Voltammetric Peak Heights of the ProtonâHydrogen Redox Couple: A

Article pubs.acs.org/JPCC

Voltammetric Peak Heights of the Proton−Hydrogen Redox Couple: A Comprehensive Analysis Enno Kaẗ elhön, Christopher Batchelor-McAuley, and Richard G. Compton* Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford University, Oxford OX1 3QZ, United Kingdom ABSTRACT: We present a comprehensive analysis of the voltammetric peak heights of the proton−hydrogen redox couple. To this end, we first reconcile conflicting literature that provides deviating solutions for the one-dimensional problem of a macrodisk electrode under reversible conditions, and numerically verify the correct solution. We then compare one-dimensional theory with two-dimensional simulations of commonly used, standard-size macroelectrodes and establish the loss of electrochemical reversibility as a result of high diffusion coefficients. We further explore the magnitude of the standard electrochemical rate constant needed for “Nernstian” behavior in such systems.

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INTRODUCTION The Randles−Ševčiḱ equation provides an exact expression for the voltammetric peak heights in electrochemically reversible systems that feature a 1:1 reaction stoichiometry studied at macroelectrodes. Initially described independently by Randles1 in Britain and Ševčiḱ 2 in Czechoslovakia in their 1948 publications, it has since then become an analytical standard tool in electrochemical sciences. However, as its applicability is limited to 1:1 stoichiometries, there is importance in expanding this theory to include arbitrary reaction stoichiometries. This more complicated problem was initially attacked by Shuman3 in 1969 and simulated by Jaworski4 et al. in 1999, and exact analytical expressions were provided by Ito et al. in 20025 and 20116 using the Crank−Nicolson method and a Laplace approach, respectively. Herein, we present a detailed analysis of voltammetric peak heights in electrochemical systems that feature a 2:1 stoichiometry. We first discuss previous conflicting literature that suggests different analytical solutions of the one-dimensional problem of a macrodisk electrode and complex stoichiometries in analogy with the Randles−Ševčiḱ equation. Since previous publications did not provide full derivations, we provide a detailed derivation and additionally verify the obtained results through finite difference simulations. We then compare analytical findings with two-dimensional simulations of commonly used macrodisk electrodes and quantify the applicability of onedimensional theory to such two-dimensional systems in view of the relatively fast diffusion coefficients of protons and hydrogen in aqueous solution. We further investigate the differences between Nernstian voltammetry and Butler−Volmer reaction kinetics and show which standard rate constants must be reached to allow an approximation of Butler−Volmer kinetics through Nernstian reactions in this context.

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for clarity, a list of all variables and their meanings is additionally provided in Table 1. We investigate reactions of the form mA + ne− ⇋ qB

where m entities of species A react to q entities of species B in an electrochemically reversible n-electron-transfer reaction. Assuming convection and migration is negligible,8 both species’ mass transport can be individually modeled via a diffusive process as described by Fick’s second law. In the one-dimensional case, the respective diffusion equations are given by9 ∂ci( r ⃗ , t ) = DiΔci( r ⃗ , t ) ∂t

(2)

where ci and Di represent the concentrations and diffusion coefficients of species i (= A or B), respectively, and the differential operator Δ depicts the Laplace operator. This set of transport equations is solved subject to a number of boundary conditions listed below. At the beginning of the simulated time of the experiment, the concentrations are set to cA( r ⃗ , t = 0) = cA* cB( r ⃗ , t = 0) = cB* = 0 for all r ⃗ ∈ 0, +

(3)

which represent homogeneous initial concentrations of both species equal to the corresponding species’ bulk concentration ci*, while the initial concentration of A adopts a finite nonzero value and the initial concentration of B is set to zero. Both bulk concentrations are further defined to be constant at any time:

THEORETICAL MODEL

Received: June 24, 2015 Revised: July 22, 2015 Published: August 7, 2015

In the following, we define the theoretical model employed in the presented study. Variables are introduced in the text; however, © 2015 American Chemical Society

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DOI: 10.1021/acs.jpcc.5b06040 J. Phys. Chem. C 2015, 119, 23203−23210

