Fugacity defined

He chose In/i to have the right property that as/A, —oo, the natural log (In) oif also approaches minus infinity, which makes 0. The RT was needed because the natural log of fi is dimensionless and the other terms in the equation have dimension of (energy/mol). 

The fugacity (as we will see later) has the dimension of pressure, so mathematical purists would have us write the natural log as ln( /l unit of pressure), for example. 

The fugacity is a convenience function, like the enthalpy. We use the enthalpy to replace a more complex set of symbols, h = u + Pv. If we solve Eq. 7.1 for / and replace G in terms of its basic components, we find 

Physical and Chemical Equilibrium for Chemical Engineers, Second Edition. Noel de Nevers. 2012 John Wiley Sons, Inc. Published 2012 by John Wiley Sons, Inc. 

FUGACITY, IDEAL SOLUTIONS, ACTIVITY ACTIVITY COEFFICIENT 

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For nonideal gases the above simplifications are not applicable and it would appear to be best to use equation (21) directly as the chemical equilibrium condition. However, it is conventional when dealing with nonideal gases to replace p,- by the fugacity defined below. By solving equation (15) for Ni and using equation (25) to express Ni in terms of p, it is found that... 

For high pressures, it is recommended to substitute the partial pressure with the fugacity, defined as ... 

Seeing the difficulty of working numerically with the chemical potential, whose value goes to minus infinity when the concentration goes to zero, G. N. Lewis invented a new property, which he named the fugacity/ -, defined by... 

Utilizing the concept of partial fugacity which is defined by the relation J... 

One can also define an activity coefficient or fugacity coefficient y =flp, obviously... 

The values of the thermodynamic properties of the pure substances given in these tables are, for the substances in their standard states, defined as follows For a pure solid or liquid, the standard state is the substance in the condensed phase under a pressure of 1 atm (101 325 Pa). For a gas, the standard state is the hypothetical ideal gas at unit fugacity, in which state the enthalpy is that of the real gas at the same temperature and at zero pressure. 

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. 

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

Perhaps the most significant of the partial molar properties, because of its appHcation to equiHbrium thermodynamics, is the chemical potential, ]1. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equihbrium problems. The natural logarithm of the Hquid-phase activity coefficient, Iny, is also defined as a partial molar quantity. For Hquid mixtures, the activity coefficient, y, describes nonideal Hquid-phase behavior. 

Another property which can be represented by generalized charts is fugacity, ( ). The fugacity of a substance can be regarded as a corrected vapor pressure. At low pressures (below atmospheric) the use of pressure in the place of fugacity leads to tittle error in calculations. The fugacity coefficient is defined by... 

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 i = J, all equations reduce to the appropriate values for a pure species. When i j, these equations define a set of interaction parameters having no physical significance. For a mixture, values of By and dBjj/dT from Eqs. (4-212) and (4-213) are substituted into Eqs. (4-183) and (4-185) to provide values of the mixture second virial coefficient B and its temperature derivative. Values of and for the mixture are then given by Eqs. (4-193) and (4-194), and values of In i for the component fugacity coefficients are given by Eq. (4-196). 

A generally apphcable alternative to the gamma/phi approach results when both the hquid and vapor phases are described by the same equation of state. The defining equation for the fugacity coefficient, Eq. (4-79), may be applied to each phase ... 

Activity coefficients are equal to 1.0 for an ideal solution when the mole fraction is equal to the activity. The activity (a) of a component, i, at a specific temperature, pressure and composition is defined as the ratio of the fugacity of i at these conditions to the fugacity of i at the standard state [54]. 

The fugacity of a component i in a gas mixture is related to the total pressure P and to its mole fraction yt through the fugacity coefficient [Pg.144]

At constant temperature, the activity coefficient depends on both pressure and composition. One of the important goals of thermodynamic analysis is to consider separately the effect of each independent variable on the liquid-phase fugacity it is therefore desirable to define and use constant-pressure activity coefficients which at constant temperature are independent of pressure and depend only on composition. The definition of such activity coefficients follows directly from either of the exact thermodynamic relations... 

If we define the standard-state fugacity f° at a fixed pressure, then the second term on the left side of Eq. (50) vanishes and we obtain Eq. (42). However, if we define f° at the total pressure of the system, we obtain Eq. (43). 

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. 

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. 

So far we have defined fugacity for a single component gas. We will first see how fugacities are determined for a pure gas before we expand to include... 

The defining equation for fugacity fc in a condensed phase (solid or liquid) is the same as in the gas phase... 

For a component in a mixture, the fugacity is defined by the same equation as for a pure substance, except that partial molar quantities are substituted for molar quantities. Thus,... 

Fugacity of a Component in a Gaseous Mixture One could guess that the determination of fugacities, /, for the individual components in a gaseous mixture can become complicated as one takes into account the different types of interactions that are present. The mathematical relationship that applies is obtained by starting with the defining equations... 

Equation (6.38) defines fugacity in a mixture through the relationship... 

However, as can be seen in Figure 6.15, which is a graph of the fugacity of HC1 against molality in dilute aqueous solutions of HC1 (near. i = 1), f2 approaches the m axis with zero slope. This behavior would lead to a Henry s law constant, kn.m = 0. given the treatment we have developed so far. Since the activity with a Henry s law standard state is defined as a —fi/kwnu this would yield infinite activities for all solutions. 

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. 

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. 

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