Fugacities

The fugacity was introduced by G.N. Lewis in 1901, and became widely used after the appearance of Thermodynamics, a very influential textbook by Lewis and Randall in 1923. Lewis describes the need for such a function in terms of an analogy with temperature in the attainment of equilibrium between phases. Just as equilibrium requires that heat must flow such thaf temperature is the same in all parts of the system, so matter must flow such thaf chemical potentials are also equalized. He referred to the flow of matter from one phase to another as an escaping tendency, such as a liquid escaping to the gas form to achieve an equilibrium vapor pressure. He pointed out that in fact vapor pressure is equilibrated between phases under many conditions (and in fact is the basis for the isopiestic method of activity determinations, 5.8.4), and could serve as a good measure of escaping tendency if it behaved always as an ideal gas. 

The chemical potential is of course another measure of escaping tendency, but Lewis pointed out that there are certain respects in which this function is awkward. This refers to the fact that /r — oo as a 0, because activity is defined as /r — u.° = RT na. Lewis defined a function which would be much like a vapor pressure, which would be equilibrated between phases at equilibrium, even in nonideal cases, and even if no vapor phase actually existed. It is an absolute property, in the sense that it does not depend on a standard state, and it has units of pressure. Lewis and Randall (1923) called it a kind of ideal or corrected vapor pressure.  

Fugacity has proved useful in a number of ways. One way is to provide a relatively simple way to evaluate the integral / VdP. In 5.7.1 we saw one way to do this. That is, for solids, we often assume that the molar volume is constant, making the integration very simple. Another way, for gases, is to assume the ideal gas law (see below). This is actually a special case of the most general method, which is to develop an equation of state for the system (Chapter 13), from which you can generate all its thermodynamic properties. 

Lacking an equation of state, how do we evaluate the pressure integral for a fluid such as H2O or CO2, either in the pure form, mixed with other fluid 

Fortunately, thanks to the insight of Lewis, we can proceed in a simpler and completely different fashion. To see how the inspiration for such a function might have arisen, consider the form of the volume integral f V dP for an ideal 

We preface our application of thermodynamics to chemical reactions by defining fugacity, a measure of the nonideality of real gases. First, let us justify the need to define such a quantity. 

In developing theory, we assume ideal materials, and we have done just that in thermodynamics. For example, the use of the ideal gas is common throughout these chapters. However, there is no such thing as a truly ideal gas. Real gases do not obey the ideal gas law and have more complex equations of state. 

As expected, the chemical potential of a gas varies with pressure. By analogy to equation 4.46  

We can write both of these equations in a different fashion, by recognizing that the A signs on G and jx represent a change so we can write AG or A/x as 

Suppose that for both equations, the initial state is some standard pressure, like 1 atmosphere or 1 bar. (1 atm = 1.01325 bar, so very little error is introduced by using the non-SI-standard 1 atm.) We will denote the initial conditions with a ° symbol and bring the initial energy quantity over to the right side of the equation. The final subscripts are deleted, and the equations are now written as G or /x at any pressure p, calculated with respect to G° and fx° at some standard pressure (that is, 1 atm or 1 bar)  

To avoid the aforementioned uncertainties associated with the evaluation of the chemical potential G.N. Lewis introduced the concept of fugacity in 1901. 

Consider first the change in chemical potential of a pure ideal gas from pressure Pj to pressure P2 at a constant temperature T. From Eq. 9.9.2  

This expression indicates that, for an ideal gas, the isothermal change in chemic potential can be determined through its initial and final pressure. 

Turning to real systems, Lewis introduced a new function called fuga-city (f)y such that the isothermal change in the chemical potential for any compound in any system (gas, liquid, or solid) is given by an expression similar to Eq.9.10.1  

Consider now the ratio of the fugacity of a pure compound at a pressure P divided by P. This ratio is called the fugacity coefficient.  

The chemical potential of component i in a fluid may be derived on the basis of Equation 1.6. By holding T and constant, then T, P and rij (j i) constant, the following relationships may be written  

If Hj is differentiated with respect to P and the reciprocity rnle applied, the following is obtained  

Defining the partial pressure of component i as p, = (n/n)P, and combining this definition with the ideal gas equation to eliminate P, the ideal gas equation for component i may be written as 

for an ideal gas, the derivative of the volume with respect to the number of moles of component i is 

For real fluids the partial pressure is replaced by the fugacity, a defined property, 

An ideal gas is an ideal solution. Thus, the chemical potential of species a in an ideal gas mixture at temperature T, pressure p, and species mole fractions j/i, 2/2-- is given by 

The difference in the chemical potential of an ideal gas at pressure p and mole fractions yi, y, and an ideal gas at pressure p and mole fractions y, y, ... is given by 

Note that we have made use of the relation In analogy with ideal gases, we define the fugacity of a real gas as 

This gives us the difference between the chemical potential of the system and the chemical potential of an ideal gas at the same temperature, pressure, and composition. That is  

The residual chemical potential is equal to the difference between the chemical potential of the system and the chemical potential of an ideal gas at the same temperature, molar volume, and composition. The difference given in Eq. (11.20) is between the system and an ideal gas at the same temperature, pressure, and composition. This is not the same as the residual chemical potential. However, the two quantities are related. If we consider an ideal gas with the same molar volume as the system, its pressure will be equal to p = RT/V, which will, in general, not be equal to the actual pressure p of 

Now if we represent the gas phase by prime and the liquid phase by double-primes, and the composition of the gas and liquid phases by y and Xj, respectively, 

Equation (1.114) relates the vapor-liquid equilibrium ratio, Ki, to the ratio of fugacity coefficients. The fugacity coefficients can be obtained from the volumetric properties given by an EOS. However, as Eq. (1.109) demands, the volumetric data are required from zero pressure to pressure P of the system at constant temperature and composition. Therefore, the EOS should represent the volumetric behavior over the whole range. 

