Activity coefficient relationships

Table 6.2 Activity coefficient relationships for electrolyte solutions (single electrolyte)... 

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. 

Activity coefficient relationships for the different types of strong electrolyte solutions are summarized in Table 11.3. 

Isothermal or isobaric activity coefficient relationships are modeled from experimental data by a variety of equations, all with a thermodynamic basis. The more useful of these equations are Van... 

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... 

To illustrate the Raoultian activity-activity coefficient relationship we use activity coefficients defined by Equations (8.31), shown in Figure 8.2. In real systems these are measured quantities with associated uncertainties, and the shape of the activity curve may not fit any simple function. Figure 8.2 shows... 

Activity Coefficient Relationships Using the Cihhs—Duhem Equation... 

Derive the equation of state, that is, the relationship between t and a, of the adsorbed film for the case of a surface active electrolyte. Assume that the activity coefficient for the electrolyte is unity, that the solution is dilute enough so that surface tension is a linear function of the concentration of the electrolyte, and that the electrolyte itself (and not some hydrolyzed form) is the surface-adsorbed species. Do this for the case of a strong 1 1 electrolyte and a strong 1 3 electrolyte. 

Provided that the ratio of activity coefficients is invariant over the range of acidity concerned, a linear relationship with unit slope between logic Aaobs. 2nd +logic % o) i expected. However, there is... 

The accurate determination of relative retention volumes and Kovats indices is of great utility to the analyst, for besides being tools of identification, they can also be related to thermodynamic properties of solutions (measurements of vapor pressure and heats of vaporization on nonpolar columns) and activity coefficients on polar columns by simple relationships (179). 

A solution which obeys Raoult s law over the full range of compositions is called an ideal solution (see Example 7.1). Equation (8.22) describes the relationship between activity and mole fraction for ideal solutions. In the case of nonideal solutions, the nonideality may be taken into account by introducing an activity coefficient as a factor of proportionality into Eq. (8.22). 

In thermodynamics the formal way of dealing with nonideality is to introduce an activity coefficient 7 into the relationship between activity and mole fraction ... 

The quantitative relationship between the degree of adsorption at a solution iaterface (7), G—L or L—L, and the lowering of the free-surface energy can be deduced by usiag an approximate form of the Gibbs adsorption isotherm (eq. 9), which is appHcable to dilute biaary solutions where the activity coefficient is unity and the radius of curvature of the surface is not too great ... 

Many additional consistency tests can be derived from phase equiUbrium constraints. From thermodynamics, the activity coefficient is known to be the fundamental basis of many properties and parameters of engineering interest. Therefore, data for such quantities as Henry s constant, octanol—water partition coefficient, aqueous solubiUty, and solubiUty of water in chemicals are related to solution activity coefficients and other properties through fundamental equiUbrium relationships (10,23,24). Accurate, consistent data should be expected to satisfy these and other thermodynamic requirements. Furthermore, equiUbrium models may permit a missing property value to be calculated from those values that are known (2). 

A.queous Solubility. SolubiHty of a chemical in water can be calculated rigorously from equiHbrium thermodynamic equations. Because activity coefficient data are often not available from the Hterature or direct experiments, models such as UNIFAC can be used for stmcture—activity estimations (24). Phase-equiHbrium relationships can then be appHed to predict miscibility. Simplified calculations are possible for low miscibiHty however, when there is a high degree of miscibility, the phase-equiHbrium relationships must be solved rigorously. 

Based on Hquid—Hquid equiHbrium principles, a general model of octanol—water partitioning is possible if accurate activity coefficients can be determined. First, phase equiHbrium relationships based on activity coefficients permit Hquid—Hquid equiHbrium calculations for the biaary octanol—water system. Because the two components are almost immiscible ia each other, two phases form an octanol-rich phase containing dissolved water, and a water-rich phase containing dissolved octanol. 

A sampling of appHcations of Kamlet-Taft LSERs include the following. (/) The Solvatochromic Parameters for Activity Coefficient Estimation (SPACE) method for infinite dilution activity coefficients where improved predictions over UNIEAC for a database of 1879 critically evaluated experimental data points has been claimed (263). (2) Observation of inverse linear relationship between log 1-octanol—water partition coefficient and Hquid... 

This relationship depends on the assumption that two similar stationary phases, irrespective of their polarity, can be considered to differ by measuring the ratio of the activity coefficients of two noncomplexing solutes (this basically implies the solute is nonpolar and will only interact with the stationary phase by dispersion forces). If this were true then. 

There is a third experimental design often used for studies in electrolyte solutions, particularly aqueous solutions. In this design the reaction rate is studied as a function of ionic strength, and a rate variation is called a salt effect. In Chapter 5 we derived this relationship between the observed rate constant k and the activity coefficients of reactants l YA, yB) and transition state (y ) ... 

Because it is impossible to vary single ion concentrations independently, the activity coefficient of an electrolyte is a function of activity coefficients of the cation and anion of the electrolyte. For example, for 1 1 electrolytes the relationship is... 

Without some additional relationship it is impossible to resolve y into and "y. By introducing an extrathermodynamic assumption as this additional relationship, it becomes possible to estimate single ion transfer activity coefficients. A widely used assumption is that the transfer activity coefficients of the cation and anion of tetraphenylarsonium tetraphenylboride, Ph4As BPh4, are equal, i.e.,... 

The quantity on the left side of the equation is referred to as the logarithm of the rate constant corrected for protonation often the correction term is negligible. If the activity coefficient term on the right side is negligible, Eq. (8-104) predicts a linear relationship between the corrected log kobs and —//o, the slope being unity. A similar treatment of the A-Se2 mechanism also predicts a linear plot of log fcobs against —Ho-... 

Wilson s [77] equation has been found to be quite accurate in predicting the vapor-liquid relationships and activity coefficients for miscible liquid systems. The results can be expanded to as many components in a multicomponent system as may be needed without any additional data other than for a binary system. This makes Wilson s and... 

For solutions that do not follow Raoult s law, it is convenient to define a quantity called the activity coefficient, 7k. ]. that is a measure of the deviation of the activity from the concentration, through the relationship... 

So far we have considered only symmetrical 1 1 electrolytes such as HC1, K.CI, or MgS04. For unsymmetrical electrolytes, the limiting law takes a different form, and different relationships between activity, molality and activity coefficient are obtained. For example, for the 2 1 electrolyte, Na SO,, the dissociation reaction is... 

A similar relationship relating the two activity coefficients can also be derived. We defined activity coefficients such that a =7i-V and a2 — f2x2 where the activities and activity coefficients are established for the standard state that corresponds to and p2, respectively. For both components, changes in the activity at constant temperature and pressure are given by... 

The osmotic coefficient 4> and activity coefficient are related in a simple manner through the Gibbs-Duhem equation. We can find the relationship by writing this equation in a form that relates a and 2-... 

Amongst the various relations given above, the last is the one which is actually used to calculate activity coefficients. For calculating the activity coefficient of the component, A, when the activity coefficient versus composition relationship for the other component, B, is known, the equation is used in the following form ... 

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