Debye Huckel equation table

Table 8.3 Constants of the Debye-Huckel Equation from 0 to 100°C 8.5...

Various attempts have been made to increase the valid range of the Debye- Huckel equation to regions of high ionic strength by the use of empirically fitted parameters. Several of these equations are listed in table I. 

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 ... 

Table 1.2. Ion size parameters for the Debye-Huckel equation.

TABLE 2.2. Values of the Parameter a Used in the Extended Debye-Huckel Equation (Eq. 2.3a) for Selected Ions... 

TABLE 2.3. Single-Ion Activity Coefficients Calculated from the Extended Debye-Huckel Equation at 25°C... 

Table 1.1 gives the coefficients in the Debye-Huckel equation, and in the equations for Af Gy °(7) and Af 7/y (7) that have been calculated by Clarke and Glew (10). 

Either (2-20) or (2-21) is referred to as the extended Debye-Huckel equation (EDHE) this pair of equations gives results appreciably different from the DHLL when H > 0.01 (that is, V/i > 0.1). For comparison, some ionic activity coefficients calculated from (2-15) and (2-20) are listed in Table 2-1. 

In dilute solutions (/ < 10 M), that is, in fresh waters, our calculations are usually based on the infinite dilution activity convention and thermodynamic constants. In these dilute electrolyte mixtures, deviations from ideal behavior are primarily caused by long-range electrostatic interactions. The Debye-Huckel equation or one of its extended forms (see Table 3.3) is assumed to give an adequate description of these interactions and to define the properties of the ions. Correspondingly, individual ion activities are estimated by means of individual ion activity coefficients calculated with the help of the Guntelberg or Davies (equations 3 and 4 of Table 3.3) or it is often more convenient to calculate, with these activity coefficients, a concentration equilibrium constant valid at a given /,... 

Some ion activity coefficients at 25°C computed with the Debye-Huckel equation as a function of ionic strength, ion size, and charge, are shown in Table 4.2. Debye-Huckel ion activity coefficients up to 0.1 mol/kg ionic strength, are plotted in Fig. 4.3 for some monovalent and divalent ions. The Debye-Huckel equation can be used to compute accurate activity coefficients for monovalent ions up to about / = 0.1 mol/kg, for divalent ions to about / = 0.01 mol/kg, and for trivalent ions up to perhaps / = 0.001 mol/kg. 

TABLE 4.2. Individual ion activity coefficients at 25°C for different ion sizes (a ) in angstroms (1 A = 10 cm) as a function of ionic strength, computed using the extended Debye-Huckel equation with A = 0.5091 and B = 0,3286... 

This semiempiiical equation utilizes two parameters, d and B, to evaluate log 7+ or log 7. While B has a value of about 0.33, d is approximated by the size of the hydrated cation (if 7+ is being evaluated) or anion (if 7 is being evaluated), expressed in Angstrom units. Values of d for common ions are given in Table 1.1. Since an infinitely dilute solution of electrolyte is considered to be ideal (no electrostatic interactions), the Debye-Huckel equation predicts that 7+ = 7 = 1 for / =... 

Table 1.4 Constants A and B in Debye-Huckel equations in molal concentration scale calculated from equations (1.70) and (1.74) (Physical chemistry, 2001).

Sulfuric acid is a 2 1 electrolyte, and so (by using the data in Table 3.1) the ionic strengthlis three times the concentration, i.e.l = 0.03 moldm f Next, from the Debye-Huckel extended law equation (3.15), we can obtain the mean ionic activity coefficient y as follows ... 

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.

In this appendix, we summarize the coefficients needed to calculate the thermodynamic properties for a number of solutes in an electrolyte solution from Pitzer s equations.3 Table A7.1 summarizes the Debye-Huckel parameters for water solutions as a function of temperature. They provide the leading terms for Pitzer s equations, and can also be used to calculate the Debye-Huckel limiting law values from the equations... 

The Debye-Huckel-Onsager equation has been tested against a large body of accurate experimental data. A comparison of theory and experiment is shown in Fig. 4.93 and Table 4.21 for aqueous solutions of true eiectroiytes, i.e., substances that consisted of ions in their crystal lattices before they were dissolved in water. At very low concentrations (< 0.(X)3 N), the agreement between theory and experiment is very good. There is no doubt that the theoretical equation is a satisfactory expression for the limiting tangent to the experimentaiiy obtained/ versus curves. 

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. 

Table 3.4 lists some activity coefficients as calculated from the extended Debye-Huckel limiting law for various values of /. In dilute solutions, such calculated values agree well with experimental data for mean activity coefficients of simple electrolytes. At higher concentrations, the Davies equation usually represents the experimental data better. 

Approximately the same result would be obtained from the extended Debye-Huckel limiting law (equation 1, Table 3.3) p"H = —log [H ] + 0.113 = 3.II3. 

This equation can be used when no experimental information is available. In some cases it can give usable results for activity coefficients up to ionic strengths of 0.5 mol kg or beyond, but it is ordinarily in error by several percent in this region. Table A. 11 in Appendix A gives experimental values of the mean ionic activity coefficients of several aqueous electrolytes at various concentrations. It also gives the predictions of the Debye-Huckel formula with fia taken equal to 1.00kg / mol /, and of the Davies equation. 

Most vulnerable in equation (1.73) is r. value (Table 1.5). Debye and Huckel defined it as average distance at which ions are capable of approaching one another. It is associated with the diameter of a hydrated ion. It is determined experimentally by selecting it so that activities coefficients calculated from equation (1.73) coincided with the experimental. Besides, vulnerability of this value is caused by its dependence on temperature. In case of NaCl solution it may range between 3-10 and 4-10 cm. 

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