes and the steady state condition jA/cA = jB/cB = vb... 

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 ... 

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). 

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

A consistency test described by Chueh and Muirbrook (C4) extends to isothermal high-pressure data the integral (area) test given by Redlich and Kister (Rl) and Herington (H2) for isothermal low-pressure data. [A similar extension has been given by Thompson and Edmister (T2)]. For a binary system at constant temperature, the Gibbs-Duhem equation is written... 

An expression for V can be obtained from equation (5.29) by integration of the Gibbs-Duhem equation. Starting with the Gibbs-Duhem equation equation (5.23) applied to volume gives... 

L can also be obtained from Lj by integration of the Gibbs Duhem equation... 

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. 

Figure 3.9 Graphical integration of Gibbs-Duhem equation.

The quantities appearing in Eq. (16.2) are not independent. They are related by a Gibbs-Duhem equation, which is obtained in the same way as in the ordinary thermodynamics of bulk phases integrating with respect to the extensive variables results in Ua —TSa — pVa + 7Aa + E/if Nf. Differentiating and comparing with Eq. (16.2) gives ... 

The same type of polynomial formalism may also be applied to the partial molar enthalpy and entropy of the solute and converted into integral thermodynamic properties through use of the Gibbs-Duhem equation see Section 3.5. 

We may also integrate the Gibbs-Duhem equation using an Henrian reference state for B ... 

An alternative method of integrating the Gibbs-Duhem equation was developed by Darken and Gurry [10]. In order to calculate the integral more accurately, a new function, a, defined as... 

A graphical integration of the Gibbs-Duhem equation is not necessary if an analytical expression for the partial properties of mixing is known. Let us assume that we have a dilute solution that can be described using the activity coefficient at infinite dilution and the self-interaction coefficients introduced in eq. (3.64). 

For the ternary solution, the Gibbs-Duhem equation can be easily integrated to calculate the activity coefficient of water when the expressions for the activity coefficients of the electrolytes are written at constant molality. For Harned s rule, integration of the Gibbs-Duhem equation gives the activity of water as ... 

An alternative approach is to estimate activity coefficients of the solvents from experimental data and correlate these coefficients using, for example, the Wilson equation. Rousseau et al. (3) and Jaques and Furter (4) have used the Wilson equation, as well as other integrated forms of the Gibbs-Duhem equation, to show the utility of this approach. These authors found it necessary, however, to modify the definitions of the solvent reference states so that the results could be normalized. 

Table III presents the values of the constants used in the calculations. The t/o data have been obtained from the variation of Pq with Z by numerical integration of the Gibbs-Duhem equation using the Runge-Kutta method (22,23). The comparison of the Pq values of Table I with those obtained by some previous workers (24, 25) shows that our results are at most higher by 0.5-1 Torr.

The differentials of Equations 5 and 6 are introduced in the Gibbs-Duhem equation, the terms mid In (mj7 ) are replaced by Bjerrum s terms d(mi0-)), and the integration is carried out. The resulting equation is... 

To calculate the relative partial enthalpy of the salts in sea water it was necessary to integrate the Gibbs-Duhem equation graphically. 

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... 

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... 

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. 

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 ... 

An example drawn from Deitrick s work (Fig. 2) shows the chemical potential and the pressure of a Lennard-Jones fluid computed from molecular dynamics. The variance about the computed mean values is indicated in the figure by the small dots in the circles, which serve only to locate the dots. A test of the thermodynamic goodness of the molecular dynamics result is to compute the chemical potential from the simulated pressure by integrating the Gibbs-Duhem equation. The results of the test are also shown in Fig. 2. The point of the example is that accurate and affordable molecular simulations of thermodynamic, dynamic, and transport behavior of dense fluids can now be done. Currently, one can simulate realistic water, electrolytic solutions, and small polyatomic molecular fluids. Even some of the properties of micellar solutions and liquid crystals can be captured by idealized models [4, 5]. 

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],... 

---