Pursuit of the Elusive Single-Ion Activity - ACS Symposium Series

Chapter 10

Pursuit of the Elusive Single-Ion Activity Downloaded by UNIV OF MISSOURI COLUMBIA on September 24, 2013 | http://pubs.acs.org Publication Date: January 1, 1989 | doi: 10.1021/bk-1989-0390.ch010

Roger G. Bates Department of Chemistry, University of Florida, Gainesville, F L 32611 Although not accessible to direct measurement, the single electrode potential and the activities of individual ionic species nonetheless appear explicitly in many formulations of electrochemical metrology. Proposals for evaluating these quantities fall into two general categories. The first includes attempts to determine experimentally the potential of a single electrode through elimination or evaluation of phaseboundary potentials. The second depends on ionic solution theory for a guide to a reasonable separation of the mean activity coefficients of electrolytes into their ionic contributions. The result is a conventional basis for a scale of ionic activities which, despite its nonthermodynamic nature, satisfies the requirements of modern electroanalytical chemistry. Major developments of the past 75 years in these two directions are traced. The absolute potentials of single electrodes have been a subject of interest since Ostwald, Nernst, and their contemporaries formulated the beginnings of modern electrochemistry in the nineteenth century. The twentieth century brought new methods of applying electrochemical measurements to analytical problems. For these, a knowledge of electrode potentials, or, alternatively, the activities of individual species of ions, could provide simplicity and accuracy not hitherto attainable. Nevertheless, ordinary thermodynamic procedures are incapable of measuring these quantities. This paper reviews efforts to establish single ion activities for aqueous electrolytes. Nevertheless, a closely related problem, that of the energies of transfer of single ionic species from one solvent to another, has received much attention. Among the chief approaches on which these efforts are based are the following: choice of a reference electrode the potential of which may be independent of the solvent, such as Rb+/Rb or the ferrocinium/ferrocene couple; assumption of the equality of the transfer energies of certain large ions such as tetraphenylarsonium and tetraphenylborate; and efforts to nullify the liquid-junction potential between ionic solutions in different solvents. 0097-6156/89/0390-0142$06.00/0 © 1989 American Chemical Society

In Electrochemistry, Past and Present; Stock, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Downloaded by UNIV OF MISSOURI COLUMBIA on September 24, 2013 | http://pubs.acs.org Publication Date: January 1, 1989 | doi: 10.1021/bk-1989-0390.ch010

10. BATES

The Elusive Single-Ion Activity

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The past 75 years have seen the expenditure of considerable ef fort to solve the problem of individual ion activities in aqueous solutions. Two general approaches have been pursued. The first, experimental in nature, sought a means of nullifying or evaluating the space-charge energy barrier that accompanies the transfer of an ionic species into or out of solution. The second accepted the im possibility of defining exactly the single ion activity operational ly and tried to find a conventional definition that appeared reason able in relation to measured mean ionic activities and that would be consistent with modern electrolyte solution chemistry. The concept of activity of a single ionic species is somewhat obscure and complex, as Noyes (1) has shown. Conway emphasizes that interactions between both like kinds and different kinds of ions are clearly involved, as are ion-solvent interactions (2,3). In thermodynamic terms, taking cognizance of internationally ac cepted sign conventions, one can formulate the following expressions for the single electrode potential E, the cell potential difference V, the activity a, and the phase-boundary or liquid-junction poten tial Фb. For the half reaction M n + + ne = M,

where a M is the activity of the ion M . To attempt a measurement of E, one must have recourse to a reference electrode in a cell such as

where the double vertical line marks a phase boundary. is given by

The emf (V)

Thus,

Alternatively, one may elect to measure cells without transfer ence. For simplicity, we consider solutions of the uniunivalent electrolyte MX in a cell composed of electrodes reversible to the cation M+ and the anion X-:

for which the cell reaction is

and the emf is given by

where the subscripts r and 1 refer to the right and left electrodes, respectively. A cell of this type avoids the complication of the boundary potential but introduces the potential of a second electrode



