Activity coefficient mean definition

This is an equation for calculating the activity coefficient of an individual ion m (i.e., a parameter inaccessible to experimental determination). Let us, therefore, change to the values of mean ionic activity. By definition [see Eq. (3.27)],... 

In view of the definition of the mean activity coefficient and of the electroneutrality condition, v+z+ = -v z, the limiting law also has the form... 

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. 

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 ... 

With this definition, the relationship for the mean activity coefficient... 

For solutions of ions, departures from ideality can be large even in quite dilute solutions because of the strong electrostatic attractions or repulsions between the ions. Furthermore, the simple definition of activity coefficient given in Eq. 2.3 fails for electrolytes because we can never measure the activity of, say, a cation Mm+ without anions Xx being present at the same time instead, we usually define a mean ionic activity a and coefficient /y as... 

This definition of x and y is more realistic at low and moderate salt concentrations and is in agreement with that of Sada and Morisue (17). Broul and Hala also assumed complete salt dissociation. The assumption of full dissociation of the salt may not be entirely valid at high salt concentrations, especially where the concentration of the nonaqueous solvent is also high. However, even in those instances where the assumption of full dissociation of the salt may be invalid, it appears to describe the system better than ignoring salt ionization completely. The terms x/ and y/ are referred to hereafter as ionic mole fraction and ionic activity coefficient, respectively. These should not be confused with the mean ionic terms used by Hala which are also based on complete salt dissociation, but are defined differently. No convergence problems were encountered when the ionic quantities were employed. 

The reference state of the electrolyte can now be defined in terms of thii equation. We use the infinitely dilute solution of the component in the solvent and let the mean activity coefficient go to unity as the molality or mean molality goes to zero. This definition fixes the standard state of the solute on the basis of Equation (8.184). We find later in this section that it is neither profitable nor convenient to express the chemical potential of the component in terms of its molality and activity. Moreover, we are not able to separate the individual quantities, and /i . Consequently, we arbitrarily define the standard chemical potential of the component by... 

We cannot determine values of the activity coefficients of the individual ions, but by definition of the mean activity coefficients (Eq. (11.182)), we have... 

Other definitions of chemical diffusion coefficients were also suggested for various particular cases (e.g., see [iii, vi-viii]). In all cases, however, their physical meaning is related either to the ambipolar diffusion or to diffusion in non-ideal systems where the activity coefficients differ from unity. 

From the context of the current discussion, it should be evident that the mean activity coefficient cited above is related to molarity. On the other hand, as is to be proved in Exercise 4.2.2, the definition of S remains virtually unaltered by switching from molarity to molality in aqueous solutions at ordinary conditions of temperature and pressure. Thus, the quantity specified by Eq. (4.2.3a) may be considered to represent either 7 (T,P,c) 7 (c) or 7 (T,P,m) 7 nonaqueous solvents are employed, or whenever T and P deviate greatly from standard conditions, the two preceding quantities cannot be used interchangeably Eq. (4.2.3a) specifies 7 [Pg.392]

Activity Coefficients from Solubility Measurements.—The activity coefficient of a sparingly soluble salt can be determined in the presence of other electrolytes by making use of the solubility product principle. In addition to the equations already given, this principle may be stated in still another form by introducing the definition of the mean ionic concentration, i.e., c , which is equal to c+clr, into equation (109) this equation then becomes... 

Unfortunately, the calculated values of yt cannot be confirmed by direct experiment, because in principle all experimental methods yield the mean activity coefficient y rather than the individual ionic values. By use of the definition given in (2-16), the experimentally determined value can be apportioned to give nd y. This procedure is theoretically justified only at high dilution, where the DHLL is valid because the limiting slope of log y plotted against /n is found experimentally to be O.SZ Zb, as required by (2-17). At higher values of n the ion-size parameter a must be introduced. 

