Activity coefficients Debye-Hiickel

Among other applications of electrolyte solution theory to defect problems should be mentioned the application of the Debye-Hiickel activity coefficients by Harvey32 to impurity ionization problems in elemental semiconductors. Recent reviews by Anderson7 and by Lawson45 emphasizing the importance of Debye-Hiickel effects in oxide semiconductors and in doped silver halides, respectively, and the book by Kroger41 contain accounts of other applications to defect problems. However, additional quantum-mechanical problems arise in the treatment of semiconductor systems and we shall not mention them further, although the studies described below are relevant to them in certain aspects. 

Substituting analytical values of molality for the terms on the left of Equation 5, Debye-Hiickel activity coefficients in the right-hand terms, and neglecting mH+... 

Numerous models predict the activity coefficient of individual ions in solution. The one by Debye and Hiickel [8] considers only electrostatic (columbic) interactions between cations and anions in a dilute solution of a single, completely dissociated salt. It is assumed that ion-ion interactions (as opposed to other phenomena such as ion-solvent interactions, ion solvation effects, and variations in the solvent dielectric constant with salt concentration) cause the ion activity coefficients to deviate from 1.0. From a practical point, only the Debye-Hiickel activity coefficient relationship is needed, along with some knowledge of the theory s shortcomings, which restrict its application. For a dilute electrolytic solution containing a binary salt (i.e., a salt with one type each of cation and anion species), the ion activity coefficient from Debye-Hiickel theory is given by... 

Debye-Hiickel Activity Coefficient Parameters for Aqueous Solutions of Any Binary Salt... 

FIG. 5.5 Convergence of the iteration to very small residuals in the reduced basis method, for systems contrived to have large positive and negative residuals at the start of the iteration. The tests assume Debye-Hiickel activity coefficients ( ) and an ideal solution in which the activity coefficients are unity (O). 

Equation (12-3) is plotted in Figure 12-1 for the case 0.1 mol (10.787 g) silver in contact with one liter of solution. Silver metal, at unit activity (remember this is independent of amount of silver) is present until the (Ag" ) reaches the 0.1 M value. Since the estimated Debye-Hiickel activity coefficient is 0.75, this is... 

First, the Debye-Hiickel activity coefficient equation (57) can be used to write explicit expressions for the individual ion activity coefficients ... 

When the permittivities of solvents S and R are very different, this affects the De-bye-Hiickel activity coefficient of M+ (9) in Chapter 2). If necessary, the effect should be estimated (or eliminated) using the Debye-Hiickel theory. 

K is not the true dissociation constant, however, as the ions are not ideal solutes, and allowance must be made for their activity coefficients. These can be calculated from the Debye-Hiickel activity equation t, according to which,... 

If a mixture of finely divided metallic tin and lead is shaken with a solution of their perchlorates and the equilibrium concentrations of tin and lead in solution are measured, Kc can be calculated. If it is assumed that the activity coefficients of the two species are equal, Ktherm is giveu by the ratio of concentrations. Corrections can be applied for the activity coefficients using the Debye-Hiickel activity equation—this correction is most necessary for standard redox potentials. For this system Ktherm = 2.98 at 298 K. Therefore... 

Debye-Hiickel theory The activity coefficient of an electrolyte depends markedly upon concentration. Jn dilute solutions, due to the Coulombic forces of attraction and repulsion, the ions tend to surround themselves with an atmosphere of oppositely charged ions. Debye and Hiickel showed that it was possible to explain the abnormal activity coefficients at least for very dilute solutions of electrolytes. 

Although it is not possible to measure an individual ionic activity coefficient,, it may be estimated from the following equation of the Debye-Hiickel theory ... 

At moderate ionic strengths a considerable improvement is effected by subtracting a term bl from the Debye-Hiickel expression b is an adjustable parameter which is 0.2 for water at 25°C. Table 8.4 gives the values of the ionic activity coefficients (for Zi from 1 to 6) with d taken to be 4.6A. 

At sufficiently low ionic strengths the activity coefficient of each electrolyte in a mixture is given by the Debye-Hiickel limiting law... 

