Fugacity description

It should be evident from the examples in Chapters 10, 11, and 12 that the evaluation of species fugacities or partial molar Gibbs energies (or chemical potentials) is central to any phase equilibrium calculation. Two different fugacity descriptions have been used, equations of. state and activity coefficient models. Both have adjustable parameters. If the values of these adjustable parameters are known or can be estimated, the phase equilibrium state may be predicted. Equally important, however, is the observation that measured phase equilibria can be used to obtain these parameters. For example, in Sec. 10.2 we demonstrated how activity coefficients could be computed directly from P-T-x-y data and how activity coefficient models could be fit to such data. Similarly, in Sec. 10.3 we pointed out how fitting equation-of-state predictions to experimental high-pressure phase equilibrium data could be used to obtain a best-fit value of the binary interaction parameter.. /"... 

In Section I, we indicated that significant progress in understanding high-pressure thermodynamics of mixtures requires a quantitative description of the variation of fugacity with pressure as given by Eq. (3). To obtain the effect of pressure on activity coefficient we substitute as follows ... 

Activity ax is termed the rational activity and coefficient yx is the rational activity coefficient This activity is not directly given by the ratio of the fugacities, as it is for gases, but appears nonetheless to be the best means from a thermodynamic point of view for description of the behaviour of real solutions. The rational activity corresponds to the mole fraction for ideal solutions (hence the subscript x). Both ax and yx are dimensionless numbers. 

The method of using fugacity calculations will be discussed later in this symposium, therefore a detailed description will not be given in this paper. The description of equilibrium models using chemical equilibrium expressions will be discussed with the recognition that the two approaches are very much the same. 

Mainly, the available models have been developed based on the fugacity approach, which use the fugacity as surrogate of concentration, for the compilation and solution of mass-balance equations involved in the description of chemicals fate. However, a new... 

This choice of basis follows naturally from the steps normally taken to study a geochemical reaction by hand. An aqueous geochemist balances a reaction between two species or minerals in terms of water, the minerals that would be formed or consumed during the reaction, any gases such as O2 or CO2 that remain at known fugacity as the reaction proceeds, and, as necessary, the predominant aqueous species in solution. We will show later that formalizing our basis choice in this way provides for a simple mathematical description of equilibrium in multicomponent systems and yields equations that can be evaluated rapidly. 

Most applications in materials science are carried out under pressures which do not greatly exceed 1 bar and the difference between/and/ is small, as can be seen from the fugacity of N2(g) at 273.15 K [15] given in Figure 2.11. Hence, the fugacity is often set equal to the partial pressure of the gas, i.e./ p. More accurate descriptions of the relationship between fugacity and pressure are needed in other cases and here equations of state of real, non-ideal gases are used. 

For concentrated solutions, the activity coefficient of an electrolyte is conveniently defined as though it were a nonelectrolyte. This is a practical definition for the description of phase equilibria involving electrolytes. This new activity coefficient f. can be related to the mean ionic activity coefficient by equating expressions for the liquid-phase fugacity written in terms of each of the activity coefficients. For any 1-1 electrolyte, the relation is ... 

The empirical description of dilute solutions that we take as the starting point of our discussion is Henry s law. Recognizing that when the vapor phase is in equUibrium with the solution, p,2 in the condensed phase is equal to p,2 g, we can state this law as follows For dilute solutions of a nondissociating solute at constant temperature, the fugacity of the solute in the gas phase is proportional to its mole fraction in the condensed phase That is. 

From a thermodynamic viewpoint, we may imagine that, in an actinide metal, the model of the solid in which completely itinerant and bonding 5 f electrons exist and that in which the same electrons are localized, constitute the descriptions of two thermodynamic phases. The 5f-itinerant and the 5 f-localized phases may therefore have different crystal properties a different metallic volume, a different crystal structure. The system will choose that phase which, at a particular T and p (since we are dealing with metals, the system will have only one component) has the lower Gibbs free-energy. A phase transition will occur then the fugacity in the two possible phases is equal e.g. the pressure. To treat the transition, therefore, the free energies and the pressures of the two phases have to be compared. We recall that ... 

Physical state and its description. The physical state of each substance is indicated, in the text and in column 2 of the table, as gaseous, liquid, crystal, glass, colloidal, or in aqueous or other solution. (See the list of abbreviations on page 14.) All states are for a pressure, or a fugacity, of one atmosphere and a temperature of 18°, unless otherwise indicated. 

It has been said chemists have solutions 3 Solutions are involved in so many chemical processes1 that we must have the mathematical tools to comfortably work with them, and thermodynamics provides many of these tools. Thermodynamic properties such as the chemical potential, partial molar properties, fugacities, and activities, provide the keys to unlock the description of mixtures. 

In equilibrium, the quantity N of a given sorbate, which is absorbed on a given sorbent, depends on its partial pressure (fugacity) P in the gas phase and the temperature T. A basic phenomenological description is specification of the functional dependence between N, P, and T. Both experimental observations and theoretical or thermodynamic descriptions are often the case in univariant functional descriptions the relation between N and P at constant T (an isotherm), between N and T at constant P (an isobar), or between P and T at constant N (an isostere). 

