Infinite-dilution activity coefficient

State is the ionic medium (i.e., infinitely diluted with respect to HCl only). In such a medium /hci (solid line, right ordinate) is very nearly constant, that is,/Hci = 1- Both activity coefficients are thermodynamically equally meaningful. (Adapted from P. Schindler.) (b) A comparison of activity coefficients (infinite dilution scale) of electrolytes and nonelectrolytes as a function of concentration (mole fraction of solute) m = moles of solute per kg of solvent (molality) = number of moles of ions formed from 1 mol of electrolyte 1 kg solvent contains 55.5 mol of water. (From Robinson and Stokes, 1959. Reproduced with permission from Butterworths, Inc., London.)... 

Figure 4. Solubilities of halite (NaCl) in water to 350°C. The curve represents values calculated using the Margules expansion model for activity coefficients (infinite dilution reference state), and standard state Gibbs energies for NaCl(aq) derived from the equations of Pitzer et al. to 300°C, and of Tanger and Helgeson above 300 C.

For the i surface active components, infinite dilution (x 0) is experimentally better accessible than the pure state. It should be mentioned that setting the activity coefficients to 1 at infinite dilution is not necessarily consistent with setting the activity coefficient for pure components to unity. Therefore, for the case of infinite dilution of a multicomponent system, an additional normalisation of the potentials of the components should be performed [49]. This yields unity for the activity coefficient of pure components, while the activity coefficients at infinite dilution, in general should not be equal to 1. Indicating parameters at infinite dilution by the subscript (0), and those in the pure state by the superscript 0, the two standard potentials are interrelated by... 

Both activity coefficients are larger than 1. In real systems, however, there always exists some mutual miscibility of the substances and, in turn, the activity coefficients of diluted systems (x 0) are always smaller than infinite. These activity coefficients, called border activity coefficient y, very clearly reflect the intermo-lecular forces between the species involved. 

Activity-coefficient data at infinite dilution often provide an excellent method for obtaining binary parameters as shown, for example, by Eclcert and Schreiber (1971) and by Nicolaides and Eckert (1978). Unfortunately, such data are rare. 

Figure 4-11. Activity coefficients for noncondensable solutes at infinite dilution.

GAMMA calculates activity coefficients for N components (N 20) at system temperature. For noncondensable components effective infinite-dilution activity coefficients are calculated. 

Activity coefficients for condensable components are calculated with the UNIQUAC Equation (4-15)/ and infinite-dilution activity coefficients for noncondensable components are calculated with Equation (4-22). ... 

The Debye-Htickel limiting law predicts a square-root dependence on the ionic strength/= MTLcz of the logarithm of the mean activity coefficient (log y ), tire heat of dilution (E /VI) and the excess volume it is considered to be an exact expression for the behaviour of an electrolyte at infinite dilution. Some experimental results for the activity coefficients and heats of dilution are shown in figure A2.3.11 for aqueous solutions of NaCl and ZnSO at 25°C the results are typical of the observations for 1-1 (e.g.NaCl) and 2-2 (e.g. ZnSO ) aqueous electrolyte solutions at this temperature. 

If the mutual solubilities of the solvents A and B are small, and the systems are dilute in C, the ratio ni can be estimated from the activity coefficients at infinite dilution. The infinite dilution activity coefficients of many organic systems have been correlated in terms of stmctural contributions (24), a method recommended by others (5). In the more general case of nondilute systems where there is significant mutual solubiUty between the two solvents, regular solution theory must be appHed. Several methods of correlation and prediction have been reviewed (23). The universal quasichemical (UNIQUAC) equation has been recommended (25), which uses binary parameters to predict multicomponent equihbria (see Eengineering, chemical DATA correlation). 

Eor given biaary pair, two-phase behavior likely when infinite dilution activity coefficient of either component ia other component is >7.5. 

Experimentally deterrnined equiUbrium constants are usually calculated from concentrations rather than from the activities of the species involved. Thermodynamic constants, based on ion activities, require activity coefficients. Because of the inadequacy of present theory for either calculating or determining activity coefficients for the compHcated ionic stmctures involved, the relatively few known thermodynamic constants have usually been obtained by extrapolation of results to infinite dilution. The constants based on concentration have usually been deterrnined in dilute solution in the presence of excess inert ions to maintain constant ionic strength. Thus concentration constants are accurate only under conditions reasonably close to those used for their deterrnination. Beyond these conditions, concentration constants may be useful in estimating probable effects and relative behaviors, and chelation process designers need to make allowances for these differences in conditions. 

Terminal activity coefficients, 7°, are noted in Figure 3. These are often called infinite dilution coefficients and for some systems are given in Table 1. The hexane—heptane mixture is included as an example of an ideal system. As the molecular species become more dissimilar they are prone to repel each other, tend toward liquid immiscihility, and have large positive activity coefficients, as in the case of hexane—water. 

Hquid-phase activity coefficient (eq. 6) terminal activity coefficient, at infinite dilution constant in Wilson activity coefficient model (eq. 13)... 

Fiend s Constant. Henry s law for dilute concentrations of contaminants ia water is often appropriate for modeling vapor—Hquid equiHbrium (VLE) behavior (47). At very low concentrations, a chemical s Henry s constant is equal to the product of its activity coefficient and vapor pressure (3,10,48). Activity coefficient models can provide estimated values of infinite dilution activity coefficients for calculating Henry s constants as a function of temperature (35—39,49). 

For low miscibility, the solubiHty of a substance in water is often estimated as the inverse of the chemical s infinite dilution activity coefficient ... 

