Activity coefficient in dilute solution

The Bronsted-Guggenheim equations provide a highly satisfactory description of the activity coefficients in dilute solutions however, their empirical extension to concentrated solutions (Harned s rule) introduces several serious problems. 

The Debye-Hiickel term, which is the dominant term in the expression for the activity coefficients in dilute solution, accounts for electrostatic, nonspecific long-range interactions. At higher concentrations, short-range, nonelectrostatic interactions have to be taken into account. This is usually done by adding ionic strength dependent terms to the Debye-Hiickel expression. This method was first outlined by Bronsted [5,6], and elaborated... 

In order to overcome the problem with activity coefficients in dilute solutions, Fprland (1964) suggested the addition of four additive binary terms... 

The activity coefficient in dilute solution, i, is expressed as i = 1 — A V Na where A is the Debye-Hiickel constant (1.18 at 25° C). Similar equations are given for the surface excess concentration of Na and Cl" ions. Equation 1.11 reduces to Equation 1.10 when Tq- = 0 and i = 1. Application of this theoretical treatment to NaDS solutions containing varying amounts of added NaCl showed that the Tq- value was indeed zero (or slightly negative) (see Fig. 1.5), thus confirming the validity of the assumptions of the Matijevic and Pethica equation (Equation 1.10). Additional proof of the validity of this equation came from the excellent agreement between predicted values of F s- and experimental values determined by a radiotracer technique (see Fig. 1.5). 

In 1922, Johannes Nicolaus Br0nsted estabhshed an empirical relation for the activity coefficients in dilute electrolyte solutions ... 

Other than specific effects that result from conventional chemical interactions (such as acid-base or complex formation), the main factors to be considered are hydration of ions, electrostatic effects, and change in dielectric constant of the solvent. For example, hydration of ions of added salt effectively removes some of the free solvent, so that less is available for solution of the nonelectrolyte. The Setschenow equation probably best represents the activity coefficient of dilute solutions (less than 0.1 M) of nonelectrolytes in aqueous solutions of salts up to relatively high concentrations (about 5 M) ... 

Nonelectrolytes in nonaqueous solvents Activity coefficients of dilute solutions of solutes can be studied experimentally by liquid-liquid chromatography as well as techniques such as solvent extraction, light scattering, vapor pressure, and freezing point depression. 

Table 1.4 Raoult Law and Henry Law Activity Coefficients for Dilute Solutions of Methanol in...

The Debye-Hiickel theory is also used to estimate activity coefficients for dilute solutions on the molality scale. In this case, equation (3.8.32) becomes... 

Lee, B.-C. and Danner, R.P., Prediction of infinite dilution activity coefficients in polymer solutions comparison of prediction models. Fluid Phase Equilibria, 128, 97, 1997. 

Eqn. 3.5, taking fl/=Y/ i(), the reduction in a species chemical potential is reflected by a decreased value (relative to one in an ideal solution) for its activity coefficient. By Coulomb s law, electrostatic forces vary inversely with the square of the distance of ion separation. For this reason, activity coefficients in dilute fluids decrease as concentration increases because the coulombic forces become stronger as ions pack together more closely. 

The model of Debye and Hiickel enabled the construction of a relatively simple equation for the determination of activities coefficients in diluted electrolyte solutions. In first approximation their equation, which is called Debye-Huckel equation, looks as follows ... 

Since the value of an activity coefficient depends on the standard state, an activity coefficient based on (10.2.21) will differ numerically from one that is based on a pure-component standard state. To emphasize that difference, we make a notational distinction between the two we use y for an activity coefficient in a pure-component standard state and use y for an activity coefficient in the solute-free infinite-dilution standard state. Then for y, the generic definition of the activity coefficient (5.4.5) gives... 

In order to solve these equations, the parameter Y = [A J tK ] ) was introduced and the activity coefficients for dilute solutions were taken to be 1. Hence... 

The theory of Peter Debye and Erich Hiickel (1923) provides theoretical expressions for single-ion activity coefficients and mean ionic activity coefficients in electrolyte solutions. The expressions in one form or another are very useful for extrapolation of quantities that include mean ionic activity coefficients to low solute molality or infinite dilution. 

Most models use dilute solution assumptions and former models [93] assumed unit values [2,19,93,94] of all activity coefficients (except in some cases for H+ ions). Bernhardsson et al. [4] reached a compromise by using two sets of equilibrium constants, one for dilute solutions and the other for concentrated solutions. Other authors used Debye-Huckel, the truncated Davies model [96,98], or the B-dot Debye-Huckel model [98] to derive the activity coefficients in concentrated solutions. 

Experimentally deterrnined equiUbrium constants are usually calculated from concentrations rather than from the activities of the species involved. Thermodynamic constants, based on ion activities, require activity coefficients. Because of the inadequacy of present theory for either calculating or determining activity coefficients for the compHcated ionic stmctures involved, the relatively few known thermodynamic constants have usually been obtained by extrapolation of results to infinite dilution. The constants based on concentration have usually been deterrnined in dilute solution in the presence of excess inert ions to maintain constant ionic strength. Thus concentration constants are accurate only under conditions reasonably close to those used for their deterrnination. Beyond these conditions, concentration constants may be useful in estimating probable effects and relative behaviors, and chelation process designers need to make allowances for these differences in conditions. 

It follows that die separation of cadmium must be carried out in a distillation column, where zinc can be condensed at the lower temperamre of each stage, and cadmium is preferentially evaporated. Because of the fact that cadmium-zinc alloys show a positive departure from Raoult s law, the activity coefficient of cadmium increases in dilute solution as the temperature decreases in the upper levels of the still. The separation is thus more complete as the temperature decreases. 

Examples of this procedure for dilute solutions of copper, silicon and aluminium shows the widely different behaviour of these elements. The vapour pressures of the pure metals are 1.14 x 10, 8.63 x 10 and 1.51 x 10 amios at 1873 K, and the activity coefficients in solution in liquid iron are 8.0, 7 X 10 and 3 X 10 respectively. There are therefore two elements of relatively high and similar vapour pressures, Cu and Al, and two elements of approximately equal activity coefficients but widely differing vapour pressures. Si and Al. The right-hand side of the depletion equation has the values 1.89, 1.88 X 10- , and 1.44 X 10 respectively, and we may conclude that there will be depletion of copper only, widr insignificant evaporation of silicon and aluminium. The data for the boundaty layer were taken as 5 x lO cm s for the diffusion coefficient, and 10 cm for the boundary layer thickness in liquid iron. 

The Change of Solubility with Temperature. The solubilities of various salts have been measured in aqueous solution at various temperatures. But from these measurements we cannot derive values of L as a function of temperature, until the activity coefficients in the various saturated solutions have been accurately measured. In dilute solutions... 

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 ... 

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