Mass transport of charged species in solution

One of the most fundamental parts of electrochemistry is that devoted to the study of the production of electrical energy and current by means of the transformations of chemical species that are able to take part in electron transfers with charged interfaces [1,2,3,4]. Heterogeneous electron transfer processes are strongly affected by the presence of mass transport of the electroactive species from/to the bulk of the solution to/from the electrode surface due to the appearance of a concentration gradient because of their disappearance/appearance at the electrode surface. This situation gives rise to the fact that the study of the electroactive species fluxes is crucial for the comprehension of these processes.

Let us consider a chemical species i with charge \(z_{{\text{i}}}\) and concentration \(c_{{\text{i}}}\) in an electrolytic solution that moves under the influence of a driving force. The overall flux of this species, \(J_{{\text{i}}}^{{}}\), at constant temperature and pressure, is equal to the minus gradient of the electrochemical potential plus an additional term related to the motion of the solution with a speed v,

$$J_{{\text{i}}}^{{}} = - u_{{\text{i}}}^{{\text{c}}} \frac{RT}{{\left| {z_{{\text{i}}} } \right|F}}\nabla c_{{\text{i}}} - c_{{\text{i}}} \frac{{z_{{\text{i}}} }}{{\left| {z_{{\text{i}}} } \right|}}u_{{\text{i}}}^{{\text{c}}} \nabla \phi + c_{i} v$$
(1)

where \(u_{{\text{i}}}^{{\text{c}}}\) is the charge mobility and \(\nabla c_{{\text{i}}}\) and \(\nabla \phi\) are the gradients of the concentration of species i and the electric potential in solution, respectively [1,2,3,4]. Taking into account the Nernst-Einstein relationship between the ionic charge mobility and the diffusion coefficient of an ionic species \(D_{{\text{i}}}\),

$$u_{{\text{i}}}^{{\text{c}}} = \frac{{\left| {z_{{\text{i}}} } \right|F}}{RT}D_{{\text{i}}}$$
(2)

it is possible to re-write Eq. (1) as

$$J_{{\text{i}}}^{{}} = - D_{{\text{i}}} \nabla c_{{\text{i}}} - c_{{\text{i}}} \frac{{z_{{\text{i}}} }}{{\left| {z_{{\text{i}}} } \right|}}u_{{\text{i}}}^{{\text{c}}} \nabla \phi + c_{i} v$$
(3)

The relationship between the current (I) and the total flux of N charged species is given by:

$$I = FA\sum\limits_{{i{ = 1}}}^{N} {z_{i} J_{i} }$$
(4)

where A is the electrode area and F the Faraday constant. Equations (1) and (4) clearly show that the current has three components. The first one, originated from a gradient of concentration, is called the diffusive flux (first term on the right-hand side of Eq. (3)); the second one is due to a gradient of electric potential, and it is called migrational flux (second term on the right-hand side of Eq. (3); and the third one is the so called convective flux due to the motion of the solution with speed v.

The migration contribution to the flux of species “i” can be minimized by the addition of an inert (or supporting) electrolyte in a concentration of 2 orders of magnitude higher to than that of species “i.” These electrolyte ions should be electrochemically and chemically inert but increase the ionic strength of the solution. Moreover, the electrolyte must show high mobility so that the contribution to the migrational current density due to the target species “i” decreases considerably. This effect can be clearly seen if we take into account the transport number of species i, defined as [2]

$$t_{i} = \left| {z_{{\text{i}}} } \right|F\frac{{c_{{\text{i}}} u_{{\text{i}}}^{{\text{c}}} }}{\kappa }$$
(5)

with \(\kappa\) being the electrical conductivity of the solution.

The convective contribution to the overall current can be neglected if the solution is in total rest and not very long times are considered.

Under the above conditions, the overall flux (and therefore the current) is practically governed by the diffusive mass transport. The diffusive flux appears as a consequence of a concentration gradient and is given by:

$$J_{{\text{diffusion,i}}} = - D_{{\text{i}}} \nabla c_{{\text{i}}}$$
(6)

which is known as Fick’s first law. By inserting Eq. (6) for the flux into the continuity equation for mass transport, an expression for the time evolution of the concentration of species i is obtained,

$$\frac{{\partial c_{i} }}{\partial t} = - {\text{div}}J_{{\text{i}}} = D_{{\text{i}}} \nabla^{2} c_{{\text{i}}}$$
(7)

with \({\text{div}}\,J_{{\text{i}}}\) being the divergence of the flux and \(\nabla^{2}\) the Laplacian operator for mass transport by diffusion, the expression of which for usual electrode geometries is given in Table 1.

Table 1 Expression of the Laplacian operator \(\nabla^{2}\) for different more usual electrode geometries [5]
Full size table

Fast (Nernstian) heterogeneous charge transfer reactions: the diffusion layer

The mathematical problem of the diffusive current for a fast charge transfer process as the following,

$$O+{e}^{-}\rightleftharpoons R$$
(I)

when a planar electrode is considered (linear diffusive mass transport) is given by:

$$\begin{gathered} \frac{{\partial c_{{\text{O}}}^{{}} (x,t)}}{\partial t} = D_{\rm{O}} \left( {\frac{{\partial^{2} c_{{\text{O}}}^{{}} (x,t)}}{{\partial x^{2} }}} \right) \hfill \\ \frac{{\partial c_{{\text{R}}}^{{}} (x,t)}}{\partial t} = D_{\rm{R}} \left( {\frac{{\partial^{2} c_{{\text{R}}}^{{}} (x,t)}}{{\partial x^{2} }}} \right) \hfill \\ \end{gathered}$$
(8)

subject to the conditions corresponding to semi-infinite diffusion, conservation of matter and reversible (Nernstian) electrode reaction:

$$\left. \begin{gathered} t = {0,}\;x \ge {0}\, \hfill \\ t > {0},\;x \to \infty \hfill \\ \end{gathered} \right\}\; \, c_{{\text{O}}} {(}x,t{)} = c_{{\text{O}}}^{*} \;, \, c_{{\text{R}}} {(}x,t{)} = c_{{\text{R}}}^{*} \quad$$
(9)
$$\begin{aligned} t > {0},&\;x = {0:} \\ \, D_\mathrm{O}\left( {\frac{{\partial c_{{\text{O}}} }}{\partial x}} \right)_{{x = {0}}} &+ D_\mathrm{R}\left( {\frac{{\partial c_{{\text{R}}} }}{\partial x}} \right)_{{x = {0}}} = {0} \end{aligned}$$
(10)
$${c}_{\mathrm{O}}^{s}={e}^{\upeta }{c}_{\mathrm{R}}^{\mathrm{s}}$$
(11)

with

$$\eta =\frac{F({E-E}^{0{\prime}})}{RT}$$
(12)

