Robert S. Hansen and C. A. Smolders
Institute for Atomic Research Iowa State University Ames
Colloid and Surface Chemistry in the Mainstream of Modern Chemistry
colloid and surface chemistry is a field with a large research frontier. Opportunities at this frontier exist for many talents: skilled mathematicians, clever experimentalists, and those with penetrating intuitive insight may all iind problems to challenge their abilities. ks is true in many other fields, recognition of colloid and surface chemistry as a rapidly developing area of physical chemistry with extensive unsolved research problems is not often adequately conveyed in undergraduate and graduate instruction. In undergraduate courses in physical chemistry, fundamentals (thermodynamics, kinetics, equilibria) are almost universally emphasized; but however important they may he, their essential elements have been well understood for a t least 50 years and instruction in them is less likely to convey an impression of the research frontiers of any part of physical chemistry. In specialized colloid and surface chemistry courses, overattention to descriptive material and a wealth of applications may also easily mask the developmental character of the science. This review is intended to emphasize that developmental character. For this reason it is divided into three main sections: (I) major problems of 30 years ago which now appear to be solved in principle, (11) unsolved major problems receiving active and productive current study, and (111) unsolved problems apparently needing different research approaches if fundamental understanding is to be achieved. Some subjectivity is probable in selection of problems as "major" as well as in the designation of some of these problems as %olved in principle."
Stability of Lyophobic Colloids" by Verwey and Overbeek (S), respectively. Brunauer's book contains the first problem in only partially solved form, but focused attention on the problem so well that it can be considered a major landmark in its solution. The works of Flory and of Verwey and Overbeek contain the solutions in principle to the second and third problems. The Interaction of Gases with Solids at a Distance Brunauer, Emmett, and Teller (4) initiated a major resurgence of interest in physical adsorption with their theory of multimolecular adsorption presented in 1938, which supposed that an infinite number of molecules could adsorb on a given site. A modification by Brunauer, Deming, Deming, and Teller (6) s u p posed adsorption restricted to any finite number n of molecules per site; with the additional flexibility thus ohtained, it was successful in representing adsorption isotherms very well over a broad intermediate pressure range. Figure 1, taken from Brunauer (I), illustrates thc representation obtained. Furthermore, the BET theory provided a tool for estimating surface areas of adsorbents which has remained the major source of surface area information. The origin of multimolecular adsorption according to this theory is to be found in the energy of liquefaction of the adsorbing gas and essentially in an extra entropy resulting from a variety of equivalent sites on which the outer molecules might adsorb. The justification for this extra entropy was shown by Halsey (6) to he tenuous, and in retrospect it appears that the BET theory was a
Major Problems Recently Solved in Principle
We consider three major problems in colloid and surface chemistry to have been solved in principle during the past 30 years. These are (a) the nature of the interaction of molecules with solids at distances greater than one molecular diameter, including the nature of multimolecular adsorption; (b) the nature of lyophilic colloids; and (c) the nature of lyophobic sols. In each case a hook can be cited which was a landmark in the problem: Brunauer's "Physical Adsorption of Gases and Vapors" (I), Flory's "Principles of Polymer Chemistry" (S), and "Theory of the Presented as part of the Symposium on the Teaching of Colloid and Surface Chemistrv before the Divisions of Colloid and Surface Chemistry and "chemical Eduation at the 140th ACS Meeting, Chicego, September, 1961. Contribution No. 1098 from the Ames Laboboretory of the U. 5. Atomic Energy Commission. The author and editor gratefully acknowledge permission to reproduce figures, granted by authors of books and editors of journals in which the figures were first printed.
Figure 1. Admrption of nitrogen on o singly promoted iron cotalyrt (from 1 , p . 160.Fig.71.)