Mean activity coefficients have been evaluated for hydrochloric acid by potential measurements in alcohols. The salt-effect activity coefficient (left) and its product with the transfer activity coefficient (right) are shown in Figure 4-1. The values of are lower than would be calculated from the appropriate modification of the Debye-Hiickel equation (2-21) applied in the usual way to account for interionic interactions. The low values result from significant ion pairing due to the low dielectric constant. Thus, 0.1 M hydrochloric acid in 95% ethanol is about half in the form of ion pairs rather than being completely dissociated. As shown in Figure (4-1), at low concentrations the salt-effect activity coefficients approach unity, as they must by definition, whereas at moderate concentrations they are somewhat less than unity. On... 

From the context of the current discussion, it should be evident that the mean activity coefficient cited above is related to molarity. On the other hand, as is to be proved in Exercise 4.2.2, the definition of S remains virtually unaltered... 

The product 7 + 7- is experimentally measurable. The quantity (7 referred to as the mean molal activity coefficient The mean ionic molality is defined as im+mJ) and is simply m for a univalent-univalent electrolyte. Summarizing these definitions for a nonideal, univalent-univalent solution, where the solute is component 2. 

A corresponding definition is used for the mean mole fraction. The mean ionic activity is the product of the mean concentration and the mean activity coefficient. 

We cannot measure the individual ion activity coefficients here, only their total effect on /f,p. It is convenient to lump this total effect in the geometric mean of the product of the individual activity coefficients and to call this the mean ion activity coefficient of the salt. Thus for a K2SO4 solution, by definition... 

In general, for an electrolyte that dissociates into a total of n ions, the mean ion activity coefficient equals, by definition,... 

It should be noted that the above definitions of pH are based on an assumption that the solution is behaving in an ideal nature meaning that the thermodynamic activity is equal to concentration (e.g., what happens when the dilution is infinite). However, as the concentration increases, ionic attraction and incomplete hydration results in a decrease in the effective concentration (or the activity). This activity is defined as the "apparent concentration" of an ionic species, which is due to the attraction that ions exert on one another, and the incomplete hydration of ions in solutions that are too concentrated. The lower the concentration, the less is the interaction. At infinite dilution, activity coefficients approach unity. 

This table gives mean activity coefficients at 25°C for molalities ences, and data over a wider concentration range, in the range 0.1 to 1.0. See the following table for definitions, refer-... 

The other source of uncertainty in the definition of pH arises from assumptions that must be made in the use of various modifications of the Debye-Hiickel equation [44,45,59] which is used to estimate the mean ionic activity coefficient [Eqs. (13-17) Section 2.2.5]. The geometric mean ionic activity coefficient (/ = (/+/ )) is used, as it is relatively easy to estimate with reasonable precision, whereas current methods for estimation of individual ionic activity coefficients still require significant computational effort, even for small inorganic ions [62]. The mean ionic activity coefficient [see Eqs. (13-17) in Section 2.2.5] is required to estimate hydronium ion concentrations from measured pH values. The hydronium ion concentrations are required, in turn, for calculations of the pKg values. 

From this expression comes the definition of the mean activity coefficient. Y+. in terms of the ionic activity coefficients Yc and Ya The mean activity coefficient is the property which is determined or calculated from experimental measurements. A similar expression results for the mean molality m, which is not generally used in reporting experimental measurements ... 

Equations (5.6) and (5.7) ignore any possible cation-cation or anion-anion Interaction and any higher order interactions. The activity coefficient of an electrolyte in a multicomponent solution is, by combining equations (5.5), (5.6) and (5.7) and remembering the definition of a mean activity coefficient, equation (2.26) ... 

Recalling the definitions of the mean activity coefficient and mean molality from Chapter II ... 

In Chapters IV and V considerable effort was spent describing strong electrolytes and alternative formulations for their corresponding mean and/or ionic activity coefficients. A strict definition of a strong electrolyte is a species which completely dissociates in water. In reality, very few species fit this definition of strong electrolytes. The following definitions are offered in order to provide a practical classification of electrolytes ... 

Equation 12.9.2 is an example of a mixed equilibrium constant—one using more than one kind of standard state. From the definition of the mean ionic activity coefficient (Eq. 10.3.7), we can replace the product y+Y- by y . where y is the mean ionic activity coefficient of aqueous Ca(HC03)2 ... 

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