Standard States Using Fugacities

We can also see from the same equation why it often proves convenient to choose a standard state for i, which is not only not its most stable state, but one that is extremely hypothetical. If f° is set to 1.0, equation (12.1) becomes 

To put this in other terms, if you have the fugacity of some substance i in some system, then RT In is the difference in Gibbs free energy per mole of i in the system at T and i as an ideal gas at T. Whether i could ever come close to existing as an ideal gas is irrelevant. Other examples of hypothetical standard states are discussed below. 

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Before a reaction-equilibrium calculation can be performed, we must select an appropriate standard state for each species. Moreover, we must clearly distinguish quantities, such as fugacities and activities, that depend on the final equilibrium state (T, P, x ), from those quantities, such as equilibrium constants, that depend only on the equilibrium temperature T, the standard-state pressures P , and the phase. Typically, the standard-state pressure and phase are chosen according to whether the real substance is gas, liquid, or solid at the equilibrium conditions. Those three possibilities are discussed, in turn, here, and each discussion culminates with a particular expression for the activity. Those expressions can be used either in the stoichiometric development, via (10.3.14), or in the nonstoichiometric development, via (10.3.38). We emphasize that when we use the stoichiometric approach, the standard states used for the fugacities must be consistent with those associated with the equilibrium constant. 

Finally, comparing Equations (8.32) and (8.19) we see that it is quite possible for the activity of i to be numerically identical to the fugacity of i. It just requires that the standard state chosen for i is ideal gas i at T and 1 bar. We will find (Chapter 9) that this is a very common situation when i is a gas or a gaseous component, and in fact it is the standard state used in program supcrt92, as shown in Table 8.1. In this table, note that at each of the three temperatures... 

When the same standard-state fugacity is used in both phases. Equation (5) can be rewritten... 

It is strictly for convenience that certain conventions have been adopted in the choice of a standard-state fugacity. These conventions, in turn, result from two important considerations (a) the necessity for an unambiguous thermodynamic treatment of noncondensable components in liquid solutions, and (b) the relation between activity coefficients given by the Gibbs-Duhem equation. The first of these considerations leads to a normalization for activity coefficients for nonoondensable components which is different from that used for condensable components, and the second leads to the definition and use of adjusted or pressure-independent activity coefficients. These considerations and their consequences are discussed in the following paragraphs. 

For such components, as the composition of the solution approaches that of the pure liquid, the fugacity becomes equal to the mole fraction multiplied by the standard-state fugacity. In this case,the standard-state fugacity for component i is the fugacity of pure liquid i at system temperature T. In many cases all the components in a liquid mixture are condensable and Equation (13) is therefore used for all components in this case, since all components are treated alike, the normalization of activity coefficients is said to follow the symmetric convention. ... 

In a binary liquid solution containing one noncondensable and one condensable component, it is customary to refer to the first as the solute and to the second as the solvent. Equation (13) is used for the normalization of the solvent s activity coefficient but Equation (14) is used for the solute. Since the normalizations for the two components are not the same, they are said to follow the unsymmetric convention. The standard-state fugacity of the solvent is the fugacity of the pure liquid. The standard-state fugacity of the solute is Henry s constant. 

The use of Henry s constant for a standard-state fugacity means that the standard-state fugacity for a noncondensable component depends not only on the temperature but also on the nature of the solvent. It is this feature of the unsymmetric convention which is its greatest disadvantage. As a result of this disadvantage special care must be exercised in the use of the unsymmetric convention for multicomponent solutions, as discussed in Chapter 4. 

The standard-state fugacity of any component must be evaluated at the same temperature as that of the solution, regardless of whether the symmetric or unsymmetric convention is used for activity-coefficient normalization. But what about the pressure At low pressures, the effect of pressure on the thermodynamic properties of condensed phases is negligible and under such con-... 

In some cases, the temperature of the system may be larger than the critical temperature of one (or more) of the components, i.e., system temperature T may exceed T. . In that event, component i is a supercritical component, one that cannot exist as a pure liquid at temperature T. For this component, it is still possible to use symmetric normalization of the activity coefficient (y - 1 as x - 1) provided that some method of extrapolation is used to evaluate the standard-state fugacity which, in this case, is the fugacity of pure liquid i at system temperature T. For highly supercritical components (T Tj,.), such extrapolation is extremely arbitrary as a result, we have no assurance that when experimental data are reduced, the activity coefficient tends to obey the necessary boundary condition 1... 

