Activity coefficients Debye-Huckel equation

Provided the ionic strength is not too high, this equation is obeyed as well as (but no better than) the Debye-Huckel equation for activity coefficients. One can expect deviations at higher ionic strength, and they are in general more serious the higher... 

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... 

Changes in activity coefficients (and hence the relationship between concentration and chemical activity) due to the increased electrostatic interaction between ions in solution can be nicely modeled with well-known theoretical approaches such as the Debye-Huckel equation ... 

And (b) the extended Debye-Huckel equation for the approximation of the activity coefficient yj of the j-th ion. It needs the charge Zi and the ionic radius ay... 

Very few generalized computer-based techniques for calculating chemical equilibria in electrolyte systems have been reported. Crerar (47) describes a method for calculating multicomponent equilibria based on equilibrium constants and activity coefficients estimated from the Debye Huckel equation. It is not clear, however, if this technique has beep applied in general to the solubility of minerals and solids. A second generalized approach has been developed by OIL Systems, Inc. (48). It also operates on specified equilibrium constants and incorporates activity coefficient corrections for ions, non-electrolytes and water. This technique has been applied to a variety of electrolyte equilibrium problems including vapor-liquid equilibria and solubility of solids. 

If you use a spreadsheet for this exercise, compute activity coefficients with the extended Debye-Huckel equation and compute many more points. (You can look up the results of a similar titration to compare with your calculations.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 ... 

The BMREP and SDM currently use the Davies technique for activity coefficient prediction. The Davies technique is a combination of the extended Debye-Huckel equation (6) and the Davies equation (7). The Davies technique (and hence both equilibrium models) is accurate up to ionic strengths of 0.2 molal and may be used for practical calculations up to ionic strengths of 1 molal (8). Ion-pair equilibria are incorporated for species that associate (e.g., 1-2 and 2-2 electrolytes). The activity coefficients (y ) are calculated as a simple function of ionic strength (I) and are represented as ... 

Given the ionic strength of the solution from the chemical analysis, the activity coefficient can be computed using several approximation equations. All of them are inferred from the DEBYE-HUCKEL equation and differ in the range of the ionic strength they can be applied for. 

The second term in the DAVIES and extended DEBYE-HUCKEL equations forces the activity coefficient to increase at high ionic strength. This is owed to the fact, that ion interactions are not only based on Coulomb forces any more, ion sizes change with the ionic strength, and ions with the same charge interact. 

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

Cupric ion activities and cupric ion concentrations were determined using the Nernst equation from the differences in potential between the test solutions and a standard solution consisting of 10-5 M CUSO4 and 0.01 M KNO3 at pH 5.4 + 0.3. Values of cupric ion activity in test solutions were based on a cupric ion activity coefficient of 0.68 in the standard solution as calculated from the extended Debye-Huckel equation. For measurements in defined solutions containing 0.01 M KNO3 or 0.01 M NaHC03 cupric ion concentrations could be directly computed via the Nernst equation because activity coefficients were the same in both test and standard solutions. 

Values of K, the thermodynamic association constants are given at 25°. The concentrations of ionic species in the solutions at any time can be determined from mass balance, electroneutrality, and the appropriate equilibrium constants as described previously (19, p, 85-92) by successive approximations for the ionic strength. The activity coefficients of Z-valent ionic species may be calculated from an extended form of the Debye-Huckel equation such as that proposed by Davies (20, p. 34-53). 

FIGURE 2-3 Activity coefficients calculated by the limiting Debye-Huckel equation (dotted lines) and those observed experimentally. Left, electrolytes of three charge types in water. Right, hydrochloric acid in water-dioxane mixtures with bulk dielectric constants as indicated. Adapted from Homed and Ow . )... 

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 /,... 

In addition to the short-range interactions between species in all solutions, long-range electrostatic interactions are found in electrolyte solutions. The deviation from ideal solution behavior caused by these electrostatic forces is usually calculated by some variation of the Debye-Huckel theory or the mean spherical approximation (MSA). These theories do not include terms for the short-range attractive and repulsive forces in the mixtures and are therefore usually combined with activity coefficient models or equations of state in order to describe the properties of electrolyte solutions. 

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. 

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. 

The effect of temperature on ion activity coefficients is largely predicted by changes in the value of A, which is proportional to - log y, in the Debye-Huckel equation. The value of A increases from 0.492 to 0.534 between 0 and 50°C. Thus, activity coefficients become smaller with increasing temperature. Because A is multiplied by z in the Debye-Hiickel equation, the effect of temperature on activity coefficients is greatest for multivalent ions. 

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... 

In view of these uncertainties, it may prove more advantageous provisionally to work with an empirical relationship between activity coefficients and ionic strength of more concentrated solutions, instead of the Debye-Huckel equation. Bjerrum has found from experience that the following equation holds within wide limits ... 

We must state, however, that the thermodynamic dissociation constants have a relatively small practical importance. It is true that they remain constant with changing ionic strength but to use them it is necessary to know the activity coefficients of the several components at different electrolyte concentrations. The simple Debye-Huckel equation for computing activity coefficients is valid only at very small ionic strengths. At larger ionic strengths it is preferable to determine empirically the stoichiometric dissociation constants for various types of electrolytes. 

---