Next we define ideal and nonideal fluids and the representation of the corresponding chemical potentials. 

Ideal gas. An ideal gas is defined as the fluid that obeys the equation 

Note that the partial molar volume, V, of component i in an ideal gas mixture is simply 

It is a derived thermodynamic property, unlike the measured thermodynamic properties, temperature and pressure, that provide the criteria for thermal and mechanical equilibrium, respectively Although the chemical potential is an abstract concept, it is useful since it provides a simple criterion for chemical equilibria of each species i. 

Unfortunately, in application, it turns out the chemical potential has some inconvenient mathematical behaviors (which we will see shortly). Consequently, it is convenient to define a new derived thermodynamic property that is mathematically better behaved but provides just as simple a criterion for equilibrium the fugacity 

The definition oifugacity can be attributed to the thermodynamics giant G. N. Lewis. Unlike the other concepts we have seen so far in this text, it developed inductively rather than deductively. In fact, fugacity is undoubtedly but one of many ways to get around the mathematical anomalies of the chemical potential however, it is the way that is used in practice, and we will learn about it next. 

This equation is vahd only at constant temperature. We will begin by restricting ourselves to an ideal gas. We will remove this restriction and include real systems when we introduce fugacity. In the development of fugacity that follows, we will always be at constant temperature. With this restriction, we can write  

Since energies never have absolute values, we need a reference state for the partial molar Gibbs energy. The reference state is indicated by a superscript o . In choosing a reference state, we must specify the appropriate number of thermodynamic properties as prescribed by the state postulate the rest of the properties of the reference state are then constrained. The reference chemical potential, juf, is the chemical potential at the reference pressure, P°, and at the same temperature as the chemical potential of interest, T. The latter constraint derives from our stipulation of constant temperature. Integrating between a reference state and the state of the system, we get  

---

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

In Chapter 2 we discuss briefly the thermodynamic functions whereby the abstract fugacities are related to the measurable, real quantities temperature, pressure, and composition. This formulation is then given more completely in Chapters 3 and 4, which present detailed material on vapor-phase and liquid-phase fugacities, respectively. 

The calculation of vapor and liquid fugacities in multi-component systems has been implemented by a set of computer programs in the form of FORTRAN IV subroutines. These are applicable to systems of up to twenty components, and operate on a thermodynamic data base including parameters for 92 compounds. The set includes subroutines for evaluation of vapor-phase fugacity... 

The data base contains provisions for a simple augmentation by up to eight additional compounds or substitution of other compounds for those included. Binary interaction parameters necessary for calculation of fugacities in liquid mixtures are presently available for 180 pairs. 

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 activity coefficient y relates the liquid-phase fugacity... 

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

A rigorous relation exists between the fugacity of a component in a vapor phase and the volumetric properties of that phase these properties are conveniently expressed in the form of an equation of state. There are two common types of equations of state one of these expresses the volume as a function of... 

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. ... 

According to Equation (14), the fugacity of component i becomes equal to the mole fraction multiplied by the standard-... 

Henry s constant is the standard-state fugacity for any component i whose activity coefficient is normalised by Equation (14). ... 

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-... 

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... 

We can now consider the most convenient form for writing the liquid-phase fugacity of component i. First we consider a condensable component and write... 

We choose this form because we want to express the fugacity f ... 

We find that the standard-state fugacity fV is the fugacity of pure liquid i at the temperature of the solution and at the reference pressure P. ... 

Standard-State Fugacity for a Noncondensable Component For a noncondensable component we write... 

Chapter 3 discusses calculation of fugacity coefficient < ). Chapter 4 discusses calculation of adjusted activity coefficient Y fugacity of the pure liquid f9 [Equation (24)], and Henry s constant H. 

In the calculation of vapor-liquid equilibria, it is necessary to calculate separately the fugacity of each component in each of the two phases. The liquid and vapor phases require different techniques in this chapter we consider calculations for the vapor phase. 

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. 

The fugacity fT of a component i in the vapor phase is related to its mole fraction y in the vapor phase and the total pressure P by the fugacity coefficient ... 

The fugacity coefficient is a function of temperature, total pressure, and composition of the vapor phase it can be calculated from volumetric data for the vapor mixture. For a mixture containing m components, such data are often expressed in the form of an equation of state explicit in the pressure... 

The fugacity coefficient can be found from the equation of state using the thermodynamic relation (Beattie, 1949) ... 

It is important to be consistent in the use of fugacity coefficients. When reducing experimental data to obtain activity coefficients, a particular method for calculating fugacity coefficients must be adopted. That same method must be employed when activity-coefficient correlations are used to generate vapor-liquid equilibria. 

---