Fugacity of pure substances

Appendix C shows the mathematics of the fugacity of pure substances and of mixrnres. We may summarize the findings... 

Equations 7.5 to 7.9 are the whole story on the fugacity of pure substances. They are derived here and copied into the main text. 

The procedure of Beutier and Renon as well as the later on described method of Edwards, Maurer, Newman and Prausnitz ( 3) is an extension of an earlier work by Edwards, Newman and Prausnitz ( ). Beutier and Renon restrict their procedure to ternary systems NH3-CO2-H2O, NH3-H2S-H2O and NH3-S02 H20 but it may be expected that it is also useful for the complete multisolute system built up with these substances. The concentration range should be limited to mole fractions of water xw 0.7 a temperature range from 0 to 100 °C is recommended. Equilibrium constants for chemical reactions 1 to 9 are taken from literature (cf. Appendix II). Henry s constants are assumed to be independent of pressure numerical values were determined from solubility data of pure gaseous electrolytes in water (cf. Appendix II). The vapor phase is considered to behave like an ideal gas. The fugacity of pure water is replaced by the vapor pressure. For any molecular or ionic species i, except for water, the activity is expressed on the scale of molality m ... 

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 ... 

The fugacity can be calculated from Equation 7-15 once the P-V-T behavior of the fluid is known from an equation of state. The fugacity coefficient is the ratio of the fugacity of a substance to its pressure. For a pure substance. 

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. 

Fugacity is expressed as a function of the molar volume, the temperature, the parameters for pure substances Oj and h, and the binary interaction coefficients )... 

The calculation of vapor pressure of a pure substance consists of finding the pressure for which the fugacities of the liquid and vapor are equal. 

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. 

With this definition, T is the numerical value of the activity for the substance under some pressure p. It is also the ratio of the fugacity of the pure condensed phase under pressure p to that of the phase under 1 bar pressure. 

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 the multimedia models used in this series of volumes, an air-water partition coefficient KAW or Henry s law constant (H) is required and is calculated from the ratio of the pure substance vapor pressure and aqueous solubility. This method is widely used for hydrophobic chemicals but is inappropriate for water-miscible chemicals for which no solubility can be measured. Examples are the lower alcohols, acids, amines and ketones. There are reported calculated or pseudo-solubilities that have been derived from QSPR correlations with molecular descriptors for alcohols, aldehydes and amines (by Leahy 1986 Kamlet et al. 1987, 1988 and Nirmalakhandan and Speece 1988a,b). The obvious option is to input the H or KAW directly. If the chemical s activity coefficient y in water is known, then H can be estimated as vwyP[>where vw is the molar volume of water and Pf is the liquid vapor pressure. Since H can be regarded as P[IC[, where Cjs is the solubility, it is apparent that (l/vwy) is a pseudo-solubility. Correlations and measurements of y are available in the physical-chemical literature. For example, if y is 5.0, the pseudo-solubility is 11100 mol/m3 since the molar volume of water vw is 18 x 10-6 m3/mol or 18 cm3/mol. Chemicals with y less than about 20 are usually miscible in water. If the liquid vapor pressure in this case is 1000 Pa, H will be 1000/11100 or 0.090 Pa m3/mol and KAW will be H/RT or 3.6 x 10 5 at 25°C. Alternatively, if H or KAW is known, C[ can be calculated. It is possible to apply existing models to hydrophilic chemicals if this pseudo-solubility is calculated from the activity coefficient or from a known H (i.e., Cjs, P[/H or P[ or KAW RT). This approach is used here. In the fugacity model illustrations all pseudo-solubilities are so designated and should not be regarded as real, experimentally accessible quantities. 

The fugacity of the pure substance can be derived by means of equation 2.70,... 

Combination of Equation 15-1 with 15—3 results in the defining equation for the fugacity of a pure substance. 

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... 

For a pure substance, the ratio of fugacity to pressure, f/p, is called... 

The Peng-Robinson equation of state, Equation 4-35, will be used with Equation 15-7 to develop a procedure for calculating of the vapor pressure of a pure substance.3 The vapor pressure is simply the pressure, points e on Figure 15-2, for which the fiigacity of the liquid equals the fugacity of the gas. 

The procedure to calculate the vapor pressure of a pure substance involves Equations 15-9 through 15-17. Once temperature is selected, the results of Equations 15-9 through 15-12 are fixed. The problem then is to find a pressure for use in Equations 15-14 through 15-16 which will give values of z-factors for gas and liquid which will result in equal values of fugacities of gas and liquid from Equation 15-17. 

Chemical Potential—The Fugacity—Fugacity Coffi-cient—Example of State Calculation for a Pure Substance Mixtures 425... 

Water solubility and vapor pressures of PFOS and PFOA are given in Table 2. These data were obtained from products that were not refined and as a result may contain more than one PFA such that these data may not be representative of the pure compounds, especially in environmental media. Due to the lack of accurate information on the physico-chemical properties, accurate prediction of the environmental fate and transport of most perfluoroalkyl substances has not yet been possible. The prediction of the distribution and ultimate fates of perfluoroalkyl substances is further complicated by their hydrophobic and lipophobic properties, such that the fugacity approach that has been useful in describing the environmental fates of organochlorines is less useful for describing the environmental fate of PFAs and their precursors. The bulk of the available physical and chemical information is for PFOS... 

Real gases are usually non-ideal. Thermodynamics describes both ideal and non-ideal gases with the same type of formulas, except that for non-ideal gas mixtures the fugacity f is substituted in place of the pressure pi and that the activity at is substituted in place of the molar fraction xi or concentration c, of constituent substance i. We have already seen that in the ideal gas of a pure substance the chemical potential is expressed by Eq. 7.5. By analogy, we write Eq. 7.9 for the non-ideal gas of a pure substance i ... 

The fugacity of a pure liquid or solid can be defined by applying Eq. si.4 to the vapor in equilibrium with the substance in either condensed phase. Usually, the volume of the vapor will follow the ideal gas equation of state very closely, and the fugacity of the vapor may be set equal to the equilibrium vapor pressure. The thermodynamic basis of associating the fugacity of a condensed... 

For the predominant component of a solution, i. e. the solvent, the 3tate of the pure liquid at the temperature of the systom and the pressure of 1 atm. is chosen as the standard state.. In so far as sufficiently diluted solutions are concerned (i. e. such solutions the composition of which differs but slightly from the pure solvent) the solvent can be considered to follow approximately Raoult s law valid for ideal solutions, according to which the fugacity / of the solvent in a solution can be expressed as the product of its molar fraction N-, ) and of the fugacity of the pure liquid substance f at the same temperature, thus ... 

To avoid some possible difficulties in determining chemical potentials, Lewis proposed a new property called the fugacity /. At low pressure and concentration, the fugacity is a well-behaved function. The fugacity function can define phase equilibrium and chemical equilibrium. For an ideal gas, the fugacity of a species in an ideal gas mixture is equal to its partial pressure. As the pressure decreases to zero, pure substances or mixtures of species approach an ideal state, and we have... 

Similarly, the term activity, a is defined as the ratio of its fugacity, / to its fugacity at standard state,/°. Standard state is generally taken to be pure substance at standard temperature (298 °K) and pressure (one atmosphere pressure). 

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