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`
`If”.
`
`'L.1:"U'_:.II.:in.r-.L.1Hun-{ML
`
`T
`
`f' .‘c'grrr l¢_'.’.'{’HJH .2} \rli'li’f‘C
`
`
`n—J-fi'gilx
`’34,;- :.
`
`V. S. BALU'ISK‘L
`
`Page 1
`
`OWT Ex. 2011
`
`Tennant Company v. OWT
`IPR2021-00602
`
`

`

`FUNDAMENTALS OF
`ELECTROCHEMISTRY
`
`Second Edition
`
`V. S. BAGOTSKY
`A. N. Frumkin Institute of Physical Chemistry and Electrochemistry
`Russian Academy of Sciences
`Moscow, Russia
`
`Sponsored by
`
`THE ELECTROCHEMICAL SOCIETY, INC. Pennington, New Jersey
`
`A JOHN WILEY & SONS, INC., PUBLICATION
`
`

`

`Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved
`
`Published by John Wiley & Sons, Inc., Hoboken, New Jersey
`Published simultaneously in Canada
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`Library of Congress Cataloging-in-Publication Data:
`
`Bagotsky, V. S. (Vladimir Sergeevich)
`Fundamentals of electrochemistry / V. S. Bagotsky—2nd ed.
`p.
`cm.
`Includes bibliographical references and index.
`ISBN-13 978-0-471-70058-6 (cloth : alk. paper)
`ISBN-10 0-471-70058-4 (cloth : alk. paper)
`1. Electrochemistry I. Title.
`QD553.B23 2005
`541⬘.37—dc22
`
`2005003083
`
`Printed in the United States of America
`
`10 9 8 7 6 5 4 3 2 1
`
`

`

`60
`
`MASS TRANSFER IN ELECTROLYTES
`
`It can be seen that the ohmic potential drop ϕ
`ohm differs from the overall potential
`drop ϕσ in the electrolyte as given by Eq. (4.25). The difference between these two
`values corresponds exactly to the diffusional potential drop ϕ
`d for the given concen-
`tration ratio that was given in Eq. (4.19).
`Thus, the potential difference in electrolytes during current flow is determined by
`two components: an ohmic component ϕ
`ohm proportional to current density and a
`diffusional component ϕ
`d, which depends on the concentration gradients. The latter
`arises only when the Dj values of the individual ions differ appreciably; when they
`are all identical, ϕ
`d is zero. The existence of the second component is a typical fea-
`ture of electrochemical systems with ionic concentration gradients. This component
`can exist even at zero current when concentration gradients are maintained artificially.
`When a current flows in the electrolyte, this component may produce an apparent
`departure from Ohm’s law.
`As the diffusional field strength Ed depends on the coordinate x in the diffusion
`layer, the diffusion flux density (in contrast to the total flux density) is no longer con-
`stant and the concentration gradients dcj/dx will also change with the coordinate x.
`
`4.3.3 The General Case
`
`Generally, an electrolyte may contain several ionic reactant species but no obvious
`excess of a foreign electrolyte. Then, as already mentioned, a calculation of the
`migration currents [or coefficients α in equations of the type (4.22)] is very complex
`and requires computer use.
`Often, we need only a qualitative estimate; that is, we want to know whether the
`limiting current is raised or lowered by migration relative to the purely diffusion-
`limited current, or whether α
`
`j is larger or smaller than unity. It is evident that αj will
`be larger than unity when migration and diffusion are in the same direction. This is
`found in four cases: for cations that are reactants in a cathodic reaction (as in the exam-
`ple above) or products in an anodic reaction, and for anions that are reactants in an
`anodic reaction or products in a cathodic reaction. In the other four cases (for cations
`that are reactants in an anodic or products in a cathodic reaction, and for anions that
`are reactants in a cathodic or products in an anodic reaction), we have α
`⬍1, a typical
`example being the cathodic deposition of metals from complex anions.
`
`j
`
`4.4 CONVECTIVE TRANSPORT
`
`Convective transport is the transport of substances with a moving medium (e.g., the
`transport of a solute in a liquid flow). The convective flux is given by
`
`⫽ (cid:2)cj,
`Jkν, j
`where (cid:2)is the linear velocity of the medium and cj is the concentration of the sub-
`stance. In electrolyte solutions, the convective flux is always electroneutral because
`of the medium’s electroneutrality.
`
`(4.31)
`
`