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side branrh in the development of our understanding of interactions of solids with gases a t a distance; nevertheless it provided a major stimulus to the main tree. The main tree, in this case, seems to have started with Polanyi's (7) potential theory of adsorption. The potential field postulated by Polaoyi mas initially of uncertain origin, but in 1930 London and Pclanyi (8) pointed out that London's (9)theory of van der Waals forces implied that a molecule must interact with a plane solid surface with an energy inversely proportional to the third power of the distance from it. This together with Polanyi's potential theory of adsorption implied a multimolecular adsorption isotherm, but this line of attack was not pursued further for 18 years, after which Halsey (6) and Hill (10) both presented the implied isotherm. The most elegant demonstration of the character of the interaction between molecules of a gas phase and solid surfaces has been due to Halsey and co-workers (II), who showed that imperfections of gases in contact with solid surfaces must depend on temperature in a manner which could be calculated if the dependence of potential energy of molecules in the gas on distance from the surface were known. Such calculations w-ere made and shown to agree with observations of great precision within the limits of uncertainty of parameters of the calculation (surface area, van der Waals coefficients),all of which could be established independently and a priori. Figure 2 shows the character of agreement between theory and experiment. Our present understanding of the interaction of molecules with solids a t distances greater than one molecular diameter from the surface may be summarized in the following way. Two spherically symmetric molecules with centers a distance r apart have an interaction energy - or/r6; the coefficient u can be calculated from properties of the molecules such as polarizability and diamagnetic susceptibility (12). If a molecule in a gas is a t a distance from a solid surface appreciable compared to the distance between closestneighboring molecules in the solid its total energy of interaction with the solid can be obtained by integrating its energy of interaction with a volume element of the solid over the whole solid; for a solid with plane surface and large thickness the total interaction energy is -p/d3, where d is the distance of the gas molecule from the solid surface and p is a known constant. For
dilut,e gases this energy will lead to a concentration of gas molecules at a distance d from a surface larger than its concentration a t great (infinite) distance by a factor exp (p/dSlcT), and for condensible gases liquefaction will occur at all distances less than that satisfying p/da = IcT in p,/p (p = pressure and p, = saturated vapor pressure of the liquid; the formula means that the chemical potential of the liquid a t d is lowered by p/d3 by interactiou with the solid). All of these conclusions are in general agreement with experiments for spherical molecules suitably distant from the surface. For non-spherical and polar molecules and for molecules distant from solid surfaces by amounts not large compared t c distances between molecules in the solid, complex theoretical problems remain; but the complexity appears to he more in theoretical techniques than in concepts. Statistical thermodynamic theories of adsorption imply a large amount of thermodynamic information as well as adsorption isotherms. Hill's paper (IS) in 1948 did much to clarify the manner in which thermodynamic informatioh useful for testing such theories could he obtained from experiment; Figures 3 and 4 illustrate dependence of integral and differential energies, enthalpies, and entropies of adsorption on quantity of material adsorbed. Analyses of this sort have provided much more severe tests of theories of adsorption, and more penetrating insights concerning the nature of physical adsorption, than had been provided by studies limited to the character of adsorption isotherms. The Nature of lyophilic Calloids
Our present understanding of lyophilic colloids seems to have benefited greatly from the efforts of protein chemists and polymer chemists. Particularly important among these efforts are Svedberg's development (14) of the ultracentrifuge and the attendant demonstration that many common proteins had reproducible and measurable molecular weights in solution; the development of organic macromolecular chemistry by Staudinger and Caruthers; and the development of polymer and protein physical chemistry by
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8=vlvm. Figure 2. Excess volumes of helium, neon, m d argon on carbon block os functionrof l / T I f r o m 11, Fig.3).
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Figure 3. Differential and integral enthaipier and energies of odmrplion of nitrogenon graphon lfrom 83, Fig. 71.
Flory, Scatchard, and others. Flory's hook (op. eit.) is an excellent review of polymer physical chemistry, and protein physical chemistry has been well summarized in a recent book by Edsall and Wymann (15). Most of those systems once referred t o as lyophilic colloids are now thought of as true solutions of m a c m molecules: that such solutions differ in important respects from solutions of small molecules can be explained in broad statistical thermodynamic terms using a quasi-lattice model. The essential feature of the model is that a macromolecule is recognized as requiring many lattice sites, each of a size adequate to accommodate a solvent molecule, and the macrcmolecule is assumed capable of more or less random kinking and coiling. Figure 5 illustrates the dependence of osmotic pressure on concentration in a polymer solution; Figures 6 and 7 illustrate binary and ternary phase diagrams in systems involving polymers.
Figure 4. Differential and integral entropies of adsorption of nitrogen on graphon (from 83, Fig. 3L
Figure 6. Binary phase diogramr (temperature-composition. expressed in volume fraction polymer V 2 ) for three polyimbvtylene frostions in di-isobvtyl ketone. Dashed curves oretheoreticol (from 85, Fig. 2).
BENZENE
Figure 7.
Ternary isotherm for system benzene-rubber-polystyrene (from
86, Fig. 51.