with E being the applied potential and \(E^{0^{\prime}}\) the formal potential of the redox couple O/R. Under the above conditions, it is demonstrated that the surface concentrations of the electroactive species are independent of time (see below) and the following solutions apply to the concentration profiles of species O and R as a function of the distance to the electrode surface (x) and time (t) [1,2,3,4]

$$\left. \begin{gathered} c_{{\text{O}}} \left( {x,t} \right) = c_{{\text{O}}}^{*} + \left( {c_{{\text{O}}}^{{\text{s}}} - c_{{\text{O}}}^{*} } \right){\text{erfc}}\left( {\frac{x}{{2\sqrt {D_{{\text{O}}} t} }}} \right) \hfill \\ c_{{\text{R}}} \left( {x,t} \right) = c_{{\text{R}}}^{*} + \left( {c_{{\text{R}}}^{{\text{s}}} - c_{{\text{R}}}^{*} } \right){\text{erfc}}\left( {\frac{x}{{2\sqrt {D_{{\text{R}}} t} }}} \right) \hfill \\ \end{gathered} \right\}$$
(13)

where erfc(x) is the complementary error function [6], \(c_{{\text{i}}}^{*}\) the bulk concentrations of species i, and \(c_{{\text{i}}}^{{\text{s}}}\) is their time-independent surface concentrations given by:

$$\left. \begin{gathered} c_{{\text{O}}}^{{\text{s}}} = {(}c_{{\text{O}}}^{*} + c_{{\text{R}}}^{*} {)}\left( {\frac{{e^{\eta } }}{{1 + e^{\eta } }}} \right) \hfill \\ c_{{\text{R}}}^{{\text{s}}} = {(}c_{{\text{O}}}^{*} + c_{{\text{R}}}^{*} {)}\left( {\frac{1}{{1 + e^{\eta } }}} \right) \hfill \\ \end{gathered} \right\}$$
(14)

An example of the concentration profile of the oxidized species O (\(c_{{\text{O}}} \left( {x,t} \right)\)), calculated for different time values upon the application of a constant potential under linear diffusion conditions is shown in Fig. 1. The electrode reaction at the interface leads to the depletion of species O at the solution region adjacent to the electrode surface, the thickness of which increases with time. An estimation of the magnitude of such thickness is given by the so-called linear diffusion layer, which is determined from the linearized concentration profile (dashed lines) as shown in Fig. 1.

Fig. 1
figure 1

Concentration profiles of species O at a planar electrode calculated from Eq. (13) for the application of a potential pulse a at a fixed time value t = 0.1 s and different values of the applied potential \({E-E}^{0{\prime}}\) (in mV) (indicated on the curves) and b at different values of time (in second) (indicated on the curves) for a fixed potential \({E-E}^{0{\prime}}=-500 \, \rm{ mV}\). Dashed lines correspond to their linear concentration profiles: \(c_{{\text{i}}} \left( {x \le \delta_{{\text{i}}} ,t} \right) = c_{{\text{O}}}^{{\text{s}}} + c_{{\text{O}}}^{*} \frac{x}{{\delta_{{\text{i}}} }}\), \(c_{{\text{i}}} \left( {x > \delta_{{\text{i}}} ,t} \right) = c_{{\text{O}}}^{*}\). \(D_{{\text{O}}} = 10^{ - 5} {\text{ cm}}^{2} {\text{ s}}^{ - 1}\). Reproduced with permission from [2]

Full size image

The surface gradient of species O or R can be easily obtained by differentiating the expressions of the concentration profiles given in Eq. (13), obtaining

$$\left( {\frac{{\partial c_{{\text{i}}} }}{\partial x}} \right)_{x = 0} = \frac{{c_{{\text{i}}}^{*} - c_{{\text{i}}}^{{\text{s}}} }}{{\sqrt {\pi D_{{\text{i}}} t} }}{\text{ (i}} \equiv {\text{O, R)}}$$
(15)

The linear diffusion layer (hereafter, the diffusion layer) can be defined from the expression of the surface concentration gradient on the basis of Eq. (15) as the zone adjacent to the electrode surface where the concentration of electroactive species is disturbed. Therefore, from the above equation, it is immediately deduced that its expression is given by [2]:

$$\delta_{{\text{i}}} = \sqrt {\pi D_{{\text{i}}} t}$$
(16)

As can be inferred from Fig. 1a and b, the thickness of the diffusion layer of a fast electrode reaction is independent of the applied potential (Fig. 1a) and dependent only on time (Fig. 1b).

Note that for the case under study here (fast or Nernstian electrode reaction), the thickness of the diffusion layer is practically coincident (with the exception of a small constant value) with that obtained from the Einstein–Smoluchowski equation for a stochastic displacement (random walk). In the case of linear diffusion, from which the mean squared displacement of a particle, obtained as a result of many microscopic displacements in random directions due to collisions is proportional to the elapsed time, this is given by [7]

$$\left\langle {x^{2} } \right\rangle = 2D_{{\text{i}}} t$$
(17)

It should be taken into account that the expression obtained in Eq. (16) is rigorous and based on the linear diffusion model. Nevertheless, as it is logical to expect, the diffusion layer is strongly affected by the geometry of the diffusion field under study [8]. Thus, in Table 2, the expression for the diffusion layer are given for fast charge transfer reactions without chemical complications for different types of diffusion fields.

Table 2 Expressions for the thickness of the linear diffusion layers under transient (\(\delta\)) and stationary (\(\delta_{{{\text{ss}}}}\)) conditions for several electrode geometries for an electroactive species of diffusion coefficient D [2, 11]
Full size table

It is also important to consider that the diffusion layer would be affected by the rate of the electrode process since, for slow charge transfer processes, the diffusive mass transport and the electron transfer process take place at the same time scale so they compete with each other. This situation will be not considered in this work, and it is analysed in detail in references [9, 10].

Homogeneous first-order chemical reactions coupled to the heterogeneous charge transfer: the reaction layer

A historical perspective

Let us now consider a more complicated situation as it is the case of a first-order or pseudo-first-order chemical reaction coupled to a charge transfer process. For this type of processes, it is convenient to introduce the concept of reaction layer. This concept was initially presented by Wiesner in 1943 in an intuitive form in different polarographic studies devoted to the study of kinetic currents associated to the oxidation of leucoforms of some dyes in solution in presence of colloidal palladium saturated with molecular hydrogen [12], together with different studies of the reduction of undissociated acidic forms [13], and the electrochemical behaviour of hydrogen peroxide in presence of ferric ions [14, 15].