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0 where E⊖ f is the formal potential, ci are the surface concentrations ⊖ at the electrode, c is the standard concentration, and R, T, and F have their usual significance. Our study hence focuses on electrochemical systems near the reversible limit. In recent publications,11,12 it was, for instance, shown that this model closely approximates the electrochemistry of the proton− hydrogen redox couple at platinum electrodes. At low electrolyte concentrations, voltammograms may however be distorted through the electrode’s electric field,8and depending on the electrode material and the solvent, the applicability of the assumption of electrochemical reversibility may vary. One-Dimensional Geometry of a Macroelectrode. In the one-dimensional case of a large planar macroelectrode, we substitute the corresponding Laplace operator and Fick’s second law (eq 2) turns into

Table 1. Explanation of the Variables Used in This Study variable

explanation

m, n, q ci Di c*i n⃗ E E⊖ f R T F c0i c⊖ Einit v S = e−at nF a= v RT

stoichiometry of the considered reaction concentration of species i diffusion coefficient of species i bulk concentration of species i normal vector of the electrode surface electrode potential formal potential of the reaction shown in eq 1 universal gas constant temperature Faraday constant surface concentrations unit concentration initial electrode potential sweep rate of the electrode potential parameter substitution7 parameter substitution

⎛ nF ⎞ θ = exp⎜ (E − Ef⊖)⎟ ⎝ RT ⎠

3t α Ci Ai(s), Bi(s) Je C0i je χ(at)

γ=

DA DB

I

∂ci(x , t ) ∂ 2c(x , t ) = Di ∂t ∂x 2

(8)

7

We further set the electrode surface to x = 0, while the space x > 0 is the bulk reservoir. Two-Dimensional Geometry of a Disk Electrode. In the two-dimensional case of a disk electrode, the Laplace operator in eq 2 can be written in cylindrical coordinates. The equation can then be expressed as

parameter substitution7 Laplace transformation with respect to t limit of convergence in the Laplace image Laplace-transformed concentration ci integration constants resulting from the integration with respect to t electron flux at the electrode surface in the Laplace image Laplace-transformed surface concentrations electron flux at the electrode surface substitution7 parameter substitution7

⎛1 ∂ ∂ ∂ci(r , z , t ) ∂2 ⎞ r + 2 ⎟c(r , z , t ) = Di⎜ ∂t ⎝ r ∂r ∂r ∂z ⎠

(9)

We further consider an electrode surface that expands in the area 0 < r < re and z = 0, where hence boundary conditions 5 and 7 apply, and assume that the electrode is surrounded by an electrochemically inactive surface (see Figure 1). The concentration gradient at this inactive surface must therefore be

electrode current

∂ci(r , z , t ) ∂z

cA(| r |⃗ → ∞ , t ) = cA* cB(| r |⃗ → ∞ , t ) = cB* = 0 for all t ∈ 0, +

(10)

Due to the system’s rotational symmetry with respect to the zaxis, the concentration gradient must also vanish at r = 0:

(4)

The total mass within the system is conserved throughout the experiment. Hence, the flux at the electrode must satisfy the condition ⎛ ⎞ ⎜ 1 DA ∇⃗cA( r ⃗ , t ) + 1 DB∇⃗cB( r ⃗ , t ) ⎟ ·n ⃗( r ⃗) = 0 ⎜q ⎟ m el. ⎠ ⎝ el.

=0 z = 0, r > re

∂ci(r , z , t ) ∂r

=0 r=0

(11)

(5)

where the reaction stoichiometry (eq 1) is considered through the factors m and q, and n ⃗ ( r ⃗) = ⎧ Normal vector of the electrode surface at the electrode ⎨ ⎩ 0 everywhere else

(6)

As the considered theoretical model examines an electrochemically reversible process, kinetic processes at the electrode surface are modeled as Nernstian reactions10 E(t ) = Ef⊖ + for all t ∈ 0, +

⎞ (c 0 (t ))m RT ⎛ ln⎜ 0 Aq ⊖ m − q ⎟ nF ⎝ (cB(t )) (c ) ⎠ Figure 1. Simulated geometry of a disk electrode in cylindrical coordinates.

