CHEMICAL KINETICS
`
`EDITED BY
`
`C.H. BAMFORD
`
`M.A., Ph.D., Sc.D. (Cantab.), F.R.I.C., F.R.S.
`Formerly Campbell-Brown Professor of Industrial Chemistry,
`University of Liverpool
`
`AND
`
`R.G. COMPTON
`
`M.A., D.Phil. (Oxon.)
`University Lecturer in Physical Chemistry
`and Fellow, St. John’s College, Oxford
`
`VOLUME 26
`
`ELECTRODE KINETICS:
`PRINCIPLES AND METHODOLOGY
`
`
`
`ELSEVIER
`AMSTERDAM—OXFORD—NEW YORK—TOKYO
`1986
`
`InnoPharma Exhibit 1104.0001
`
`
`
`

`

`ELSEVIER SCIENCE PUBLISHERS B.V.
`
`Sara Burgerhartstraat 25
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`
`Distributors for the United States and Canada
`
`ELSEVIER SCIENCE PUBLISHING COMPANY INC.
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`
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`
`
`
`ISBN 0-444-41631-5 (Series)
`ISBN 0-444-42550—0 (Vol. 26)
`
`with 120 illustrations and 55 tables
`
`© Elsevier Science Publishers B.V., 1986
`
`All rights reserved. No part of this publication may be reproduced: stored in a retrieval
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`
`Printed in the Netherlands
`
`InnoPharma Exhibit 1104.0002
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`
`This material may be protected by Copyright law (Title 17 U.S. Code)
`
`
`
`Chapter 2
`
`Mass Transport to Electrodes
`
`KEITH B. OLDHAM and CYNTHIA G. ZOSKI
`
`1. Introduction
`
`The investigation of electrode kinetics has one paramount advantage
`over other kinetic studies: the rate of the electron transfer reaction
`
`'
`
`Reactants + n e
`
`Products
`
`(1)
`
`can be measured directly rather than needing to be inferred from concen-
`tration changes. This advantage is a consequence of Faraday’s law, which
`asserts the proportionality of the electron-transfer rate
`
`1'
`nAF
`
`2
`
`)
`
`(
`
`Reaction rate =
`
`to the faradaic current 1' divided by the elctrode area A. In eqn. (2), n
`is the number of electrons and F is Faraday’s constant.
`On the other hand, electrode kinetic studies are at a disadvantage
`compared with investigations of homogeneous kinetics because con-
`centrations are not uniform and surface concentrations can rarely be
`measured directly (optical methods can sometimes provide direct measure-
`ment of the product concentration [1]). This means that the converse
`situation to that in classical homogeneous kinetics exists in electrode
`kinetics: concentration information needs to be inferred from reaction
`rates.
`
`the electrode surface requires a
`
`To calculate concentrations at
`knowledge of
`(a) the stoichiometry of the electrode reaction;
`(b) the bulk concentrations of the species involved;
`(c) the rate of the reaction [or equivalently, because of relationship
`(2), the faradaic current] since the onset of the experiment;
`(d) the laws governing mass transport
`for the particular electrode
`geometry; and
`(e) the prevailing experimental conditions.
`This chapter is concerned with how one uses items (a)-(e) to calculate
`concentrations at the electrode surface. In the electrochemical literature,
`expressions for surface concentrations are seldom regarded as the end
`result of a transport prediction. Instead, one usually assumes that the
`surface concentrations of the species involved in the electrode reaction
`
`References pp. 1 41—1 43
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`80
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`thermodynamic relation (nernstian conditions) or a
`a
`obey either
`particular kinetic expression (volmerian conditions), so that the result of
`the analysis of mass transport can be presented as a relationship between
`the experimentally observable variables: current, cell potential, and time.
`In this chapter, we shall primarily report relationships involving concentra-
`tions, since these are the kinetically significant variables.
`
`1.1 SPECIES INVOLVED IN TRANSPORT
`
`specifically exclude electrode
`shall
`this chapter, we
`Throughout
`reactions that consume or generate insoluble species. Thus,
`the most
`general electrode reaction is
`
`VA A(soln) + VB B(soln). .
`
`. i n e——> VzZ(soln) + VY Y(soln) +. .
`
`.
`
`(3)
`
`. are reactant species
`.
`where the 12s are stiochiometric coefficients, A, B. .
`and Z, Y. .
`. are product species. Usually, all the species are dissolved in
`the electrolyte solution, but an important exception occurs in the
`reduction of certain metal
`ions at a mercury electrode to produce an
`amalgam
`
`M"+(soln) + n e (Hg)———>M(amal)
`
`(4)
`
`Cases in which one of the reactants or products is the material of the
`electrode itself, as in
`
`2 Hg(l) _ 2 e (Hg)—_’ Hg? (301)
`
`or
`
`CuCl§‘(aq) + 2 e (Cu) ——>Cu(s) + 4 Cl‘(aq)
`
`are not excluded.
`
`(5)
`
`(6)
`
`There is, of course, always an ample supply of the electrolytic solvent
`(often water) and of the electrode material (often a metal) at
`the
`interface, but all other species must be transported to and from the
`electrode surface as illustrated in Fig. 1.
`Usually,
`there is no significant impediment to the transport of elec-
`trons,
`through this may not be true of some semiconductor electrodes
`[2—7]. When there is more than one reactant, it is usually possible to
`adjust bulk concentrations (or adopt other experimental strategies such as
`buffering) so that all reactants except one are in such excess that their
`transport poses no difficulty. This is an analog of the “isolation technique”
`familiar to kineticists. The same is true of product species: it is generally
`possible to arrange experimental conditions so that, at most, only one
`product species is subject to a transport restriction.
`Hence, we shall customarily ignore all but one reactant species and all
`but one product species. Moreover, we shall assume that the stoichio-
`metric coefficients of these species are both unity. This is not an essential
`
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`

