Activity coefficients variation with concentration

The activity coefficient varies with concentration. This variation is rather complex the activity coefficient of a particular ion being dependent upon the concentration of all ionic species present in the solution. As a measure of the latter, Lewis and Randall (1921) introduced the quantity called ionic strength, /, and defined it as the half sum of the products of the concentration of each ion multiplied by the square of its charge. With mathematical symbols this can be expressed as... 

Empirically, the Setchenov equation [37,39] has been found to express the variation of the neutral solute activity coefficient (7 ) with the electrolyte concentration (Ce), at least for low electrolyte concentrations (a few tens molar) ... 

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 ... 

In the HMW model, in contrast, Ca++ and SO4 are the only calcium or sulfate-bearing species considered. The species maintain equal concentration, as required by electroneutrality, and mirror the solubility curve in Figure 8.6. Unlike the B-dot model, the species activities follow trends dissimilar to their concentrations. The Ca++ activity rises sharply while that of SO4 decreases. In this case, variation in gypsum solubility arises not from the formation of ion pairs, but from changes in the activity coefficients for Ca++ and SO4 as well as in the water activity. The latter value, according to the model, decreases with NaCl concentration from one to about 0.7. 

The solubility product is equal to Ca x AI(OH)4 x [OH , where curly brackets denote species activities and [OH ] may be replaced by 2[Ca ] - [A1(0H)4"]. As the concentrations are low, activity coefficients may be calculated from simplified Debye-Hiickel theory. Solubility products may thus be obtained from experimental data (NI8,B118,BI 19.C48) (Table 10.2). The variations in solubility products with temperature may be represented by empirical equations of the form... 

In classical structure-activity studies, most of the attempts concentrated on correlating the activity with one of the molecular properties— e.g., the carcinogenic activity of polynuclear aromatic compounds with their electronic structure (18, 19), the narcotic activity with lipophilicity 20, 21), the insecticidal activity of cyclodienes with their three-dimensional molecular silhouette 22), etc. Sometimes the activity correlated well with only one of the molecuar parameters. In our approach these are special cases where other physicochemical properties do not play critical roles in determining the variation in the activity within a set of congeners so that the coefficients defining these other properties are zero. 

Rigorously speaking, the diffusion coefficient is not a constant (Table 4.3). If, however, the variation of the activity coefficient is not significant over the concentration difference that produces diffusion, then (c,/ )(3 /9c,) 1 and for all practical purposes D is a constant. This effective constancy ofD with concentration will be assumed in most of the discussions presented here. 

The general case is too difficult to solve analytically, but several special cases can be solved. For example (Fig. 4.86), the activity coefficients can be taken as unity, fi = 1—ideal conditions the transport numbers r,- can be assumed to be constant and a linear variation of concentrations with distance can be assumed. The last assumption implies that the concentration c,(a ) of the tth species atx is related to its concentration C((0) atx = 0 in the following way... 

The Ionic Strength.—In order to represent the variation of activity coefficient with concentration, especially in the presence of added electrolytes, Lewis and Randall introduced the quantity called the ionic strength, which is a measure of the intensity of the electrical field due to the ions in a solution. It is given the symbol i and is defined as half the sum of the terms obtained by multiplying the molality, or concentration, of each ion present in the solution by the square of its valence that is... 

Activities in Concentrated Solutions.—For relatively concentrated solutions it is necessary to use the complete Hiickel equation (62) by choosing suitable values for the two adjustable parameters a and it has been found possible to represent the variation of activity coefficients with concentration of several electrolytes from 0.001 to 1 molal, and sometimes up to 3 molal. The values of C seem to lie approximately between 0.05 and 0.15 in aqueous solution. At the higher concentrations it is necessary to make allowance for the difference between the rational and stoichiometric activity coefficients the latter, which is the experimentally determined quantity, is represented by an extension of equation (62) thus (cf. p. 135),... 

The polymerization of lactams initiated with carboxylic acids has been found to be first order with respect to the lactam [10, 194, 200]. However, the order of reaction with respect to the initiating acid, eqn. (100), varied from 0.5 for caprolactam [194, 196] to 0.8 for capryllactam [200]. The deviation of the apparent order of reaction from unity can be due to variation in the activity coefficients of the reacting species which are affected by the dielectric constant of the medium [197]. The different polarity and basicity of caprolactam and capryllactam is one of the reasons for the different apparent order of reaction with respect to the initiator concentration. 

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