Gibbs method

The same pseudo-ensemble concept has been used by Camp and Allen [44] to obtain a pseudo-Gibbs method in which particle transfers are substituted by volume fluctuations of the two phases. The volume fluctuations are unrelated to the ones required for pressure equality (10.7) but are instead designed to correct imbalances in the chemical potentials of some of the components detected, for example, by test particle insertions. 

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]. 

Figure 7. Possible steps in the Gibbs method for simulating the properties of fluids. The schematic illustrates the initial system configuration and three steps (a) particle displacement, (b) volume change, and (c) particle transfer. Variables are defined as in Fig. 4. Reprinted with permission from W. L. Jorgensen and J. Tirado-Rives, J. Am. Chem. Soc. 110, 1657 (1988) [8], Copyright 1988 American Chemical Society.

Gibbs method at constant pressure A = test particle method at constant volume. Horizontal and vertical lines are error bars, (b) Experimental (x) and empirical equation of state (-) results for acetone-carbon dioxide mixtures. Reprinted with the permission of Taylor Francis Ltd. from Panagiotopoulos et al. [16] and with permission from A. Z. Panagiotopoulos, U. W. Suter, and R. C. Reid, Ind. Eng. Chem. Fundam. 25, 525 (1986). Copyright 1986 American Chemical Society. 

Kohn M. J. (1993b) Uncertainties in differential thermodynamic (Gibbs Method) P-T paths. Contrib. Mineral. Petrol. 113, 249-261. 

Spear F. S. (1988) The Gibbs method and Duhem s theorem the quantitative relationships among P, T, chemical potential, phase composition and reaction progress in igneous and metamorphic systems. Contrib. Mineral. Petrol 99, 249-256. 

Spear F. S., Ferry J. M., and Rumble D. (1982) Analytical formulation of phase equilibria the Gibbs method. Rev. Mineral. 10, 105-152. 

The study of thermodynamic stability owes a very great deal to Gibbs, who first realized its importance. Gibbs method was later examined in detail by Duhemf who clarified several aspects of the treatment. 

The inconvenience of Gibbs method is that it only allows a study of stability to be made when the perturbation takes place at constant values of one of the four groups of variables S, V S, p T, V or T, p.l... 

J. Willard Gibbs, in his celebrated memoir On the equilibrium of heterogeneous substances, stated the conditions of thermodynamic equilibrium in a form which has not been surpassed either in its elegant simplicity or in its generality. There is no problem concerning thermodynamic equilibrium which cannot, in principle at least, be dealt with by Gibbs methods. 

Theophile De Donder showed that this paradox could be resolved elegantly by the explicit calculation of the uncompensated heat, or better of the entropy production, resulting from a chemical reaction. To do this it is necessary to introduce a new function of state, the affinity, characteristic of the reaction and closely related to its irreversibility. In a series of papers since 1920, De Donder has developed a new formulation of chemical thermodynamics by combining the fundamental features of both the Gibbs method and those of the vanT Hoff-Nernst school. 

Throughout the preparation of this book I have received the fullest co-operation from the original authors, not only in providing new material for the revision, but also in their readiness to allow me to make a number of changes in the presentation. To facilitate the assimilation of De Donder s thermod3mamics into the more familiar techniques of the Gibbs method, the notation employed in the French edition has been extensively modified to conform more closely with established conventions. The changes made are discussed separately below. 

It is evident that the question of determination of the size of the critical cluster is a rather non-trivial problem, it depends qualitatively on the definition of the size, i.e., for which of the dividing surfaces the parameter has to be computed and which assumptions are employed in its determination. Employing the classical Gibbs method and the capillarity approximation, we arrive at curve 1 in Fig.lc. 

Figure 2. Hyper-surface of the thermodynamic potential difference between the heterogeneous state consisting of a cluster in the otherwise homogeneous ambient phase and the homogeneous initial state (here demonstrated for the case of a binary system). The curve via the saddle corresponds to the generalized Gibbs approach while the classical Gibbs method corresponds to ridge crossing, nj and n2 are here the number of particles in the cluster.

The authors interpret their results based on the work of Feder et al. According to Feder et al., which follow in their analysis the classical Gibbs method, the temperature difference AT = Tn - Tp between the droplet and the vapor can be written in the form... 

The thermodynamic properties of a system can be expressed as averages of certain functions of the phase-space (positions and momenta) of the constituent particles. These averages are either calculated over a phase-space trajectory of one system using the Boltzmann method, or over a phase-space of single points from several systems using the Gibbs method [44]. In both... 

Larionov and co-workers (Institute of Physical Chemistry, the U.S.S.R. Academy of Sciences, Moscow) (308-312) carried out systematic theoretical and experimental investigations of the adsorption from liquid solutions of nonelectrolytes on silica adsorbents. They studied the adsorption of individual substances and binary liquid solutions (benzene/carbon tetrachloride, carbon tetrachloride/isooctane, benzene/isooctane, etc.) on Si02 samples with different degrees of porosity but identical surface chemical properties. The experimental results were compared with the theoretical calculations carried out by the Gibbs method. This procedure made it possible to calculate the dependence of the enthalpy, entropy, and free energy of wetting on the concentration and to obtain expressions describ-... 

Surface or spreading pressure using the Gibbs method. 

Number of moles of adsorbed gas (surface excess) using the Gibbs method. Surface concentration, n /a, using the Gibbs method. 

For the subject of this book, Langmuir s (1933) extension of the phase rule for adsorption under equilibrium and non-equilibrium conditions is placed at the beginning of the treatment of surface thermodynamics. In the study of heterogeneous equilibrium by Gibbs methods, the term phase is used for a homogeneous part of a system without regarding for quantity or form. 

It is important to realise that the Gibbs model does not imply that the surface excess is concentrated at the Gibbs dividing surface this is clearly physically impossible since molecules have a finite size and cannot occupy a mathematical surface. What the Gibbs method docs is to recognise the existence of concentration profiles such as those shown in Figure 5.3, whose exact form cannot yet be measured experimentally, and to provide a method of expressing the observable consequences of their existence. 

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