Later on, in 1947 [16, 17], Wiesner carried out, in a not strictly justified way, the definition of the reaction layer, \(\mu\), based on statistic reasoning, as one half of the mean displacement of a random walk:

$$\mu = \frac{1}{2}\sqrt {2D\,\tau }$$
(18)

with \(\tau\) being the mean lifetime of a first order chemical reaction, that is, the reciprocal of the rate constant of the reaction that inactivates the electroactive molecule,

$$\tau = \frac{1}{k}$$
(19)

with k being the pseudo-first order rate constant.

Also in 1947, Koutecky and Brdička [18] treated rigorously the problem concerning the recombination of reducible acid molecules with an arbitrary number of donors and acceptors. To this end, they solved the differential equations involved in this problem by using the Laplace transform method under polarographic conditions, that is, for the dropping mercury electrode under the expanding plane model. They deduce that for the particular case in which the values of the pseudo-first order rate constants are large enough and the chemical equilibrium is shifted towards the electro-inactive species, it is fulfilled that:

$$\mu = \sqrt{\frac{D}{k}}$$
(20)

As can be seen, this expression is identical to that proposed by Wiesner (Eq. (18)) except for the factor \(\sqrt 2 /2\).

Later on, Hanus [19] expressed this quantity by means of Fick’s first law. As in the original concept, he considered that the equilibrium between the electro-active and the electro-inactive forms is maintained through most of the diffusion layer and that it changes only in the vicinity of the electrode corresponding to the reaction layer. Thus, the concentration gradient for the electroactive form in this example (CE mechanism), for the free aldehydic form (or formaldehyde), may be expressed by the approximate relationship:

$$\left( {\frac{\partial f}{{\partial x}}} \right)_{x = 0} = \frac{{\left[ f \right]_{r} - \left[ f \right]_{0} }}{\mu }$$
(21)

“where \(\left[ f \right]_{r}\) stands for the concentration at the external boundary and \(\left[ f \right]_{0}\) for the concentration close to the electrode surface” [20]. Under this formalism, the reaction layer can be defined as the region in which the concentration of the electroactive product resulting from the chemical reaction (in which an electro-inactive species is involved) differs from the value corresponding to the bulk of the solution (if it has been initially added) or zero (otherwise) [20].

As a summary, the concept of “reaction layer,” is based on an approximation and it was considered as the solution zone in which the concentration of the electroactive product resulting from the chemical reaction reaches the value corresponding to the bulk of the solution (if it has been added initially) or zero (on the contrary case).

In the following sections, we will discuss this concept as well as its influence on the electrochemical response under linear diffusive transport that is logically conditioned by the geometric shape of the working electrode employed.

A more general concept of reaction layer in the study of reaction mechanisms in electrochemistry

As has been previously indicated, the reaction layer is, in general, an approximate concept. When a chemical reaction is coupled to an electron transfer, the reaction layer can be defined, in a general way, as the zone adjacent to the electrode surface in which the chemical equilibrium is disturbed, i.e., the zone in which it is fulfilled that if species B and C are related by a (pseudo) first-order chemical reaction with kinetic rate constants \(k_{1}\) and \(k_{2}\) given by scheme II with the concentration profiles of these species depending both on the chemical reaction and on the presence of mass transport. For the resolution of the mathematical problem, it is useful to define the function given by Eq. (22):

$${\text{B}}\underset{{k_{2} }}{\overset{{k_{1} }}{\longleftrightarrow}}{\text{C}}$$
(II)
$$\phi (x,t) = c_{B} (x,t) - Kc_{C} (x,t)$$
(22)

where K is the inverse of the chemical equilibrium constant

$$K = \frac{{c_{{\text{B}}}^{*} }}{{c_{{\text{C}}}^{*} }}$$
(23)

Note that the value of \(\phi (x,t)\) reflects the magnitude of the deviation of the species concentrations from chemical equilibrium conditions, so that it is non null near the electrode surface where the chemical equilibrium (Eq. (23)) is disturbed (see Fig. 2b).

Fig. 2
figure 2

Schematics of a the reaction layer in the catalytic mechanism (IV) and b the reaction and diffusion layers in the CE mechanism (III)

Full size image

The surface gradient of function \(\phi\) is equal to

$$\left( {\frac{\partial \phi }{{\partial x}}} \right)_{x = 0} = \frac{{\phi^{{\text{s}}} - \phi^{\infty } }}{\mu }$$
(24)

where \(\mu\) represents the thickness of the spatial region for which this gradient is different from zero, i.e., the zone in which the chemical equilibrium is disturbed.

As it is logical to assume, the expression of the reaction layer will vary if the geometry of the diffusive field does so, due to the fact that the homogeneous chemical kinetics is unavoidably conditioned by the mass transport. Due to this reason, it is important to highlight that as has been indicated by Saveant [21], this layer is indeed a diffusion–reaction layer. However, the “reaction layer” term will be used instead for historical reasons in honour of their first coiners.

Some authors have defined this zone as that resulting from a stationary state arising from mutual compensation between the chemical kinetics and the diffusion [21]. This definition is in agreement with the first one deduced as the previously-mentioned limiting case of a preceding chemical reaction to the electron transfer (so-called chemical electrochemical or CE mechanism), under linear diffusion conditions at the dropping mercury electrode (DME) [18].

In this work, it will be considered a more general concept of “reaction layer” based on the definition of Eq. (24), by considering the region where the chemical equilibrium is disturbed. It can be demonstrated that, in general, the expressions of the reaction layer are approximate for most mechanisms given in Sect. “More complex reaction mechanisms”. Nevertheless, in the case of a catalytic mechanism, rigorous non-stationary or transient expressions for the reaction layer can be found. In order to explain this, it is of interest to consider the comparative study of the CE and the catalytic mechanisms,

$$\text{B}\overset{k_1}{\underset{k_2}\rightleftarrows}\text{ C+e}^\text{-}\overset{}{\underset{}\rightleftarrows}\text{D} \;\;\;\;\; \text{EC Mechanism}$$
(III)
$$\text{C}+\text{e}^-\overset{}\rightleftarrows\text{ B}\left(\text{ + Z}\right)\,\overset{k'_1}{\underset{k'_2}\rightleftarrows}\text{ C}\left(\text{ + P}\right) \;\;\;\;\; \text{ Catalytic Mechanism}$$
(IV)

In the first case (CE mechanism), the chemical reaction product C is reduced at the electrode surface and converted into the new species D that diffuses towards the bulk solution. Under these conditions, the chemical kinetics between species B and C is complicated by such “escape.” For the CE reaction scheme, only an approximate expression of the reaction layer can be obtained, which is only valid under the conditions indicated by Koutecky and Brdička [14]. This is because the departure by diffusion of species D, produced by the reduction of C, and its chemical kinetic-diffusive dependence on species B, cannot be rigorously analysed separately. Thus, it is only possible to deduce an approximate reaction layer for a CE mechanism by considering a diffusive-kinetic steady state for the chemical reaction and, separately, a diffusive behaviour of both species B and C (see Sect. “Diffusive-kinetic steady state (dkss)”; Fig. 2b). Evidently, this response is only valid under the conditions indicated by ref. [14].

