Infinite dilution activity scale

The infinite dilution activity scale is useful for ionic equilibria in fresh waters, but for equilibria in sea water one gains precision by applying an ionic medium activity scale. Measuring pH in sea water gives less information than total alkalinity and total carbonate. Calculations on redox equilibria are simplified by introducing the master variable pE -----log e. ... 

In treating ionic equilibria in aqueous solution, two activity scales have proved especially useful. The first is the traditional infinite dilution activity scale, which is defined in such a way that the activity coefficient yA = A /[A] approaches unity as the solution approaches pure water. One might refer to this scale as the fresh water scale. 

Activity coefficients defined within the infinite dilution activity scale cannot be formulated theoretically for the ionic medium of seawater. Since the oceans contain an ionic medium of practically constant composition, the ionic medium activity scale might be used advantageously in studying acid-base and other equilibria in seawater (see also Appendix 6.2 in Chapter 6). 

For the reference state on the infinitely dilute solution scale, the activity coefficient of a species approaches unity as the concentrations of all the solutes approach zero ... 

The activity definition (17) on the infinite dilution scale, paH = —log- H+ > = —log yH —log[H+] was experimentally studied by Bjerrum and Unmack (4). This definition is not rigid because of the intrinsic uncertainty in defining individual ionic activities. Depending on the precision required, it is usually stated that the approximations used to estimate yH breakdown at some ionic strength between 0.01 and 0.2M. 

At finite concentrations the effect of the solvent on the ion-ion interactions are superimposed on the solvent effect discussed above for infinite dilution. The former effect can be expressed as the mean ionic activity coefficient, y again, expressed conventionally on the molal scale, relative to infinite dilution in the solvent in question, which in dilute solutions, where the extended Debye-Huckel expression is deemed to hold, is ... 

This equation relates the activity on the Henrian scale to the activity on the Raoultian scale. The Henrian standard state is sometimes called the infinitely dilute solution standard state because it is mostly used for dilute solutions. 

The inclusion of activity coefficients into the simple equations was briefly considered by Purlee (1959) but his discussion fails to draw attention to the distinction between the transfer effect and the activity coefficient (y) which expresses the non-ideal concentration-dependence of the activity of solute species (defined relative to a standard state having the properties of the infinitely dilute solution in a given solvent). This solvent isotope effect on activity coefficients y is a much less important problem than the transfer effect, at least for fairly dilute solutions. For example, we have already mentioned (Section IA) that the nearequality of the dielectric constants of H20 and D20 ensures that mean activity coefficients y of electrolytes are almost the same in the two solvents over the concentration range in which the Debye-Hiickel limiting law applies. For 0-05 m solutions of HC1 the difference is within 0-1% and thus entirely negligible in the present context. Of course, more sizeable differences appear if concentrations are based on the molality scale (Gary et ah, 1964a) (see Section IA). 

Since the Margules expansions represent a convergent power series in the mole fractions,8 they can be summed selectively to yield closed-form model equations for the adsorbate species activity coefficients. A variety of two-parameter models can be constructed in this way by imposing a constraint on the empirical coefficients in addition to the Gibbs-Duhem equation. For example, a simple interpolation equation that connects the two limiting values of f (f°° at infinite dilution and f = 1.0 in the Reference State) can be derived after imposing the scaling constraint... 

For solutes the standard state and the activity usually must be defined in terms of behavior under conditions of infinite dilution, where by definition the activity of a solute is set equal to its concentration. Thus at infinite dilution the ratio of activity to concentration (in whatever units) is unity, and y, = 1. When the value of some physical property of a solution is plotted as a function of concentration, a curve like those in Figure 2-2 is obtained. If the asymptote passing through the origin on the concentration scale is extrapolated to higher concentrations, we obtain the standard state of unit activity for the property in question. This hypothetical solution, labeled S, of unit concentration exhibits the same type of behavior as the infinitely dilute solution. The extent to which the real value of the physical property measured differs from the hypothetical value at a specific concentration is expressed by the activity coefficient, a coefficient that is simply the ratio between two measurable quantities. In Figure 2-2 the activity coefficient yj is the ratio BC/AC and is defined by... 

