Davies’ equation

The charts of Figures 8-109-112 were developed [5] from the modified Davies equation to simplify the solution of a tedious problem. The mean tray width is usually taken as average of weir length and column diameter. Special tray patterns may indicate another mean value. 

Fig. 8.1. Activity coefficients y, predicted at 25 °C for a singly charged ion with size a of 4 A, according to the Debye-Huckel (Eqn. 8.2), Davies (Eqn. 8.4), and B-dot (Eqn. 8.5) equations. Dotted line shows the Davies equation evaluated with a coefficient of 0.2 instead...

A coefficient of 0.2 is used sometimes instead of 0.3. The only variable specific to the species in question is the charge Zj, which of course is known. For this reason, the Davies equation is especially easy to apply within geochemical models designed for work at 25 °C, such as WATEQ (Ball et al., 1979) and its successors, and PHREEQE (Parkhurst et al., 1980). 

As can be seen in Figure 8.1, the Davies equation does not decrease monotoni-cally with ionic strength, as the Debye-Huckel equation does. Beginning at ionic strengths of about 0.1 molal, it deviates above the Debye-Huckel function and at about 0.5 molal starts to increase in value. The Davies equation is reasonably accurate to an ionic strength of about 0.3 or 0.5 molal. 

Helgeson (1969 see also Helgeson and Kirkham, 1974) presented an activity model based on an equation similar in form to the Davies equation. The model, adapted from earlier work (see Pitzer and Brewer, 1961, p. 326, p. 578, and Appendix 4, and references therein), is parameterized from 0°C to 300 °C for solutions of up to 3 molal ionic strength in which NaCl is the dominant solute. The model takes it name from the B-dot equation,... 

Activity coefficients in the aqueous phase, yiw, of neutral molecules are set equal to one because of the zero charge, and under the assumption that the activity coefficient of the infinitely diluted solution equals the actual activity coefficient. The activity coefficients of the charged species can be approximated with the Davies equation ... 

In terms of empirical equations, the following one provides good fits from 10-3 to 1.0 m solutions according to our work. The Davies equation (10)... 

Siveral approaches are available in the case of mixed electrolyte solutions. The Guntelberg equation can be used at very high dilutions to avoid the ambiguity in the meaning of aD, the distance of closest approach, when several electrolytes are present. This equation is empirical and has fewer terms than the Debye-Huckel extended equation. I found it to yield poor agreement with experimental results even at m = 0.01 for NaCl at 25°C (y+ caic = 0.8985 and y+ exp = 0.9024). For the Davies equation for m = 0.20 one obtains y+ calc = 0.752 andy+exo = 0.735 also for NaCl at 25°C. 

None of these extensions has been really satisfactory and they are not very useful at high ionic strength. The Davies equation (19) differs from the others in providing an additional term which alters the response of the activity coefficient to changes in ionic strength, particularly at higher values. The authors have had some success with this type of equation by replacing the. 2 factor in the second term with a variable. The variable can be determined by experiment at a particular set of conditions. 

Various empirical relations are available for calculating individual ion activity coefficients [discussed by Stumm and Morgan (1996) for natural waters and Sposito (1984a, b), for soil solutions]. In the calculations in this book I used the Davies equation ... 

The Davies equation [29] has been used extensively to calculate activity coefficients of electrolytes at fairly low ionic strengths. 

The equation has no theoretical foundation but is found to work fairly well up to ionic strengths of 0.1 mol kg It should not be used at higher ionic strengths. The Davies equation has a form similar to the B-G-S equation but with ion interaction coefficients equal to 0.153zf, i.e., 0.15, 0.61, and 1.38 for ions of charge 1, 2, and 3, respectively. These values do not agree very well with the tabulated s values. 

While several experimental techniques provide Information relating to dual phase continuity, the two most important methods Involve scanning electron microscopy and dynamic mechanical spectroscopy [16,22-2A]. Donatelll, et al [1 ] performed the first mechanical study on PB/PS IPN s. Figure 5 [ 6] illustrates the fit provided by the Davies equation [22] and the Budlansky equation [25,26], both of these equations derived on the assumption of dual phase continuity. 

Figure 5. Modulus-composition curves for crass-polybutadiene-inier-cross-polystyrene semi-I and full IPNs (16). (a) Kerner equation (upper bound) (b) Budiansky model (c) Davies equation and (d) Kerner equation (lower bound). (Reproduced from ref. 23. Copyright 1981 American Chemical Society.)...

Davies Equation Debye-Huckel Treatment Isotonic Buffers... 

DAVIES EQUATION DEBYE-HOCKEL TREATMENT Day-night cycle,... 

By considering Boi as a single adjustable parameter, data can then be fitted using only Boi and B as variables. A further simplification is to make Boi equal to unity and add a further constant ionic strength term which gives the Davies equation (Davies 1962)... 

The second semiempirical approach to the evaluation of the activity coefficient is the use of the Davies equation which modifies the Bronsted extension of the limiting Debye-Hilckel expression and is given by... 

From Equations 14, 20, and 31 it is clear that the difference in the two t values for which zd -> oo and Zi = Xo is negligible, so that Equation 31 is an accurate cutoff value. The restriction becomes t t (z) for z > xu or t — tx m( i ) for < The limits imposed on maximum particle size by both the drag-slip and Davies equation cutoff are shown in Figure 7. 

Figure 7. Particle size limits imposed by the drag-slip and the Davies equation cutoff...

An estimate of the probable errors in the correction factors and cutoff values follows. From Equation 3 one sees that the fractional errors in both are of the same order of magnitude as the fractional error in the velocity, Av/v, averaged over the region of motion. There are three main contributions to this error. One comes from the approximation to the Davies equations (8 and 10). The average fractional error is of the order of Av/v —5%, the minus sign occurring since Equations 9 and 10 underestimate the true values of Re and v. The other error contributions come from the approximations for air density and viscosity. One sees from Equations 7-9 that the first-order term in v is independent of p and has a 1/rj dependence. The second-order term is directly proportional to P. Since this term contributes a maximum of 30% to the velocity and the maximum error in p is 8%, this contribution to Av/v should be... 

Figure 13-2 Activity coefficients from extended Debye-Huckel and Davies equations. Shaded areas give Debye-Huckel activity coefficients for the range of ion sizes in Table 8-1.

Even if we know all reactions and equilibrium constants for a given system, we cannot compute concentrations accurately without activity coefficients. Chapter 8 gave the extended Debye-Huckel equation 8-6 for activity coefficients with size parameters in Table 8-1. Many ions of interest are not in Table 8-1 and we do not know their size parameter. Therefore we introduce the Davies equation, which has no size parameter ... 

For ionic species, we compute activity coefficients with the Davies equation 13-18. For the neutral species H,P04, we assume that y 1.00. 

Figure 13-3 puts everything together in a spreadsheet. Input values for FKH,P04, FNaiHPOj, pA i, pKn, pK3, and pA w are in the shaded cells. We guess a value for pH in cell H15 and write the initial ionic strength of 0 in cell Cl9. Cells A9 H10 compute activities with the Davies equation. With pi = 0, all activity coefficients are 1. Cells A13 H16 compute concentrations. [HT] in cell B13 is (10 PH)/yH = (10A-H15)/B9. Cell El 8 computes the sum of charges. 

Mixture of KH2P04 and Na2HP04 including activity coefficients from Davies equation ... 

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