Gibbs-Duhem equation integrated form

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). 

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. 

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 ... 

It is possible, as a result of cancellation, to satisfy the integral test of Eq. 10.2-13 while violating the differential form of the Gibbs-Duhem equation, Eq. 9.3-15, on which Eq. 10.2-13 is based, at some or all data points. In this case the experimental data should be rejected as thermodynamically inconsistent. Thus, the integral consistency test is a necessar). but not sufficient, condition for accepting experimental data. 

This equation is rarely used for testing the consistency of p, jr, and y data, the claim being made that it is difficult to obtain the gradients with sufficient accuracy except where there are a considerable number of points spread over the entire composition range. The most common method for testing consistency is to use an integrated form of the Gibbs-Duhem equation ... 

This equation needs to be modified when only (pyX), (p,y)y or (x,y) is used to derive or when one component is involatile. In the former case, one of the error terms drops out but a further error will be introduced because an equation in some form, or an integration, or a differentiation, is necessary to reduce the data. When component B is involatile, is calculated directly from equation (2) using only (p, at) data since the vapour phase consists only of the volatile component and hence can only be calculated by making use of the Gibbs-Duhem equation. 

Moderate temperature changes result in such minor changes in activity coefficient that constant-pressure data are ordinarily satisfactory for application of the various integrated forms of the Gibbs-Duhem equation. 

Applications of the Integrated Equations. The usefulness of the Gibbs-Duhem equation for establishing the thermodynamic consistency of, and for smoothing, data has been pointed out. The various integrated forms are probably most useful for extending limited data, sometimes from even single measurements, and it is these applications that are most important for present purposes. 

Sometimes the thermodynamic data are expressed in the form of an empirical equation. For example, the activity coefficient of a component in a solution is often expressed as a function of composition in terms of an empirical equation. In such cases, the Gibbs-Ouhem equation can be solved analytically instead of graphically. The following example illustrates the analytical integration of the Gibbs-Duhem equation. 

The thermodynamic equilibrium constant IC is then obtained by integrating the appropriate form of the Gibbs-Duhem equation to give as corresponding... 

As shown by results for the Lu + H system from Subramanian and Smith (1982) in fig. 12, the metal-cubic hydride plateaus become very short for the later elements. Lu forms the smallest of all the rare-earth hydrides and has the highest solubility in the a-phase, as noted in section 3.2. Pressure-composition isotherms are reproducible between runs, with minimal hysteresis except at the lowest temperature of824°C. Phase boundaries and pressure equations thus determined are in excellent agreement with earlier work. Gibbs-Duhem integrations of the measured data and a small extrapolation to the LuHj composition give AH/ = — 204.8 + 1.9 kJ/mol and ASf = — 136 2J/molK. 

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