Fugacity ratio, estimation

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. 

Saturation properties such as solubility in water and vapor pressure can be measured directly for solids and liquids. For certain purposes it is useful to estimate the solubility that a solid substance would have if it were liquid at a temperature below the melting point. For example, naphthalene melts at 80°C and at 25°C the solid has a solubility in water of 33 g/m3 and a vapor pressure of 10.9 Pa. If naphthalene was a liquid at 25°C it is estimated that its solubility would be 115 g/m3 and its vapor pressure 38.1 Pa, both a factor of 3.5 greater. This ratio of solid to liquid solubilities or vapor pressures is referred to as the fugacity ratio. It is 1.0 at the melting point and falls, in this case at lower temperatures to 0.286 at 25°C. 

Measurements of gas chromatographic retention time are often used as a fast and easy method of estimating vapor pressure. These estimated pressures are related to the gas/substrate partition coefficient, which can be regarded as a ratio of solubility of the substance in the gas to that in the substrate, both solubilities being of the substance in the liquid state. As a result the estimated vapor pressures are of the liquid state. To obtain the solid vapor pressure requires multiplication by the fugacity ratio. It is important to establish if the estimated and reported property is of the vapor or liquid. 

The fugacity ratio F can be estimated at temperature T (K) from the expression... 

As was discussed earlier in Section 1.2.8 a complication arises in that two of these properties (solubility and vapor pressure) are dependent on whether the solute is in the liquid or solid state. Solid solutes have lower solubilities and vapor pressures than they would have if they had been liquids. The ratio of the (actual) solid to the (hypothetical supercooled) liquid solubility or vapor pressure is termed the fugacity ratio F and can be estimated from the melting point and the entropy of fusion. This correction eliminates the effect of melting point, which depends on the stability of the solid crystalline phase, which in turn is a function of molecular symmetry and other factors. For solid solutes, the correct property to plot is the calculated or extrapolated supercooled liquid solubility. This is calculated in this handbook using where possible a measured entropy of fusion, or in the absence of such data the Walden s Rule relationship suggested by Yalkowsky (1979) which implies an entropy of fusion of 56 J/mol-K or 13.5 cal/mol-K (e.u.)... 

In studies of Lakes Superior and Michigan during spring through fall of 1997 and 1998 [50], fugacity ratios were not specifically calculated but net fluxes of toxaphene were estimated to be from water to air (Table 8), imply-... 

Jantunen and Bidleman [46] estimated monthly and annual fugacity ratios and fluxes of toxaphene in Lake Superior. Their calculations used monthly average air concentrations of toxaphene over the lake, estimated from their parameters of Eq. 1 (Sect. 2.2). The concentration of toxaphene in surface water was assumed constant over the year, since measurements in August 1996 and May 1997 were not statistically different. However, we now know... 

The ratio of the solid and liquid pressures is termed the fugacity ratio F and is usually estimated by the equation ... 

For solids, it is essential to correlate Kow with the liquid solubility, not the solid solubility thus the fugacity ratio expression must be included in any correlation. This is readily done, since melting point TM(K) is usually available and Walden s Rule can be applied. Following Yalkowsky (1979), the fugacity ratio F can be estimated at 25°C (T = 298 K) as... 

For compounds that are solids at ambient temperatures, P can be estimated by exploiting the fugacity ratio-melting point relationship discussed in the introduction to this book, namely,... 

After obtaining V and L, the activity coefficients and fugacity ratios are calculated using 1/ra as the average composition in all stages. Then Equations 3 and 5 are simultaneously solved to give the liquid phase compositions. These compositions for each stage are then summed and normalized to one. These serve as the first estimates for X2,.. Xm. 

Prausnitz, et al. (10) show how the pure component fugacity ratio (f 2 Vf 2 can be estimated from readily available data. Choi and McLaughlin... 

The use of halogen concentrations in apatite to make estimates of aqueous fluid compositions is well developed (Korzhinskiy 1981, Zhu and Sveijensky 1991,1992). Given the temperature of apatite crystallization (calculated as an AST), or some other independent estimate of temperature, and an estimate of the pressure of crystallization (although the calculation is fairly insensitive to pressure), the fugacity ratio... 

