Fugacity evaluation

Therefore, for a mixture in which the pure component and partial molar volumes are identical [i.e., V-, (T, P.x) =. v,-V,- (T, P) at all conditions], the fugacity of each species in the mixture is equal to its mole fraction times its pure-component fugacity evaluated at the same temperature and pressure as the /hixUire /, [T, P,x) — x fi T, P). However, if. as is generally the case, V-, Vj.-a hen /j and /, are related through the integral o er all pressures of the difference between the species partial molar and pure-component molar volumes. 

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 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... 

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. 

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... 

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. 

The fugacity coefficient departure from nonideaHty in the vapor phase can be evaluated from equations of state or, for approximate work, from fugacity/compressibiHty estimation charts. References 11, 14, and 27 provide valuable insights into this matter. 

The fugacityy) of pure compressed liqiiid i must be evaluated at the T and P of the equilibrium mixture. This is done in two steps. First, one calculates the fugacity coefficient of saturated vapor 9i = by an integrated form of Eq. (4-161), written for pure species i and evalu-atea at temperature T and the corresponding vapor pressure P = Equation (4-276) written for pure species i becomes... 

The second step is the evaluation of the change in fugacity of the liquid with a change in pressure to a value above or below For this isothermal change of state from saturated liquid at to liquid at pressure P, Eq. (4-105) is integrated to give... 

The binary interaction parameters are evaluated from liqiiid-phase correlations for binaiy systems. The most satisfactoiy procedure is to apply at infinite dilution the relation between a liquid-phase activity coefficient and its underlying fugacity coefficients, Rearrangement of the logarithmic form yields... 

Both and ° represent fugacity of pure hquid i at temperature T, but at pressures P and P°, respectively. Except in the critical region, pressure has little effecl on the properties of liquids, and the ratio ° is often taken as unity. When this is not acceptable, this ratio is evaluated by the equation... 

Fugacities are evaluated at the conditions at which the gases exist in the mixture. 

Equation 2-63 gives the fugacity at p and T in terms of an integral that is evaluated from experimental data. 

The chemical literature is rich with empirical equations of state and every year new ones are added to the already large list. Every equation of state contains a certain number of constants which depend on the nature of the gas and which must be evaluated by reduction of experimental data. Since volumetric data for pure components are much more plentiful than for mixtures, it is necessary to estimate mixture properties by relating the constants of a mixture to those for the pure components in that mixture. In most cases, these relations, commonly known as mixing rules, are arbitrary because the empirical constants lack precise physical significance. Unfortunately, the fugacity coefficients are often very sensitive to the mixing rules used. 

If we use the symmetric convention for normalization,/ 0 is the fugacity of pure liquid / at the temperature of the mixture and at some specified pressure, usually taken to be the total pressure of the system. Equation (69) presents no problem for subcritical components, where the pure liquid can exist at the system temperature. However, for supercritical components in the symmetric convention,/,0 is a fictitious quantity which must be evaluated by some arbitrary extrapolation. 

For components near or above their critical temperatures, the liquid volume t>j was evaluated by extrapolation with respect to temperature. For supercritical components, the fugacity f° was also evaluated by extrapolation the effect of pressure was found from the Poynting relation using the previously extrapolated liquid molar volumes. 

We illustrate these concepts by applying various fugacity models to PCB behavior in evaluative and real lake environments. The evaluative models are similar to those presented earlier (3, 4). The real model has been developed recently to provide a relatively simple fugacity model for real situations such as an already contaminated lake or river, or in assessing the likely impact of new or changed industrial emissions into aquatic environments. This model is called the Quantitative Water Air Sediment Interactive (or QWASI) fugacity model. Mathematical details are given elsewhere (15). 

