Liquid, fugacity potential

As implied by Fig. 13-2, the rate of mass transfer of a component from the liquid to the vapor phase consists of a sequence of rate processes which occur in series. First component i is transferred from the liquid phase to the interface at the liquid side. The fugacity potential or driving force for this transfer process is taken to be /f —ff, where the fugacity /f is evaluated at the bulk conditions of the liquid phase and ff is evaluated at the conditions at the liquid side of the interface. The rate of mass transfer per unit time per unit height of packing Rf for this step of the process is expressed as follows... 

Thermodynamics of Liquid—Liquid Equilibrium. Phase splitting of a Hquid mixture into two Hquid phases (I and II) occurs when a single hquid phase is thermodynamically unstable. The equiUbrium condition of equal fugacities (and chemical potentials) for each component in the two phases allows the fugacitiesy andy in phases I and II to be equated and expressed as ... 

In Chapter 5, we considered systems in which composition becomes a variable, and defined and described the chemical potential. We showed that the chemical potential provides the condition for spontaneity or equilibrium. It is the potential that drives the flow of mass in a chemical process, A useful quantity related to the chemical potential is the fugacity. It can also be thought of as a measure of the flow of mass in a chemical process, and can be used to determine the point of equilibrium. It is often known as the escaping tendency since it can be used to describe the ease with which mass flows from one phase to another, particularly the flow from a solid or liquid phase to a gas phase. 

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. 

An attractive feature of K<)A is that it can replace the liquid or supercooled liquid vapor pressure in a correlation. K,-ja is an experimentally measurable or accessible quantity, whereas the supercooled liquid vapor pressure must be estimated from the solid vapor pressure, the melting point and the entropy of fusion. The use of KOA thus avoids the potentially erroneous estimation of the fugacity ratio, i.e., the ratio of solid and liquid vapor pressures. This is especially important for solutes with high melting points and, thus, low fugacity ratios. 

At equilibrium the chemical potential should be equal in the gas and liquid phases. At uniform temperature and pressure, this leads to the same fugacities in the two phases. In the liquid, the fugacity may be related to the fugacity of a standard state, f°... 

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... 

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. 

Let us now continue with our discussion of how to relate the chemical potential to measurable quantities. We have already seen that the chemical potential of a gaseous compound can be related to pressure. Since substances in both the liquid and solid phases also exert vapor pressures, Lewis reasoned that these pressures likewise reflected the escaping tendencies of these materials from their condensed phases (Fig. 3.9). He thereby extended this logic by defining the fugacities of pure liquids (including subcooled and superheated liquids, hence the subscript L ) and solids (subscript s ) as a function of their vapor pressures, pil ... 

Using the concept of fugacity we can now, in analogy to the gaseous phase (Eq. 3-28), express the chemical potential of a compound i in a liquid solution by ... 

Remember that chemical potential for the liquid must equal chemical potential for the gas at equilibrium. For a pure substance this means that at any point along the vapor pressure line, the chemical potential of the liquid must equal the chemical potential of the gas. Thus Equation 15-3 shows that the fugacity of the liquid must equal the fugacity of the gas at equiHbliuffl ffn"thre Vaporf "pressure" line. So gas-liquid Equilibria can be calculated under the condition that... 

Remember that the chemical potential, Gj, for a component of a mixture at equilibrium must be the same in both the gas and the liquid. Thus Equation 15-18 shows that at equilibrium the fugacities of a component must be equal in both the gas and the liquid. So gas-liquid equilibria can be calculated under the condition that... 

In general, equality of component fugacities, ie, chemical potentials, in the vapor and liquid phases yields the following relation for vapor—liquid equilibrium ... 

Perhaps the most significant of the partial molar properties, because of its application to equilibrium thermodynamics, is the chemical potential, i. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equilibrium problems. The natural logarithm of the liquid-phase activity coefficient, lny, is also defined as a partial molar quantity. For liquid mixtures, the activity coefficient, y describes nonideal liquid-phase behavior. 

Although activity and fugacity are closely related, they have quite different characteristics in regard to phase equilibria. Consider, for example, the equilibrium between liquid water and water vapor in the interstices of an unsaturated soil. At a given temperature and pressure, the principles of thermodynamic equilibrium demand that the chemical potentials and fugacities of water in the two phases be equal. However, the activities of water in the two phases will not be the same because the Standard State for the two phases is not j the same. Indeed, f° = 1 atm for the water vapor, so its activity is numerically ]... 