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which can be solved numerically for χ(at). In the case of a 1:1 stoichiometry, this result equals the expression that was, for instance, derived by Nicholson and Shain.7 Aside from the use of the standard concentration c⊖, eq 17 is further identical to the result published by Ito et al.6 in 2011; it, however, does not match the expression found by Shuman.3 [The derivation presented in ref 3 provides an incorrect expression for the convolution integral (eq 17) (see eq 16 in the original publication).] Equation 17 was solved numerically by Ito et al.6 for various stoichiometries, while peak currents were determined in terms of the dimensionless χ(at). On the basis of these results, analogues to the Randles− Ševčiḱ equation can be found for the case of the proton− hydrogen reaction (m = 2, n = 2, and q = 1; specifically corresponding to 2H+ + 2e− ⇌ H2). Using Faraday’s law, we can relate the electrode current I(t) and the flux of electrons at the electrode surface je(t)

In combination with eq 4, eqs 5, 7, 10, and 11 provide a full set of boundary conditions for the diffusion problem 9 in the space r, z > 0. In the simulation of the two-dimensional model, we additionally consider Butler−Volmer kinetics. The electrode boundary condition 7 is then replaced by n ∂cA(r , z , t ) m ∂z =

z = 0, r < re

k 0(cA0 )m (c ⊖)1 − m e−αcθ

− k 0(c B0)q (c ⊖)1 − q e(n − αc)θ

(12)

where k0 is the standard rate constant, αc is the cathodic transfer coefficient as defined by the latest IUPAC Technical Report,13,14 and θ is defined through θ=

■

F (E − Ef⊖) RT

(13)

I(t ) = FAje (t ) = nFAcA* πDA a χ (at )

THEORETICAL PREDICTION OF PEAK CURRENTS We consider previous theory that leads to the determination of an expression for the voltammetric peak heights of reversible reactions for the case of complex stoichiometries. To this end, we solve the above-defined theoretical model in the one-dimensional case via Laplace transformation and the convolution theorem. The presented derivation is along the lines of the works by Nicholson and Shain7 and Ito et al.6 However, in contrast to their works, we provide a more detailed derivation to clarify inconsistencies (Ito et al.6 note that their result differs from Shuman’s result3 and state that “the reason for this difference is not clear”) in the literature that may present incorrect results. During the forward sweep of a cyclic voltammogram, the electrode potential E(t) is parametrized as E(t ) = E init − vt

where A is the electrode surface area. Substituting the numerically found dimensionless peak currents χmax(at) from ref 6, we obtain * Ipeak,ox = 0.3843·2FAcox

Ipeak,re = −0.1766·2FAcre*

nF v RT (15)

The Nernst equation (eq 7) can then be rewritten as = θS(t )

where X0 is the initial dimensionless grid spacing and ω is the grid expansion factor. In the two-dimensional case, we use a grid that expands from z = 0, r = 0, and r = re. Grid points are defined through

(16)

while all other equations remain unchanged. The resulting system of two partial differential equations (eq 2) and six boundary conditions (eqs 3, 4, 5, and 16) can be transformed via Laplace transformation, solved in the Laplace image, and then inversely transformed back into the desired form as a function of time. As is discussed in the Appendix, this approach yields the following equation ⎛ ⎜cA* − mcA* ⎝

∫0

at

R i ,j,k

m χ (z ) ⎞ dz ⎟ at − z ⎠

⎛ = θS(at )⎜qcA*γ ⎝

∫0

at

dz

q χ (z ) ⎞ ⊖ m − q ⎟ (c ) at − z ⎠

(20)

COMPUTATIONAL METHODS The principles of the computational methods used in this paper have previously been described fully for the one-dimensional15 and two-dimensional11 cases, respectively. In short, the theoretical model detailed above is transferred into dimensionless coordinates, as described in the textbook by Compton et al.16 In the next step, we discretize the simulated space, which in dimensionless coordinates Xi, Ri, and Zi is normalized to the electrode radius re, through an expanding grid. In one dimension, this is done according to the expression x Xi = i = X 0 ∑ ωi re (21) i

S(t ) = e−at

(cB0(t ))q (c ⊖)m − q

2vFDre RT

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In analogy with the notation used in previous works, we make three simplifying substitutions when solving the diffusion problem:

(cA0 (t ))m

(19)

for the peak height of the oxidative and reductive peaks, Ipeak,ox and Ipeak,re, for the proton−hydrogen reaction, respectively.