Figure 5. Plots of ratio of osmotic presure to concentrotion (rr/C) agoinst concentration C (gms1100 c d of polyimbutylene in cyclohexane (filled circle4 ond in benzene (open circler) for a rerier of polyirobutylener. Curves are theoretical (from 84, Fig. 5).
For comparison, the phase diagram for waterisoelectric gelatin-gum arabic is shown in Figure 8. This system is referred t o as showing simple coacervation; the similarity between Figures 7 and 8 is evident. A more complex phase diagram involving positively charged gelatin, negatively charged gum arabic, and water is shown in Figure 9; this system shows complex coacervation. It should be noted that Figures 5 and 6 compare experimental data with theoretical curves. While this is not the case with the ternary diagrams Volume 39, Number 4, April 1962
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shown in Figures 7 and 8, such systems are tractable in terms of the present theoretical framework. Phase equilibria involving charged species, such as that represented in Figure 9, have been subjected to much less extensive theoretical study; Voorn (16) has indicated a profitable approach to this problem. Sorption of vapors by cross-linked polymers is another phenomenon which can be considered fairly well understood. Ionization and binding of cations by macromolecules appear similar to dissociation of simple polybasic acids and complexing by polydentate complexing agents; they apparently are understood in principle, although much work remains to be done in their quantitative representation. Reversible formation of molecular aggregates in the colloidal range, notably micelles, is another phenomenon in this area of which we now have a reasonably good understanding. Debye (17) has shown that the van der Waals attraction between tails and electrical repulsion of heads in micelles composed of long chain organic ions must lead to an aggregate size of maximum stability. The size of aggregate depends on length of chain and ionic strength of solution. Debye has been able to rationalize dependence of micelle size and critical micelle concentration on tail length and ionic strength very well on this basis. Micellar shapes other than t,hat suggested by Debye also appear capable of rationalizing micellar behavior in these respects; the problem of micellar size distributions was not considered by Debye, and the theory of solubilization is as yet imcomplete. None of these problems appears insurmountable a t present. The Nature of Lyophobic Colloids
Lyophobic colloids, on the ot,her hand, are now understood to depend for their stability on an extremely slow flocculation rate; i.e., they are not thermodynamically stable systems. In 1916 von Smoluchowski (18) presented a theory of flocculation based on mutual diffusion of colloidal particles in the absence of potential barriers. Roughly, this theory leads to a rate of disappearance of colloidal particles which is second order in the concentration of these particles and with a rate constant capable of a priori calculation; experimental results on rates of rapid flocculation of sols and smokes are in satisfact0r.y agreement with the theory. In 1934 Fuchs (19) showed how t,his theory must he modified to account for a potential energy barrier to flocculation. The modification amounts to a decrease in the second order rate constant by a factor which depends on the shape, especially as to height and
Figure 8. Simple coacervotion ryrtem water (W1-iroelectric gelatin (GI-gum arabic (A1 (fmm 87, Fig. 1 2 a ) Ibelowl.
breadth, of the potential energy barrier. If the stability of lyophobic colloids were to be explained through an extremely slow flocculation rate, there remained a t this point the problem of explaining physically why the colloidal system should flocculate, and why a potential barrier of character suitable to account for observed flocculation rates existed. This problem was solved by Derjaguin and co-workers (ZO), and subsequently and independently by Verwey and Overbeek (5). We now understand this barrier t o result from a combination of London dispersion forces (attractive) and double layer repulsion. The attractive dispersion interaction energies bctween molecules in two different bodies can he summed over all molecules in the bodies (integrated) and lead to a total energy which is large compared to t.herma1 energy, if the distance between the surfaces of the two bodies is sufficiently small. The predicted forces are sufficiently great for direct measurement of them to be possible. Such measurements have been made and are in satisfactory agreement with t,heory; Figure 10, taken from Derjagiun, Titijevsk i a , and Abricossova (21), illustrates such a direct measurement.
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Figure 9. Complex coocervotion ryttey water (WI-positively chorged gelatm (GI-negatively charged gum orobis ( A )(from 87, Fig. 12bl (above).
Journal of Chemical Education
Figure 10. The energy of attraction of two quartz surfaces per cma as o function of their gap width (charge% removed from rurfocer) (from 2 1 , bil Fig. 7.1.