`

`CONVECTIVE TRANSPORT
`
`61
`
`In electrochemical cells we often find convective transport of reaction compo-
`nents toward (or away from) the electrode surface. In this case the balance equation
`describing the supply and escape of the components should be written in the general
`form (1.38). However, this equation needs further explanation. At any current den-
`sity during current flow, the migration and diffusion fluxes (or field strength and con-
`centration gradients) will spontaneously settle at values such that condition (4.14) is
`satisfied. The convective flux, on the other hand, depends on the arbitrary values
`selected for the flow velocity v and for the component concentrations (i.e., is deter-
`mined by factors independent of the values selected for the current density). Hence,
`in the balance equation (1.38), it is not the total convective flux that should appear,
`only the part that corresponds to the true consumption of reactants from the flux or
`true product release into the flux. This fraction is defined as the difference between
`the fluxes away from and to the electrode:
`
`ΔJkν, j
`
`⫽ nF(cid:2)(cj
`
`
`
`⫺ cj⬘),
`
`(4.32)
`
`⬘ is the concentration of substance j in the flow leaving the electrode.
`where cj
`For the present argument and in what follows, we assume that the migrational
`transport is absent (that we have uncharged reaction components or an excess of for-
`eign electrolyte).
`⫽
`Let us estimate the ratios of diffusion and maximum convective fluxes, Jd, j /Jkν, j
`
`⫻ grad cj /cj(cid:2). The order of magnitude of the concentration gradient is cj /δ. Therefore,
`
`Dj
`
`D
`
`⬇ ᎏδ(cid:2)jᎏ.
`
`(4.33)
`
`jᎏ
`d ν,
`ᎏJJ k
`
`j
`
`,
`
`⬇10⫺5 cm2/s; a typical value of δ is 10⫺2 cm. It follows that
`In aqueous solutions Dj
`the convective and diffusional transport are comparable even at the negligible linear
`velocity of 10⫺3 cm/s of the liquid flow. At larger velocities, convection will be pre-
`dominant.
`
`4.4.1 Flow-by Electrodes
`
`Flow of the liquid past the electrode is found in electrochemical cells where a liquid
`electrolyte is agitated with a stirrer or by pumping. The character of liquid flow near
`a solid wall depends on the flow velocity (cid:2), on the characteristic length L of the solid,
`and on the kinematic viscosity ν
`kin (which is the ratio of the usual rheological vis-
`cosity η and the liquid’s density ρ). A convenient criterion is the dimensionless
`parameter Re ⬅ (cid:2)L/ν
`kin, called the Reynolds number. The flow is laminar when this
`number is smaller than some critical value (which is about 103 for rough surfaces and
`about 105 for smooth surfaces); in this case the liquid moves in the form of layers
`parallel to the surface. At high Reynolds numbers (high flow velocities) the motion
`becomes turbulent and eddies develop at random in the flow. We shall only be con-
`cerned with laminar flow of the liquid.
`
`