(14)

⎛ nF ⎞ θ = exp⎜ (E − Ef⊖)⎟ ⎝ RT ⎠

2vFDox RT

and

6,7

a=

(18)

(17)

⎧ ri 1 i for 0 < R i < ⎪ = R0 ∑ ω 2 i ⎪ re ⎪ rj 1 ⎪ j = ⎨ = 1 − R 0 ∑ ω for < R j < 1 2 ⎪ re j ⎪r ⎪ k = 1 + R 0 ∑ ωk for 1 < R k ⎪ re ⎩ k

(22)

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zi = Z0 ∑ ωi re i

Table 3. Computational Parameters Used in Simulations (23)

where R0 and Z0 are the initial dimensionless grid spacings in r and z, respectively. We then employ the finite difference approach to linearize the diffusion equation (eq 2) with respect to the above-defined one- and two-dimensional grids. The system of equations is either solved directly or further simplified through the alternating directions implicit method (ADI).16 Individual equations in the so-obtained sets of equations are solved through triagonal matrix decomposition and the multidimensional Newton method,16 if quadratic boundaries are involved (i.e., all sets of equations that involve Nernstian (eq 7) or Butler−Volmer boundary conditions), or solved via the Thomas algorithm16 if no quadratic boundary conditions are involved. Values for all electrochemically relevant parameters used in the calculations and simulations are shown in Table 2. The diffusion coefficients of hydrogen and protons are chosen on the basis of ref 17 and our previous work,11 respectively. Additional simulation parameters are presented in Table 3. Our simulation software was developed in C++ and compiled via the GNU Compiler Collection (GCC). Some simulations additionally employ the OpenMP application programming interface. Further data processing and visualization is done in Python using the modules NumPy, matplotlib, and Python’s standard library.

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value 8.1 × 10−9 m2 s−1 11 5.0 × 10−9 m2 s−1 17

E⊖ f ci* re (1d) re (2d) T

0V 1 mM or 0 mM 1 cm 1 mm, 3 mm 298 K

10−10 10−4 10−4 1.02 1.04

additionally plot the peak heights predicted by Ito et al.5,6 and Shuman3 as well as the result of the Randles1−Ševčiḱ 2 equation for a 1:1 stoichiometry for direct comparison. Herein, we find a good agreement between Ito’s result and the simulated data. Shuman’s result, however, overestimates the simulated value by a factor of exactly 2. We further observe that the reductive peak height in the considered 2:1 stoichiometry exceeds the peak height of a 1:1 stoichiometry by about 12%. Figure 3 depicts the absolute voltammetric peak heights obtained from multiple simulations at different scan rates for the oxidative and reductive peaks in comparison to the result of Ito et al. and the Randles−Ševčiḱ equation. Our simulation and theory show excellent agreement over 4 orders of magnitude, while the absolute peak height of the oxidative peak is found below the peak height in a 1:1 stoichiometry. Nernstian Voltammetry in Two-Dimensional Systems. In analogy with Figure 2, Figure 4 shows voltammograms that are simulated for a Nernstian reaction at a disk electrode at various scan rates between 10 and 100 mV/s in comparison to the theoretical prediction for a planar one-dimensional geometry, as described by eq 20. The plot illustrates that the results of the finite difference simulation for an electrode of the radius re = 1 mm deviate significantly from the one-dimensional model, which may appear to be a counterintuitive finding: Electrodes in the millimeter range are usually referred to as macroelectrodes, and experimental results are commonly compared with one-dimensional theoretical models. In many cases, particularly if the electrodes feature radii of several millimeters, such comparisons are relatively good approximations as edge effects become

Table 2. Electrochemical Parameters Used in Theory and Simulation

D H+ D H2

value

ΔX0 ΔR0 ΔZ0 ω (1d) ω (2d)

Figure 2. Simulated voltammogram of the one-dimensional geometry of a macrodisk electrode featuring a radius of re = 1 cm under Nernstian conditions. The scan rate v is set to 1 V s−1, and the initial concentration of protons is 1 mM. The picture additionally depicts the peak heights predicted by Ito et al.5,6 and Shuman.3 For comparison, we also plot the expected peak height of a one-electron transfer reaction in a 1:1 stoichiometry as it is described by the Randles1−Ševčiḱ 2 equation.