During the past three decades, studies of the electrical double layer have reached a state of considerable refinement due largely to the efforts of D. C. Grahame. Results have demonstrated the excellence of the Gouy-Chapman model as applied to the diffuse double layer, and have further shown that in many cases of interest the compact double layer could be treated as charge free. Figures 11 and 12 are taken from work of Grahame (Zf?),who has also written an excellent general review of electrical double layer properties (23). The surface potential of a given solid is established in many cases by the concentration of a potential-determining ion in the solution bathing it, and a double layer is set up whose structure depends on the surface potential and the concentrations and charges of ions in the solution. Double layers of different particles repel each
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02 0 -0.4 -0.8 -12 -1.6 POTENTIAL RELATIVE TO N CALOMEL ELECTRODE
Figure 1 1 . Differential capacity of mercury in contact with aqueous 0.1 M NoF solution at 25OC as function of polarization (fmm 22, Fig. 3).
other on overlapping, giving rise to a potential energy of repulsion. For simple models, e.g., the GouyChapman model, the dependence of this potential energy on ionic strength, surface potential, and distance of separation can be calcnlated. The net potential rnwgy (attractive and repulsive) can therefore also be
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Figure 12. Differential capacity of mercury in contoct with aqueous 0.001 M NaF solution at 2S°C or functionof poloriration(from 22, Fig. 51.
calculated, and a representative result is shown in Figure 13. In the Gony-Chapman model the surface potential, as simply calculated from the concentration of the potential-determining ion, is the parameter needed for calculations such as those illustrated by Figure 13; Grahame's work implies that in real systems t,he potential of the outer Helmholtz plane should be used. This will depend on the capacity of compact double layer as well as on the concentration of the potential-determining ion, and its calculation will be more complex but not int,ractahle. The potential harrier theory here summarized, when coupled with the theory of Fnchs, appears to explain the major features of stability and flocculation of hydrophobic sols, including dependence of flocculation rate on concentration of potent,ial determining ion, on P O T E N T I A L ENERO* OF INTERACTION
Figure 13. Potential energy d interaction of two spherical particle, as function of disfance IS is proportional to distance between centerr) and electrolve concentrationIK ir proportional to CL/4 (from 88, Fig. 241.
concentration of principal oppositely charged ion in solution, and on charge of this ion (Schulze-Hardy Rule). Sols in organic media (organosols) have been far less extensively investigated than hydrosols, but there seems to be evidence that the same theory suffices to explain t,he stability of most such systems. The low dielectric constant (which would lead to small double layer thicknesses a t electrolyte concentrations common in aqueous systems) is compensated by extremely small electrolyte concentrations resulting from small electrolyte solubilities in these solvents. Major features in the stability of soap films can also be explained in these terms (24). A non-electrical factor, the so-called entropy stabilization due to adsorbed flexible molecules, may also be important to the stability of some organosols (25). While stability of lyophobic systems may be considered understood in principle, it should be noted that quant,itative aspects of emulsion stability treatments have not so far been satisfactory. Major Problems Receiving Current Productive Study
There seem to he three major areas in colloid and surface chemistry receiving active and productive cnrrent study, which are not yet satisfactorily understood in principle. These are the nature of the first adsorbed layer, especially in chemisorption; the mechanism of electrochemical reactions; and the nature of the wetting process. The Nature of the First Adsorbed Layer
Our understanding of the first adsorbed layer in physical adsorption has not improved appreciably in the past 40 years. It cannot really be considered sat,isfactory, and therefore our progress in this area must be considered unsatisfactory. I t is known that the adsorbed molecules are relatively mobile a t temperatures not far below the normal boiling point of the adsorbed material, that energies of interaction between neighboring molecules are appreciable fractions of the energy of vaporization, and that most common adsorhents have heterogeneous surfaces. Enthalpies of adsorption generally decrease appreciably with increasing amounts adsorbed within the first layer; this seems to indicate that the variation in adsorption energy with position (presumably the first molecules adsorbed go to positions where they are most strongly bound) is not compensated completely by the lateral interaction energy developing on increasing coverage. There is no adequate a priori theory of the variation of heat of adsorption with coverage, and correspondingly no isotherm equations for this region can he considered theoretically adequate. Because of the partial compensation of heterogeneity and lateral interaction effects referred to previously, Langmuir and Brunauer-Emmett-Teller isotherms r e p resent adsorption in the first molecular layer better than one might expect. Basically, variations in enthalpy of adsorption and lateral interaction energy with coverage are comparable in magnitude to the enthalpy of adsorption in the first layer; experimental work on uniform surfaces has so far been too limited to permit separate assessment of the lateral interaction effect, and so one is somewhat in the position of trying to establish two unknowns with one equation. Progress in the study of chemisorption has been Volume 39, Number 4, April 1962
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much more satisfactory. Powerful new techniques have been applied to it with gratifying results. Information derived from the infrared ahsorption spectra of adsorbed molecules has been especially impressive. Thus, for example, it has hccn possiblc to estahlish that earbon monoxide chemisorhed on transition metals gives rise t,o stretching frequencies characteristic of the corresponding metal carbonyls, and t,hat the CH ~t~retching frequencies of olefins chemisorbed on t,hesemetals are characteristic of sahrated, not olefinic, hydrocarhoiis.
