Activity coefficients practical

We will see in chapter 8 that, in the case of aqueous solutions, it is convenient to adopt the condition of hypothetical 1-molal solution at P = 1 bar and T = 298.15 K as the standard state. Most experimental data on aqueous solutions conform to this reference condition. In this case, the resulting activity coefficient is defined as the practical activity coefficient and must not be confused with the rational activity coefficient of general relation 2.80. 

The activity coefficient fz is sometimes called the rational activity coefficient, since it gives the most direct indication of the deviation from the ideal behavior required by Raoult s law. It is, however, not often used in connection with measurements on solutions of electrolytes, and so the coefficients/c and/, which are commonly employed, are described as the practical activity coefficients. The coefficient / , from which the... 

Attention should be drawn to the fact that the activity coefficients given by the Debye-Hiickel treatment are the so-called rational coefficients (p. 135) to express the values in the form of the practical activity coefficients, it is necessary to make use of equation (26). If the solvent... 

Xi, Ci and mi are the mole fraction, molar concentration and molal concentration respectively, yx is known as the rational activity coefficient while y SiXt practical activity coefficients. (Often, the symbols / , ,are used for molar and molal activity coefficients, but to avoid confusion y will be used throughout with an appropriate suffix to indicate the concentration units.)... 

In practice activity coefficients initially decrease with increasing concentration of electrolyte (Fig. 2.3). Such behaviour is entirely consistent with both the Debye-Hiickel equation and its limiting form. However, in practice, activity coefficients show a turning point at some value of p, after which they progressively increase. It is thus seen to be necessary to modify Equation (2.33) by the addition of a further term which is an increasing function of p, i.e.. 

When the pressure is low and mixture conditions are far from critical, activity coefficients are essentially independent of pressure. For such conditions it is common practice to set P = P in Equations (18) and (19). Coupled with the assumption that v = v, substitution gives the familiar equation... 

Figure 4-2 displays plots of AH, AS, and AG as functions of composition for 6 binary solutions at 50°C. The corresponding excess properties are shown in Fig. 4-3 the activity coefficients, derived from Eq. (4-119), appear in Fig. 4-4. The properties shown here are insensitive to pressnre, and for practical pnrposes represent sohition properties at 50°C (122°F) and low pressnre (P 1 bar [14.5 psi]).

When Eq. (4-282) is applied to XT E for which the vapor phase is an ideal gas and the liquid phase is an ideal solution, it reduces to a veiy simple expression. For ideal gases, fugacity coefficients and are unity, and the right-hand side of Eq. (4-283) reduces to the Poynting factor. For the systems of interest here this factor is always veiy close to unity, and for practical purposes <1 = 1. For ideal solutions, the activity coefficients are also unity. Equation (4-282) therefore reduces to... 

Standard potentials Ee are evaluated with full regard to activity effects and with all ions present in simple form they are really limiting or ideal values and are rarely observed in a potentiometric measurement. In practice, the solutions may be quite concentrated and frequently contain other electrolytes under these conditions the activities of the pertinent species are much smaller than the concentrations, and consequently the use of the latter may lead to unreliable conclusions. Also, the actual active species present (see example below) may differ from those to which the ideal standard potentials apply. For these reasons formal potentials have been proposed to supplement standard potentials. The formal potential is the potential observed experimentally in a solution containing one mole each of the oxidised and reduced substances together with other specified substances at specified concentrations. It is found that formal potentials vary appreciably, for example, with the nature and concentration of the acid that is present. The formal potential incorporates in one value the effects resulting from variation of activity coefficients with ionic strength, acid-base dissociation, complexation, liquid-junction potentials, etc., and thus has a real practical value. Formal potentials do not have the theoretical significance of standard potentials, but they are observed values in actual potentiometric measurements. In dilute solutions they usually obey the Nernst equation fairly closely in the form ... 

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... 

The formal Galvani potential, described by Eq. (22), practically does not depend on the concentration of ions of the electrolyte MX. Since the term containing the activity coefficients of ions in both solutions is, as experimentally shown, equal to zero it may be neglected. This results predominantly from the cross-symmetry of this term and is even more evident when the ion activity coefficients are replaced by their mean values. A decrease of the difference in the activity coefficients in both phase is, in addition, favored by partial hydration of the ions in the organic phase [31 33]. Thus, a liquid interface is practically characterized by the standard Galvani potential, usually known as the distribution potential. 

In practical electrochemistry, however, the molality m or molar concentration c is used more often than the mole fraction. Thus, the molal activity amy molal activity coefficient ym, molar activity ac and molar activity coefficient yc are introduced. The adjective molal is sometimes replaced by practical . 