`

`62
`
`MASS TRANSFER IN ELECTROLYTES
`
`FIGURE 4.4 Schematic of a flow-by electrode.
`
`In the flow, the thin layer of liquid that is directly adjacent to the solid is retained
`by molecular forces and does not move. The liquid’s velocity relative to the solid
`increases from zero at the very surface to the bulk value (cid:2)which is attained some
`distance away from the surface. The zone within which the velocity changes is called
`the Prandtl or hydrodynamic boundary layer.
`Hydrodynamic theory shows that the thickness, δ
`b, of the boundary layer is not
`constant but increases with increasing distance y from the flow’s stagnation point at
`the surface (Fig. 4.4); it also depends on the flow velocity:
`
`δ
`b
`
`⬇ ν
`kin
`
`1/2 y1/2 (cid:2)⫺1/2.
`
`(4.34)
`
`It is important to note that even in a strongly stirred solution, a thin layer of stag-
`nant liquid is present directly at the electrode surface, within which convection is
`absent so that substances involved in the reaction are transported in it only by
`diffusion and migration. Here the concentration gradient (grad cj)x ⫽ 0 is steepest and
`(in the absence of convection) determined by the balance equation
`
`(4.35)
`
`⫽ ⫺Dj(grad cj)x⫽0.
`
`Fjᎏ
`nν
`
`i ᎏ
`
`In the bulk, to the contrary, concentration gradients are leveled only as a result of
`convection, and diffusion has practically no effect. In the transition region we find
`both diffusional and convective transport. The concentration gradient gradually falls
`to zero with increasing distance from the surface.
`Diffusion in a convective flow is called convective diffusion. The layer within
`which diffusional transport is effective (the diffusion layer) does not coincide with
`the hydrodynamic boundary layer. It is an important theoretical problem to calculate
`the diffusion-layer thickness δ. Since the transition from convection to diffusion is
`gradual, the concept of diffusion-layer thickness is somewhat vague. In practice, this
`thickness is defined so that Δcj /δ ⫽ (dcj/dx)x⫽0. This calculated distance δ (or the
`value of κ
`j) can then be used to find the relation between current density and con-
`centration difference.
`
`

`

`CONVECTIVE TRANSPORT
`
`63
`
`FIGURE 4.5 Distributions of flow velocities and concentrations close to the surface of a
`flow-by electrode.
`
`An analogy exists between mass transfer (which depends on the diffusion
`coefficient) and momentum transfer between the sliding liquid layers (which depends
`on the kinematic viscosity). Calculations show that the ratio of thicknesses of the
`diffusion and boundary layer can be written as
`
`⬇冢ᎏν
`
`i
`
`j nᎏ
`D k
`
`冣1/3
`
`⫽ Pr⫺1/3.
`
`(4.36)
`
`δ bᎏ
`
`ᎏδ
`
`The dimensionless ratio ν
`kin/Dj is called the Prandtl number, Pr. In aqueous solutions
`⬇ 10⫺5 cm2/s and ν
`⬇ 10−2 cm2/s (i.e., Pr ⬇ 10⫺3). Thus, the diffusion layer is
`Dj
`kin
`approximately 10 times thinner than the boundary layer. This means that in the major
`part of the boundary layer, motion of the liquid completely levels the concentration
`gradients and suppresses diffusion (Fig. 4.5).
`Allowing for Eqs. (4.34) and (4.36), we obtain
`
`(4.37)
`
`δ ⬇ Dj
`1/3 ν
`1/6 y1/2 (cid:2)⫺1/2.
`kin
`The gradual increase in thickness δ that occurs with increasing distance y leads to a
`decreasing diffusion flux. It follows that the current density is nonuniform along the
`electrode surface.
`It is important to note that the diffusion-layer thickness depends not only on hydro-
`dynamic factors but also (through the diffusion coefficient) on the nature of the diffusing
`species. This dependence is minor, of course, since the values of Dj differ little among
`the various substances, and in addition are raised to the power one-third in Eq. (4.37).
`It follows that convection of the liquid has a twofold influence: It levels the con-
`centrations in the bulk liquid, and it influences the diffusional transport by govern-
`ing the diffusion-layer thickness. Slight convection is sufficient for the first effect,
`but the second effect is related in a quantitative way to the convective flow velocity:
`The higher this velocity is, the thinner will be the diffusion layer and the larger the
`concentration gradients and diffusional fluxes.
`
`