Once the composition of each equiHbrium phase is known, infinite dilution activity coefficients for a third component ia each phase can then be calculated. The octanol—water partition coefficient is directly proportional to the ratio of the infinite dilution activity coefficients for a third component distributed between the water-rich and octanol-rich phases (5,24). The primary drawback to the activity coefficient approach to estimation is the difficulty of the calculations involved, particularly when the activity coefficient model is complex. 

A sampling of appHcations of Kamlet-Taft LSERs include the following. (/) The Solvatochromic Parameters for Activity Coefficient Estimation (SPACE) method for infinite dilution activity coefficients where improved predictions over UNIEAC for a database of 1879 critically evaluated experimental data points has been claimed (263). (2) Observation of inverse linear relationship between log 1-octanol—water partition coefficient and Hquid... 

Rutan, The SPACE Predictor for Infinite Dilution Activity Coefficients, unpubhshed, presented at AlChE 1992 Annual Meeting, Nov. 3, Miami Beach, Fla. For information, write to Chades. A. Eckert, School of Chemical Engineering, Georgia Institute of Technology, Adanta, Ga. 30332-0100. 

The symmetrical nature of these relations is evident. The infinite-dilution values of the activity coefficients are In yF = Iri JT = B. 

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... 

Outlined below are the steps required for of a X T.E calciilation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of for each pq binaiy. The Wilson, NRTL (with a fixed), and UNIQUAC equations are of general applicabihty for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binaiy. 

Concentrated, Binary Mixtures of Nonelectrolytes Several correlations that predict the composition dependence of Dab. re summarized in Table 5-19. Most are based on known values of D°g and Dba- In fact, a rule of thumb states that, for many binary systems, D°g and Dba bound the Dab vs. Xa cuiwe. CuUinan s equation predicts dif-fusivities even in hen of values at infinite dilution, but requires accurate density, viscosity, and activity coefficient data. 

The solvent and the key component that show most similar liquid-phase behavior tend to exhibit little molecular interactions. These components form an ideal or nearly ideal liquid solution. The ac tivity coefficient of this key approaches unity, or may even show negative deviations from Raoult s law if solvating or complexing interactions occur. On the other hand, the dissimilar key and the solvent demonstrate unfavorable molecular interactions, and the activity coefficient of this key increases. The positive deviations from Raoult s law are further enhanced by the diluting effect of the high-solvent concentration, and the value of the activity coefficient of this key may approach the infinite dilution value, often aveiy large number. 

The effect of solvent concentration on the activity coefficients of the key components is shown in Fig. 13-72 for the system methanol-acetone with either water or methylisopropylketone (MIPK) as solvent. For an initial-feed mixture of 50 mol % methanol and 50 mol % acetone (no solvent present), the ratio of activity coefficients of methanol and acetone is close to unity. With water as the solvent, the activity coefficient of the similar key (methanol) rises slightly as the solvent concentration increases, while the coefficient of acetone approaches the relatively large infinite-dilution value. With methylisopropylketone as the solvent, acetone is the similar key and its activity coefficient drops toward unity as the solvent concentration increases, while the activity coefficient of the methanol increases. 

Gmehhng and Onken (op. cit.) give the activity coefficient of acetone in water at infinite dilution as 6.74 at 25 C, depending on which set of vapor-liquid equilibrium data is correlated. From Eqs. (15-1) and (15-7) the partition ratio at infinite dilution of solute can he calculated as follows ... 

Another method to determine infinite dilution activity coefficients (or the equivalent FFenry s law coefficients) is gas chromatography [FF, F2]. In this method, the chromatographic column is coated with the liquid solvent (e.g., the IL). The solute (the gas) is introduced with a carrier gas and the retention time of the solute is a measure of the strength of interaction (i.e., the infinite dilution activity coefficient, y7) of the solute in the liquid. For the steady-state method, given by [FF, F2] ... 

J. Gmehling, J. Menke, M. Schiller, Activity Coefficients at Infinite Dilution DECHEMA, Frankfurt, 1994. 

In Chapter 7 we found it convenient to distinguish between proton transfers involving a solvent molecule and those involving only solute particles but this difference will lose its significance when the distinction between solvent and solute begins to break down. We recall that in Sec. 54 the mole fraction of the solvent did not differ appreciably from unity and could be omitted from (72). In investigating concentrated solutions, however, there is no question of extrapolating to infinite dilution the mole fraction of the solvent will differ from unity and will have to be retained in all formulas. At the same time each of the mole fractions will need to be multiplied by its activity coefficient. 

For those dilute mixtures where the solute and the solvent are chemically very different, the activity coefficient of the solute soon becomes a function of solute mole fraction even when that mole fraction is small. That is, if solute and solvent are strongly dissimilar, the relations valid for an infinitely dilute solution rapidly become poor approximations as the concentration of solute rises. In such cases, it is necessary to relax the assumption (made by Krichevsky and Kasarnovsky) that at constant temperature the activity coefficient of the solute is a function of pressure but not of solute mole fraction. For those moderately dilute mixtures where the solute-solute interactions are very much different from the solute-solvent interactions, we can write the constant-pressure activity coefficients as Margules expansions in the mole fractions for the solvent (component 1), we write at constant temperature and at reference pressure Pr ... 

The osmotic coefficient is often used as a measure of the activity of the solvent instead of a because a is nearly unity over the concentration range where 7 is changing, and many significant figures are required to show the effect of solute concentration on a. The osmotic coefficient also becomes one at infinite dilution, but deviates more rapidly with concentration of solute than does a. ... 

In the reference state the activity coefficients are, by definition, unity. The reference state may be that in the limit of infinite dilution, but the more conventional reference state is C° = 1 M. With the -y s = 1,... 

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