Equation Hiickel

Debye-Hiickel equation Debye-length Condenser capacity... 

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

The values were calculated from the modified Debye-Hiickel equation utilizing the modifications proposed by Robinson and by Guggenheim and Bates ... 

It can be seen from Figure 7.8(b) that the curved lines predicted by the extended form of the Debye-Hiickel equation follow the experimental results to higher ionic strengths than do the limiting law expressions for the (1 1) and (2 1) electrolytes. However, for the (2 2) electrolyte, the prediction is still not very good even at the lowest measured molality.0... 

Experience shows that solutions of other electrolytes behave in a manner similar to the examples we have used. The conclusion we reach is that the Debye-Hiickel equation, even in the extended form, can be applied only at very low concentrations, especially for multivalent electrolytes. However, the behavior of the Debye-Hiickel equation as we approach the limit of zero ionic strength appears to give the correct limiting law behavior. As we have said earlier, one of the most useful applications of Debye-Hiickel theory is to... 

This is known as the Brpnsted-Debye-Hiickel equation. It is convenient to rewrite it as... 

The nature of the Debye-Hiickel equation is that the activity coefficient of a salt depends only on the charges and the ionic strength. The effects, at least in the limit of low ionic strengths, are independent of the chemical identities of the constituents. Thus, one could use N(CH3)4C1, FeS04, or any strong electrolyte for this purpose. Actually, the best choices are those that will be inert chemically and least likely to engage in ionic associations. Therefore, monovalent ions are preferred. Anions like CFjSO, CIO, /7-CIC6H4SO3 are usually chosen, accompanied by alkali metal or similar cations. 

The accelerating effect of ionic strength on the reaction between Cu and Co(NH3)5C12 is illustrated. The axes are the quantities specified by the Brpnsted-Debye-Hiickel equation—log k and ju. 1/2/( 1 + fc n). 

The experimental constant ki is either K k or K k. Now each of the equilibrium constants has a predictable ionic strength effect. If the Debye-Hiickel equation is applied to them, we have... 

This composite rate constant is predicted to have a different ionic strength dependence for the two schemes. According to the Br0nsted-Debye-Hiickel equation, the composite rate constant for Eq. (9-76) will be independent of ionic strength if Scheme ... 

Finally, it must be recalled that the transport properties of any material are strongly dependent on the molecular or ionic interactions, and that the dynamics of each entity are narrowly correlated with the neighboring particles. This is the main reason why the theoretical treatment of these processes often shows similarities with models used for thermodynamic properties. The most classical example is the treatment of dilute electrolyte solutions by the Debye-Hiickel equation for thermodynamics and by the Debye-Onsager equation for conductivity. 

Intense ion-ion interactions which are characteristic of salt solutions occur in the concentrated aqueous solutions from which AB cements are prepared. As we have seen, in such solutions the simple Debye-Hiickel limiting law that describes the strength goes up so the repulsive force between the ions becomes increasingly important. This is taken account of in the full Debye-Hiickel equation by the inclusion of a parameter related to ionic size and hence distance of closest approach (Marcus, 1988). 

Fig. 2.3 was constructed using a K2-3 value at 250°C extrapolated from high-temperature data by Orville (1963), liyama (1965) and Hemley (1967). Ion activity coefficients were computed using the extended Debye-Hiickel equation of Helgeson (1969). The values of effective ionic radius were taken from Garrels and Christ (1965). In the calculation of ion activity coefficients, ionic strength is regarded as 0.5 im i ++mci-) (= mc -)- The activity ratio, an-f/aAb, is assumed to be unity. 

Fig. 1.8 Dependence of the mean activity coefficient y tC of NaCl on the square root of molar concentration c at 25°C. Circles are experimental points. Curve 1 was calculated according to the Debye-Hiickel limiting law (1.3.25), curve 2 according to the approximation aB = 1 (Eq. 1.3.32) curve 3 according to the Debye-Hiickel equation (1.3.31), a = 325nm curve 4 according to the Bates-Guggenheim approximation (1.3.33) curve 5 according to the Bates-Guggenheim approximation + linear term 0.1 C curve 6 according to Eq. (1.3.38) for a = 0.4nm, C = 0.055dm5-mor ...

Can the species activity coefficients be calculated accurately An activity coefficient relates each dissolved species concentration to its activity. Most commonly, a modeler uses an extended form of the Debye-Hiickel equation to estimate values for the coefficients. Helgeson (1969) correlated the activity coefficients to this equation for dominantly NaCl solutions having concentrations up to 3 molal. The resulting equations are probably reliable for electrolyte solutions of general composition (i.e., those dominated by salts other than NaCl) where ionic strength is less than about 1 molal (Wolery, 1983 see Chapter 8). Calculated activity coefficients are less reliable in more concentrated solutions. As an alternative to the Debye-Hiickel method, the modeler can use virial equations (the Pitzer equations ) designed to predict activity coefficients for electrolyte brines. These equations have their own limitations, however, as discussed in Chapter 8. 

