Fugacity practical

Equation (1) is of little practical use unless the fuga-cities can be related to the experimentally accessible quantities X, y, T, and P, where x stands for the composition (expressed in mole fraction) of the liquid phase, y for the composition (also expressed in mole fraction) of the vapor phase, T for the absolute temperature, and P for the total pressure, assumed to be the same for both phases. The desired relationship between fugacities and experimentally accessible quantities is facilitated by two auxiliary functions which are given the symbols (f... 

The determination of equilibria is done theoretically via the calculation of free energies. In practice, the concept of fugacity is used for which the unit of measurement is the bar. The equation linking the fugacity to the free energy is written as follows >... 

Both convention and convenience suggest use of the fugacity in practical calculations in place of the chemical potential ]1. Equation 218 is then replaced by the equal fugacity criterion which follows directiy from equation 160 ... 

The chemical potential pi plays a vital role in both phase and chemical-reaction equilibria. However, the chemical potential exhibits certain unfortunate characteristics which discourage its use in the solution of practical problems. The Gibbs energy, and hence pi, is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. Moreover, pi approaches negative infinity when either P or Xi approaches zero. While these characteristics do not preclude the use of chemical potentials, the application of equilibrium criteria is facilitated by introduction of the fugacity, a quantity that takes the place of p. but which does not exhibit its less desirable characteristics. 

When Eq. (4-282) is applied to XT E for which the vapor phase is an ideal gas and the liquid phase is an ideal solution, it reduces to a veiy simple expression. For ideal gases, fugacity coefficients and are unity, and the right-hand side of Eq. (4-283) reduces to the Poynting factor. For the systems of interest here this factor is always veiy close to unity, and for practical purposes <1 = 1. For ideal solutions, the activity coefficients are also unity. Equation (4-282) therefore reduces to... 

Equilibrium Constants For practical application, Eq. (4-336) must be reformulated. The initial step is elimination of the in favor of fugacities. Equation (4-74) for species i in its standard state is subtracted from Eq. (4-77) for species i in the equilibrium mixture, giving... 

When the fuel gas is not pure hydrogen and air is used instead of pure oxygen, additional adjustment to the calciJated cell potential becomes necessary. Since the reactants in the two gas streams practically become depleted between the inlet and exit of the fuel cell, the cell potential is decreased by a term representing the log mean fugac-ities, and the operating cell efficiency becomes ... 

Finally, it is necessary to observe that the values of activities and fugacities calculated are thermodynamic quantities that cannot always be realised in practice, e.g. very high activities of metal ions cannot be attained because of solubility consideration and very low activities have no physical significance. 

Since Eqs. (5) and (6) are not restricted to the vapor phase, they can, in principle, be used to calculate fugacities of components in the liquid phase as well. Such calculations can be performed provided we assume the validity of an equation of state for a density range starting at zero density and terminating at the liquid density of interest. That is, if we have a pressure-explicit equation of state which holds for mixtures in both vapor and liquid phases, then we can use Eq. (6) to solve completely the equations of equilibrium without explicitly resorting to the auxiliary-functions activity, standard-state fugacity, and partial molar volume. Such a procedure was discussed many years ago by van der Waals and, more recently, it has been reduced to practice by Benedict and co-workers (B4). 

Air-soil diffusion thus appears to be much slower than air-water diffusion because of the slow migration in the soil matrix. In practice, the result will be a nonuniform composition in the soil with the surface soil (which is much more accessible to the air than the deeper soil) being closer in fugacity to the atmosphere. 

Mackay, D., Paterson, S. (1990) Fugacity models. In Practical Applications of Quantitative Structure-Activity Relationships (QSAR) in Environmental Chemistry and Toxicology. Karcher, W., Devillers, J., Eds., pp. 433 -60, Kluwer Academic Publishers, Dordrecht, The Netherlands. 

In a fixed activity path, the activity of an aqueous species (or those of several species) maintains a constant value over the course of the reaction path. A fixed fugacity path is similar, except that the model holds constant a gas fugacity instead of a species activity. Fixed activity paths are useful in modeling laboratory experiments in which an aspect of a fluid s chemistry is maintained mechanically. In studying reaction kinetics, for example, it is common practice to hold constant the pH of... 

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 ... 

