Debye activity coefficient

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. 

The situation for electrolyte solutions is more complex theory confimis the limiting expressions (originally from Debye-Htickel theory), but, because of the long-range interactions, the resulting equations are non-analytic rather than simple power series.) It is evident that electrolyte solutions are ideally dilute only at extremely low concentrations. Further details about these activity coefficients will be found in other articles. 

The Debye-Htickel limiting law predicts a square-root dependence on the ionic strength/= MTLcz of the logarithm of the mean activity coefficient (log y ), tire heat of dilution (E /VI) and the excess volume it is considered to be an exact expression for the behaviour of an electrolyte at infinite dilution. Some experimental results for the activity coefficients and heats of dilution are shown in figure A2.3.11 for aqueous solutions of NaCl and ZnSO at 25°C the results are typical of the observations for 1-1 (e.g.NaCl) and 2-2 (e.g. ZnSO ) aqueous electrolyte solutions at this temperature. 

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. 

For gases, pure solids, pure liquids, and nonionic solutes, activity coefficients are approximately unity under most reasonable experimental conditions. For reactions involving only these species, differences between activity and concentration are negligible. Activity coefficients for ionic solutes, however, depend on the ionic composition of the solution. It is possible, using the extended Debye-Htickel theory, to calculate activity coefficients using equation 6.50... 

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

If the activity coefficients are estimated from the Debye-Huckel theory in dilute regions of simple electrolyte systems, we have for aqueous solutions at 25 °C,... 

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

Equation (7.45) is a limiting law expression for 7 , the activity coefficient of the solute. Debye-Htickel theory can also be used to obtain limiting-law expressions for the activity a of the solvent. This is usually done by expressing a in terms of the practical osmotic coefficient

electrolyte solute, it is defined in a general way as... 

Figure 9.6 Mean ionic activity coefficients for HCl(aq) at T = 298.15 K obtained from the emf results of G. A. Linhart, J. Am. Chem. Soc.. 41, 1175-1180 (1919). The dashed line is the Debye-Huckel limiting law prediction.

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 ... 

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... 

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. 

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... 

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. 

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 ... 

The question of the relationship between activity and concentration arises. Here the Debye-Huckel theory of activity coefficients, although valid only below 0.01 M, has proved to be most helpful, either for establishing an acid concentration from its H+ activity or for calculating H+ activity from its previously known acid concentration. 

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. 

Fig. 4.5. 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-Huckel activity coefficients ( ) and an ideal solution in which the activity coefficients are unity (o).

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. 

Fig. 8.1. Activity coefficients y, predicted at 25 °C for a singly charged ion with size a of 4 A, according to the Debye-Huckel (Eqn. 8.2), Davies (Eqn. 8.4), and B-dot (Eqn. 8.5) equations. Dotted line shows the Davies equation evaluated with a coefficient of 0.2 instead...

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