In each case, we use program spece8 or react and employ an extended form of the Debye-Hiickel equation for calculating species activity coefficients, as discussed in Chapter 8. In running the programs, you work interactively following the general procedure ... 

Geochemical modelers currently employ two types of methods to estimate activity coefficients (Plummer, 1992 Wolery, 1992b). The first type consists of applying variants of the Debye-Hiickel equation, a simple relationship that treats a species activity coefficient as a function of the species size and the solution s ionic strength. Methods of this type take into account the distribution of species in solution and are easy to use, but can be applied with accuracy to modeling only relatively dilute fluids. 

Here, i, j, and k are subscripts representing the various species in solution and /dh is a function of ionic strength similar in form to the Debye-Hiickel equation. The terms Xy and Hijk are second and third virial coefficients, which are intended to account for short-range interactions among ions the second virial coefficients vary with ionic strength, whereas the third virial coefficients do not. 

It is interesting to compare the Debye-Hiickel and virial methods, since each has its own advantages and limitations. The Debye-Hiickel equations are simple to apply and readily extensible to include new species in solution, since they require few coefficients specific to either species or solution. The method can be applied as well over the range of temperatures most important to an aqueous geochemist. There is an extensive literature on ion association reactions, so there are few limits to the complexity of the solutions that can be modeled. 

Unlike the Debye-Hiickel equations, the virial methods provide little or no information about the distribution of species in solution. Geochemists like to identify the dominant species in solution in order to write the reactions that control a system s behavior. In the virial methods, this information is hidden within the complexities of the virial equations and coefficients. Many geochemists, therefore, find the virial methods to be less satisfying than methods that predict the species distribution. The information given by Debye-Hiickel methods about species distributions in concentrated solutions, however, is not necessarily reliable and should be used with caution. 

Since we have no direct information about the chemistry of the Fountain fluid, we assume that its composition reflects reaction with minerals in the evaporite strata that lie beneath the Lyons. We take this fluid to be a three molal NaCl solution that has equilibrated with dolomite, anhydrite, magnesite (MgCC>3), and quartz. The choice of NaCl concentration reflects the upper correlation limit of the B-dot (modified Debye-Hiickel) equations (see Chapter 8). To set pH, we assume a CO2 fugacity of 50, which we will show leads to a reasonable interpretation of the isotopic composition of the dolomite cement. 

Activity coefficients can be determined by experimental observations. Since they are functions of ionic strength, temperature and pressure, marine scientists typically estimate values at the environmental conditions of interest from semi-empirical equations. In dilute solutions, the activity coefficient of a monoatomic ion can be calculated from the Debye-Hiickel equation ... 

Figure 8JO Comparison between individual ionic activity coefficients obtained with Debye-Hiickel equation and with mean salt method for various ionic strength values. Reprinted from Garrels and Christ (1965), with kind permission from Jones and Bartlett Publishers Inc., copyright 1990.

The Extended Debye-Hiickel Equation. This exercise reminds us that the Debye-Hiickel limiting law is not sufficiently accurate for most physicochemical studies. To estimate the calculated activity coefficient more accurately, one must consider the fact that ions are not point charges. To the contrary, ions are of finite size relative to the distance over which the ions interact electrostatically. This brings us to the extended Debye-Hiickel equation ... 

Although these approaches seem to be similar, the former method has several limitations it should be recognized that intrinsic mobility (yu,A ) depends on the ionic strength of the buffer, as shown in Ref. 17. Also, the activity coefficients of the zwitterions could not be calculated via the De-bye-Hiickel equation, and other methods, such as melting point depression, should be used to obtain the activity coefficients of the zwitterions (31). On the other hand, the latter method is directly applicable to the zwitterions to obtain their acidity constants. 

Let us now examine how we can obtain an estimate of /q from the measured electromobility of a colloidal particle. It turns out that we can obtain simple, analytic equations only for the cases of very large and very small particles. Thus, if a is the radius of an assumed spherical colloidal particle, then we can obtain direct relationships between electromobility and the surface potential, if either Kit > 100 or Kd < 0.1, where K" is the Debye length of the electrolyte solution. Let us first look at the case of small spheres (where Kd < 0.1), which leads to the Hiickel equation. 

Temperature was set at 35 °C. Brine-rock mass ratio was set to 0.4 10, which corresponds to a porosity of approximately 10%. The mineral content and brine compositions were set to measured values (Table 1). Debyc-Hiickel equations were used to correct activity coefficients for saline solutions. The brine was allowed to come to equilibrium with the C02, then the 10 kg of sandstone was added and equilibrium assemblages were computed a second time. 

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