Activity coefficient average

The activity coefficient (y) based corrector is calculated using any applicable activity correlating equation such as the van Laar (slightly polar) or Wilson (more polar) equations. The average absolute error is 20 percent. 

An alternate method for binary concentrated liquid systems where activity coefficients are not available or estimable is the method of Leffler and Cullinan previously given in Eq. (2-156). Absolute errors average 25 percent. 

The activity product Q ave corresponding to the averaged analysis (ignoring variation in activity coefficients) equals the equilibrium constant K only when fluids A and B are identical otherwise Qmc exceeds K and anhydrite is reported to be supersaturated. To demonstrate this inequality, we can assume arbitrary values for aCa++ and so4 that satisfy Equations 6.4—6.5 and substitute them into Equation 6.6. 

The model calculated in this manner predicts that two minerals, alunite [KA13(0H)6(S04)2] and anhydrite (CaSC>4), are supersaturated in the fluid at 175 °C, although neither mineral is observed in the district. This result is not surprising, given that the fluid s salinity exceeds the correlation limit for the activity coefficient model (Chapter 8). The observed composition in this case (Table 22.1), furthermore, actually represents the average of fluids from many inclusions and hence a mixture of hydrothermal fluids present over a range of time. As noted in Chapter 6, mixtures of fluids tend to be supersaturated, even if the individual fluids are not. 

The value of y is even more difficult to predict because solutes contain both anions and cations. In fact, it is impossible to differentiate between the effects of each, so we measure a weighted average. Consider a simple electrolyte such as KC1, which has one anion per cation. (We call it a 1 1 electrolyte .) In KC1, the activity coefficient of the anions is called y(a-) and the activity coefficient of the cations is 7(k+)- We cannot know either y+ or y we can only know the value of y . Accordingly, we modify Equation (7.25) slightly by writing... 

Meeron60 62 first pointed out how the terms in S(Jt> in the solution theory can be arranged in a form much more compact than that above, which is of the form of a virial expansion in which the coefficients involve the Debye Hiickel potential of average force rather than the unscreened potential. Similar manipulations can be made in the present case, but we shall omit the details, which are very simple, and quote only the final result. It is found using Meeron s form of S that the activity coefficient of defect number s can be written... 

The discussion of the defect distribution functions and potentials of average force follows along rather similar fines to that for the activity coefficient. The formal cluster expansions, Eqs. (90)-(91), individual terms of which diverge, must be transformed into another series of closed terms. This can clearly be done by... 

Chapter 17 - Vapor-liquid equilibrium (VLE) data are important for designing and modeling of process equipments. Since it is not always possible to carry out experiments at all possible temperatures and pressures, generally thermodynamic models based on equations on state are used for estimation of VLE. In this paper, an alternate tool, i.e. the artificial neural network technique has been applied for estimation of VLE for the binary systems viz. tert-butanol+2-ethyl-l-hexanol and n-butanol+2-ethyl-l-hexanol. The temperature range in which these models are valid is 353.2-458.2K at atmospheric pressure. The average absolute deviation for the temperature output was in range 2-3.3% and for the activity coefficient was less than 0.009%. The results were then compared with experimental data. 

Table VI summarizes values of the activity coefficient ratio Ygr-/YC].- in the saturated solution for each average solid composition (as calculated from the model of Table II), the calculated provisional equilibrium distribution coefficient (Equation 12) and the provisional equilibrium aqueous solution activity ratio of Br to Cl- (Equation 13) based on the data of Table V.

It may be conjectured that collective behavior implies that the surfactants that make up the mixture are not too different, the presence of an intermediate being a way to reduce the discrepancy. When the activity coefficient is calculated from non-ideal models it is often taken to be proportional to the difference in solubihty parameters [42,43], which in case of a binary is the difference (3i - if the system is multicomponent, then the dil -ference is - Sm) y which is often less, because the mean value exhibits an average lower deviation. In other terms, it means that for a ternary in which the third term is close to the average of the two first terms, then the introduction of the third component reduces the nonideahty because (5i - 53) + ( 2 - < (5i - 52) -... 

For most of the systems with alcohols, the description of SLE was given by the average standard mean deviation (oj) < 2 K for UNIQUAC ASM and NRTL 1 equations. The procedure of correlation has been described in many articles [52-54,79,84-88,91-94]. Using GE models the solute activity coefficients in the saturated solution, y, were described. 

Here, c is the total concentration of MA, KA is the association constant, a is the degree of dissociation of the ion-pair M2+A2h and y is the average activity coefficient of free ions of concentration ca. Because the ion-pairs do not conduct electricity, the molar conductivity A in the presence of ion association is less than in its absence. In Fig. 7.1, the difference between the experimental molar conductivity (A) and the value calculated from Eq. (7.1), /lcai, are plotted against c1/2 for lithium halides in sulfolane [la]. For Lil, the difference between A and /lcai is small because ion association is not appreciable (KA=5.(> mol-11). For LiBr (KA = 278 mol-11) and LiCl (KA= 13860 mol-11), however, A is much smaller than... 