On the other hand, in the catalytic scheme, species B has difficulty escaping to the bulk solution because it is “trapped” by the catalytic cycle close to the electrode surface (see Scheme (IV)), so that the cyclic situation shown in Fig. 2a arises, giving rise to a rigorous expression for the reaction layer. Such cyclic behaviour, joint to the reversible character of the electrode reaction, leads to that the surface concentrations of both species B and C remain independent of time. Moreover, as the sum of the concentrations of species B and C remain constant over the experiment, the electrochemical response, that is the current, of a catalytic mechanism will only depend on function \(\phi\), given by Eq. (22), which takes a time-independent value at the electrode surface. Therefore, from the derivative of \(\phi\) at the electrode surface (Eq. (24)), the expressions of the rigorous transient or approximate stationary reaction layer can be easily extracted. Even more important, the stationary limit also corresponds to the reaction layer for a CE mechanism, giving rise to an approximate reaction layer only valid under the conditions indicated by ref. [18] (see also reference [20]). In order to clearly point out the differences between the CE and the catalytic reaction schemes, as well as the key influence of their different behaviours on the reaction layer, a more detailed discussion of the two mechanisms will be carried out.

Catalytic mechanism

For a catalytic mechanism (Scheme (IV)) at a macroelectrode, the variation of the concentrations of the chemical species when applying a potential-controlled perturbation is described by Fick’s second law modified with the corresponding kinetic terms (linear diffusion is considered for the sake of simplicity):

$$\begin{gathered} \frac{{\partial c_{C}^{{}} (x,t)}}{\partial t} = D_{{}} \left( {\frac{{\partial^{2} c_{C}^{{}} (x,t)}}{{\partial x^{2} }}} \right) + k_{1} c_{B}^{{}} (x,t) - k_{2} c_{C}^{{}} (x,t) \hfill \\ \frac{{\partial c_{B}^{{}} (x,t)}}{\partial t} = D_{{}} \left( {\frac{{\partial^{2} c_{B}^{{}} (x,t)}}{{\partial x^{2} }}} \right) - k_{1} c_{B}^{{}} (x,t) + k_{2} c_{C}^{{}} (x,t) \hfill \\ \end{gathered}$$
(25)

with the following boundary conditions corresponding to a reversible electrode reaction:

$$\left. \begin{gathered} t = {0,}\;x \ge {0}\, \hfill \\ t > {0},\;x \to \infty \hfill \\ \end{gathered} \right\}\; \, c_{{\text{C}}} {(}x,t{)} = c_{{\text{C}}}^{*} \;, \, c_{{\text{B}}} {(}x,t{)} = c_{{\text{B}}}^{*} \quad$$
(26)
$$\begin{gathered} t > {0},\;x = {0:} \hfill \\ \, D\left( {\frac{{\partial c_{{\text{C}}} }}{\partial x}} \right)_{{x = {0}}} + D\left( {\frac{{\partial c_{{\text{B}}} }}{\partial x}} \right)_{{x = {0}}} = {0} \hfill \\ \end{gathered}$$
(27)
$$c_{C}^{s} = \,\,e^{\eta } \,c_{B}^{s}$$
(28)

where it has been assumed that all the diffusion coefficients are equal. To simplify the resolution of the problem given by Eqs. (25)–(28), it is convenient to introduce the following variable changes:

$$\zeta {(}x,t{)} = c_{{\text{C}}} {(}x,t{)} + c_{{\text{B}}} {(}x,t{)}$$
(29)
$$\phi {(}x,t{)} = c_{{\text{B}}} {(}x,t{)} - K\,c_{{\text{C}}} {(}x,t{)}$$
(30)

with:

$$K = \frac{{c_{{\text{B}}}^{*} }}{{c_{{\text{C}}}^{*} }}$$
(31)

By inserting Eqs. (29)–(30) into (25), the following is fulfilled:

$$\frac{\partial \zeta(x,t) }{{\partial t}} - D\,\nabla^{2} \zeta(x,t) = 0$$
(32)
$$\frac{\partial \phi (x,t)}{{\partial t}} - D\,\nabla^{2} \phi (x,t) = - \,(k_{1} + k_{2} )\,\phi (x,t)$$
(33)

Then, when the diffusion coefficients of the participating species are equal, from Eqs. (26), (27), and (32), it can be easily deduced that the total concentration of electroactive species does not vary with time and it is the same at any point in solution:

$$c_{{\text{C}}}^{{}} \left( {x,t} \right) + c_{{\text{B}}}^{{}} \left( {x,t} \right) = c_{{\text{C}}}^{*} { + }c_{B}^{*} \, \forall \;\;\;\;\; x, \, t$$
(34)

Note that Eq. (34) holds independently of the reversibility of the electron transfer process and of the geometry of the electrode (i.e., of the form of the diffusion operator). Furthermore, when the electrode reaction is reversible, as in this case, it can be deduced from Eqs. (28) and (34) that the surface concentrations are time-independent and their values are defined by the value of the applied potential, \(E\), as follows:

$$\begin{gathered} {\text{c}}_{C}^{s} = ({\text{c}}_{C}^{*} + {\text{c}}_{B}^{*} )\left( {\frac{{{\text{e}}^{{\upeta }} }}{{1 + {\text{e}}^{{\upeta }} }}} \right) \hfill \\ {\text{c}}_{B}^{s} = ({\text{c}}_{C}^{*} + {\text{c}}_{B}^{*} )\left( {\frac{1}{{1 + {\text{e}}^{{\upeta }} }}} \right) \hfill \\ \end{gathered}$$
(35)

so that the variable \(\phi\) is also constant at the electrode surface (whatever the electrode shape), being

$$\phi^{{\text{s}}} = \zeta^{*} \left( {\frac{{1 - K{\text{e}}^{{\upeta }} }}{{1 + {\text{e}}^{{\upeta }} }}} \right)$$
(36)

where \({\zeta }^{*}={c}_{\mathrm{c}}^{*}+{c}_{B}^{*}\) (see Eq. (34)). The rigorous solution of equation under linear diffusion conditions (25) leads to [22]:

$$\phi \left( {x,t} \right) = \phi^{{\text{s}}} \frac{1}{2}\left[ {e^{{\left( { - x\sqrt {\frac{\kappa }{D}} } \right)}} erfc\left( {\frac{x}{{2\sqrt {Dt} }} - \sqrt {\kappa t} } \right) + e^{{\left( {x\sqrt {\frac{\kappa }{D}} } \right)}} erfc\left( {\frac{x}{{2\sqrt {Dt} }} + \sqrt {\kappa t} } \right)} \right]$$
(37)

where \(\kappa\) is the sum of the rate constants for the forward and reverse chemical reactions:

$$\kappa = k_{1} + k_{2}$$
(38)