Enthalpies are referred to the ideal vapor. The enthalpy of the real vapor is found from zero-pressure heat capacities and from the virial equation of state for non-associated species or, for vapors containing highly dimerized vapors (e.g. organic acids), from the chemical theory of vapor imperfections, as discussed in Chapter 3. For pure components, liquid-phase enthalpies (relative to the ideal vapor) are found from differentiation of the zero-pressure standard-state fugacities these, in turn, are determined from vapor-pressure data, from vapor-phase corrections and liquid-phase densities. If good experimental data are used to determine the standard-state fugacity, the derivative gives enthalpies of liquids to nearly the same precision as that obtained with calorimetric data, and provides reliable heats of vaporization. 

Correlations for standard-state fugacities at 2ero pressure, for the temperature range 200° to 600°K, were generated for pure fluids using the best available vapor-pressure data. 

At temperatures above those corresponding to the highest experimental pressures, data were generated using the Lyckman correlation all of these were assigned an uncertainty of 5% of the standard-state fugacity at zero pressure. Frequently, this uncertainty amounts to one half or more atmosphere for the lowest point, and to 1 to 5 atmospheres for the highest point. 

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. 

Only those components which are gases contribute to powers of RT. More fundamentally, the equiUbrium constant should be defined only after standard states are specified, the factors in the equiUbrium constant should be ratios of concentrations or pressures to those of the standard states, the equiUbrium constant should be dimensionless, and all references to pressures or concentrations should really be references to fugacities or activities. Eor reactions involving moderately concentrated ionic species (>1 mM) or moderately large molecules at high pressures (- 1—10 MPa), the activity and fugacity corrections become important in those instances, kineticists do use the proper relations. In some other situations, eg, reactions on a surface, measures of chemical activity must be introduced. Such cases may often be treated by straightforward modifications of the basic approach covered herein. 

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). 

If, for any component, the standard-state fugacity in phase a is the same as that in phase fl, Eq. (118) can be rewritten in the more useful form... 

IUPAC suggests fi for the activity coefficient with a Raoult s law standard state. We will use instead so as not to confuse the activity coefficient with the fugacity. which is also represented by the symbol f-... 

In Chapter 6, fugacity and activity are defined and described and related to the chemical potential. The concept of the standard state is introduced and thoroughly explored. In our view, a more aesthetically satisfying concept does not occur in all of science than that of the standard state. Unfortunately, the concept is often poorly understood by non-thermodynamicists and treated by them with suspicion and mistrust. One of the firm goals in writing this book has been to lay a foundation and describe the application of the standard state in such a way that all can understand it and appreciate its significance and usefulness. 

At equilibrium, all components of a mixture have the same molar free energy, i.e., the same chemical potential, in any phase in which they are present, and they have the same chemical potential as all other components. However it is not always convenient to use the same standard state for all components or even for the same component in all phases. Just as Equation 6 defines fugacity, Equation 7 or 8 defines activity. Furthermore, Equations 6-8 define / and a for all substances, not just gases. However we should keep in mind that we do not use the same standard state for a substance in all the phases, mixtures, or pure states in which it may occur or for all components of a mixture. 

In mixtures of real gases the ideal gas law does not hold. The chemical potential of A of a mixture of real gases is defined in terms of the fugacity of the gas, fA. The fugacity is, as discussed in Chapter 2, the thermodynamic term used to relate the chemical potential of the real gas to that of the (hypothetical) standard state of the gas at 1 bar where the gas is ideal ... 

Vapor-liquid equilibrium data for the two binary systems (11) were used to calculate the standard-state fugacities required in Equations 6 and 20. In the temperature range 0-50°C, there fugacities can be expressed by ... 

The thermodynamic reaction equilibrium constant K, is only a function of temperature. In Equation 4.18, m, the activity of the guest in the vapor phase, is equal to the fugacity of the pure component divided by that at the standard state, normally 1 atm. The fugacity of the pure vapor is a function of temperature and pressure, and may be determined through the use of a fugacity coefficient. The method also assumes that an, the activity of the hydrate, is essentially constant at a given temperature regardless of the other phases present. 

In order to solve for thermodynamic equilibrium, the fugacity of water in the hydrate must be known. This method follows the common approach of solving for fugacity, using the standard state of the ideal gas of the pure component at 1 bar. 

Use has here been made in steps 2 through 4 of the fact that there is no change in the Gibbs energy for processes carried out under conditions of membrane and chemical-reaction equilibrium. This explains why the value of AG° is related directly to the ratios of the equilibrium-state and standard-state fugacities (ft = 1). 

Standard (electrode) potential — (E ) represents the equilibrium potential of an electrode under standard-state conditions, i.e., in solutions with the relative activities of all components being unity and a pressure being 1 atm (ignoring the deviations of fugacity and activity from pressure and concentration, respectively) at temperature T. A pressure of 1 bar = 105 Pa was recommended as the standard value to be used in place of 1 atm = 101,325 Pa (the difference corresponds to 0.34 mV shift of potential). If a component of the gas phase participates in the equilibrium, its partial pressure is taken as... 

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