For most of the situations encountered in solvent extraction the gas phase above the two liquid phases is mainly air and the partial (vapor) pressures of the liquids present are low, so that the system is at atmospheric pressure. Under such conditions, the gas phase is practically ideal, and the vapor pressures represent the activities of the corresponding substances in the gas phase (also called their fugacities). Equilibrium between two or more phases means that there is no net transfer of material between them, although there still is a dynamic exchange (cf. Chapter 3). This state is achieved when the chemical potential x as... 

In our kinetic calculations, we refer to the directly observed partial pressure of propylene, rather than to its fugacity, because over the temperature and pressure range examined, we can assume that partial pressures and fugacities are practically proportional. In fact, from the literature data, the variation in propylene fugacity coefficient, in the range of our kinetic tests, is small (about 0.97 at 30° and 2700 mm. Hg about 0.99 at 70° and 450 mm. Hg of propylene partial pressure). 

Figures 1 to 3 present calculated equilibrium molar ratios of products to reactants as a function of temperature and total pressure of 1 and 100 atm. for the gas-carbon reactions (4), (7), and (5), (6), (4), (7), respectively. Up to 100 atm. over the temperature range involved, the fugacity coefficients of the gases are close to 1 therefore, pressures can be calculated directly from the equilibrium constant. From Fig. 1, it is seen that at temperatures above 1200°K. and at atmospheric pressure, the conversion of carbon dioxide to carbon monoxide by the reaction C - - COj 2CO essentially is unrestricted by equilibrium considerations. At elevated pressures, the possible conversion markedly decreases hence, high pressure has little utility for this reaction, since increased reaction rate can easily be obtained by increasing reaction temperature. On the other hand, for the reaction C -t- 2H2 CH4, the production of methane is seriously limited at one atmosphere pressure and practical operating temperatures, as seen in Fig. 2. Obviously, this reaction must be conducted at elevated pressures to realize a satisfactory yield of methane. For the carbon-steam reaction.

T is temperature, P is pressure, and / is the fugacity of the component. In Equation 3 subscript k refers to each component of the system. In the present discussion the fugacity 42) is employed in preference to the chemical potential 21). Earlier in the history of the petroleum industry, Raoult s 55) and Dalton s laws were applied to equilibrium at pressures considerably above that of the atmosphere. These relationships, which assume perfect gas laws and additive volumes in the gas phase and zero volume for the liquid phase, prove to be of practical utility only at low pressures. Henry s law was found to be a useful approximation only for gases which were of low solubility and at reduced pressures less than unity. 

Nevertheless, as noted by Lewis and Randall, certain post-Gibbsian addenda appeared, which will be discussed in the present section. Some of these innovations, such as activity and fugacity (Section 5.8.1), were designed primarily to satisfy practical needs of representing experimental thermochemical data, with no deeper claims on the underlying structure of the theory. In contrast, the developments initiated by Nemst s heat theorem, culminating in what became widely known as the third law of thermodynamics, appear to call into question the structural completeness of the Gibbsian formalism. These developments will be critically discussed in Section 5.8.2. 

The Henry constant J Cis a function of T but not P. (In some theoretical treatments, the Henry constant is the ratio of fugacity to quantity adsorbed, i.e., the inverse of the sense used here.) It is generally expected that adsorption will be governed by Henry s law at sufficiently low pressures. It is possible to construct theoretical models for adsorption in which an isotherm does not reduce to Henry s law, Equation (2.3), even in the limit P —> 0, but it is not clear that such situations obtain in practice and doubtful that they are important in noble gas geochemistry. 

Although fugacity is theoretically and computationally significant, it is the pressure that we need for practical applications. To find the relation between / and P, we write, from Eqs. (31) and (51),... 

These equations are restatements of Eqs. (6.37) and (6.38) wherein the restriction of the derivatives to constant composition is shown explicitly. They lead to Eqs. (6.40), (6.41), (6.42), and (11.20), which allow calculation of residual properties and fugacity coefficients from PVT data and equations of state. It is through the residual properties that this kind of experimental information enters into the practical application of thermodynamics. 

Fugacity is relevant to imperfect water substance, but not to perfect carbon dioxide at practical temperatures. At 160 US dollars the tables are a good investment. 

---