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and the activity of a second ion. These relationships demonstrate 1) that the single ion activity and the potential Фb at the boundary in dicated by the double vertical line are interdependent, and 2) that only mean activities of neutral combinations of ions are accessible. In the words of Harned (1924), "We are confronted with the in teresting perplexity that it is not possible to compute liquid-junc tion potentials without a knowledge of individual ion activities, and it is not possible to determine individual ion activities without an exact knowledge of liquid-junction potentials." (4_) The statement "This is a dilemma from which there is apparently no escape" is per tinent . Experimental Approaches Reference electrodes of mercury have been used by several investiga tors in an attempt to measure single electrode potentials. Stastny and Strafelda (5) concluded that the zero charge potential of such an electrode in contact with an infinitely dilute aqueous solution is -0.1901V referred to the standard hydrogen electrode. Hall (6) states that the potential drop across the double layer under these conditions is independent of solution composition when specific ad sorption is absent. Daghetti and Trasatti (7,8) have used mercury reference electrodes to study the absolute potential of the fluoride ion-selective electrode and have compared their estimates of ion ac tivities in NaF solutions with those provided by other methods. Their method is based on the assumption that the potential drop across the mercury|solution interface is independent of the electrolyte concen tration once the diffuse layer effects are accounted for by the GouyChapman theory. Although the mercury reference electrode has given promising results, other electrodes, such as the common saturated calomel ref erence, are more convenient and have found extensive application. In such an arrangement the boundary is formed between two liquid phases of different composition, and the potential across this liquid junc tion must be evaluated theoretically. The Planck and Henderson for mulas apply to boundaries of different structures, neither of which may be satisfactorily realized in practice. Although Morf et al (9,10) and Harper (11) have examined carefully the calculation and have made suggestions for improving its accuracy, most investigators have used the Henderson equation, together with limiting ionic mo bilities, to arrive at reference electrode potentials, or have even neglected the boundary potential entirely (12). In a series of papers, Hurlen (13,14) has reported "convenient" single ion activities derived from cells with transference. The liquid-junction potentials involved were estimated by the Henderson equation. In addition, Shatkay's ion-selective electrode measure ments of the activities of Na + and Ca2+ ions in NaCl and CaCl2 solu tions, based on the Henderson equation, appear eminently reasonable in comparison with other estimates (15,16). Pitzer and Brewer (17) have pointed out that the space-charge energy accompanying the transfer of ions out of a solution might, in principle, be minimized such that the net electrical charge of the solution could be measured or controlled. Perhaps the ideal electro chemical method would be realized if mass transfer at the boundary



10. BATES


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could be eliminated entirely. Indeed, in the late 19th century Kenrick (18) suggested the use of a "vertical stream method" in which an air gap exists between two half cells. This technique was devel oped by Randies (19) in 1956 and later by others (20-23). In the ex perimental arrangement, flowing streams of the two electrolytes are brought in such close proximity that potential measurements of the electrode assembly are possible with electrometers of high input impedance. In Gomer and Tryson's treatment of this air-gap cell (24) , the absolute half-cell potential is regarded as VMS-ФM, where VMS is the potential difference between a metal electrode M and the solution while VM is the work function of the metal in the solution. This quantity is equated to V R S - Ф R , where V R s is the electrostatic poten tial between the reference electrode and air above the solution while ФR is the work function of this same reference in air. From sensitive potential measurements, combined with photoelectric deter minations, they concluded that the standard hydrogen half cell has an absolute potential of 4.73 ± 0.05V. Reiss and Heller (25) agree that absolute potentials cannot be obtained by measurement alone. A residual theoretical assessment, usually relating to the interfacial dipole layer, is always necessary. For the standard hydrogen elec trode, they assign a value of 4.43V. Related procedures involving different estimates of interfacial potentials have given results in substantial agreement with these (26,27) . In the opinion of Pitzer (28) , this experimental procedure, with improvements in sensitivity, may provide useful data in the future for the thermodynamic functions of single ionic species. That this goal may be achieved in the near future is suggested by the study of Farrell and McTigue (23) . These investigators obtained a precision of ±0.1 mV in the potential differences between a mercury jet and Pt|H2 and AgCl|Ag electrodes in aqueous solutions of HC1. A variety of other approaches to scales of absolute electrode potentials of an experimental nature have been suggested. Oppenheim (29) mentions the possibility, apparently not tried, of measuring the quadrupole radiation emitted by an electrode made to execute harmonic motion by mechanical or ultrasonic means. Goldberg and Frank, in a novel approach to liquid-junction potentials, noted that the time rise of the calculated potential is sensitive to the choice of single ion activity (30) . Similarly, Leckey and H o m e (31) have studied cell potentials as a function of time and have identified a compli cated interdependence of electric transport and single ion activity coefficients which may reveal the magnitude of individual ionic prop erties. Cells involving a thermal gradient, the possible utility of which was suggested by Szabo in 1938 (32) , have been studied exten sively by Milazzo and his co-workers (33,34) . In later work, Szabo and his associates (35) used a 3-electrode polarographic method, fol lowing a suggestion from the work of Janata et al (36,37) . Poten tials were referred to the half-wave potential of a ferrocinium/ferrocene reference electrode, and the cell contained no liquid junction. An accuracy of 5 mV was estimated. Elsemongy (38) proposes to obtain absolute electrode potentials for the hydrogen electrode by extrap olating standard potentials for Pt;H2|AgX;Ag cells as a function of the radius of the halide ion X-. This procedure, however, does not appear to be generally applicable.


