Activity coefficient of ionic species

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... 

Reactions in solution proceed in a similar manner, by elementary steps, to those in the gas phase. Many of the concepts, such as reaction coordinates and energy barriers, are the same. The two theories for elementary reactions have also been extended to liquid-phase reactions. The TST naturally extends to the liquid phase, since the transition state is treated as a thermodynamic entity. Features not present in gas-phase reactions, such as solvent effects and activity coefficients of ionic species in polar media, are treated as for stable species. Molecules in a liquid are in an almost constant state of collision so that the collision-based rate theories require modification to be used quantitatively. The energy distributions in the jostling motion in a liquid are similar to those in gas-phase collisions, but any reaction trajectory is modified by interaction with neighboring molecules. Furthermore, the frequency with which reaction partners approach each other is governed by diffusion rather than by random collisions, and, once together, multiple encounters between a reactant pair occur in this molecular traffic jam. This can modify the rate constants for individual reaction steps significantly. Thus, several aspects of reaction in a condensed phase differ from those in the gas phase ... 

Recently, there have been a number of significant developments in the modeling of electrolyte systems. Bromley (1), Meissner and Tester (2), Meissner and Kusik (2), Pitzer and co-workers (4, ,j5), and" Cruz and Renon (7j, presented models for calculating the mean ionic activity coefficients of many types of aqueous electrolytes. In addition, Edwards, et al. (8) proposed a thermodynamic framework to calculate equilibrium vapor-liquid compositions for aqueous solutions of one or more volatile weak electrolytes which involved activity coefficients of ionic species. Most recently, Beutier and Renon (9) and Edwards, et al.(10) used simplified forms of the Pitzer equation to represent ionic activity coefficients. 

The following are the practical procedures for obtaining the Gibbs energies of transfer and transfer activity coefficients of ionic species based on the extrathermodynamic assumptions (i), (ii) and (iii) described above ... 

A commonly used approximate form of the Debye-Hiickel theory for the activity coefficients of ionic species at 25°C is ... 

The program calculates activity coefficients of ionic species by calling a subroutine called ACTCF which calculates activity coefficients using the Davies revision of the Debye-Hiickel equation ... 

The formalism of the Eqs. (3.67), (3.78), and (3.79) is suited to explicitly showing that the (electric) activity coefficient of ionic species accounts for deviations from the ideal (unscreened) Coulomb behavior. It will be demonstrated below that an analogous formalism describes nonidealities in the interactions between dipolar species. 

Bromley s (S) method of calculating the activity coefficients of ionic species in multicomponent solutions, detailed in Chapter V. leads to the following equations ... 

Activity coefficients of ionic species are usually dependant on the ionic strength of the solution, y, y (I) I, ionic strength of the solution, the Debye-Hiickel equation being the simplest one in order to predict the values of the activity coefficients. Working with dilute metallic solutions it is likely to assume that In... 

Here a is used in place of r (the distance between ions at closest approach). The variable a is the sum of the effective radii of ions in solution and is the same for all pairs of ions (a rather bold assumption). Equation (7.25) is the Debye-Huckel equation for dilute electrolyte solutions, and is the fundamental equation for evaluating the activity coefficients of ionic species in solution. 

In this paper, two new models for the activity coefficients of ionic and molecular species in electrolyte systems are presented. The first is an extension of the Pitzer equation and is covered in more detail in Chen, et al. (11). 

The ionic strength of the solution influences the activity coefficients of the species present and the values of the liquid-junction potentials in the system and thus must be held constant in all related measurements, within a range of about 0.1 to 2m. 

One method takes into account the individual characteristics of the ionic media by using a medium-dependent expression for the activity coefficients of the species involved in the equilibrium reactions. The medium dependence is described by virial or ion interaction coefficients as used in the Pitzer equations and in the specific ion interaction model. 

The activity coefficient of a species is related to the ionic strength by the Debye-Hiickel Eq. 