The part of the activity coefficients depending on k (Equation 33) can be simplified further if suitable average values are introduced. If the restriction d a is sufficiently well fulfilled, the term in y is small compared with the term in <5. The latter therefore represents the main influence caused by the presence of dipoles since the terms in are from ionic charges. They are identical with terms of corresponding order in the Debye-Hiickel theory. 

A colleague of yours who works in oceanography bets you that both the solubility as well as the activity coefficient of naphthalene are larger in seawater (35%o salinity) at 25°C than in distilled water at 5°C. Is this not a contradiction How much money do you bet Estimate C and for naphthalene in seawater at 25°C and in distilled water at 5°C. Discuss the result. Assume that the average enthalpy of solution (A wsHh Fig. 5.1) of naphthalene is about 30 kJmol-1 over the ambient temperature range. All other data can be found in Tables 5.3 and 5.7 and in Appendix C. 

In order to calculate the aqueous concentration of compound / at equilibrium, one needs to know its mole fraction, jcimix, in the mixture (or its molar concentration, Cimix, and the molar volume, Vmix, of the mixture), as well as its activity coefficients in the organic (ymix) and the aqueous (yiw) phases. Very often, when dealing with complex mixtures, V is not known and has to be estimated. At a first approximation, this can be done from the density, pmix, of the liquid mixture, and by assuming an average molar mass, M, of the mixture components ... 

It is assumed that l is independent of the solution concentration. About the same way was followed by Manecke and Bonhoeffer (SP), who introduced the average activity coefficient of the salt in the mem-brane-... 

For the above reasons, the IFCC recommendations on activity coefficients [19] and the measurement of and conventions for reporting sodium and potassium [21] and chlorides [25] by ISEs were developed. At the core of these recommendations is the concept of the adjusted active substance concentration (mmol/L), as well as a traceable way to remove the discrepancy between direct and indirect determinations of these electrolytes in normal sera. Extensive studies of sodium and potassium binding to inorganic ligands and proteins, water binding to proteins, liquid-junction effects and the influence of ionic strength have demonstrated that the bias between sodium and potassium reports obtained from an average ISE-based commercial... 

The total net surface charge density is proportional to net valence times concentration for each reactant adsorbent species and product adsorbent-adsorbate species in Eq. 4.15. Its presence in the equation for the activity coefficient reflects a model concept, that these charged surface species create an average electric field that influences ion adsorption. See, for example, G. Sposito, op. cit.7... 

Calculated using polymer density = 0.85., (L) = liquid phase activity coefficient, (P) = polymer phase activity coefficient, - not possible to calculate a result AAR = average absolute ratio for calculated (calc.) > experimental (exp.) = calc./exp. for calc. < exp. = exp./calc. 

The quantity given in equation (V-58) and expressing the average distance between the nearest ions in the solution (i. e. equalling the total of radii of both ions with opposite charges and being within the range of 3—5 x 10-8 cm) determines the specific influence of the electrolytes on the activity coefficient. Because this quantity iH not directly measurable, verification of the validity of the Debye - Hiickel theory is carried out in such a manner that a value is substituted for which conforms best with the values of y+c obtained by experiments. 

When calculating the activity coefficient from the Debye-Hiickel equation the average value 3 X 10 8 cm might be substituted for eq into the equation (V-58) for solutions with concentration not exceeding c — 0.1 with uni-univalent electrolytes (c = 0.05 with bi-univalent ones). Because in aqueous solutions at 25° C A — 0.509 and B — 0.329 X 10-8, the Debye -Hiickel equation for the mentioned type of solutions has quite a simple form ... 

Several vapor pressure osmometers are now commercially available. Although they are mainly used for determining number-average molecular weights in aqueous and organic solvents, they can also be employed to evaluate the total osmolality of biological solutions or dissociation and activity coefficients. Each model has its own technical characteristics. However, all are comparable in terms of general measurement procedure and sensitivity. 

The extension of ideal phase analysis of the Maxwell-Stefan equations to nonideal liquid mixtures requires the sufficiently accurate estimation of composition-dependent mutual diffusion coefficients and the matrix of thermodynamic factors. However, experimental data on mutual diffusion coefficients are rare, and prediction methods are satisfactory only for certain types of liquid mixtures. The thermodynamic factor may be calculated from activity coefficient models such as NRTL or UNIQUAC, which have adjustable parameters estimated from experimental phase equilibrium data. The group contribution method of UNIFAC may also be helpful, as it has a readily available parameter table consisting of mam7 species. If, however, reliable data are not available, then the averaged values of the generalized Maxwell-Stefan diffusion coefficients and the matrix of thermodynamic factors are calculated at some mean composition between x0i and xzi. Hence, the matrix of zero flux mass transfer coefficients [k ] is estimated by... 

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