`81
`
`919C
`
`thong
`
`electrode
`
`[25
`
`reacton
`
`solution
`
`(amalgam)
`
`Product-.5
`
`T
`91 90 trade
`sur‘Face
`
`Fig. 1. Transport to and from the electrode surface.
`
`assumption, but it does serve to simplify our arguments and covers the
`majority of practical examples. Thus, for a reduction experiment, the
`electrode reaction may be abbreviated to
`
`O(soln) + n e —> R(soln)
`
`(7)
`
`where O (for oxidized species) is the single reactant species we need
`consider and R (for reduced species) is the sole product species under
`consideration. Of course, either 0 or R or both may be ions.
`Sometimes, we shall address an even simpler class of electrode reaction
`in which there is only a single electroactive species of a varible activity.
`The simplest
`instance of this class is the reduction of metal
`ions on
`a cathode composed of that metal, for example
`
`M"+(soln) + n e(M)—>M(s)
`
`(8)
`
`This reaction is the only one treated in Sect. 4 of this chapter.
`It will be our custom to deal with cathodic electrode reactions, i.e.
`with reductions like (7 ) and (8), rather than with the equally important
`oxidation processes. For this reason, cathodic currents will be treated as
`positive*.
`
`1.2 THE ELECTRODE SURFACE
`
`The region extending from the phase boundary out to about 3nm is
`quite unlike the solution beyond. Generalizations valid elsewhere in the
`solution do not necessarily apply here. In this inner zone, the so-called
`double-layer region [9], we may encounter a violation of the electro-
`neutrality condition (see' Sect. 4.1) and large electric fields. Concen-
`trations may be enhanced or depleted compared with the adjacent
`solution.
`
`* This is the usual convention in electroanalytical chemistry, though it is at variance
`with the more logical IUPAC convention [8].
`
`References pp. 141—143
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`82
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`Through phenomena inside the double-layer region may have profound
`effects on the kinetics of electrode reactions [10] , the zone is fortunately
`of little or no consequence in discussions of transport from the solution
`to the electrode or vice versa. To appreciate why this is so, consider
`a typical electrochemical experiment in which species R is being generated
`at an electrode by a current of density 1.0 A m'z. After 1 s of electrolysis,
`the transport zone (the region into which R is being carried) will extend
`into the solution a distance of some 10 ,um, in comparison with which the
`double-layer
`thickness of 3 nm is negligible, Moreover, each electro-
`generated R molecule will
`transit the double-layer region in less than
`200 us.
`in this chapter, concentrations “at the electrode
`When we discuss,
`surface”, we will generally ignore the effect of the double layer. Hence,
`“at the electrode surface” really means “outside the double-layer” or,
`more precisely, “extrapolating to the electrode surface -the concentration
`profile from beyond the double layer”, as illustrated in Fig. 2.
`The narrow double-layer zone is the site of all the chemical and physico-
`chemical processes that attend the electrode reaction. These may include,
`in addition to the electron transfer itself, adsorption and desorption steps,
`as well as chemical transformation between 0 and R (which are the
`species stable in the bulk of the solution) and modified species (with less
`solvation, perhaps, or with different configurations) o and r which are
`adsorbable. Thus, the complete train of events may be
`
`O(soln)
`
`R(soln)
`double-1a er region
`trans-
`transport
`y
`port
`O(soln) fi O(soln) <——> o(ads)X—:r(ads) = r(soln) # R(soln)
`
`or some even more elaborate scheme. Most of these complexities need not
`concern us in this chapter, but should be noted.
`Let [‘0 and PR (with units of mol m'z) denote the amounts of O and
`R (or their modified forms 0 and r) that are adsorbed. The shaded area in
`
`concentration
`
`" surface
`concentration "
`\a
`
`
`> distance
`H————9
`double layer
`
`Fig. 2. Illustration of electrode “surface concentration”.
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`83
`
`Fig. 2 illustrates the significance of F. The validity of certain equations
`that we shall develop in Sect. 2.2 requires that F0 and PR remain
`constant. Certain experimental methods, both thermodynamic and
`kinetic*, can be used to investigate the extent of adsorption on electrodes,
`but these are beyond the scope of the present discussion.
`
`1.3 ELECTRODE GEOMETRY
`
`We stipulate the electrode to be smooth (though not necessarily flat)
`and of constant area A. By “smooth” we mean that any undulations in
`the electrode surface should not exceed the thickness of the double layer.
`For an electrode that is less smooth than this, the concept of electrode
`area is somewhat vague and the “effective electrode area” may change
`with time. By prescribing a constant electrode area, we exclude one of
`the most practical electrodes: the dropping mercury electrode treated in
`.Chap. 5.
`Where it is possible to define the coordinate unambiguously, we shall
`use x to denote distance measured normal to the electrode into the trans-
`
`port medium, with x = 0 corresponding to the electrode surface. We
`may distinguish three simple types of electrode geometry: planar, convex,
`and concave, as shown in Fig. 3. Transport to a planar electrode occurs
`along parallel
`lines. For a convex electrode, transport to the electrode
`is convergent, whereas transport away from the electrode is divergent.
`The opposite is
`true for a concave electrode. The case of a convex
`mercury electrode at which reaction (4) occurs is unusual but important;
`there, the transport of both 0 and R occurs convergently.
`is
`The region around the electrode, through which transport occurs,
`filled with solution and should usually be unimpeded by cell walls, other
`electrodes, etc. for a sufficient distance. How far is “sufficient” depends
`upon the mode of transport and on the duration of the experiment. For
`transport by diffusion alone, the requirements are very modest indeed,
`as indicated in Table 1.
`The electrode at which the reaction under study is proceeding is called
`the “working electrode”. It is with this one electrode that we are solely
`concerned. There is, however, an important exception: the experiments
`
`/<:
`
`:C
`/C
`
`(a)
`
`/
`
`/<:
`/V§
`
`/
`
`:<:
`/ fl/
`
`(b)
`
`'
`
`(c)
`
`Fig. 3. Types of electrode geometry. (a) Planar; (b) convex; (c) concave. The arrows
`indicate transport lines to the electrode surface.
`
`* Especially chronocoulometry [11—14].
`
`References pp. 141—143
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`84
`
`TABLE 1
`
`Unimpeded distances from the electrode surface
`
`
`
` Duration of experiment/s Unimpeded distance/mm
`
`0.06
`0.1
`0.2
`1.0
`0.6
`10.0
`
`100.0 2.0
`
`discussed in Sect. 4, in which there are two working electrodes. In those
`cases,
`it is important that the two working electrodes be parallel and
`separated by a rather narrow gap.
`
`1.4 FARADAIC AND NON-FARADAIC CURRENTS
`
`Two distinctly different types of current may flow at the electrode
`surface. One kind includes those in which electrons are transferred
`across the interface between the electrode and the solution. This electron
`
`transfer causes a chemical reaction, either oxidation or reduction (Sect.
`1.1)
`to occur. These reactions are governed by Faraday’s law, which
`asserts the proportionality of the electron-transfer rate to the current
`[eqn. (2)] . Accordingly, this current is called a faradaic current.
`A second type of current arises due to the presence of the electro-
`chemical double layer (Sect. 1.2). Additionally, a current may flow due
`to the adsorption or desorption (Sect. 1.2) or species 0 and R as well as
`electroinactive species. In these instances, no chemical reaction occurs
`and consequently electrons are not transferred across the electrode—
`solution interface. However, a current may flow elsewhere and this
`current is called a non-faradaic current.
`Both faradaic and non-faradaic currents may flow when an electrode
`reaction occurs. Thus, the total current which flows is often the sum of
`the faradaic and non-faradaic contributions to the current. Most often,
`it is the faradaic current that is of interest. Many electrochemical tech-
`niques have been developed which minimize or eliminate this non-faradaic
`contribution to the current, but discussion of these is beyond the scope
`of the present chapter.
`In Sect. 1.2, we chose to ignore the effects of the double layer as well
`as complexities due to adsorption and desorption processes. Similarly,
`we will also choose to ignore the presence of non-faradaic currents,
`though in practice they may be important. Hence, throughout this chapter
`only faradaic currents will be considered.
`
`1. 5 KINETIC STRATEGY
`
`Electrochemical studies are performed for many reasons, but in this
`chapter our preoccupation is with experiments carried out for the purpose
`
`InnoPharma Exhibit 1104.0008
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`