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Assumptions and Conventions


In his classic study of liquid-junction potentials in concentration cells (1919), MacInnes (39) suggested that the activity coefficients of K + and Cl- ions be assumed equal to the mean activity coefficient of KC1 at all ionic strengths:

This assumption seemed reasonable in view of the similarity of the sizes and limiting conductances of the two ions. It gave rise to the MacInnes convention, which enables the activity coefficients of other ionic species to be evaluated from the known mean activity coefficients of selected electrolytes. This process, called the "mean salt method", has been used extensively and with apparent success in studies of the seawater medium. Garrels and Thompson (40,41) used a glass electrode reversible to Na + to establish a reference point for the activity coefficient of that ion in seawater. The MacInnes convention was then applied to obtain data for the other ions present. The procedure proposed by Maronny and Valensi (42,43) for the determination of standard pH values utilizes Equation 6, where vKCl is the mean activity coefficient of KC1 in each particular medium, rather than in aqueous KC1 alone. Nesbitt (44,45) has pointed out that ratios of the activity coefficients of ions of the same charge in mixtures can be obtained without ambiguity from mean activity coefficients of electrolytes with a common anion or cation. If HC1 is one of the electrolytes, a pH measurement might provide a reference point for calculating the activity coefficient of a second cation as well as that of the anion involved. Equilibrium theory suggests that pH measurements of saturated solutions of a metal hydroxide or carbonate might also lead to the activity coefficient of the metal ion concerned (46). In these cases, a convention is necessary to provide numerical values of the pH. The Debye-Hückel theory, which appeared in 1923, predicted that the activity coefficients of singly charged cations and anions would be equal at ionic strengths so low that departures from ideal behavior are caused solely by long-range electrostatic interactions. Guggenheim (47) proposed a formula applicable to mixed electrolytes, more general and complex than that of MacInnes, in which the numerical value of the single ion activity coefficient is weighted according to the contribution of that ion to the total ionic strength. For a single symmetrical electrolyte MX, Guggenheim's formula reduces to

The ionic interaction theories of BrФnsted-Guggenheim (48) and Pitzer (49,50) have been conspicuously successful in accounting for the mean activity coefficients and other thermodynamic properties of electrolytes, singly and in mixtures of ionic solutes. They have proved especially useful in salt mixtures such as seawater (51,52). Unfortunately, specific parameters characteristic of single ions do not appear in the theory. For a single 1:1 electrolyte, the equations lead to equality of the activity coefficients of cation and anion, as in Equation 7.


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The Debye-Hückel equation and its semi-empirical extensions have constituted the framework for a number of conventions. A useful ver sion of this equation,

represents rather satisfactorily the mean activity coefficients (v+) of many unassociated electrolytes up to ionic strengths (I) of 0.05 to 0.1 mol kg-1. In this equation, A and B are constants dependent on the temperature, density, and dielectric constant of the medium, z+ and z- are the charges of the cation and anion, and a is an adjust able parameter representing, in the original theory, the closest distance of approach of the cation and anion. An empirical term linear in I is often added to the right of Equation 8 in order to account for the minimum in curves of -1n v± as a function of I, thus extending the applicability to higher con centrations:

where c is an adjustable parameter. As c is unknown, Equation 8 rather than Equation 9 has been used for estimating activity coeffic ients of ionic species. For this purpose, z+z- becomes z2 while a is regarded formally as the ionic diameter. Kielland (53) examined critically varied data for unassociated electrolytes and recommended values of a for a series of single cat ions and anions. This convenient procedure has found wide use. Tamamushi (54) has calculated activity coefficients for single ionic species by identifying ă in Equation 8 with the distance parameter in the Debye-Hückel-Onsager equations for electrolytic conductance. Neff (55) prefers to express ionic activity coefficients as a func tion of the sums of the activities of all species rather than the ionic strength. In 1960, the "pH convention" was adopted by the International Union of Pure and Applied Chemistry, in response to the need for a consistent series of standards to define the pH scale (56). It was agreed that the activity coefficient of chloride ion in aqueous buf fer solutions at ionic strengths no greater than 0.1 would be defined by

at all temperatures. The activity coefficients of chloride ion at 25°C defined in this way are nearly the same as the mean activity co efficients of NaCl in its pure aqueous solutions. Although this con vention is nonthermodynamic in nature, its adoption removed a source of ambiguity and placed pH measurements on a common basis. A propos al to extend this convention to ionic strengths higher than 0.1, as needed for the standardization of many ion-selective electrode meas urements, proved impractical (57).