This notional definition of the pH scale can, however, not be used for practical measurements, as it contains the activity coefficients of the individual ions, y(H30+). 

It has been emphasized repeatedly that the individual activity coefficients cannot be measured experimentally. However, these values are required for a number of purposes, e.g. for calibration of ion-selective electrodes. Thus, a conventional scale of ionic activities must be defined on the basis of suitably selected standards. In addition, this definition must be consistent with the definition of the conventional activity scale for the oxonium ion, i.e. the definition of the practical pH scale. Similarly, the individual scales for the various ions must be mutually consistent, i.e. they must satisfy the relationship between the experimentally measurable mean activity of the electrolyte and the defined activities of the cation and anion in view of Eq. (1.1.11). Thus, by using galvanic cells without transport, e.g. a sodium-ion-selective glass electrode and a Cl -selective electrode in a NaCl solution, a series of (NaCl) is obtained from which the individual ion activity aNa+ is determined on the basis of the Bates-Guggenheim convention for acr (page 37). Table 6.1 lists three such standard solutions, where pNa = -logflNa+, etc. 

Equation 7.1.2 characterizes what is known as the primary salt effect (i.e. the influence of ionic strength on the reaction rate through the activity coefficients of the reactants and the activated complex). Much early work on ionic reactions is relatively useless because this effect was not understood. Now it is common practice in studies of ionic reactions to add a considerable... 

If the substrate is predominantly unprotonated in the acidity range covered, the protonation correction term (second on the left in equation (35)) will be zero, and if the activity coefficient term on the right cancels to zero, log values will be linear in —Ho with unit slope (the Zucker-Hammett hypothesis).146 In practice, linearity is usually observed,23 e.g. for the trioxane depolymerization in equation (31),144 although the slopes are seldom exactly unity. For more strongly basic substrates that are predominantly protonated in the acidity range covered, equation (36) is easily derived this is acidity independent and should have zero slope against — H0 if the substrate is fully protonated and the last term cancels.145... 

The other variables in Equations 3.32-3.34 are either known values, such as the equilibrium constants K and reaction coefficients v, or, in the case of the activity coefficients y, yj and activities aw, a, values that can be considered to be known. In practice, the model updates the activity coefficients and activities during the numerical solution so that their values have been accurately determined by the time the iterative procedure is complete. 

The following sections describe the two estimation techniques. The discussion here leans toward the practical aspects of estimating activity coefficients. For an... 

The previous derivation was made under the implicit assumption that the activity coefficients of A and B are both equal to unity. This assumption matches the definition of E° as a standard potential. There are two cases of practical interest, where these conditions are not fulfilled. One is when the activity coefficients differ from unity but do not depend on the relative amounts of A and B in the film. This type of situation may arise when the interactions between the reactants are weak but the presence of the supporting electrolyte decreases the activity coefficients of A and/or B, yA and yB, to below 1 while they remain constant over the entire voltammo-gram. The only change required is thus to replace the standard potential by the formal potential ... 

This change has two attractive features (1) It eliminates the separation of activity coefficients into short- and long-range interactions, which cannot be evaluated separately in practice, and (2) implicitly incorporates an expected effect of surface potential on solution activity through the activity coefficient relationship of Equation 22. Table II summarizes the relevant reaction and activity coefficient terms based on the above modifications of the TLM. 

The formal potential is the quantity determined from the analysis of a volta-mmogram, but the true thermodynamic quantity (the standard potential) can be derived by obtaining 0/(R/R ) for different bulk concentrations (c) and extrapolating to c = 0 (unit activity coefficients). The procedure is, however, seldom adopted in practice, (R/R-) is identified with the standard potential. The lower the concentration of the electroactive species, the better the assumption. 

For concentrated solutions, the activity coefficient of an electrolyte is conveniently defined as though it were a nonelectrolyte. This is a practical definition for the description of phase equilibria involving electrolytes. This new activity coefficient f. can be related to the mean ionic activity coefficient by equating expressions for the liquid-phase fugacity written in terms of each of the activity coefficients. For any 1-1 electrolyte, the relation is ... 

Whilst the fundamental driving force for crystallisation, the true thermodynamic supersaturation, is the difference in chemical potential, in practice supersaturation is generally expressed in terms of solution concentrations as given in equations 15.1-15.3. Mullin and Sohnel(19) has presented a method of determining the relationship between concentration-based and activity-based supersaturation by using concentration-dependent activity-coefficients. 

Stability (or binding) constants Ks are often used instead of dissociation constants (Ks = 1/Kd). These equilibrium constants are concentration quotients as the corresponding activity coefficients are given the value 1. However, in many practical situations, other... 

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