`
`surFace
`
`
`
`
`
`F1
`electric
`
`current
`UXQS
`
`
`initial
`conditions
`
`kinetic relations
`
`trons ort relations
`P
`
`85
`
`surFoce
`
`electric
`Potential
`
`concentrations
`O; 0 and R
`
`(boundary
`
`,
`
`gametrlc
`5
`conditions
`
`condition)
`
`Fig. 4. Linkage between kinetics and mass transport. Quantities enclosed by a single
`frame are (generally) functions of time only, while those enclosed by a double frame
`are functions of spatial variables as well as time.
`
`serve, Current “\53231\\$fl9
` predict
`
`
`
`
`current
`
`
`
`
`otential
`
`JM 032
`
`
`
`P
`
`
`
`
`ob
`
`Experiment
`
`initial and
`
`geometric
`conditions
`
`
`interrelationship 0F
`surFace concentrations
`
`
`
`and surFace Fluxes
`surFace Fluxes
`
`
`
`concentration
`
`
`transport
`
`and
`Volmerion
`problem
`
`kinetics
`
`
`
`Flux profiles
`
`
`Fig. 5. Indirect method for determining electrode kinetics. In this method, a particular
`kinetic law has been assumed.
`
`of determining the kinetics of the electrode reaction. At first sight, it is
`not evident how mass transport,
`i.e.
`the motion of the electroactive
`species 0 and R through the solution,
`interacts mathematically with
`the elctrode kinetics, which manifests itself only at the interface between
`the solution 'and- the electrode. The interaction is not direct but occurs
`
`via the surface concentrations (see Sect. 1.2) and surface fluxes (see
`Sect. 2.2), as illustrated in Fig. 4.
`The only variables that are generally accessible in the scheme portrayed
`in Fig. 4 are the current and the potential; in a typical electrochemical
`experiment, one of these variables is imposed on the cell and the other is
`observed. How,
`then, does one learn anything about the kinetics by
`making use of the known laws of transport? There are two strategies for
`doing this.
`The most common strategy is illustrated in Fig. 5. A potential (constant
`or varying) is imposed on the cell and the current—time relationship is
`monitored.
`In the theoretical segment of the study, one assumes a
`
`References pp. 1 41—1 43
`
`InnoPharma Exhibit 1104.0009
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`

