Fugacity and pressure

The limiting behavior ensures that the fugacities of real gases approach those of the ideal gas in the limit of low pressure. Since at low pressures the fugacity and pressure become the same, it should be clear that fugacities will be expressed in the same units as pressure, Pa, MPa, atm, Torr, etc. 

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. 

Simplified forms of Eq. 9 can be obtained for various systems. For gas reactions standard states are usually chosen at a pressure of 1 atm. At this low pressure the deviation from ideality invariably is small hence fugacity and pressure are identical and /° = == 1 atm. Thus... 

The first term on the right-hand side is the idea gas limit, and the remaining -logarithmic terms express the successive virial corrections for the real gas behavior. It is evidently most convenient for this problem to choose the standard state pressure as P° = 0, where all gases are ideal. With this choice, we can write the relationship between fugacity and pressure as... 

Kress V. C. and Carmichael I. S. E. (1991) The compressibility of silicate liquids containing Fc203 and the effect of composition, temperature, oxygen fugacity and pressure on their redox states. Contrib. Mineral. Petrol. 108, 82-92. 

First consider the vapour-liquid equilibrium below the critical point. The equilibrium condition isTemperature or pressure may be specified. Because the relation between fugacity and pressure is implicit, the solution is obtained as follows ... 

Because of the close relationship between fugacity and pressure, it is convenient to define their ratio for a pure substance as... 

Figure 9.3 Along subcritical isotherms for pure fluids, the fugacity passes through stable, metastable, and unstable regions just as does the pressure. Here we have plotted the subcritical isotherm TlT = 0.863 for a van der Waals fluid. Each point (a-f) on the fugacity plot corresponds to the point of the same label on the Pv diagram. Points b and e have the same fugacity and pressure (P /= 0.539) and therefore locate the vapor-liquid equUibrium state. Points c and d are on the spinodal. Line segment be locates metastable liquid states segment de locates metastable vapor states segment cd locates unstable states.

Conversions provide the following relation between fugacity and pressure ... 

Hence, given a model for how a guest molecule interacts with its water cage (i.e. Ufj), Eqs. (2)-(4) can be used to calculate the conditions for hydrate stability. Note that very good quantitative models exist for linking fugacity and pressure in the gas phase [18,19], and so this scheme can predict hydrate pressures. 

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 pressure at which standard-state fugacities are most conveniently evaluated is suggested by considerations based on the Gibbs-Duhem equation which says that at constant temperature and pressure... 

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

Unfortunately, the ideal-gas assumption can sometimes lead to serious error. While errors in the Lewis rule are often less, that rule has inherent in it the problem of evaluating the fugacity of a fictitious substance since at least one of the condensable components cannot, in general, exist as pure vapor at the temperature and pressure of the mixture. 

For condensable components, we use the symmetric normaliza-L as x - 1 therefore, the quantity in brackets is the fugacity of pure liquid i at system temperature and pressure. 

Standard-state fugacities at zero pressure are evaluated using the Equation (A-2) for both condensable and noncondensable components. The Rackett Equation (B-2) is evaluated to determine the liquid molar volumes as a function of temperature. Standard-state fugacities at system temperature and pressure are given by the product of the standard-state fugacity at zero pressure and the Poynting correction shown in Equation (4-1). Double precision is advisable. 

One of the simplest cases of phase behavior modeling is that of soHd—fluid equilibria for crystalline soHds, in which the solubility of the fluid in the sohd phase is negligible. Thermodynamic models are based on the principle that the fugacities (escaping tendencies) of component are equal for all phases at equilibrium under constant temperature and pressure (51). The soHd-phase fugacity,, can be represented by the following expression at temperature T ... 

The heart of the question of non-ideality deals with the determination of the distribution of the respective system components between the liquid and gaseous phases. The concepts of fugacity and activity are fundamental to the interpretation of the non-ideal systems. For a pure ideal gas the fugacity is equal to the pressure, and for a component, i, in a mixture of ideal gases it is equal to its partial pressure yjP, where P is the system pressure. As the system pressure approaches zero, the fugacity approaches ideal. For many systems the deviations from unity are minor at system pressures less than 25 psig. 

The ratio f/f° is called activity, a. Note This is not the activity coefficient. The activity is an indication of how active a substance is relative to its standard state (not necessarily zero pressure), f°. The standard state is the reference condition, which may be anything however, most references are to constant temperature, with composition and pressure varying as required. Fugacity becomes a corrected pressure, representing a specific component s deviation from ideal. The fugacity coefficient is ... 

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). 

In Section HI, we discussed the relation between fugacities and activity coefficients in liquid mixtures, and we emphasized that we have a fundamental choice regarding the way we wish to relate the fugacity of a component to the pressure and composition. This choice follows from the freedom we have in choosing a convention for the normalization of activity coefficients. 

Fugacity, like other thermodynamics properties, is a defined quantity that does not need to have physical significance, but it is nice that it does relate to physical quantities. Under some conditions, it becomes (within experimental error) the equilibrium gas pressure (vapor pressure) above a condensed phase. It is this property that makes fugacity especially useful. We will now define fugacity, see how to calculate it, and see how it is related to vapor pressure. We will then define a related quantity known as the activity and describe the properties of fugacity and activity, especially in solution. 

We now have the foundation for applying thermodynamics to chemical processes. We have defined the potential that moves mass in a chemical process and have developed the criteria for spontaneity and for equilibrium in terms of this chemical potential. We have defined fugacity and activity in terms of the chemical potential and have derived the equations for determining the effect of pressure and temperature on the fugacity and activity. Finally, we have introduced the concept of a standard state, have described the usual choices of standard states for pure substances (solids, liquids, or gases) and for components in solution, and have seen how these choices of standard states reduce the activity to pressure in gaseous systems in the limits of low pressure, to concentration (mole fraction or molality) in solutions in the limit of low concentration of solute, and to a value near unity for pure solids or pure liquids at pressures near ambient. 

As mentioned above, activities are correctly represented by fugacities and not by partial pressures. Hence, the correct form of Eq. (22) is ... 

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