As the current corresponding to the catalytic mechanism is given by:

$$\frac{I}{FAD} = \left( {\frac{{\partial c_{{\text{C}}} }}{\partial x}} \right)_{x = 0} = - \;\frac{1}{1 + K}\left( {\frac{\partial \phi }{{\partial x}}} \right)_{x = 0}$$
(39)

one obtains that:

$$\frac{I}{FAD} = \frac{{\zeta^{*} }}{1 + K}\frac{{1 - K{\text{e}}^{\eta } }}{{1 + {\text{e}}^{\eta } }}f_{{}}^{{{\text{cat}}}} \left( t \right)$$
(40)

where:

$$f_{{}}^{{{\text{cat}}}} \left( t \right) = \sqrt {\frac{\kappa }{D}} \,\left[ {\frac{{{\text{e}}^{ - \kappa t} }}{{\sqrt {\pi \kappa t} }} + \,{\text{erf(}}\sqrt {\kappa t} {)}} \right]$$
(41)

Hence, the current is given by the product of a potential-dependent function and a time dependent function, \(f_{{}}^{{{\text{cat}}}} \left( t \right)\). By comparing Eqs. (36), (40), and (41) with Eq. (24), it is concluded that the transient reaction layer \(\mu^{{{\text{tr}}}}\) under linear diffusion conditions is given by:

$$\mu^{{{\text{tr}}}} = \frac{1}{{f_{{}}^{{{\text{cat}}}} \left( t \right)}} = \sqrt {\frac{D}{\kappa }} \,\left[ {\frac{{{\text{e}}^{ - \kappa t} }}{{\sqrt {\pi \kappa t} }} + \,{\text{erf(}}\sqrt {\kappa t} {)}} \right]^{ - 1}$$
(42)

For \(\kappa t\) > 1.5 [2] (see Eqs. (41) and (42)), corresponding to sufficiently fast chemical kinetics relative to the time scale of the measurement, a steady state expression for the reaction layer \(\mu^{ss}\) is reached and the above expression is simplified to

$$\mu^{{{\text{ss}}}} = \sqrt {\frac{D}{\kappa }}$$
(43)

that is coincident with the expression found by Koutecky and Brdička in [18]. In Fig. 3, the transient and stationary reaction layers given by expression (42) and (43) are indicated in the plots.

Fig. 3
figure 3

a Schematic of the reaction layer in the catalytic mechanism (III) and b its definition on the basis of the linearized profile (dotted lines) of function \(\phi {(}x,t{)}\) (solid line) under transient (black) and steady state (red) conditions. Graph b adapted from [22]

Full size image

As \(\phi^{{\text{s}}}\) is time-independent regardless of the electrode geometry, the mathematical form of the current response given by (40) holds for any other electrode shape with the appropriate expression for \(f_{{}}^{{{\text{cat}}}} \left( t \right)\). In Table 3, \(f_{{}}^{{{\text{cat}}}} \left( t \right)\) for common electrode shapes is given. As will be shown below, the stationary reaction layer expressions given in this table can be used for any of the variety of reaction mechanisms considered in Sects. “CE mechanism” and “More complex reaction mechanisms”, since it is obtained as a time-independent solution deduced from the differential equation of function \(\phi\), which is only dependent on spatial variables:

$$\nabla^{2} \phi = - \,\kappa \,\phi$$
(44)
Table 3 Expressions for the transient \(\mu_{{}}^{{{\text{tr}}}}\) and steady state \(\mu_{{}}^{{{\text{ss}}}}\) linear reaction layer at planar, spherical and disc electrodes [23]
Full size table

CE mechanism

For the CE reaction (see Scheme (III)), the following three differential equations, together with their corresponding boundary conditions, describe the variations of the concentrations of the participating species with time and distance to the electrode:

$$\left. \begin{gathered} \frac{{\partial c_{{\text{B}}} \left( {x,t} \right)}}{\partial t} = D\frac{{\partial^{2} c_{{\text{B}}} \left( {x,t} \right)}}{{\partial x^{2} }} - k_{1} c_{{\text{B}}} \left( {x,t} \right) + k_{2} c_{{\text{C}}} \left( {x,t} \right) \hfill \\ \frac{{\partial c_{{\text{C}}} \left( {x,t} \right)}}{\partial t} = D\frac{{\partial^{2} c_{{\text{C}}} \left( {x,t} \right)}}{{\partial x^{2} }} + k_{1} c_{{\text{B}}} \left( {x,t} \right) - k_{2} c_{{\text{C}}} \left( {x,t} \right) \hfill \\ \frac{{\partial c_{{\text{D}}} \left( {x,t} \right)}}{\partial t} = D\frac{{\partial^{2} c_{{\text{D}}} \left( {x,t} \right)}}{{\partial x^{2} }} \hfill \\ \end{gathered} \right\}$$
(45)
$$\left. \begin{gathered} t = 0, \, x \ge 0 \hfill \\ t > 0, \, x \to \infty \hfill \\ \end{gathered} \right\}$$
$$c_{{\text{B}}} \left( {x,t} \right) = c_{{\text{B}}}^{*} , \,\, c_{{\text{C}}} \left( {x,t} \right) = c_{{\text{C}}}^{*} {, }\,\,\, c_{{\text{D}}} \left( {x,t} \right) = 0$$
(46)
$$\left. {t > 0, \, x = 0} \right\}$$
$$D\left( {\frac{{\partial c_{{\text{C}}} \left( {x,t} \right)}}{\partial x}} \right)_{x = 0} + D\left( {\frac{{\partial c_{{\text{D}}} \left( {x,t} \right)}}{\partial x}} \right)_{x = 0} = 0$$
(47)
$$\left( {\frac{{\partial c_{{\text{B}}} \left( {x,t} \right)}}{\partial x}} \right)_{x = 0} = 0$$
(48)
$$c_{C}^{s} (t) = \,\,e^{\eta } \,c_{D}^{s} (t)$$
(49)