`

`64
`
`MASS TRANSFER IN ELECTROLYTES
`
`FIGURE 4.6 Rotating-disk electrode (arrows in the space below the electrode indicate the
`directions of liquid flow).
`
`4.4.2 Rotating-Disk Electrode
`
`At the rotating-disk electrode (RDE; Fig. 4.6), it is the solid electrode and not the
`liquid that is driven; but from a hydrodynamic point of view this difference is unim-
`portant. Liquid flows, which in the figure are shown by arrows, are generated in the
`solution when the electrode is rotated around its vertical axis. The liquid flow
`impinges on the electrode in the center of the rotating disk, then is diverted by cen-
`trifugal forces to the periphery.
`Let ω be the angular velocity of rotation; this is equal to 2πf, where f is the disk fre-
`quency or number of revolutions per second. The distance r of any point from the cen-
`ter of the disk is identical with the distance from the flow stagnation point. The linear
`velocity of any point on the electrode is ωr. We see when substituting these quantities
`into Eq. (4.34) that the effects of the changes in distance and linear velocity mutually
`cancel, so that the resulting diffusion-layer thickness is independent of distance.
`The constancy of the diffusion layer over the entire surface and thus the uniform cur-
`rent-density distribution are important features of rotating-disk electrodes. Electrodes
`of this kind are called electrodes with uniformly accessible surface. It is seen from the
`quantitative solution of the hydrodynamic problem (Levich, 1944) that for RDE to a
`first approximation
`
`δ ⫽ 1.616 Dj
`
`kin
`
`1/3 ν1/6 ω⫺1/2,
`
`(4.38)
`
`and hence,
`
`2/3 ν
`FDj
`
`⫺1/6ω⫺1/2
`kin
`
`n jᎏ
`
`i ⫽ 0.62 ᎏν
`(the Levich equation). A more exact calculation leads to complex expressions with a
`number of correction terms; however, some of these corrections mutually cancel, so
`
`(4.39)
`
`

`

`252
`
`SOME ASPECTS OF ELECTROCHEMICAL KINETICS
`
`FIGURE 14.7 Polarization curves for the anodic dissolution of (1) p-type and (2) n-type
`germanium in 0.1 M HCl solution.
`
`and the slope changes to RT/F (i.e., is about twice as small as in the first case). In
`general, intermediate values of the slope are possible.
`A typical feature of reactions involving the minority carriers are the limiting cur-
`rents developing when the surface concentration of these carriers has dropped to zero
`and they must be supplied by slow diffusion from the bulk of the semiconductor. A
`reaction of this type, which has been studied in detail, is the anodic dissolution of
`germanium. Holes are involved in the first step of this reaction Ge → Ge(II), and
`electrons in the second Ge(II) → Ge(IV). The overall reaction equation can be writ-
`ten as
`
`Ge ⫹ 3H2O ⫹ 2h⫹ → H2GeO3
`
`⫹ 4H⫹ ⫹ 2e⫺.
`
`(14.26)
`
`It can be seen from Fig. 14.7 that the polarization curve for this reaction involv-
`ing p-type germanium in 0.1 M HCl is the usual Tafel straight-line plot with a slope
`of about 0.12 V. For n-type germanium, where the hole concentration is low, the
`curve looks the same at low current densities. However, at current densities of about
`50 A/m2 we see a strong shift of potential in the positive direction, and a distinct lim-
`iting current is attained. Thus, here the first reaction step is inhibited by slow supply
`of holes to the reaction zone.
`Under the effect of illumination, new phenomena arise at semiconductor elec-
`trodes, which are discussed in Chapter 29.
`
`14.5 REACTIONS PRODUCING A NEW PHASE
`
`14.5.1
`
`Intermediate Stages in the Formation of New Phases
`
`In applied electrochemistry, reactions are very common in which a new phase is
`formed (i.e., gas evolution, cathodic metal deposition, etc.). They have a number of
`special features relative to reactions in which a new phase is not formed and in which
`the products remain part of the electrolyte phase.
`The first step in reactions of the type to be considered here is the usual electro-
`chemical step, which produces the primary product that has not yet separated out to
`
`

