Debye-Huckel equation applications

It should be mentioned that the results of the above extrapolation can be improved when its application is extended to concentrations even higher than 0.01 M, by means of an extended Debye-Huckel equation, viz., introduction of the factor 1/(1 + ba) (cf., eqns. 2.60 and 2.60a) into the final expression 2.62 for log f and its general form 2.64 leads to... 

Marshall s extensive review (16) concentrates mainly on conductance and solubility studies of simple (non-transition metal) electrolytes and the application of extended Debye-Huckel equations in describing the ionic strength dependence of equilibrium constants. The conductance studies covered conditions to 4 kbar and 800 C while the solubility studies were mostly at SVP up to 350 C. In the latter studies above 300°C deviations from Debye-Huckel behaviour were found. This is not surprising since the Debye-Huckel theory treats the solvent as incompressible and, as seen in Fig. 3, water rapidly becomes more compressible above 300 C. Until a theory which accounts for electrostriction in a compressible fluid becomes available, extrapolation to infinite dilution at temperatures much above 300 C must be considered untrustworthy. Since water becomes infinitely compressible at the critical point, the standard entropy of an ion becomes infinitely negative, so that the concept of a standard ionic free energy becomes meaningless. 

Equation 6-33 suggests that extrapolation of equilibrium constants to infinite dilution is done appropriately by plotting log Kc vs-01. For example, Fig. 6-1 shows plots of pK a for dissociation of H2P04-, AMP, and ADP2-, and ATP3 vs fp. The variation of pK a with- p at low concentrations (Eq. 6-35) is derived by application of the Debye-Huckel equation (Eq. 6-33) ... 

Electrode response is related to analyte activity rather than analyte concentration. We are usually interested in concentration, however, and the determination of this quantity from a potentiometric measurement requires activity coefficient data. Activity coefficients are seldom available because the ionic strength of the solution either is unknown or else is so large that the Debye-Huckel equation is not applicable. 

In most soil solutions, the ionic strength I is low (<0.01), so that the extended Debye-Huckel equation is applicable for the correction of ionic concentrations to the more thermodynamically meaningful activities. Typically, conductivity measurements are used to estimate the ionic strength, a much less laborious procedure than measurement of each cation and anion present in solution. More problematic, however, is the detection and measurement of complexing anions and molecules (ligands). As will be shown later in this chapter, their presence can result in activities of metal ions in soil solution being much lower than measured concentrations would suggest. 

The above procedure has been coded in FORTRAN as the program EQBRM and a copy suitable for personal computers is included in this book as Appendix E, along with an example showing the proper format for input data. For different applications it is necessary to choose among the available methods of estimating activity coefficients. For example, the Debye-Huckel equation can often be used for dilute systems such as rivers and groundwater, but concentrated brines will require the Pitzer equations or measured coefficients if they are available. For this reason, a subroutine should be written to calculate activity coefficients for your application. 

A knowledge of a solution s ionic strength enables a determination of the activity coefficient to be made. This can occur through the application of the Debye-Huckel equation according to... 

Summarizing, we may state that the application of the. Debye-Huckel equations is only feasible for spherical particles when their dimensions and the electrolyte concentrations are small enough to satisfy the relation m< l in that case it will give reasonable results, even when the potential is no longer small. For larger particles and (or) larger ionic concentrations, however, except for the rare cases in which the... 

As a result of these electrostatic effects aqueous solutions of electrolytes behave in a way that is non-ideal. This non-ideality has been accounted for successfully in dilute solutions by application of the Debye-Huckel theory, which introduces the concept of ionic activity. The Debye-Huckel Umiting law states that the mean ionic activity coefficient y+ can be related to the charges on the ions, and z, by the equation... 

Tanford examined the application of Debye-Huckel theory and found the theory not to be valid because the high charge density generatedby the closely spaced head groups leads to substantial charge neutralization by counter ions Alternatively, he equated the work of... 

