Activity coefficient, variation with ionic strength

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 ... 

Figure 19.10. Variation of solubility of AgCl with ionic strength, from which activity coefficients can be calculated. Data from Ref. 3.

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... 

Fig. B.2 The variation of the mean activity coefficient with ionic strength according to the extended Debye-Hiickel theory, (a) The limiting law for a 1,1-electrolyte, (b) The extended law with B = 0.5. (c) The extended law, extended further by the addition of a term CI-, in this case with C = 0.02. The last form of the law reproduces the observed behavior reasonably well.

Variation of mean ion activity coefficients with ionic strength (log 7+ versus Vm)-... 

It has been reported for both anionic [14,20,28] and non-ionic surfactants [2,15,23] that sorption increases with the number of carbon atoms in the hydrophobic chain (Table 5.4.5). The sorption coefficient of LAS in activated sludges [22] increases by 2.8 times with each methylene group, and a similar variation has been observed for river and marine sediments (Table 5.4.5). The partition coefficients obtained for the marine medium are slightly higher than those for river sediment, as a consequence of the higher ionic strength of seawater [14]. 

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... 

The values plotted in Figure 2 display the expected linear variation with molality of hydrochloric acid at constant ionic strength. The intercept measures the trace activity coefficient, yHci ", the limit of y in pure (acid-free) seawater. At 25°C, ynci " = 0.731 as compared with 0.728 in... 

Figure 2. Variation of the activity coefficient of hydrochloric acid with molality in seawater I at a constant ionic strength of 0.66...

The various forms of equation (40.15), referred to as the Debye-HUckel limiting law, express the variation of the mean ionic activity coefficient of a solute with the ionic strength of the medium. It is called the limiting law because the approximations and assumptions made in its derivation are strictly applicable only at infinite dilution. The Debye-Hfickel equation thus represents the behavior to which a solution of an electrolyte should approach as its concentration is diminished. 

At any given ionic strength, the activity coefficients of ions of the same charge are approximately equal. The small variations that do exist can be correlated with the effective diameter of the hydrated ions. 

Eq. (B.l) will allow fairly accurate estimates of the aetivity coefficients in mixtures of electrolytes if the ion interaction coefficients are known. Ion interaction coefficients for simple ions can be obtained from tabulated data of mean activity coefficients of strong electrolytes or from the corresponding osmotic coefficients. Ion interaction coefficients for complexes can either be estimated from the charge and size of the ion or determined experimentally from the variation of the equilibrium constant with the ionic strength. 

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