`86
`
`
`
`observe
`
`current
`
`impose
`.
`wak%9
`potential
`
`
`Experiment
`
`initial and
`current
`
`
`
`geometric
`conditions
`
`
`
`
`interrelationship 0F
`PFEdIct
` surFace Fluxes
`SUFFOCQ concentrations
`
`
`
`concentration
`
`and
`
`
`Flux proFiles
`
`
`Nernst
`
`
`equation
`
`
`
`transport
`problem
`
`
`
`Fig. 6. Indirect method in which the electrode reaction is assumed to be reversible.
`
`
`
`1 MPOSQ
`
`otentiol
`Experiment
`
`
`
`
`seek kinetic low
`
`
`observe
`
`
`current
`
`semianQgrate
`surface
`concentrations
`
`
`
`Fig. 7. Direct method for determining electrode kinetics. Contrary to the indirect
`method portrayed in Fig. 5, no particular kinetic law has been assumed.
`
`particular kinetic law (usually volmerian kinetics, discussed in Sect. 3.5)
`by means of which an equation linking current, surface concentrations,
`and time is evolved. With the aid of this equation, the transport problem
`can be solved, generating expressions for the concentration and flux
`profiles of O and R. Then, the fluxes of O and R at the electrode surface,
`and hence the current, can be predicted. Finally, comparison of the
`measured current with the predicted current
`is used to verify the
`adequacy of the kinetic assumption and to provide values of the kinetic
`parameters.
`Commonly, the electrode reaction is assumed to be “reversible”, which
`simplifies the mathematics considerably because a direct prediction of
`surface concentrations is possible from the potential alone, as in Fig. 6.
`However, kinetic information is entirely lacking from experiments
`conducted under reversible conditions since the electrode reaction is then
`
`an equilibrium.
`The second strategy which may be used to learn about the kinetics of
`an electrode reaction is illustrated in Fig. 7. As before, a potential
`(constant or varying) is imposed on the cell and a current—time relationship
`is monitored. However, instead of assuming a particular kinetic law, one
`processes the experimental current by semi-integration (see Sects. 5.2 and
`5.4), thus enabling the surface concentrations to be calculated directly.
`Hence, the kinetics can be elucidated by a study that involves only the
`
`InnoPharma Exhibit 1104.0010
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`