Thirdly, another corollary of the first limitation, is the inconsistency and inadequacy of activity coefficient equations. Some models use the extended Delbye-Huckel equation (EDH), others the extended Debye-Huckel with an additional linear term (B-dot, 78, 79) and others the Davies equation (some with the constant 0.2 and some with 0.3, M). The activity coefficients given in Table VIII for seawater show fair agreement because seawater ionic strength is not far from the range of applicability of the equations. However, the accumulation of errors from the consideration of several ions and complexes could lead to serious discrepancies. Another related problem is the calculation of activity coefficients for neutral complexes. Very little reliable information is available on the activity of neutral ion pairs and since these often comprise the dominant species in aqueous systems their activity coefficients can be an important source of uncertainty. 

Activity coefficients in en Schaap and others reviewed the application of equations of the Debye-Huckel type to solutions of electrolytes in en. The appropriate modification of the limiting law (DHLL) becomes... 

The various forms of equation (40.15), referred to as the Debye-HUckel limiting law, express the variation of the mean ionic activity coefficient of a solute with the ionic strength of the medium. It is called the limiting law because the approximations and assumptions made in its derivation are strictly applicable only at infinite dilution. The Debye-Hfickel equation thus represents the behavior to which a solution of an electrolyte should approach as its concentration is diminished. 

Another policy in writing the book has been the attempt to base the deduction of all equations on first principles. What actually constitutes such principles is, to an extent, a matter of individual preference. Any attempt at definition would immediately lead one into the field of the professional philosopher. Such an intrusion the author is, above everything, anxious to avoid. Fie feels, however, that the attempt to build from the ground up has been accomplished in most of the subjects considered. Exceptions are, however, the extension of the Debye-Huckel theory, and the application of the interionic attraction theory to electrolytic conductance. In the latter case the fundamentals lie in the field of statistical mechanics, which cannot be adequately treated short of a book the size of this one, and which, in any case, would not be written by the author. 

A very interesting application of Eq. 1.7-12 is the Br nsted-Bjerrum equation for rate constants in solutions where the Debye-Huckel theory is applicable. The latter provides an equation for the activity coefficient, Rutgers [38] ... 

To test the application of Debye-Huckel theory, take the logs of equation (10-2) and arrange for linear plotting,... 

Both start out with the Debye-HUckel term which depends primarily on ionic strength but that term alone is applicable only to very dilute solutions. Guggenheim adds an essentially empirical first-order correction and this is sufficient for ionic strengths to about 1 or 2 molar. For more concentrated solutions, Pitzer has proposed a semi-theoretical equation which, however, has many parameters and all of these depend on temperature (22). 

Attempts to improve the theory by solving the Poisson-Boltzmann equation present other difficulties first pointed out by Onsager (1933) one consequence of this is that the pair distribution functions g (r) and g (r) calculated for unsymmetrically charged electrolytes (e.g., LaCl or CaCl2) are not equal as they should be from their definitions. Recently Outhwaite (1975) and others have devised modifications to the Poisson-Boltzmann equation which make the equations self-consistent and more accurate, but the labor involved in solving them and their restriction to the primitive model electrolyte are drawbacks to the formulation of a comprehensive theory along these lines. The Poisson-Boltzmann equation, however, has found wide applicability in the theory of polyelectrolytes, colloids, and the electrical double-layer. Mou (1981) has derived a Debye-Huckel-like theory for a system of ions and point dipoles the results are similar but for the presence of a... 

The derivation of the Debye— Hiickel equation for the activity coefficient is based on the linearized Boltzmann equation for electrostatic charge distribution around an ion. This limits the applicability of Eq. (57) to solutes with low surface potentials, which occurs for solution concentrations of monovalent ions of < 0.01 M. However, it is important to note that die method used for deriving activity coefficient equation (25) is based on rigorous thermodynamics and is not limited by the Debye—Huckel theory. If, for example, the Gouy—Chapman equation [22] was... 

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