As in the previous case, it is convenient to define the variables:

$$\zeta \left( {x,t} \right) = c_{{\text{B}}} \left( {x,t} \right) + c_{{\text{C}}} \left( {x,t} \right)$$
(50)
$$\phi \left( {x,t} \right) = c_{{\text{B}}} \left( {x,t} \right) - K\,c_{{\text{C}}} \left( {x,t} \right)$$
(51)

so that the differential equation system becomes into

$$\left. \begin{gathered} \frac{{\partial \zeta \left( {x,t} \right)}}{\partial t} = D\frac{{\partial^{2} \zeta \left( {x,t} \right)}}{{\partial x^{2} }} \, \hfill \\ \frac{{\partial \phi \left( {x,t} \right)}}{\partial t} = D\frac{{\partial^{2} \phi \left( {x,t} \right)}}{{\partial x^{2} }} - \kappa \phi \left( {x,t} \right) \hfill \\ \frac{{\partial c_{{\text{D}}} \left( {x,t} \right)}}{\partial t} = D\frac{{\partial^{2} c_{{\text{D}}} \left( {x,t} \right)}}{{\partial x^{2} }} \hfill \\ \end{gathered} \right\}$$
(52)

where \(\kappa\) is given by Eq. (38) and the boundary conditions:

$$\left. \begin{gathered} t = 0, \, x \ge 0 \hfill \\ t > 0, \, x \to \infty \hfill \\ \end{gathered} \right\}$$
$$\zeta \left( x, t \right) = \zeta^{*} = c_{{\text{B}}}^{*} + c_{{\text{C}}}^{*} , \,\,\,\, \phi \left( x,t \right) = 0, \,\,\,\,\, c_{{\text{D}}} \left( x,t \right) = 0$$
(53)
$$\left. {t > 0, \, x = 0} \right\}$$
$$\left( {\frac{{\partial \zeta \left( {x,t} \right)}}{\partial x}} \right)_{x = 0} = - \left( {\frac{{\partial c_{{\text{D}}} \left( {x,t} \right)}}{\partial x}} \right)_{x = 0}$$
(54)
$$- K\left( {\frac{{\partial \zeta \left( {x,t} \right)}}{\partial x}} \right)_{x = 0} = \left( {\frac{{\partial \phi \left( {x,t} \right)}}{\partial x}} \right)_{x = 0}$$
(55)
$$\zeta^{{\text{s}}} \left( t \right) - \phi^{{\text{s}}} \left( t \right) = \left( {1 + K} \right)e^{\eta } c_{{\text{D}}}^{{\text{s}}} \left( t \right)$$
(56)

with the current response being given by:

$$\frac{I}{nFAD} = \left( {\frac{\partial \zeta }{{\partial x}}} \right)_{x = 0} = - \frac{1}{K}\left( {\frac{\partial \phi }{{\partial x}}} \right)_{x = 0}$$
(57)

A key difference with respect to the catalytic mechanism is that now the variable \(\zeta\) is not constant. Moreover, at the electrode surface, \(\zeta\) and \(\phi\) are not constant either, the extraction of the time-dependent reaction layer from Eq. (55) would require also to know the surface value of \(\phi\) at each time. As a summary, the rigorous solution of this problem is complicated, given the relationship between \(\zeta\), \(\phi\), and \(c_{{\text{D}}}\) through the surface conditions in such a way that it is not possible to extract an expression for the reaction layer from the expression for the current, as in the catalytic mechanism (see also references [24, 25]).

Kinetic steady state (kss)

In order to obtain a more manageable solution for the CE mechanism, an approximate solution can be deduced by introducing the assumption that the perturbation of the chemical equilibrium (given by the variable \(\phi\)) is independent of time, so that:

$$\frac{{\partial \phi^{ss} \left( {x,t} \right)}}{\partial t} = 0$$

which is true for fast chemical kinetics, that is, for large enough values of \(\chi\)\(( = \kappa t)\). This approach was introduced for the first time in the literature by Koutecky [26] and it leads to the following solution for the current response:

$$\frac{I}{{I_{d} (t)}} = \frac{{F\left( {\chi_{{{\text{CE}}}} \left( E \right)} \right)}}{{1 + \left( {1 + K} \right)e^{\eta } }}$$
(58)

with

$$\chi_{{{\text{CE}}}} \left( E \right) = \chi_{{{\text{CE}}}} \left[ {1 + \left( {1 + K} \right)e^{\eta } } \right]$$
(59)
$$\chi_{{{\text{CE}}}} = \frac{{2\sqrt {\kappa t} }}{K}$$
(60)

and \(F(\chi_{{{\text{CE}}}} (E))\) being the Koutecký function given by

$$F\left( x \right) = \sum\limits_{j = 0}^{\infty } {\frac{{\left( { - 1} \right)^{j} x^{j + 1} }}{{\prod\limits_{i = 0}^{j} {p_{i} } }}} = \sqrt \pi \frac{x}{2}\exp \left( \frac{x}{2} \right)^{2} erfc\left( \frac{x}{2} \right)$$
(61)

where \(\pi = 3.14159... \,\,\) and \(I_{{\text{d}}} (t)\) is given by \(I_{{\text{d}}} = FAD\zeta^{*} /\sqrt {\pi Dt}\). Equation (58) can be applied if K = 1, when \(\chi = \kappa t_{{}} \ge 3.0\).

Although notably simpler than the rigorous solution, the expression deduced by Koutecky under the kss approach (Eq. (58)) is also not possible to extract a practical definition of the reaction layer.

Diffusive-kinetic steady state (dkss)

In addition to the kss restriction, \(\partial \phi^{{{\text{ss}}}} /\partial t = 0\), one can assume that \(\zeta\) and \(c_{{\text{D}}}\) variables are given by

$$\zeta \left( {x,t} \right) = \zeta^{*} - \left( {\zeta^{*} - \zeta^{{\text{s}}} \left( t \right)} \right)erfc\left( {\frac{x}{{2\sqrt {Dt} }}} \right)$$
(62)
$$c_{{\text{D}}} \left( {x,t} \right) = c_{{\text{D}}}^{{\text{s}}} \left( t \right)erfc\left( {\frac{x}{{2\sqrt {Dt} }}} \right)$$
(63)

which implies that both (pseudo)species diffuse in an independent way. Under this consideration (so-called the dkss approach [24, 25]), their surface gradients simplify to:

$$\left( {\frac{{\partial \zeta \left( {x,t} \right)}}{\partial x}} \right)_{x = 0} = \frac{1}{{\sqrt {\pi Dt} }}\left( {\zeta^{*} - \zeta^{{\text{s}}} \left( t \right)} \right)$$
(64)
$$\left( {\frac{{\partial c_{{\text{D}}} \left( {x,t} \right)}}{\partial x}} \right)_{x = 0} = - \frac{1}{{\sqrt {\pi Dt} }}c_{{\text{D}}}^{{\text{s}}} \left( t \right)$$
(65)

and the following very simple expression of the current is deduced:

$$I/I_{d} = \frac{{{{\sqrt {\pi Dt} } \mathord{\left/ {\vphantom {{\sqrt {\pi Dt} } {\left( {K\sqrt {D/\kappa } } \right)}}} \right. \kern-0pt} {\left( {K\sqrt {D/\kappa } } \right)}}}}{{1 + \left( {\sqrt {\pi Dt/K\sqrt {D/\kappa } } } \right)\left( {1 + \left( {1 + K} \right)e^{\eta } } \right)}}$$
(66)