As in the case for adsorption (see Section 2.2), in equilibrium, the quantity N of a given solute which is dissolved in a given solvent depends on its gas phase partial pressure (fugacity) P and on the temperature T, and a basic phenomenological description of the equilibrium is specification of the functional relationship between N, P, and T. At sufficiently low pressures, it is expected that the pressure dependence is linear (Henry s law) ... 

Fugacity Determinations of the Products of Detonation were determined by M.A. Cook for PETN, RDX, LNG, Tetryl and 60% Straight Dynamite, by employing the equation of state derived from die hydrodynamic theory and observed velocities of detonation. The so-called reiteration method was developed for solving simultaneously as many equilibria as is necessary to define completely the composition of the products of detonation. Detailed description, together with 14 references is given-in the original article ... 

The three seminal ideas in this early work of Temkin are quite general. The first is that adsorption of nitrogen is rate determining. The second is the virtual pressure or fugacity of adsorbed nitrogen, a concept of great importance to the understanding of catalytic cycles at the steady state. The third idea is the kinetic description of the catalytic surface as a nonuniform one. The last was systematized later by Temkin s school, both in theory and in application, to a... 

The description of vapour-liquid equilibrium behaviour can be obtained from analytical equations and generalized correlations. The generalized conelations are generally for the equilibrium ratio, K, and the fugacity coefficients. 

Since then further progress has extended the field of applicability of Gibbs chemical thermodynamics. Thus the introduction of the ideas of fugacity and activity by G. N. Lewis enabled the thermodynamic description of imperfect gases and of real solutions to be expressed with the same formal simplicity as that of perfect gases and ideal solutions. These results were completed when N. Bjerrum and E. A. Guggenheim introduced osmotic coefficients. 

One complication with this description is that a species can be present in a liquid mixture, though at the temperature and pressure of the mixture the substance would be a vapor or a solid as a pure component. This is especially troublesome if the compound is below its melting point, so that it is the solid sublimation pressure rather than the vapor pressure that is known, or if the compound is above its critical temperature, so that the vapor pressure is undefined. In the first case one frequently ignores the phase change and extrapolates the liquid vapor pressure from higher temperatures down to the temperature of interest using, for example, the Antoine equation, eqn. (2.3.11). For supercritical components it is best to use an EOS and compute the fugacity of a species in a mixture, as described in Section 2.5. 

For the description of inter-diffusion in dense gases and liquids the expression for ds is further modified introducing a fugacity (i.e., a corrected pressure) or an activity (i.e., a corrected mole fraction) function [76]. The activity as(T,ct 2, 3,a q i) for species s is defined by ... 

The pure-component fugacity is a substance-specific function of [T,p]. It follows from Equation (4.474) that ideal K values are substance-specific functions of [T,p], but they are also independent of the composition of the mixture. Ideal K values provide an approximate description of mixtures of isomers, mixtures of near-neighbor homologues, and mixtures of isomers of nearneighbor homologues. 

The one-field model gives a description of the approximate grand canonical ensembles introduced in Chapter 11, Section 5. These ensembles, called equilibrium ensembles , depend on only two fugacities, which determine respectively the average number of polymers and the average number of monomers (constituting the polymers). In the continuous case, a connected partition function J(/, a) can be associated with this ensemble it is defined by... 

We will have occasion to use the Henry s law descriptions (on both a mole fraction and a molality basis) and the associated activity coefficients several times in this book. The immediate disadvantage of these choices is that / (T, P,x — 1) and ff T, P, M = ]) can be obtained only by extrapolation of experimental information for very dilute solutions. However, this information may be easier to obtain and more accurate than that obtained by estimating the pure liquid fugacity of a species whose equilibrium state is a supercritical gas or a solid below its triple-point temperature. 

The starting point for the description of this phase equilibrium problem is again the equality of fugacities of each species in all phases in which that species appears,... 

The equation (6.36) combines the fligacity description for the vapour phase with liquid activity modelling, but needs the pure liquid fugacity as function of more accessible properties. In Chapter 5 the following relation was demonstrated ... 

More simple models, requiring only an approximate description of the main driving forces as input data, produce less precise results but their versatility allows their application to relatively non-homogeneous areas and, therefore, on a larger scale. Simple runoff models derived from the original fugacity approach , were developed at the University of Toronto by Mackay and co-workers, but too technical to be described here. (Mackay,... 

The thermodynamic behavior of real solutions, such as those in which most reactions take place, is based on a description of ideal solutions. The model of an ideal solution is based on Raoult s law. While we can measure the concentration of a species in solution by its mole fraction, X the fact that the solution is not ideal tells us that thermodynamic behavior must be based on fiigacity, fi. In this development, we will use L as the fugacity of the pure component i and f, as the fugacity of component i in the solution. When Xi approaches unity, its fugacity is given by... 

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