The evaluative fugacity model equations and levels have been presented earlier (1, 2, 3). The level I model gives distribution at equilibrium of a fixed amount of chemical. Level II gives the equilibrium distribution of a steady emission balanced by an equal reaction (and/or advection) rate and the average residence time or persistence. Level III gives the non-equilibrium steady state distribution in which emissions are into specified compartments and transfer rates between compartments may be restricted. Level IV is essentially the same as level III except that emissions vary with time and a set of simultaneous differential equations must be solved numerically (instead of algebraically). 

The QUASI fugacity model was then run for a trichlorobiphenyl in a lake the size of Lake Michigan, being approximately 60,000 times the size of the evaluative environment. A detailed justification for the selection of D values is beyond our scope here but in selecting values, we have relied on recent reports by Neely (11), Rogers (15), Armstrong and Swackhamer (16), Thomann (17), and Andren (18). 

Besides the fugacity models, the environmental science literature reports the use of models based on Markov chain principle to evaluate the environmental fate of chemicals in multimedia environment. Markov chain is a random process, and its theory lies in using transition matrix to describe the transition of a substance among different states [39,40]. If the substance has all together n different kinds of states,... 

The current version of CalTOX (CalTOX4) is an eight-compartment regional and dynamic multimedia fugacity model. CalTOX comprises a multimedia transport and transformation model, multi-pathway exposure scenario models, and add-ins to quantify and evaluate variability and uncertainty. To conduct the sensitivity and uncertainty analyses, all input parameter values are given as distributions, described in terms of mean values and a coefficient of variation, instead of point estimates or plausible upper values. 

Mackay D, Paterson S (1991) Evaluating the multimedia fate of organic chemicals a level III fugacity model. Environ Sci Technol 25 427-436... 

Paterson and Mackay (1985), Mackay and Paterson (1990, 1991), and a recent text (Mackay 2001). Only the salient features are presented here. Three evaluations are completed for each chemical, namely the Level I, II and III fugacity calculations. These calculations can also be done in concentration format instead of fugacity, but for this type of evaluation the fugacity approach is simpler and more instructive. The mass balance models of the types described below can be downloaded for the web site www.trentu.ca/cemc... 

Mackay, D., Paterson, S., Chung, B., Neely, W.B. (1985) Evaluation of the environmental behavior of chemicals with a level III fugacity model. Chemosphere 14, 335-374. 

This choice of basis follows naturally from the steps normally taken to study a geochemical reaction by hand. An aqueous geochemist balances a reaction between two species or minerals in terms of water, the minerals that would be formed or consumed during the reaction, any gases such as O2 or CO2 that remain at known fugacity as the reaction proceeds, and, as necessary, the predominant aqueous species in solution. We will show later that formalizing our basis choice in this way provides for a simple mathematical description of equilibrium in multicomponent systems and yields equations that can be evaluated rapidly. 

In these cases, the equation in question is evaluated to give the mole number Mw, and so on, of the corresponding component. In the presence of a gas buffer, the values of one or more fugacities fm are fixed. Now, the mole number Mm of the gas component remains to be determined. In general, the value of either Mw or nw needs to be set to evaluate Equation 3.32, and either M, or m, is required for each Equation 3.33. Each Equation 3.34 can be solved knowing either M or ny, whereas Equation 3.35 is generally evaluated directly for Mm. 

Here the aw and ak are the activities of water and minerals, and the fm are gas fugacities. We assume that each ak equals one, and that aw and the species activity coefficients y can be evaluated over the course of the iteration and thus can be treated as known values in posing the problem. 

To calculate a fixed activity path, the model maintains within the basis each species At whose activity at is to be held constant. For each such species, the corresponding mass balance equation (Eqn. 4.4) is reserved from the reduced basis, as described in Chapter 4, and the known value of a, is used in evaluating the mass action equation (Eqn. 4.7). Similarly, the model retains within the basis each gas Am whose fugacity is to be fixed. We reserve the corresponding mass balance equation (Eqn. 4.6) from the reduced basis and use the corresponding fugacity fm in evaluating the mass action equation. 

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