`

`87
`
`electrical variables: potential and current, together with the semi-integral
`of the current. This is the direct method of elucidating electrode kinetics
`and is discussed in Sects. 3.5 and 5.5.
`
`1.6 SYMBOLS AND UNITS
`
`The symbols listed below are used throughout this chapter. In most
`cases, the usage follows the recommendations of the IUPAC Commission
`on Electrochemistry [8] ; however, there are exceptions.
`
`electrode area (m2)
`constant of the j th component (Table 6)
`constant (Table 6)
`capacitance (F)
`capacitance at the input of an operational amplifier (F)
`capacitance of capacitors j and j — 1 (F)
`concentration (mol m'3)
`concentration of electroactive species at the cathode
`surface (mol m'3)
`bulk concentration of the jth ionic species (mol m‘3)
`total initial bulk ionic concentration (mol m"3 )
`bulk concentrations of species 0 and R (mol m"3)
`concentration of species 0 and R at the electrode surface
`(mol m'3)
`'
`concentration at distance x and time t (mol m‘3)
`total
`ionic concentration at distance x and time t
`
`(mol m'3)
`con3centration of species j at distance x and time t(mol
`m" )
`concentration of species 0 and R at distance x and time
`t (mol m‘3)
`Laplace transform of co (x, t) and cR (x, t) (mol m"3 s“)
`steady-state total
`ionic concentration at distance x
`(mol m'3)
`steady-state concentration of species j at distance x
`(mol m'3)
`surface concentrations of species 0 and R at time t (mol
`m'3)
`diffusion coefficient (m2 s' 1)
`diffusion coefficient of species j (m2 s“)
`diffusion coefficient of species 0 and R (m2 s“)
`electrode potential (vs. some reference electrode) (V)
`null or equilibrium potential (V)
`standard potential (V)
`interelectrode potential (V)
`voltage input and output of a circuit (V)
`
`i
`
`cJ-(x, t)
`
`00 (x, t), CK (x, t)
`
`50 (x, s), 51105: s)
`0105)
`
`0,-(x)
`
`080), 0130)
`
`D D
`
`i
`DO ,DR
`
`E E
`
`n
`
`Eo
`
`Ea —Ec
`Ein:Eout
`
`References pp. 141—1 43
`
`InnoPharma Exhibit 1104.0011
`
`

