Gibbs integral method

Considerable use has been made of the thermodynamic perturbation and thermodynamic integration methods in biochemical modelling, calculating the relative Gibbs energies of binding of inhibitors of biological macromolecules (e.g. proteins) with the aid of suitable thermodynamic cycles. Some applications to materials are described by Alfe et al. [11]. 

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. 

Once a state point of coexistence is established, additional state points can be determined expeditiously through application of the Gibbs-Duhem integration method [48,85,86]. In this approach a differential equation for the coexistence line is used to guide the establishment of state points away from the known coexistence point. The most well known such formula is the Clapeyron equation [41]... 

These data can be studied in two ways. The first is to use the Gibbs-Duhem equation and numerical integration methods to calculate the vapor-phase mole fractions, as considered in Problem 10.2-6. A second method is to choose a liquid-phase activity coefficient model and determine the values of the parameters in the model that give the best fit of the experimental data. We have, from Eq. 10.2-2b, that at the jth experimental point... 

In Figure 1 b a different construction is described. In this situation two coexistence lines are being traced by the Gibbs-Duhem method, each line having one phase (a) in common. The point where the fi-ct coexistence coincides with the -a. coexistence again forms a triple point. Subsequent integration then follows coexistence between the and p phases. It might be... 

Summary of Applications of Gibbs-Duhem Integration Method"... 

The inverse Kirkwood-Buff integral method (IKBI) does not involve a model such as the QLQC method does, hence is rigorous (Ben-Naim 1988). Note, again, the differences with respect to the treatment of binary solvent mixtures in Sect. 1.2.4. It requires the derivatives of the Gibbs energy of transfer ... 

The Gibbs-Duhem integration method excels in calculations of solid-fluid coexistence [48,49], for which other methods described in this chapter are not applicable. An extension of the method that assumes that the initial free energy difference between the two phases is known in advance, rather than requiring it to be zero, has been proposed by Meijer and El Azhar [51]. The procedure has been used in [51] to determine the coexistence lines of a hard-core Yukawa model for charge-stabilized colloids. 

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... 

As noted above, it is very difficult to calculate entropic quantities with any reasonable accmacy within a finite simulation time. It is, however, possible to calculate differences in such quantities. Of special importance is the Gibbs free energy, as it is the natoal thermodynamical quantity under normal experimental conditions (constant temperature and pressme. Table 16.1), but we will illustrate the principle with the Helmholtz free energy instead. As indicated in eq. (16.1) the fundamental problem is the same. There are two commonly used methods for calculating differences in free energy Thermodynamic Perturbation and Thermodynamic Integration. 

Kofke, D. A., Gibbs-Duhem integration a new method for direct evaluation of phase coexistence by molecular simulation, Mol. Phys. 1993, 78, 1331-1336... 

Most methods for the determination of phase equilibria by simulation rely on particle insertions to equilibrate or determine the chemical potentials of the components. Methods that rely on insertions experience severe difficulties for dense or highly structured phases. If a point on the coexistence curve is known (e.g., from Gibbs ensemble simulations), the remarkable method of Kofke [32, 33] enables the calculation of a complete phase diagram from a series of constant-pressure, NPT, simulations that do not involve any transfers of particles. For one-component systems, the method is based on integration of the Clausius-Clapeyron equation over temperature,... 

While the main driving force in [43, 44] was to avoid direct particle transfers, Escobedo and de Pablo [38] designed a pseudo-NPT method to avoid direct volume fluctuations which may be inefficient for polymeric systems, especially on lattices. Escobedo [45] extended the concept for bubble-point and dew-point calculations in a pseudo-Gibbs method and proposed extensions of the Gibbs-Duhem integration techniques for tracing coexistence lines in multicomponent systems [46]. 

The value of this standard molar Gibbs energy, p°(T), found in data compilations, is obtained by integration from 0 K of the heat capacity determined by the translational, rotational, vibrational and electronic energy levels of the gas. These are determined experimentally by spectroscopic methods [14], However, contrary to what we shall see for condensed phases, the effect of pressure often exceeds the effect of temperature. Hence for gases most attention is given to the equations of state. 

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... 

One method would be to use Eq. (3.62) and utilise a Newton-Raphson technique to perform a Gibbs energy minimisation with respect to the composition of either A or B. This has an advantage in that only the integral function need be calculated and it is therefore mathematically simpler. The other is to minimise the difference in potential of A and B in the two phases using the relationships... 

The electromotive force (EMF) generated by electrochemical cells can be used to measure partial Gibbs energies which, like vapour pressure measurements, distinguishes these methods from other techniques that measure integral thermodynamic quantities. Following Moser (1979), a typical cell used to obtain results on Zn-ln-Pb is represented in the following way ... 

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.

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