RESULTS AND DISCUSSION The following section is divided into three subsections: In the first part, we present one-dimensional simulations and validate the above theoretical findings. Second, we compare twodimensional simulations of standard-size macroelectrodes with one-dimensional theory and determine under which conditions and to what extent this theory remains applicable. In the last part, we compare simulations using Nernstian voltammetry and Butler−Volmer reaction kinetics, and investigate the magnitude of the standard electrochemical rate constant that is required to allow an approximation of Butler−Volmer kinetics through Nernstian voltammetry. Nernstian Voltammetry in One-Dimensional Systems. We first use our one-dimensional simulation of a macroelectrode to numerically determine the peak heights of the first voltammetric sweeps at various scan rates. To this end, we model voltammograms in solutions that initially contain homogeneous concentrations of either hydrogen or protons in the cases of an anodic or a cathodic sweep, respectively, which react according to a 2:1 stoichiometry (n = 2, m = 2, and q = 1; specifically corresponding to 2H+ + 2e− ⇌ H2). Examples of such simulated voltammograms can be found in Figure 2, in which we

parameter

parameter

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The deviations between the theoretical one-dimensional model and the two-dimensional simulations are shown in greater detail in Figure 5, which presents the peak heights over a wide range of sweep rates. We find that at fast sweep rates above 10 V s−1 the one-dimensional model approximates the two-dimensional model well, while at the slower sweep rates likely to be employed there are significant deviations. This is further emphasized in Figure 6, which shows the deviations of the simulated current from the theoretical result calculated according to ⎛ Ipeak,2d ‐ sim ⎞ Δ% = 100⎜⎜ − 1⎟⎟ ⎝ Ipeak,1d ‐ theo ⎠

(24)

In the plot, it can be seen that, even at relatively fast sweep rates of about 0.1 V s−1, we find differences of 5−10% for the case of an electrode featuring a radius of 1 mm, while slower scan rates may result in deviations of more than 50%. Comparison with Butler−Volmer Reaction Kinetics in Two-Dimensional Systems. In order to quantify to which extent experimental results may differ from the one- and twodimensional Nernstian models, we additionally model the same system using Butler−Volmer reaction kinetics. Using the boundary condition given in eq 12, we calculate voltammograms for various standard electrochemical rate constants k0 in the quasi-reversible range. Simulated absolute peak heights are obtained from two-dimensional simulations and shown in Figure 7, where voltammograms for different α values are plotted versus fast k0 in the range between 0.5 and 10 cm s−1. We find that differences between Nernstian voltammetry and Butler−Volmer reaction kinetics increase with a decrease in the standard rate constant. For better quantification of the differences, percentaged deviations are calculated via

Figure 3. Absolute peak heights of the first anodic and cathodic peaks in a voltammogram calculated for the one-dimensional geometry of a macrodisk electrode under Nernstian conditions at varied scan rates. The electrode features a radius of re = 1 cm, and the initial concentrations of protons and hydrogen were 1 and 0 mM and 0 and 1 mM in the cases of cathodic and anodic sweeps, respectively. Simulated peaks are compared with the theoretical result of Ito et al.5,6 and the Randles1−Ševčiḱ 2 equation.