`

`oo 00
`
`electron
`
`Faraday’s constant (96 485 C mol‘l)
`denotes a functional relationship
`constant given by eqn. (161)
`faradaic current (A)
`jth and (i + 1)th faradaic current values (A)
`limiting faradaic current (A)
`faradaic current during the forward and reverse branches
`of a cyclic voltammogram (A)
`exchange current (A)
`transport-free current (Sect. 5.5) (A)
`faradaic current as a function of time (A)
`flux (mol In"2 S”)
`flux of speciesj (mol m‘2 s”)
`flux at distance x at time t (mol In‘2 5—1)
`flux of species j at distance x at time‘ t (mol m‘2 S“)
`flurges of species 0 and R at distance x at time t (mol
`m"
`s‘ )
`steady-state flux of species j (mol In"2 5—1)
`counter or index,j = 1,2,3, .
`. .N
`standard heterogeneous rate constant (m 5‘1)
`heterogeneous rate constants for forward and backward
`reactions (m 5‘1)
`distance between two parallel electrodes (m)
`Laplace transformation symbol
`molecular weight (kg mol‘ 1)
`metal
`metal ion
`
`1/2)
`semi-integral of the faradaic current (A s
`faradaic semi-integral during the forward and reverse
`branches of a cyclic voltammogram (A 51/2)
`total number of ionic species (Sect. 4.0); upper limit
`of counter (Sect. 5.4)
`Avogadro’s constant (6.02205 x 1023 mol'l)
`number of faradays to reduCe one mole of the reducible
`species
`oxidized species
`modified oxidized species at the electrode surface
`arbitrary function in Laplace space [eqn. (142)]
`arbitrary function in Laplace space
`charge (C)
`unit charge (C)
`gas constant (8.31441 J mol'1 K'l)
`reduced species
`resistance (ohm)
`
`InnoPharma Exhibit 1104.0012
`
`‘3'c, + H
`
`
`
`
`
`r-57¢:-H-OnH:'11m.n
`
`l
`
`«a?
`
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`i*
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`«II-(x, t)
`J0 (x: t)! JR (x9
`
`Jj(x)
`1'
`k0
`
`kf,kb
`
`L Y
`
`

`

`89
`
`input resistance to an operational amplifier (ohm)
`jth and (j —- 1)th resistances in a circuit (ohm)
`radius (m)
`modified reduced species at the electrode surface
`denotes “solid”, as in M(s)
`dummy variable in Laplace space (s’l)
`thermodynamic temperature (K)
`time interval in a cyclic experiment (5)
`time (5)
`lower time limit of semi-integration (s)
`mobility of an ionic species (m2 s'1 V“)
`mobility of a particular ionic species j (m2 5‘1 V'l)
`average velocity (m s”)
`average velocity at time t (m s”)
`local electric field (V m'l)
`distance measured normal to the electrode surface (m)
`charge number
`charge number of a particular ionic species j
`transfer coefficient
`
`surface excess concentration (mol m_2)
`surface excess concentrations of species 0 and R (mol
`m" )
`~
`activity coefficient
`constants given by eqns. (75) and (94)
`time interval (5)
`energy required to carry an ion from distance x = 0 to
`x = x (J)
`viscosity (kg m"1 s")
`overpotential (V)
`equivalent
`ionic conductivity of an ion (m2 ohm“l
`equiv. '1)
`order of a derivative (Table 6)
`electrochemical potential (J mol‘l)
`standard electrochemical potential (J mol'l)
`local ionic strength at distance x and time t (mol m'3)
`local ionic strength at steady state (mol m‘3)
`order of a derivative (Table 6)
`stoichiometric coefficient of species j
`support ratio
`integration variable (5)
`time of sudden change in imposed conditions after an
`experiment commences;
`transition time in a chrono-
`potentiometric experiment (s)
`local potential (V)
`reference potential (V)
`
`a. v
`.5. :U
`
`-.
`
`I
`
`p.-
`
`Suqqmmwwww
`
`GQ! A N V
`
`figiflmax
`
`"1O "1 pa
`
`7
`7, 71
`A
`do —> x)
`
`V33
`
`)1
`fl
`u
`u(x, t)
`M(x)
`V
`V,
`,0
`
`T 7
`
`.
`
`as
`
`References pp. 141—143
`
`InnoPharma Exhibit 1104.0013
`
`