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Figurer 16, 17, and 18. Mobility of oxygen phyrically a d r x b e d on top of oxygen chemirorbed on tungsten at 27°K. Field electron emission microscope picture, after 0.43, 3.57, ond 9.30 seconds. Oxygen is trapped or it rpillr aver the edge of the chemirorbed layer lfrom 31, Figs. 64.67, and 691.
edgc of the chcmisorhed layer. Figure 19 shows a striking ficld ion micrograph of a tungsten surfacc ohtained by Miiller. These studies h a w led increasingly to the treat,ment of ehemisorption as a chemical reactiou differing from corresponding hulk chemical reactions chiefly because the solid is not ohliged to surrcnder its lattice energy to react and, in case of metals, hecause of the delocalization of electrons in the metal. Studehaker and Huffman (93) were among t,he first to treat chcmisorpt,ion on carhon in straight organic chemical language, and this treatment has been much extended by others, notably Garten and Weiss (39). The approach seems to he hoth stimulating and reasonable. h very interesting
32 3.3 3.4 3.5 3.6 WAVELENGTH Figure 14. Spectrum of ethylene chemimrbed en bare nickel before IAI and after IBI treatment with Hr lfrom 26, Fig. 31.
Eischens has pionecred in this work, and has co-authored with Pliskin an authoritative review (26). Figures 14 and 1.5 present illustrative spectra; Yang and Garland, by comparing spectra of adsorbed CO with spectra of rhodium carhonyls, were able to estahlish thc character of bonding in chemisorption of CO in important respects (27). Magnetic susc~pt~ibilitymeasurements have been developed by Selwood (28) especially as important indicators of valence in adsorbed species.
Fig. 19. Field ion pattern of a tungsten tip of radiuz 750A at 2I0K. 01 1 planeincenter, 1 1 2 planer in four cornerr lfrom 89, Fig.5bl.
suggestion hy Walling (34, prompted by similar thinking, has led to an ohjertive characterization of the acidity of cracking catalysts by use of Hammett indicators. Figure 20, from an article by Benesi (35). illustrates the use of this idea in establishing "titration curves" for the acid sites on crarking catalysts.
Figure 15. Spectro of C O chemirorbed on on 8% rintered Rh romple for incremring coverages (from 27, Fig. 31.
Low energy electron diffraction provides an opportunity for establishment of surface structures in chemisorption (29). Finally, Muller's (30) inventionof field e m i s sion microscopy offers possibilities in thc detailed study of chemisorbed species, crystal face by crystal face, which are just beginning to be explored. Figures 16-18, taken from Gomer's (Sf)recent hook, show the mobility of oxygen physically adsorbed on top of chemisorbed oxygen, and its subsequent trapping as it reaches the 172 / Journal of Chemical Education
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INCREASINO (
ACID STRENGTH-+ HOUNITSl
Figure 20. Butyiamine titers versus acid strengths for crocking cotolyttr calcined at 5 5 0 ° C (from 35, Fig. 31.