⎛ Ipeak,BV ⎞ Δ% = 100⎜⎜ − 1⎟⎟ ⎝ Ipeak,Nernst ⎠

(25)

Figure 4. Simulated voltammogram of the two-dimensional geometry of a disk electrode featuring a radius of re = 3 mm. The voltammogram is simulated using Nernstian boundary conditions and an initial proton concentration of 1 mM. The picture additionally shows the peak heights predicted by Ito et al.5,6 and the expected peak height of a one-electron transfer reaction in a 1:1 stoichiometry as it is described by the Randles1−Ševčiḱ 2 equation.

negligible. For diffusion coefficients of about 10−10 m2 s−1 and a scan rate of 100 mV s−1, this was, for instance, recently shown by Ngamchuea et al.,18 who investigated the deviations of twodimensional systems from one-dimensional calculations in great detail. However, in the here-investigated case of the hydrogen− proton couple, diffusion coefficients exceed 10−10 m2 s−1, the diffusion coefficients investigated by Ngamchuea et al., by a factor of 5−8. As faster diffusion leads to a stronger impact of edge effects and to more severe deviations, this approximation must hence be reconsidered for the fast diffusion of hydrogen and protons in aqueous solution.

Figure 5. Comparison of absolute voltammetric peak heights obtained from two-dimensional simulations of disk electrodes featuring the radii re = 1 mm and re = 3 mm at varied scan rates. The initial concentrations of hydrogen and protons were c*H2 = 1 mM and c*H+ = 0 mM for the anodic sweep and cH*2 = 0 mM and cH*+ = 1 mM for the anodic sweep. Solid lines depict the result by Ito et al.5,6 for the one-dimensional case. 23207


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Figure 8. Percentaged deviations between the absolute simulated peak currents that are shown in Figure 7. Deviations are calculated via eq 25.

Figure 6. Percentaged deviations between the data of the twodimensional simulations and the one-dimensional model that is presented in Figure 5. Deviations are calculated according to eq 24.

profiles at relatively fast sweep rates and obtained simulation results may deviate significantly from one-dimensional theory. We additionally show that this is further complicated by the application of Butler−Volmer electrode kinetics that lead to additional deviations even at high standard rate constants. In this study, we provide a comprehensive analysis of the topic and quantify the obtained deviations in great detail. Presented plots may be used as references and may be a helpful guide for other researchers in this field.

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APPENDIX In this appendix, we present the derivation leading to eq 17. The Laplace transforms of eq 2 with respect to t are calculated to be ∂ 2Ci(x , s)

= sCi(x , s) − ci* (26) ∂x 2 where we implicitly use the initial concentrations specified in boundary condition 3. This equation is solved by a function of the form19 Di

Figure 7. Absolute peak heights for the case of Nernstian voltammetry and Butler−Volmer kinetics at varied standard electrochemical rate constants k0. All data was simulated for the two-dimensional system of a disk electrode of the radius 1 mm. In analogy with the previous figures, the initial concentrations of protons and hydrogen were 1 and 0 mM and 0 and 1 mM in the cases of cathodic and anodic sweeps, respectively.

⎛ ⎛ s ⎞ s ⎞ Ci(x , s) = Ai (s) exp⎜⎜ + x⎟⎟ + Bi (s) exp⎜⎜ − x⎟⎟ Di ⎠ Di ⎠ ⎝ ⎝ c* + i (27) s

where the first two terms are obtained from the solution of the homogeneous partial differential equation and the last term is the particular integral. Ai(s) and Bi(s) (i = A, B) are four integration constants in the diffusion profiles of species A and B that result from the integration with respect to x and that may be functions of s. In order to determine the integration constants Ai(s) and Bi(s), we calculate the Laplace transform of eq 4:

and plotted in Figure 8. Here, we find that depending on the cathodic transfer coefficient (α) chosen deviations may be as large as |7.5%| for the reductive peak at an α value of 0.5. When approximating proton−hydrogen reactions via Nernstian voltammetry, it is hence crucial to ensure that the standard rate constant of the modeled system is sufficiently high even if fast standard rate constants are expected.