`

`90
`
`¢(x,t)
`¢(x)
`
`local potential at a distance x and time t (V)
`local potential in the steady state (V)
`
`2. Modes of transport
`
`There are three distinct mechanisms by which electroactive solutes
`from the solution may reach the electrode or, conversely, by which
`electrogenerated solutes may be transferred into the solution. These are
`migration, diffusion, and convection.
`Migration is perhaps the easiest to understand. The presence of an
`electric field causes charged particles to move along the field lines; this is
`migration. The force experienced by the particle is proportional to its
`charge and to the electric field (i.e.
`the electrical potential gradient).
`Migration affects ions only; neutral species do not migrate.
`Diffusion does not occur in response to any physical force; it occurs
`because an inhomogeneous solution seeks to maximize its entropy.
`Explained another way, the Brownian motion that all solute particles
`undergo inevitably tends to enrich regions of low local concentration at
`the expense of neighbouring regions of greater concentration. Diffusion
`affects charged and uncharged species equally; both ions and molecules
`diffuse.
`
`the solute particle moves
`In transport by migration or diffusion,
`through a stationary solvent. Convection is a totally different process in
`which the solution as a whole is transported. Solute species reach or leave
`the vicinity of the electrode by being entrained in a moving solution.
`A concise summary of the distinctions between migration, diffusion,
`and convection is that migration occurs in response to a potential gradient,
`diffusion in response to a concentration gradient, and convection in
`response to a pressure gradient.
`
`2.1 CONVECTIVE TRANSPORT
`
`We may distinguish two kinds of convection [15, 16] : forced convection
`and natural convection. Forced convection is the result of some motion
`
`deliberately introduced by the experimenter; natural convection arises
`from changes brought about as a result of the electrolysis itself.
`Stirring the solution,
`rotating the electrode, bubbling gases, and
`pumping solution towards or across the electrode are all methods of
`inducing forced convection. Such methods yield irreproducible results
`unless careful attention is paid to the geometry of the system and to
`ensuring uniformity of the imparted motion. Rotating disks and wall-
`jet devices are among the most reproducible examples of forced con-
`vection . These are the subject of Chap. 5 and will not be further discussed
`here.
`
`InnoPharma Exhibit 1104.0014
`
`

`

`91
`
`The most usual sequence of events leading to natural convection starts
`when electrolysis generates a region close to the electrode in which a
`concentration is significantly enhanced or depleted. Usually, a solution
`that is enriched in a solute has a slightly greater density than the bulk
`solution and therefore, under the influence of gravity, there is a tendency
`for enriched regions to fall. Conversely, a region of depleted concentration
`tends to rise as a result of its slightly diminished density.
`The extent to which a solution is able to respond to these gravitational
`tendencies depends on the geometry of the cell (see Sect. 4). Even under
`the worst conditions, however, the driving force for natural convection
`is small and it takes a considerable amount of time for this small force
`
`to overcome the inertia of the solution mass. Accordingly, natural con-
`vection is unimportant in rapid experiments. In fact, its effect seldom
`appears before about 10 or 20 s, which is long in relation to most electro-
`chemical experiments.
`Henceforth in this chapter, convection will be ignored. In other words,
`we shall treat only experiments in which forced convection is absent and
`in which natural convection is inhibited either by judicious geometric
`design of the cell or by the brevity of the experiment.
`
`2.2 FLUX
`
`The number of moles of a solute being transported across unit area of
`a surface in unit time is known as the flux at that surface; it has units of
`mol m"2 S”. The flux, J, may be equated to the average velocity, v, of
`the individual solute molecules in the direction normal to the surface
`
`multiplied by their concentration, c
`
`J(x, t) = vc(x, t)
`
`(9)
`
`If the surface in question is the electrode and the solute in question is
`the electrogenerated species R,
`then the flux is equal
`to the rate of
`generation of R, i.e. the rate of the electrode reaction
`
`Rate of reaction = JR (0,t)
`
`(10)
`
`Similarly, the flux of the electroactive species 0 at the electrode surface
`is given by
`
`Rate of reaction = — Jo (0, t)
`
`(11)
`
`where the negative sign arises because of the convention that flux is
`treated as positive when it occurs in the direction of increasing x, i.e. away
`from the electrode.
`
`The third component involved in the electrode reaction
`
`O(soln) + n e —> R(soln)
`
`(12)
`
`is electrons. Their flux at the electrode surface must also be proportional
`to the rate of reaction in accordance with Faraday’s law and eqn. (2).
`
`References pp. 141—143
`
`InnoPharma Exhibit 1104.0015
`
`