`

`REACTIONS PRODUCING A NEW PHASE
`
`253
`
`form a new phase. In gas evolution, this is the step that produces gas molecules dis-
`solved in the electrolyte (possibly forming a supersaturated solution). In cathodic
`metal deposition, this is the formation of metal atoms by discharge of the ions; these
`atoms are in an adsorbed state (called adatoms) on the substrate electrode and have
`not yet become part of a new metal phase. These steps follow the usual laws of elec-
`trochemical reactions described in earlier chapters and are spread out uniformly over
`all segments of the electrode surface.
`These primary electrochemical steps may take place at values of potential below
`the equilibrium potential of the basic reaction. Thus, in a solution not yet saturated
`with dissolved hydrogen, hydrogen molecules can form even at potentials more pos-
`itive than the equilibrium potential of the hydrogen electrode at 1 atm of hydrogen
`pressure. Because of their energy of chemical interaction with the substrate, metal
`adatoms can be produced cathodically even at potentials more positive than the
`equilibrium potential of a given metal–electrolyte system. This process is called the
`underpotential deposition of metals.
`Subsequent steps are the formation of nuclei of the new phase and the growth of
`these nuclei. These steps have two special features.
`
`1. The nuclei and the elements of new phase generated from them (gas bubbles,
`metal crystallites) are macroscopic entities; their number on the surface is lim-
`ited (i.e., they emerge not at all surface sites but only at a limited number of these
`sites). Hence, the primary products should move (by bulk or surface diffusion)
`from where they had been produced to where a nucleus appears or grows.
`2. The process as a whole is transient; nucleation is predominant initially, and
`nucleus growth is predominant subsequently. Growth of the nuclei usually
`continues until they have reached a certain mean size. After some time a quasi-
`steady state is attained, when the number of nuclei that cease to grow in unit
`time has become equal to the number of nuclei newly formed in unit time.
`
`Any of the steps listed can be rate determining: formation of the primary product,
`its bulk or surface diffusion, nucleation, or nucleus growth. Hence, a large variety of
`kinetic behavior is typical for reactions producing a new phase.
`Two types of reactions producing a new phase can be distinguished: (1) those pro-
`ducing a noncrystalline phase (gas bubbles; liquid drops as, e.g., in the electrolytic
`deposition of mercury on substrates not forming amalgams), and (2) those produc-
`ing a crystalline phase (cathodic metal deposition, anodic deposition of oxides or
`salts having low solubility).
`Features common to these two reaction types are the sequence of steps above,
`particularly the step producing nuclei of small size (e.g., in the nanometer range).
`The excess surface energy (ESE) contributes significantly to the energy of these
`highly disperse entities (with their high surface-to-volume ratio). The thermody-
`namic properties of highly disperse (extremely small) particles differ from those of
`larger ones.
`When crystal structure is involved, it gives rise to special features in the reac-
`tions and makes their mechanisms more complex. Therefore, at first we consider
`
`

`

`254
`
`SOME ASPECTS OF ELECTROCHEMICAL KINETICS
`
`the common behavior of reactions producing a new phase in the instance . gas evo-
`lution reactions (Section 14.2), then we discuss the special features linked to crystal
`structure (Section 14.3).
`
`14.5.2 Formation of Gas Bubbles
`Nucleation Consider an idealized spherical nucleus of a gas with the radius rnucl on
`the surface of an electrode immersed in an electrolyte solution. Because of the small
`size of the nucleus, the chemical potential, μ
`nucl, of the gas in it will be higher than
`that (μ
`0) in a sufficiently large phase volume of the same gas. Let us calculate this
`quantity.
`At the curved surface of the sphere, a force is acting that is directed toward the
`center of the sphere and tends to reduce its surface area. Hence, the gas pressure pnucl
`in the nucleus will be higher than the pressure p0 in the surrounding medium. An
`infinitely small displacement dr of the surface in the direction of the sphere’s center
`is attended by a surface-area decrease dS (⫽ 8πr dr) and a volume decrease dV
`(⫽ 4πr2 dr). The work of compression of the nucleus is given by (pnucl
`⫺ p0) dV. It
`should be equal to the energy gain, σ dS, resulting from surface shrinkage, where σ
`is the ESE of the gas–solution interface. Hence, we find that
`
`(14.27)
`
`σ u
`2 n
`
`
`r
`
`⫽ ᎏᎏ
`cl
`
`d VSᎏ
`σ d
`
`⫽ ᎏ
`
`pnucl
`
`⫺ p0
`
`(the Laplace equation, 1806). This equation is valid for any curved phase boundary,
`⬍ p0 and the radius of curvature is conventionally
`also concave ones (for which pnucl
`⬅ pnucl
`⫺ p0 is called the capillary pressure of
`regarded as negative). Parameter pc
`this curved surface.
`We know from thermodynamics that when the pressure changes at constant tem-
`perature, we have
`
`⫽ Vj.
`
`(14.28)
`
`冢ᎏ
`
`μ p
`d d
`
`jᎏ冣
`
`T
`
`We shall integrate this equation between limits given by the pressures pnucl and p0:
`
`(14.29)
`
`(14.30)
`
`Δμ
`nucl
`
`⬅ μ
`nucl
`
`⫺ μ
`0.
`
`Using Eq. (14.27), we finally find that
`nucl ⫽ᎏ2σ
`rn
`
`lu
`cn
`V u
`
`clᎏ
`
`Δμ
`
`[the Thomson (Kelvin) equation, 1870].
`
`