It is important to highlight that the above equation can be directly deduced from Eq. (58) by making \(\chi_{CE} > > 1\) in Eq. (61) so that function \(F\left( x \right)\) reduces to

$$F\left( {\chi_{CE} } \right) \simeq \frac{{{{\chi_{CE} \sqrt \pi } \mathord{\left/ {\vphantom {{\chi_{CE} \sqrt \pi } 2}} \right. \kern-0pt} 2}}}{{1 + {{\chi_{CE} \sqrt \pi } \mathord{\left/ {\vphantom {{\chi_{CE} \sqrt \pi } 2}} \right. \kern-0pt} 2}}}$$
(67)

From Eq. (66), the diffusion and the reaction layers can be immediately identified as:

$$\delta = \sqrt {\pi Dt}$$
(68)
$$\mu^{ss} = \sqrt {\frac{D}{\kappa }}$$
(69)

The limiting current (\(e^{\eta } \to 0\)) derived from the dkss treatment is given by

$$I_{\lim } /I_{d} = \frac{{{{\sqrt {\pi Dt} } \mathord{\left/ {\vphantom {{\sqrt {\pi Dt} } {\left( {K\sqrt {D/\kappa } } \right)}}} \right. \kern-0pt} {\left( {K\sqrt {D/\kappa } } \right)}}}}{{1 + \left( {\sqrt {\pi Dt} /\left( {K\sqrt {D/\kappa } } \right)} \right)}} = \frac{{\sqrt {\pi \chi } /K}}{{1 + \sqrt {\pi \chi } /K}}$$
(70)

and the current–potential response can be linearized as follows

$$E = E_{1/2} + \frac{RT}{{nF}}\ln \left( {\frac{{I_{\lim } - I}}{I}} \right)$$
(71)

where \(E_{1/2}\) is the half-wave potential of the CE mechanism:

$$E_{1/2} = E^{0^{\prime}} + \frac{RT}{{nF}}\ln \left( {\frac{{{{I_{d} \left( t \right)} \mathord{\left/ {\vphantom {{I_{d} \left( t \right)} {I_{\lim } }}} \right. \kern-0pt} {I_{\lim } }}}}{{\left( {1 + K} \right)}}} \right)$$
(72)

Expressions (66) and (70) hold with errors below 5% for \(\chi_{{{\text{CE}}}} ( = 2\sqrt \chi /K) \ge 19.4\), i.e., \(\sqrt \chi /K \ge 9.7\).

It is worth noting that for the CE reaction scheme, both the diffusion and reaction layers appear as a consequence of the non-cyclic nature of this mechanism due to the conversion of species C to D. Of course, the expression derived for the reaction layer is the same as that obtained for the catalytic mechanism under kss conditions (Eq. (43)), since the mathematical expression for \(\mu\) is obtained from the approximate time-independent diffusive-chemical Eq. (44). Hence, \(\mu^{ss}\)(= \(\sqrt {D/\kappa }\)) is independent of the reaction mechanism considered [23, 27]. However, in non-catalytic mechanisms, as the CE scheme and others considered in Sect. “More complex reaction mechanisms”, \(\mu^{{{\text{ss}}}}\) can only be found directly from the current expression when the chemical reaction is fast and shifted towards the electroinactive species (K ≫ 1), as discussed in the seminal works by Wiesner and Koutecký [12, 13, 18], so that Eq. (70) becomes into:

$$\left( {I_{\lim } /I_{d} } \right)_{\begin{subarray}{l} K > > 1 \\ \kappa > > 1 \end{subarray} } = \frac{{\sqrt {\pi \chi } }}{K}$$
(73)

The discussion in this section on the diffusion and stationary reaction layers applies to other electrode geometries. Thus, their expressions for the most frequent geometries are given in Tables 2 and 3.

More complex reaction mechanisms

The dkss approximation can be directly applied to other reaction mechanisms provided that there is only one chemical reaction chemical coupled to the charge reaction step and that it follows a pseudo-first order chemical kinetics [23, 27], for example:

$$\text{B''+e}^-{\longleftrightarrow}\text{ B}\overset{k_1}{\underset{k_2}{\longleftrightarrow} \text{C}}\,\,\,\, \text{ EC Mechanism}$$
(V)
$$I_{EC}^{{}} = \frac{{FAD\,c_{{B^{\prime\prime}}}^{*} \left( {1 + K} \right)}}{{\delta \left( {1 + K} \right) + e^{\eta } \left( {\delta \,K + \mu^{ss} } \right)}}$$
(74)
$$\mathrm B''+\mathrm e^-\overset{{}_{}}{\underset{}\longleftrightarrow}\mathrm B\overset{k_1}{\underset{k_2}\longleftrightarrow}+\mathrm C+\mathrm e^-\overset{}{\underset{}\longleftrightarrow}\mathrm C'\;\,\,\,\,\mathrm{ECE}\;\mathrm{mechanism}$$
(VI)
$$I_{ECE}^{{}} = \, \frac{{FAD\,c_{{B^{\prime\prime}}}^{*} }}{\delta }\frac{{2\left( {\frac{K - 1}{2} + \frac{\delta }{{\mu^{ss} }}} \right) + e^{{\eta_{C/C^{\prime}} }} \frac{\delta }{{\mu^{ss} }}\left( {K + 1} \right)}}{{\left( {1 + e^{{\eta_{C/C^{\prime}} }} } \right)\left( {\frac{\delta }{{\mu^{ss} }} + e^{{\eta_{{B^{\prime\prime}/B}} }} } \right) + K\left( {e^{{\eta_{{B^{\prime\prime}/B}} }} + 1} \right)\left( {1 + \frac{\delta }{{\mu^{ss} }}e^{{\eta_{C/C^{\prime}} }} } \right)}}$$
(75)
$$\mathrm A''\overset{k_1}{\underset{k_2}\longleftrightarrow}\mathrm B''+\mathrm e^-\longleftrightarrow\mathrm B\overset{k_1}{\underset{k_2}\longleftrightarrow}\mathrm C^\prime\;\,\,\,\,\mathrm{CEC}\;\mathrm{mechanism}$$
(VII)
$$I_{{{\text{CEC}}}} = FAD\left( {c_{{{\text{B}^{{\prime\prime}}}}}^{*} + c_{{{{{C}^{\prime\prime}}}}}^{*} } \right)\frac{{\left( {1 + K} \right)/\delta^{\prime\prime}_{{}} }}{{1 + K + K^{\prime\prime}\left( {1 + K} \right)\frac{{\mu^{{\prime\prime}{{\text{ss}}}} }}{{\delta^{\prime\prime}}} + \left( {\frac{{\mu^{{\prime\prime}{{\text{ss}}}} }}{{\delta^{\prime\prime}}} + K} \right)\left( {1 + K^{\prime\prime}} \right){\text{e}}^{\eta } }}$$
(76)
$$\begin{array}{ccc}\mathrm B''&+\mathrm e^-\overset{}{\underset{}\longleftrightarrow}&\mathrm B\\\updownarrow&&\updownarrow\\\mathrm C''&+\mathrm e^-\overset{}{\underset{}\longleftrightarrow}&\mathrm C\end{array}\;\,\,\,\,\mathrm{Square}\;\mathrm{mechanism}$$
(VIII)
$$I_{{{\text{SQ}}}}^{{}} = \, \frac{{{\text{F}}AD\,\left( {c_{{{\text{B}^{{\prime\prime}}}}}^{*} + c_{{{\text{C}^{{\prime\prime}}}}}^{*} } \right)}}{\delta }\;\left\{ {\frac{{K^{\prime\prime}\left( {{\text{e}}^{{\eta_{{{\text{C}^{{\prime\prime}/C}}}} }} - {\text{e}}^{{\eta_{{{{B^{\prime\prime}/B}}}} }} } \right) + \left( {{1} + K^{\prime\prime}} \right)\left( {\frac{{\mu^{{\prime\prime}{{\text{ss}}}} }}{{\mu^{{{\text{ss}}}} }} + {\text{e}}^{{\eta_{{{\text{B}^{{\prime\prime}/B}}}} }} } \right)}}{{\left( {{1} + K^{\prime\prime}} \right)\left( {{1} + \left( {\frac{{{1} + K^{\prime\prime}_{{}} }}{{{1} + K_{{}} }}} \right){\text{e}}^{{\eta_{{{\text{C}^{{\prime\prime}/C}}}} }} } \right)\left( {\frac{{\mu^{{\prime\prime}{{\text{ss}}}} }}{{\mu^{{{\text{ss}}}} }} + {\text{e}}^{{\eta_{{{{B^{\prime\prime}/B}}}} }} } \right) + \left( {{\text{e}}^{{\eta_{{{\text{C}^{{\prime\prime}/C}}}} }} - {\text{e}}^{{\eta_{{{\text{B}^{{\prime\prime}/B}}}} }} } \right)\left\{ {K}^{{\prime\prime}} + \frac{{\mu^{{\prime\prime}{{\text{ss}}}} }}{\delta } + \left( {\frac{{{1} + K^{\prime\prime}}}{{{1} + K}}} \right)\left( {K{\text{e}}^{{\eta_{{{\text{C}^{{\prime\prime}/C}}}} }} - \frac{{\mu^{{\prime\prime}{{\text{ss}}}} }}{\delta }} \right) \right\}}}} \right\}$$
(77)
$$\begin{array}{ccc}\mathrm B''&+\mathrm e^-\overset{}{\underset{}\longleftrightarrow}&\mathrm B\\\updownarrow&&\\\mathrm C''&+\mathrm e^-\overset{}{\underset{}\longleftrightarrow}&\mathrm C\end{array}\;\mathrm{Open}-\mathrm{Square}\;\mathrm{mechanism}$$
(IX)
$$I_{{{\text{oSQ}}}} = \frac{{FAD\,\left( {c_{{{\text{B}^{{\prime\prime}}}}}^{*} + c_{{{\text{C}^{{\prime\prime}}}}}^{*} } \right)}}{{\delta^{\prime\prime}_{{}} }}\frac{{\frac{{\delta^{\prime\prime}_{{}} }}{{\mu^{{\prime\prime}{{\text{ss}}}} }}\left( {{\text{e}}^{{\eta_{{{\text{B}^{{\prime\prime}/B}}}} }} + K{\text{e}}^{{\eta_{{{\text{C}^{{\prime\prime}/C}}}} }} } \right) + \left( {{1} + K^{\prime\prime}} \right){\text{e}}^{{\eta_{{{\text{B}^{{\prime\prime}/B}}}} }} {\text{e}}^{{\eta_{{{\text{C}^{{\prime\prime}/C}}}} }} }}{{{\text{e}}^{{\eta_{{{\text{C/B}^{{\prime}}}}} }} \left( {K}^{{\prime\prime}} + \frac{{\delta^{\prime\prime}_{{}} }}{{\mu^{{\prime\prime}{{\text{ss}}}} }} \right) + {\text{e}}^{{\eta_{{{\text{C}^{{\prime\prime}/C}}}} }} \left( {{1} + K^{\prime\prime}\frac{{\delta^{\prime\prime}_{{}} }}{{\mu^{{\prime\prime}{{\text{ss}}}} }}} \right) + {\text{e}}^{{\eta_{{{\text{B}^{{\prime\prime}/B}}}} }} {\text{e}}^{{\eta_{{{\text{C}^{{\prime\prime}/C}}}} }} \left( {{1} + K^{\prime\prime}} \right) + \frac{{\delta^{\prime\prime}_{{}} }}{{\mu^{{\prime\prime}{{\text{ss}}}} }}\left( {{1} + K^{\prime\prime}} \right)}}$$
(78)

where \(\eta_{{\text{j}}}\) refers to the dimensionless potential (Eq. (12)) referred to the formal potential of the redox couple j indicated.

It is important to highlight that the approximate Eqs. (74)–(78) are valid for any electrode geometry, just substituting the adequate expressions for \(\delta\) and \(\mu^{{{\text{ss}}}}\) given in Tables 2 and 3, respectively. In those cases where two electrochemical reactions take place, two diffusion layers arise, which are denoted as \(\delta\) and \(\delta^{\prime\prime}\). In the same way, when the mechanism includes two chemical reactions, two stationary reaction layers can be defined referred to as \(\mu^{{{\text{ss}}}}\) and \(\mu^{{\prime\prime}{{\text{ss}}}}\), each of them accounting for the region adjacent to the electrode surface where the corresponding chemical equilibrium is broken as long as the chemical reaction is fast and the equilibrium is shifted towards the reactant (i.e., K and K’’ ≫ 1).

Conclusion

The concepts of diffusion and reaction layer have been extensively discussed and elaborated in depth.

The catalytic mechanism (Scheme (IV)) enables the natural and rigorous definition of the linear reaction layer under transient conditions and, from this, its definition under steady state conditions is obtained for any electrode geometry.

When non-catalytic first-order chemical reactions are coupled to electrode charge transfer reactions, an approximate solution for the current can be easily obtained under the conditions discussed in Sect. “Diffusive-kinetic steady state (dkss)” using the expressions for \(\delta\) and \(\mu^{{{\text{ss}}}}\) adequate to the electrode geometry (Tables 2 and 3).