Many nonionizable organic solutes in water are described thermodynamically on the mole fraction scale, although their solubilities may commonly be reported in practical units, for example, molality. [Refer to Schwarzenbach et al. (1993) and Klotz (1964) for detailed discussion of such aqueous solutions.] Here, the standard state is the pure liquid state of the organic solute, that is, Xj = 1. The reference state is Xi - 1, that is, a solution in which the organic solute molecules interact with one another entirely. Activity coefficients of solute molecules in dilute aqueous solutions are generally much greater than unity for this reference state choice, jc, 1. For example, with this reference state, aqueous benzene has an experimental infinitely dilute solution activity coefficient, T nzeno of 2400 for an infinite dilution reference state, jc, - 0, the activity coefficient would be approximately 1 (Tanford, 1991). 

In summary, thermodynamic models of natural water systems require manipulation of chemical potential expressions in which three concentration scales may be involved mole fractions, partial pressures, and molalities. For aqueous solution species, we will use the moial scale for most solutes, with an infinite dilution reference state and a unit molality standard state (of unit activity), l or the case of nonpolar organic solutes, the pure liquid reference and standard states are used. Gaseous species will be described on the partial pressure (atm — bar) scale. Solids will be described using the mole fraction scale. Pure solids (and pure liquids) have jc, = 1, and hence p, = pf. 

Figure 3.2. Activity coefficients depend on the selection of the reference state and standard state, (a) On the infinite dilution scale the reference state is an infinitely dilute aqueous solution the standard state is a hypothetical solution of concentration unity and with properties of an infinitely dilute solution. For example, the activity coeffic ent of in a HCl solution, / hcm varies with [HCl] in accordance with a Debye-Hiickel equation (dashed line, left ordinate) (see Table 3.3) only at very great dilutions does /become unity. On the ionic medium scale, for example, in I M KCl, the reference...

State is the ionic medium (i.e., infinitely diluted with respect to HCl only). In such a medium /hci (solid line, right ordinate) is very nearly constant, that is,/Hci = 1- Both activity coefficients are thermodynamically equally meaningful. (Adapted from P. Schindler.) (b) A comparison of activity coefficients (infinite dilution scale) of electrolytes and nonelectrolytes as a function of concentration (mole fraction of solute) m = moles of solute per kg of solvent (molality) = number of moles of ions formed from 1 mol of electrolyte 1 kg solvent contains 55.5 mol of water. (From Robinson and Stokes, 1959. Reproduced with permission from Butterworths, Inc., London.)... 

Constants based on activities (rather than concentrations), the activity scale being based on the infinite dilution reference state. 

In dealing with redox equilibria, we are also confronted with the problem of evaluating activity corrections or maintaining the activities under consideration as constants. The Nernst equation rigorously applies only if the activities and actual species taking part in the reaction are inserted in the equation. The activity scales discussed before, the infinite dilution scale and the ionic medium scale, may be used. The standard potential or standard pe on the infinite dilution scale is related to the equilibrium constant for / = 0 of the reduction reaction... 

Since the activities of ions and other solutes approach their concentrations as the solutions are made more and more dilute, activity coefficients must approach unity as a limit, However, to adjust the scale of activity coefficients so that they approach unity at infinite dilution, it is necessary to make non-thermodynamic assumptions as to the trend below the concentrations at which they can be measured. A basis for such assumptions is given by the Debye-Hiickel theory of interionic attractions which will be discussed in the next chapter. 

As explained in Chapter 8 the scale of activities is so chosen that gu approaches CU as the solution of A is made more dilute. In other words Ca = when f = 1. The standard state of the solute A is therefore defined as the hypothetical state in which it has a concentration of unity but with the properties associated with infinite dilution. Under these conditions... 

A choice must be made for the reference state for the solute either the pure liquid (possibly supercooled), or the solute at infinite dilution in the solvent. The latter differs from the conventional solute standard state only in the use of the mole fraction scale rather than molality units. The activity coefficient of a symmetrical salt MX is either... 

As before, the factors and are activity coefficients on the molar concentration scale referred to unity at infinite dilution in water. The indicator is commonly used in very dilute solution. If the concentrations of other solutes are also low, we may approximate these activity coefficients by medium effects... 

In Eqs. (16), jnf° is the chemical reference potential at infinite dilution, y, the corresponding activity coefficient in the molarity scale, and r/r the electric potential acting on the ions. The thermodynamic forces grad(ln y, ) will be neglected in the following discussion for the sake of simplicity. The electric potential r/r is related to the electric field E by the relationship... 

The standard state might be chosen in various ways (e.g., as the state at infinitely diluted solution). The resulting standard acidity scale is characterized by the activity of the proton solvated by the given solvent, HS, according to... 

---