`

`REACTIONS PRODUCING A NEW PHASE
`
`255
`
`Two conditions must be fulfilled for spontaneous nucleation: (1) the chemical
`potential of the primary product should be no less than μ
`nucl, and (2) conditions
`enabling the “encounter” of Nnucl particles of the primary product should exist.
`The first condition implies that the concentration, cprim
`nucl, of the primary products in
`0
`the nucleation zone should be higher than the equilibrium concentration c prim
`. Allowing
`for Eq. (3.13), we can define the required degree of supersaturation by the relation
`
`(14.31)
`
`c ml
`ru i
`0pn
`cc
`
`Δμ
`primᎏ.
`nucl ⫽ RT ln ᎏ
`
`It follows from Eqs. (14.30) and (14.31) that the required degree of supersaturation
`will be higher the smaller the size of the nuclei.
`When this supersaturation exists, the nucleation rate will be proportional to the
`probability pnucl of formation of a favorable configuration of particles of the primary
`product. According to the Boltzmann law, this probability is determined by the work
`wnucl of formation of a single nucleus:
`
`
`(14.32)
`
`⫽ B exp冢⫺ᎏuclᎏ冣,
`
`Tn
`kw
`
`pnucl
`
`where B is a normalizing factor and k is the Boltzmann constant.
`Detailed calculations show that the work of formation of a single nucleus in a
`supersaturated solution wnucl is determined by the expression
`
`(14.33)
`
`σr 2
`
`3πᎏ
`
`⫽ ᎏ4
`
`wnucl
`
`The smaller the nucleus (or higher the degree of supersaturation), the smaller will be
`work wnucl and the larger will be the probability of nucleation.
`The calculation above is valid for a spherical nucleus forming in bulk solution or
`on an electrode surface completely wetted by the liquid electrolyte, where the wetting
`angle α ⬇ 0 (Fig. 14.8a). The work of nucleation decreases markedly when wetting
`is incomplete (Fig. 14.8b), since the electrode–electrolyte contact area is smaller. The
`work also decreases when asperities, microcracks, and the like are present on the sur-
`face. Thus, Eq. (14.33) states merely the highest possible value of work wnucl.
`In an electrochemical system, gas supersaturation of the solution layer next to the
`electrode will produce a shift of equilibrium potential (as in diffusional concentra-
`tion polarization). In the cathodic evolution of hydrogen, the shift is in the negative
`direction, in the anodic evolution of chlorine it is in the positive direction. When this
`step is rate determining and other causes of polarization do not exist, the value of
`electrode polarization will be related to solution supersaturation by
`
`Δ
`⫾ΔE ⫽ ᎏ
`uclᎏ ⫽ ᎏ
`
`Fn
`nμ
`
`(14.34)
`
`cl
`ln ᎏᎏ.
`
`c
`
`
`u j0
`cjn
`
`FTᎏ
`R n
`
`