The Mechanism of Electrochemical Reoctionr
Hydrogen overvoltage furnishes the classical example of a kinetically controlled electrode phenomenon. Of all such phenomena it has been the most extensively studied; theoretical and experimental methods developed to treat it have furnished patterns for the treatment of other kinetic electrode processes now being studied in increasing numbers. The first important quantitative study of hydrogen overvoltage was due to Tafel @6), who in 1905 e s t a b lished experimentally the equation 7 = a
In i/io
(now everywhere known as the Tafel Equation). This equation relates the overvoltage 7 to the current density i; a and io are constants. Tafel also proposed a mechanism to account for his equation based on slow combination of deposited hydrogen atoms. According to Tafel's mechanism, a = RT/25, whereas experimentally a = 2RT/5, i.e. four times the expected value. No important theoretical advance in this area occurred for nearly 25 years. I n 1930 Erdey-Gruz and Volmer (87) presented a theory of hydrogen overvoltage based on a slow discharge mechanism (historically indebted to Butler's kinetic theory (33) of the Nernst Equation given in 1924) which accounted for the observed value of the Tafel constant a. This paper can be considered to mark the beginning of modern electrode kinetics. A series of important papers followed in the next decade; papers by Fnimkin (39); Okamoto, Horiuti, and Hirota (40); and Eyring, Glasstone, and Laidler (41) should be especially mentioned. These papers resulted in the incorporation of the area of electrode kinetics into the general body of absolute reaction rate theory. The chapter on Electrochemical Processes in "Theory of Rate Processes" by Glasstone, Laidler, and Eyring (42) well summarizes the theoretical framework achieved in the first decade of the modern era. At the end of thisdecade it was recognized that electrode kinetics could be treated in the transition state (activated complex) formalism; that for many electrode processes the transition state mnst occur within the electrical double layer and probably within the compact double layer; and that the existence of a potential gradient within the double layer must lead, for a charged transition state, to a free energy of activation which depends on electrode polarization. Although the paper of Erdey-Gruz and Volmer (as well as later papers) had considered the effect of polarization on both forward and reverse electrode reactions, principal interest appears to have been focused on the Tafel Eqnation and polarizations sufficient to permit neglect of the reverse reaction. Diffusion of reactants to the electrode mas assumed fast and for most purposes the entire polarization was assumed to be reflected in an increased potential difference, with uniform potential gradient, across the compact double layer. Then if the transition st,ate for an ion of charge Ze occurs a t a fraction a of the distance from solution side to electrode side of the compact donble layer the free energy of activation for the discharge step mnst be lowered by aZ5q on cathodic polarization by -7 volts. At polarizations sufficiently great to permit neglect of the reverse reaction, an increase of
by Aq should therefore increase the discharge rate (and hence the discharge current density) by a factor exp (aZSAq/RT). Hence when the reverse process occurs at negligible rate, the current density should vary exponentially with the overvoltage, or conversely the overvoltage shonld vary logarithmically with current density as found by Tafel. Further, from its definition one expected 0 a 1 ; the value a = coupled with reasonable mechanisms was found to provide a satisfactory representation of Tafel slopes then believed reliable. Developments in the area of electrode kinetics have been abundant in the posewar period. Bockris and co-workers have been especially active in developing ultrehigh purity techniques for overvoltage measnrements, studying dependence of current density on polarization over extensive polarization ranges (including very low polarizations), and on reactant concentrations and electrode materials. From such studies together with theoretical refinements developed by Parsons and Bockris, it has been possible to docnment reasonably detailed discharge mechanisms a t a variety of metal surfaces and in a variety of electrolytes. Bockris has written an extensive review of electrode kinetics through 1953 (43); the development of sophistication in this area over the period 1941-53 reflected by a comparison of this review with the chapter on Electrochemical Processes in "Theory of Rate Processes" (42) is striking. During this same period the kinetics of reactions at mercury electrodes developed into an active area both theoretically and experimentally. Interest in this area was no doubt stimulated in part by the increasingly widespread use of the polarograph and in part by the availability of new electrical instruments, including high quality oscilloscopes, permitting the study of fast electrode processes. Delahay, Ershlcr, Gerischer, Koutecky, and Randles have been particnlarly associated with these developments, and Delahay's book "New Instrumental Methods in Electrochemistry" (44) provides an excellent summary of them through 1953. From a theoretical point of view, rigorous (in some cases) and good approximate (in more complex cases) solutions of the mathematical problems posed by electrode reactions involving simultaneous diffusion and kinetic factors and at stationary and dropping mercury electrodes have been important accomplishments. Eyring, Delahay, Gerischer, and Koutecky have been important contributors in this work. I t has also been found possible to measure the rates of fast electrode reactions from the dependence of resistive and capacitative components of the double layer impedance on alternating current frequency; Ershler, Gerischer, and Randles have been leaders in this work. Figure 21 shows the dependence of resistive and capacitative components of the Faradaic impedance on the reciprocal square root of frequency for several electron exchange reactions at the dropping mercury electrode. Curves for the two components should he parallel straight lines according to theory; the distance between them is inversely proportional to the rate constant with proportionality factor also given by theory. In this and preceding cases mathematical analyses of some complexity are involved, but their results are suitable for practical use and a large amount of rate information has been obtained from them. Since publication of the
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