`

`92
`
`These results together lead to the very important relationship
`
`i(t)
`nAF
`
`JR(0,D = "Jo(0,t) =
`
`(13)
`
`in which the sign of the current i(t) reflects our choice of cathodic current
`as positive.
`Expression (13) implies that, for every n electrons that are Withdrawn
`from the cathode, one molecule (or ion) of R is transported away from
`the electrode. This may be untrue transiently if there is significant
`adsorption of R. More precisely, expression (13) will be invalid whenever
`the amounts of adsorption PR and [‘0 of R and 0, respectively, (see
`Sect. 1.2) are changing significantly with time. Correcting for this effect
`leads to
`
`J0t+d—F—Rt———JOt——£9t—‘
`
`
`i(t)
`nAF
`
`14
`
`the
`it will henceforth be assumed that
`Unless we state otherwise,
`derivatives in expression (14) are negligible, so that expression (13) can
`be used.
`
`2.3 LAWS OF MIGRATION
`
`Consider a region of solution in which the local potential ¢ changes
`along the x axis. Then, — a¢/ax denotes the local electric field X. This
`field acts upon any charge q to produce a force qX acting along the x
`axis. If the charge is that on an ion of charge number z, so that q =
`zF/NA , where NA is Avogadro’s constant, then
`
`zFX
`
`Ah
`
`(15)
`
`_
`Electrostatlc force =
`
`If the ion is in a fluid medium, then the electrostatic force seeking to
`accelerate the ion is opposed by a viscous force trying to slow the ion
`down. Though it
`strictly applies only to spheres of macroscopic
`dimensions, Stoke’s law can provide an approximate expression
`
`Viscous force = 67rnrv(t)
`
`(16)
`
`for the viscous drag, n being the coefficient of viscosity, r the ion’s radius
`and v(t) its velocity an instant t.
`If M/NA is the mass of the ion, then from Newton’s second law of
`motion
`
`M dv
`
`NA dt
`
`= electrostatic force — viscous force =
`
`
`zFX
`
`NA
`
`— 67rnrv(t)
`
`(17)
`
`InnoPharma Exhibit 1104.0016
`
`

`

`
`
`velomty v
`
` limiting velocity
`
` timet
`
` 6—)
`time
`constant
`
`Fig. 8. Velocity versus time curve for an ion in solution.
`
`On integration, one finds
`
`t
`1’0
`
`zFX
`= -——-——-—
`61rNA'qr
`
`1
`
`-—e
`
`[~61rNAnrt
`——-——.——.—
`M
`
`Xp
`
`(18
`
`)
`
`if the ion was initially stationary. This equation corresponds to an ion
`experiencing an initial acceleration of zFX/M but settling down to a
`steady limiting velocity of zFX/(67rNAnr) after times long in comparison
`with the time constant (M/61rNAnr) as shown in Fig. 8.
`For common inorganic ions in aqueous solution, one can take the
`typical values M 2 0.1 kgmol”, 17 1 1.0 x 10'3 kgm"ls"1, z = i (1 or 2),
`r2’0.3 nm and calculate a time constant of the order of 3x 10'14 s.
`
`Evidently, the limiting velocity is acquired virtually instantaneously, so
`that the adjective “limiting” may be dropped and v(t) contracted to v.
`Using the same typical values gives the (velocity/ field) ratio as
`
`(19)
`
`2 F
`= l—'— R 4x10‘3m2V'1s‘l
`67rNA77r
`
`
`
`s X
`
`
`
`This quantity is known as the mobility of the ion and is given the symbol
`u. Some experimental values are given in Table 2 for ions in dilute aqueous
`solution.
`
`Mobility is accorded a positive sign irrespective of the sign of the ion’s
`charge, though of course the direction of motion does depend on the sign,
`cations travelling with the field X (i.e. down the potential gradient a¢/ 8x)
`and anions in the opposite direction. These signs are taken into account
`in the equation
`
`qu _
`121
`
`an EMS
`~ —— ——
`lzl ax
`
`v =
`
`(20)
`
`for the velocity of migration. Util