`

`256
`
`SOME ASPECTS OF ELECTROCHEMICAL KINETICS
`
`FIGURE 14.8 Gas-bubble nuclei on an electrode with (a) complete and (b) incomplete wet-
`ting of the surface by the liquid, and (c) a gas bubble at the moment of tearing away.
`
`With Eq. (14.32) for the reaction rate and Eq. (14.34) for polarization, we obtain the
`following general form of the polarization equation:
`
`(14.35)
`
`i ⫽ A exp冤⫺ᎏᎏ冥
`
`
`(Δ
`
`)2
`
`γ E
`
`where A and γ are constants. Thus, when plotted as i vs. (ΔE)⫺2, the experimental
`data should fall onto a straight line. Such a function is actually observed in a num-
`ber of cases.
`
`Nucleus Growth After nucleation the degree of supersaturation of the solution in
`the immediate vicinity of the nucleus has fallen, and other nuclei can form only some
`distance away from the first nucleus. It follows that nucleus growth will occur (at
`least initially) not by the fusion of neighboring nuclei but by the direct addition of
`primary-product particles. For noncrystalline nuclei (bubbles or drops) no difficulties
`other than diffusional transport of particles to the nucleus are present at this stage. It
`is merely necessary that the chemical potential of these particles (or degree of super-
`saturation) not be inferior to the chemical potential in the nucleus itself, at the size
`attained. The requirements as to the needed degree of solution supersaturation
`diminish as the nucleus grows larger.
`Another question that arises is the limiting size of the gas bubbles. As the bubble
`volume Vb increases, the buoyancy force Vbg Δρ of the bubble increases (g is the
`acceleration of gravity and Δρ is the density difference between the liquid and the
`gas). The bubble will tear away from the electrode surface as soon as this buoyancy
`force becomes larger than the force fret retaining the bubbles.
`The retaining force depends on the “neck” perimeter πa along which the bubble
`is anchored on the surface (Fig. 14.8c) and on the wetting angle α; it can be formu-
`lated as πaσ sin α. It follows when the surface is readily wetted (α is small) that the
`retaining force, and hence the volume of the bubble tearing away, is considerably
`smaller than when the surface is poorly wetted. Figure 14.9 shows the relation
`between the wetting angle and the final bubble volume, which was calculated and
`confirmed experimentally in 1933 by B. Kabanov and A. Frumkin.
`
`

`

`REACTIONS PRODUCING A NEW PHASE
`
`257
`
`FIGURE 14.9 Volumes of the departing gas bubbles as a function of wetting angle (the
`solid line is the calculated function). (From Kabanov and Frumkin, 1933).
`
`FIGURE 14.10 Polarization–time relation during the formation of a new phase.
`
`The electrode’s wetting angle depends on potential; it is largest at the PZC when
`σ(S,L) is largest, and decreases with increasing distance from this point. This effect is
`the origin of a characteristic feature of hydrogen and oxygen evolution at nickel elec-
`trodes in the electrolysis close to the PZC of nickel; hence, the oxygen bubbles are
`quite large. The potential of hydrogen evolution is far from the PZC, and the gas is
`evolved in the form of very fine bubbles forming a “milky cloud.” This phenomenon
`provides the basis for technical degreasing of metal surfaces by strong cathodic or
`anodic polarization. The wetting of the surface by the aqueous solution increases
`with increasing distance from the PZC, the force with which oil droplets stick to the
`surface decreases, and they are carried away.
`When a gas bubble has torn away, usually the small nucleus of a new bubble is
`left behind in its place. Therefore, in gas evolution an appreciable supersaturation is
`needed only for creating an initial set of nuclei, and subsequent processes require
`less supersaturation. Hence, in a galvanostatic transient the electrode’s polarization
`will initially be higher but will then fall to a lower, steady-state value (Fig. 14.10).
`Such a time dependence of polarization is typical for many processes involving for-
`mation of a new phase.
`
`