The water electrolysis rest potential is determined from extrapolation to ideal conditions. Variations of the concentration, c, and pressure, p, from ideality are respectively expressed by the activity (or fugacity for a gas), as a = yc (or yp for a gas), with the ideal state defined at 1 atmosphere for a pure liquid (or solid), and extrapolated from p = 0 or for a gas or infinite dilution for a dissolved species. The formal potential, measured under real conditions of c and p can deviate significantly from the (ideal thermodynamic) rest potential, as for example the activity of water, aw, at, or near, ambient conditions generally ranges from approximately 1 for dilute solutions to less than 0.1 for concentrated alkaline and acidic electrolytes.91"93 The potential for the dissociation of water decreases from 1.229 V at 25 °C in the liquid phase to 1.167 V at 100 °C in the gas phase. Above the boiling the point, pressure is used to express the variation of water activity. The variation of the electrochemical potential for water in the liquid and gas phases are given by ... 

Equations (6.1) and (6.2) pertain to ideal systems, that is, systems where there are no interactions between the molecules. In a real system the pressure effect on p. in the vapor phase has to be modified by a fugacity coefficient < >, and the effect of mixing on the chemical potential in the liquid phase has to be modified by an activity coefficient 7,. The more general expression for equilibrium (called the 4>-y representation) then becomes... 

Activity and Activity Coefficient. —When a pure liquid or a mixture is in equilibrium with its vapor, the chemical potential of any constituent in the liquid must be equal to that in the vapor this is a consequence of the thermodynamic requirement that for a system at equilibrium a small change at constant temperature and pressure shall not be accompanied by any change of free energy, i.e., (d( )r. p is zero. It follows, therefore, that if the vapor can be regarded as behaving ideally, the chemical potential of the constituent i of a solution can be written in the same form as equation (7), where p,- is now the partial pressure of the component in the vapor in equilibrium with the solution. If the vapor is not ideal, the partial pressure should be replaced by an ideal pressure, or fugacity, but this correction need not be considered further. According to Raoult s... 

Intrapellet transport restrictions can limit the rate of removal of products, lead to concentration gradients within pellets, and prevent equilibrium between the intrapellet liquid and the interpellet gas phase. Transport restrictions increase the intrapellet fugacity of hydrocarbon products and provide a greater chemical potential driving force for secondary reactions. The rate of secondary reactions cannot be enhanced by a liquid phase that merely increases the solubility and the local concentration of a reacting molecule. Olefin fugacities are identical in any phases present in thermodynamic equilibrium thus, a liquid phase can only increase the rate of a secondary reaction if it imposes a transport restriction on the removal of reacting species involved in such a reaction (4,5,44). Intrapellet transport rates and residence times depend on molecular size, just as convective transport and bed residence time depend on space velocity. As a result, bed residence time and molecular size affect chain termination probability and paraffin content in a similar manner. 

The number of chain growth sites is assumed to be proportional to the number of exposed metal atoms in the supported catalysts. We also assume that all reactions are driven by the chemical potential or fugacity of its reactants in the liquid phase, which equals their extrapellet gas-phase values only in the absence of transport restrictions. As a result, carbon number effects on selectivity cannot arise from the higher solubility of larger hydrocarbon chains in liquid hydrocarbons. 

Tetrahydrofuran is a liquid at room temperature and boils at 66 °C. The fugacity model predicts that tetrahydrofuran will be found in the environment where it is released. Photodegradation by hydroxyl radicals in air is estimated to be rapid and the hydroxyl radical reaction half-life is estimated at 7.3 h. Tetrahydrofuran released to water could partition to the water compartment and readily biodegrade, but not hydrolyze. Tetrahydrofuran has a very low bioaccumulation potential as evidenced by its low octanol/water partition coefficient. 

Express the chemical potential as a function of concentration and activity coefficients for liquid and solid phase (Eq. 17) or as a function of partial pressure and fugacity coefficients for the gas phase (Eq. 18). Define appropriate secondary reference states. 

The theory and conditions for phase equilibrium are well established. If more than one phase is present, then the chemical potential of a component is the same in all phases present. As chemical potential is linked functionally to the concepts of fugacity and activity, models for phase behavior prediction and correlation based on chemical potentials, fugacities, and activities have been developed. Historically, phase equilibrium calculations for hydrocarbon mixtures have been fragmented with liquid-vapor, liquid-liquid, and other phase equilibrium calculations, subject to separate and diverse treatments depending on the temperature, pressure, and component properties. Many of these methods and approaches arose to meet specific needs in the chemical process industries. Poling, Prausnitz,... 

Since N2 adsorption is done at 77 K and CO2 at 273 or 298 K, the experiments cannot be directly compared, which introduces strong concerns about the similarities and differences among both adsorptives. Thus, a better way to compare the two experiments is to plot the characteristic curves [33—35, 37], These characteristic curves, obtained applying the Dubinin-Radushkevich (DR) equation [47] to the adsorption isotherms, are the plot of the logarithm of the volume of liquid adsorbed versus the square of the adsorption potential corrected for the affinity coefficient ((3) of the adsorptive ((/l//3) = (RTln(/o//)/[3), T being the temperature, / the fugacity, and/ the saturation fugacity). 

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