If the chemical of interest is a solid, it sublimates to pose its vapor pressure. The sublimation process can be viewed as the sum of processes of a hypothetical fusion (melting) to liquid and vaporization from liquid to gas at given temperature and pressure. Such liquid, of course, is hypothetical because it may not exist at the given condition. The vapor pressure of this hypothetical liquid, Pp, rather than the true vapor pressure of the solid, Pg, is often generated by various vapor pressure estimation models. Pg can be converted to liquid vapor pressure Pp using the fugacity ratio F ... 

For solid compounds that have relatively low melting points, the heat of fusion will not vary significantly, and the fugacity ratio can be estimated by the following equation (49) ... 

TABLE 1 gives the reported values or ranges of the physical-chemical properties of chlorobenzenes (CBs), polychlorinated biphenyls (PCBs) and polychlorinated dibenzo-p-dioxins (PCDDs). Fugacity ratios were obtained from a single estimated entropy of fusion of 56 J mol °K (Yalkowsky 1979), molar volumes were calculated by the Le Bas method, an additive group contribution method (Reid et al. 1977). Total surface area (TSA) values were obtained from Yalkowsky et al. (1979 a,b). Solubilities, vapour pressures and octanol/water partition coefficients (Andren et al. 1986 Shiu and Mackay 1986 Bobra et al. 1985) are also tabulated. Henry s law constants were calculated as PSl/C and the octanol solubility Q as C Kq, . 

The estimation of the two parameters requires not only conversion and head space composition data but also physical properties of the monomers, e.g. reactivity ratios, vapor pressure equation, liquid phase activity coefficients and vapor phase fugacity coefficients. 

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 coefficient ratio J can be estimated by assuming that the Lewis and Randall rule11 applies, at least approximately, for the mixture, so that each component has the same fugacity coefficient that it would have if it were a pure gas at the same total pressure. The Principle of Corresponding States can then be used to compare the fugacity coefficients of the three components. At p = 60 atm (61 bar) and in the temperature range from 900 to 1600 K, the reduced temperatures and pressures for the components of the equilibrium... 

If the particular extracting technique applied to a solution depends on the volatility of the solute between air and water, a parameter to predict this behavior is needed to avoid trial and error in the laboratory. The volatilization or escaping tendency (fugacity) of solute chemical X can be estimated by determining the gaseous, G, to liquid, L, distribution ratio, KD, also called the nondimensional, or dimensionless, Henry s law constant, If. 

Water as a Ligand As has been mentioned, ions are associated with H2O. In concentrated solutions such as seawater, the concentration of free water is considerably less than that of total water. Christenson and Gieskes (1971) estimate that only about 2.5 mol of H2O per kg are free. The activity of H2O, a o, in seawater is given by the ratio of the vapor pressure (fugacity) of seawater, p, to that of pure water, p , at the same temperature ... 

Equation (7) is obtained from Equation (2) by noting that the solid phase is pure, and therefore the mole fraction and activity coefficient in the solid phase are both unity. The ratio of pure-component fugacities can be obtained from any one of Equations ((3) to (6)), and the activity coefficient in the liquid, Ji, must be estimated. The composition-temperature behavior along the liquidus curves may then be calculated. The eutectic point is found from the intersection of the two liquidus curves. 

Equations (1.3-14) and (1.3-15) thus give the prediction from transition-state theory for the rate of a reaction in terms appropriate for an SCF. The rate is seen to depend on (i) the pressure, the temperature and some universal constants (ii) the equilibrium constant for the activated-complex formation in an ideal gas and (iii) a ratio of fugacity coefficients, which express the effect of the supercritical medium. Equation (1.3-15) can therefore be used to calcu-late the rate coefficient, if Kp is known from the gas-phase reaction or calculated from statistical mechanics, and the ratio (0a 0b/0cO estimated from an equation of state. Such calculations are rare an early example is the modeling of the dimerization of pure chlorotrifluoroethene = 105.8 °C) to 1,2-dichlor-ohexafluorocyclobutane (Scheme 1.3-2) and comparison with experimental results at 120 °C, 135 °C and 150 °C and at pressures up to 100 bar [15]. 

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