`

`15
`Reactions at Nonconsumable
`Electrodes
`
`In the present chapter we want to look at certain electrochemical redox reactions
`occurring at inert electrodes not involved in the reactions stoichiometrically. The
`reactions to be considered are the change of charge of ions in an electrolyte solution,
`the evolution and ionization of hydrogen, oxygen, and chlorine, the oxidation and
`reduction of organic compounds, and the like. The rates of these reactions, often also
`their direction, depend on the catalytic properties of the electrode employed (dis-
`cussed in greater detail in Chapter 28). It is for this reason that these reactions are
`sometimes called electrocatalytic. For each of the examples, we point out its practi-
`cal value at present and in the future and provide certain kinetic and mechanistic
`details. Some catalytic features are also discussed.
`
`15.1 SIMPLE ELECTROCHEMICAL REACTIONS
`
`Conditionally, an electrochemical reaction will be called simple when the following
`conditions hold (at least as an approximation):
`
`1. The electron transfer step is the only reaction step, which means that other par-
`allel or consecutive steps are absent.
`2. Neither the starting material nor the reaction product, nor any intermediates,
`are adsorbed on the electrode.
`3. During the reaction, chemical bonds are not broken, new chemical bonds are
`not formed, and the geometry of the reacting species remains unchanged.
`
`Electrochemical reactions only involving a change of charge of simple or com-
`plex ions but not any change in inner geometry are commonly called outer-sphere
`electron transfer reactions. For some time, the reduction and oxidation of simple and
`
`Fundamentals of Electrochemistry, Second Edition, By V. S. Bagotsky
`Copyright © 2006 John Wiley & Sons, Inc.
`
`261
`
`

`

`262
`
`REACTIONS AT NONCONSUMABLE ELECTRODES
`
`complex iron ions involving a change in valency of the iron had been regarded as an
`example of such simple reactions:
`Fe3⫹ ⫹ e⫺ (cid:2) Fe2⫹,
`[Fe(CN)6]3⫺ ⫹ e⫺ (cid:2) [Fe(CN)6]4⫺.
`
`(15.1)
`
`(15.2)
`
`Later, closer examination revealed that these reactions are attended by a marked
`change in the geometry of the hydrated ions.
`Historically, the reduction of persulfate ions has also been assumed to be a sim-
`ple reaction:
`
`
`
`S2O82⫺ ⫹ 2e⫺ → 2SO4
`2⫺.
`
`(15.3)
`
`It involves rupture of an O–O bond, and two electrons per reacting ion take part in
`it, so it also does not meet the criteria above. The reaction is simple insofar as on
`many metals, the persulfate and sulfate ions are weakly adsorbed.
`Simple electrochemical oxidation–reduction reactions do not by themselves have
`any important applications. However, results of great value for the theory of electro-
`chemical kinetics and electrocatalysis have been obtained when these reactions were
`studied experimentally in greater detail. Thus, in the instance of persulfate ion reduc-
`tion, it has been possible to quantitatively elucidate the influence of electric double-
`layer structure on electrode kinetics. In this reaction, double-layer effects (ψ⬘-effects)
`become particularly clearly evident, since divalent ions are involved. The rate of this
`reaction also depends on the metal selected for the electrode; this effect could (at
`least in part) be explained in terms of the influence exerted by the metal on the value
`of the ψ⬘-potential to be used in the kinetic equation (14.16) governing this reaction.
`In fact, the ψ⬘-potential seen at a given value of electrode potential in a solution of
`given composition depends on the metal’s zero-charge potential Ezc. Metals differ
`greatly in their zero-charge potentials. With a knowledge of Ezc, and using the the-
`ory of electric double-layer structure, values of the ψ⬘-potential can be calculated
`rather accurately. On this basis, important refinements can be introduced into kinetic
`equations of the type of Eq. (15.16), and the reaction rates that would be observed in
`the absence of double-layer effects can be calculated. Figure 15.1 shows such “cor-
`rected” polarization curves for the cathodic reduction of persulfate ions at mercury
`and at a number of metals similar to mercury. It can