Table 7.1 Debye-Hiickel parameters for the activity coefficient, volume, enthalpy, and...

The Debye-Hiickel formula for the activity coefficient of an ion was developed by a consideration of ion atmosphere effects.10 It starts with an electrostatic expression for the free energy of interaction for one ion with one mole of others ... 

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 beginning of the twentieth century also marked a continuation of studies of the structure and properties of electrolyte solution and of the electrode-electrolyte interface. In 1907, Gilbert Newton Lewis (1875-1946) introduced the notion of thermodynamic activity, which proved to be extremally valuable for the description of properties of solutions of strong electrolytes. In 1923, Peter Debye (1884-1966 Nobel prize, 1936) and Erich Hiickel (1896-1981) developed their theory of strong electrolyte solutions, which for the first time allowed calculation of a hitherto purely empiric parameter—the mean activity coefficients of ions in solutions. 

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. 

In the foregoing derivations we have assumed that the true pH value would be invariant with temperature, which in fact is incorrect (cf., eqn. 2.58 of the Debye-Hiickel theory of the ion activity coefficient). Therefore, this contribution of the solution to the temperature dependence has still to be taken into account. Doing so by differentiating ET with respect to T at a variable pH we obtain in AE/dT the additional term (2.3026RT/F) dpH/dT, which if P (cf., eqn. 2.98) is neglected and when AE/dT = 0 for the whole system yields... 

J. Bjerrum (1926) first developed the theory of ion association. He introduced the concept of a certain critical distance between the cation and the anion at which the electrostatic attractive force is balanced by the mean force corresponding to thermal motion. The energy of the ion is at a minimum at this distance. The method of calculation is analogous to that of Debye and Hiickel in the theory of activity coefficients (see Section 1.3.1). The probability Pt dr has to be found for the ith ion species to be present in a volume element in the shape of a spherical shell with thickness dr at a sufficiently small distance r from the central ion (index k). 

More rigorous Debye-Hiickel treatment of the activity coefficient... 

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 ...

In the above two equations, the former value is valid for basic SI units and the latter value for / in moles per cubic decimetre and a in nanometres. The parameter a represents one of the difficulties connected with the Debye-Hiickel approach as its direct determination is not possible and is, in most cases, found as an adjustable parameter for the best fit of experimental data in the Eq. (1.3.29). For common ions the values of effective ion radii vary from 0.3 to 0.5. Analogous to the limiting law, the mean activity coefficient can be expressed by the equation... 

In dilute solutions it is possible to relate the activity coefficients of ionic species to the composition of the solution, its dielectric properties, the temperature, and certain fundamental constants. Theoretical approaches to the development of such relations trace their origins to the classic papers by Debye and Hiickel (6-8). For detailpd treatments of this subject, refer to standard physical chemistry texts or to treatises on electrolyte solutions [e.g., that by Harned... 

If limiting forms of the Debye-Hiickel expression for activity coefficients are used, this equation becomes... 

If the Debye-Hiickel limiting law is used to evaluate the various activity coefficients in aqueous solution at 25 °C, the last equation becomes... 

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. 

In 1923, Debye and Hiickel published their famous papers describing a method for calculating activity coefficients in electrolyte solutions. They assumed that ions behave as spheres with charges located at their center points. The ions interact with each other by coulombic forces. Robinson and Stokes (1968) present their derivation, and the papers are available (Interscience Publishers, 1954) in English translation. 

The Debye-Hiickel methods work poorly, however, when carried to moderate or high ionic strength, especially when salts other than NaCl dominate the solute. In the theory, the ionic strength represents all the properties of a solution. For this reason, a Debye-Hiickel method applied to any solution of a certain ionic strength (whether dominated by NaCl, KC1, HC1, H2SO4, or any salt or salt mixture) gives the same set of activity coefficients, irregardless of the solution s composition. This result, except for dilute solutions, is, of course, incorrect. Clearly, we cannot rely... 

Helgeson, H. C. and D. H. Kirkham, 1974, Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures, II. Debye-Hiickel parameters for activity coefficients and relative partial molal properties. American Journal of Science 274, 1199-1261. 

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