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ELECTROCHEMISTRY, PAST A N D PRESENT

The MacInnes, Debye-Hückel, and pH conventions describe the ionic activity as a function only of ionic strength. It is, however, not reasonable to expect the chloride ion, for example, always to have the same activity coefficient at a fixed temperature and ionic strength, regardless of the nature of the counter cation. Bjerrum (58) showed in 1920 that the behavior of electrolytes, including the minima observed in plots of 1n v± vs. I, provides evidence for ionsolvent interactions. Hydration of the ions must be considered, and "single-parameter" conventions are inadequate from the standpoint of solution theory. Bjerrum's two-parameter description of mean activity coefficients was refined and extended by Stokes and Robinson (59) and by Glueckauf (60). Both treatments, though differing in statistical detail, consisted essentially in expressing the parameter c in Equation 9 in terms of h, a hydration index for each specific electrolyte. In a later version of the hydration equation (61) Bates and Robinson replaced the Debye-Hückel electrostatic term, that is, the first term on the right of Equation 9, by the modification proposed by Glueckauf in 1969 (62) which is valid to higher ionic strengths. In these treatments, the hydration index h was treated as a constant. It is, however, reasonable to expect h to decrease as the ionic strength increases. Indeed, Bates (1986) found support for a linear decrease with I from its limiting value (h u ) at zero ionic strength (63). His modification of the hydration equation for mean activity coefficients is as follows:

where

In Equation 11, log is the decadic logarithm, A'=0.4343A, v is the number of ions from one molecule of the electrolyte, r is the ratio of the apparent molal volume of electrolyte to the volume of a mole of water, and Ic is the ionic strength in concentration units (mol dm-3). The three parameters å, h u , and q were derived from mean activity coefficients by nonlinear least-squares procedures. The hydration indexes were found to decrease from 12 for Mg2+ and 11 for Ca2+ to 5.2, 3.8, and 2.5 for Li , Na + , and K+, respectively. Pan (64) has preferred to fix the ion-size parameter in the hydration equation at the sum of the crystallographic radii of the ions. If this is done, he has found that much larger values of h (about 21, 13, and 7 for the lithium, sodium, and potassium halides, respectively) are obtained from mean activity coefficients. Mean activity coefficients can, in theory, be determined without ambiguity. For this reason, considerable attention has been directed to the use of solution theory as a guide to separating mean activities into their cationic and anionic components. The MacInnes as-


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BATES


149

sumption described above f a l l s in t h i s category. Frank (1963) ap pears to have been the f i r s t to apply the Bjerrum hydration approach to the c a l c u l a t i o n of s i n g l e ion a c t i v i t i e s (65). From hydration numbers h + and h- for the cation and anion, estimated in the manner of Bjerrum, he derived a correction term δ± such that


and

When the parameters of the Stokes-Robinson modification of the hydration equation for simple electrolytes were examined, Bates, Staples, and Robinson (66) noted that values of h for the alkali chlorides varied in nearly linear fashion with the reciprocal of the radii of the cations. The intercept for very large cations was near h=0. On the strength of this observation, together with the assump tion that h values for cation and anion are additive, the convention that h=0 for chloride ion was advanced. From this point of depart ure, together with the Gibbs-Duhem equation, the following reasonable separation of v± for a 1:1 electrolyte MX into its ionic contribu tions was derived:

and

where m is molality and (J) is the osmotic coefficient. It has been pointed out (67) that the suggestion of hydration numbers near zero for the halide ions appears contrary to experiment al evidence. Although the physical concept of a primary hydration number is reasonably clear, the precise nature of the hydration in dex is not nearly as well defined. Furthermore, the separation of ionic activity coefficients embodied in Equations 15 and 16 is rather insensitive to the choice of hydration indexes. For example, Bagg and Rechnitz's studies of cells with liquid junction (68) lead to a value of h=0.9 for chloride ion, instead of h=0. This difference produces a change of only about 0.015 in pM, the negative logarithm of the cation activity, for Na + in 2m NaCl and less than 0.001 for Ca 2 + in 2m CaCl 2 (63). Conclusion What should one conclude from the foregoing summary of the past 75 years of research on individual ion activities? First, the inacces sibility of these quantities to thermodynamic measurement has been amply confirmed; second, that a start has been made toward the event ual establishment of absolute electrode potentials and phase-boundary potentials which may, in the future, lead to useful thermodynamic data for ionic species but which currently lacks the necessary sensi-


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tivity. Perhaps most striking is the recognition that conventional scales of ionic activity may fulfill many of the requirements of electrochemical measurement. A convention cannot be proved right or wrong, and its effectiveness depends upon its universal adoption.


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4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29. 30. 31. 32. 33.

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37. 38. 39. 40. 41.

42. 43. 44. 45. 46. 47. 48. 49. 50. 51.

52.

53. 54. 55. 56. 57.

58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68.


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Pursuit of the Elusive Single-Ion Activity - ACS Symposium Series

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