The constant-capacitance model (Goldberg, 1992) assigns all adsorbed ions to inner-sphere surface complexes. Since this model also employs the constant ionic medium reference state for activity coefficients, the background electrolyte is not considered and, therefore, no diffuse-ion swarm appears in the model structure. Activity coefficients of surface species are assumed to sub-divide, as in the triplelayer model, but the charge-dependent part is a function of the overall valence of the surface complex (Zk in Table 9.8) and an inner potential at the colloid surface exp(Z F l,s// 7). Physical closure in the model is achieved with the surface charge-potential relation ... 

Potentiometric methods have eliminated the problems that beset earlier studies, due to the high electrolyte concentrations required for ideal electrode behavior. Following the so-called constant ionic medium principle [91], a large excess of an indifferent (or inert or swamping) electrolyte is added, so that the activity coefficients of the species can be considered constant when their concentration (very low compared to that of the indifferent electrolyte) are changed over a wide range. 

Calculation of Activity Coefficients for Ionic Species with Different Charge Numbers... 

The thermal diffusion potential, td> arises if an electrochemical system is nonisothermal. This phenomenon is due to the heat transport of ionic species and can be taken into account if the individual ion entropy of transport, conductivity, and activity coefficients of the species of interest are known. Therefore, the thermal diffusion potential depends on the temperature, pressure, and composition of the electrolyte liquid junction. Also, td is a function of the temperature gradient and can be a substantial value from tens to hundreds of millivolts [19]. 

A kinetic electrolyte effect ascribable solely to the influence of the ionic strength on activity coefficients of ionic reactants and transition states is called a primary kinetic electrolyte effect. A kinetic electrolyte effect arising from the influence of the ionic strength of the solution upon the pre-equilibrium concentration of an ionic species that is involved in a subsequent rate-limiting step of a reaction is called a secondary kinetic electrolyte effect. A common case encountered in practice is the effect on the concentration of a hydrogen ion (acting as catalyst) produced from the ionization of a weak acid in a buffer solution. 

The activity coefficient of a species is a measure of the effectiveness with which that species influences an equilibrium in which it is a participant. In very dilute solutions, in which the ionic strength is minimal, this effectiveness becomes constant, and the activity coefficient is unity. Under such circum.stances, the activity and the molar concentration are identical (as are thermodynamic and concentration equilibrium constants). As the ionic strength increases, however, an ion loses some of its effectiveness, and its activity coefficient decreases. We may summarize this behavior in terms of Equations 10-2 and 10-3. At moderate ionic strengths, yx< 1 tis the solution approaches infinite dilution, however, 7x —> 1 and thus ax —> [X] and X p —> K p. At high ionic strengths (/r > 0.1 M),... 

Table IV. Activity Coefficients of Monomeric Species for 10 2 Ionic Strength Solutions...

Activity Coefficients of Aqueous Species. The original version of EQ3/6 followed Helgeson et al. (1) in using the "B-dot" equation to describe the activity coefficients of aqueous solutes and a recommended approximation for the activity of water. The "B-dot" equation represents a simple extension of the Debye-Huckel equation and is only useful in relatively dilute solutions (deviations from precise measurements can be seen at ionic strengths below 0.1 molal, and become severe above 1.0 m). Beginning with version 3245, EQ3/6 offers two alternatives, the Davies (40) equation and Pitzer s equations (21,24,20,29). 

If we make the assumption that activity coefficients of all species are unity, that is, ionic strength effects are negligible, then activities,, are equal to molar concentrations, [ ], and knowing and we can solve equations (1) to... 

For precision, the concentration [C] should be replaced by the activity of each ionic species, where P is the permeability coefficient of ionic species through the membrane. The permeability coefficient is defined as P = RTbU /h, where b is the partition coefficient of the ion at the membrane and aqueous phases, LT is the mobility of a cation, and h is the thickness of the membrane. 

---