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CONCLUSIONS We present detailed electrochemical simulations of voltammetric peak heights of the hydrogen−proton couple. While focusing on Nernstian voltammetry and the geometry of standard macrodisk electrodes that are commonly modeled as one-dimensional systems, we compare our results with such theory with rather counterintuitive findings: Due to the comparably high diffusion coefficients of H2 and H+, the system exhibits nonlinear diffusion

CA(x → ∞ , s) = CB(x → ∞ , s) =

cA* s cB* s

=0

(28)

Since Re(s) > α > 0, comparison with eq 27 gives Ai(s) = 0.19 We further calculate the Laplace transform of boundary condition 5, which accounts for the conservation of mass at 20

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the interface, and use Faraday’s law to derive a relation between the Laplace-transformed flux Je of electrons at the electrode surface and the surface concentrations of species A and B: 1 ∂CA(x , s) DA m ∂x

x=0

1 ∂C (x , s) = − DB B q ∂x

⎛q = θS(t )⎜⎜ ⎝n

x=0

1 = − Je (s) n

Using the previous result Ai(s) = 0 and the general solution (eq 27) of the above partial differential equation (eq 26) in the Laplace image, we can calculate the derivatives to be ⎞ x⎟⎟ ⎠

x=0

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(32)

ACKNOWLEDGMENTS The research leading to these results has received partial funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grand Agreement No. [320403].

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(33)

∧ F(s) = Je (s) ∧ G (s ) =

1 s

in the convolution theorem’s (3 t (f (t ) ∗g (t )) = F(t ) ·G(t )), we obtain

q n

1 πDB

1 πDA

∫0

∫0

t

dτ

t

dτ

definition

je (τ ) t−τ

je (τ ) t−τ

REFERENCES

(1) Randles, J. E. B. A cathode ray polarograph. Part II. - The currentvoltage curves. Trans. Faraday Soc. 1948, 44, 327−338. (2) Ševčík, A. Oscillographic polarography with periodical triangular voltage. Collect. Czech. Chem. Commun. 1948, 13, 349−377. (3) Shuman, M. S. Nonunity electrode reaction orders and stationary electrode polarography. Anal. Chem. 1969, 41, 142−146. (4) Jaworski, A.; Donten, M.; Stojek, Z.; Osteryoung, J. G. Conditions of Strict Voltammetric Reversibility of the H+/H2 Couple at Platinum Electrodes. Anal. Chem. 1999, 71, 243−246. (5) Ito, H.; Hasegawa, Y.; Ito, Y. Electrode behavior of hydride ion in molten alkali chlorides. J. Electrochem. Soc. 2002, 149, E273−E280. (6) Ito, H.; Hasegawa, Y.; Ito, Y. Numerical simulation of cyclic voltammetry for reversible systems with complex stoichiometry. Russ. J. Electrochem. 2011, 47, 1006−1015. (7) Nicholson, R. S.; Shain, I. Theory of stationary electrode polarography. Single scan and cyclic methods applied to reversible, irreversible, and kinetic systems. Anal. Chem. 1964, 36, 706−723. (8) Dickinson, E. J.; Limon-Petersen, J. G.; Rees, N. V.; Compton, R. G. How Much Supporting Electrolyte Is Required to Make a Cyclic Voltammetry Experiment Quantitatively ’Diffusional’? A Theoretical and Experimental Investigation. J. Phys. Chem. C 2009, 113, 11157− 11171. (9) Fick, A. Ü ber Diffusion. Ann. Phys. 1855, 170, 59−86. (10) Compton, R. G.; Banks, C. E. Understanding Voltammetry, 2nd ed.; Imperial College Press: 2011. (11) Jiao, X.; Batchelor-McAuley, C.; Kätelhön, E.; Ellison, J.; Tschulik, K.; Compton, R. G. The Subtleties of the Reversible Hydrogen Evolution Reaction Arising from the Nonunity Stoichiometry. J. Phys. Chem. C 2015, 119, 9402−9410.

where C0i (s) are the Laplace-transformed surface concentrations. The so-obtained Laplace image of the surface concentrations can be transformed into the time domain via the inverse Laplace transform. Using the convolution theorem, the linearity of the Laplace transform, and the following substitution

cB0(t ) =

AUTHOR INFORMATION

The authors declare no competing financial interest.

q Je (s) CB0(s) = n DBs

m cA0 (t ) = cA* − n

(37)

Notes

cA* m Je (s) − s n DA s

1 πt

DA DB

*E-mail: [email protected].

thus providing an expression for Bi(s) at x = 0 as a function of the Laplace-transformed flux of electrons at the electrode surface. Substitution of this result in the general solution (eq 27) at x = 0 eliminates all integration constants and yields

∧ g (t ) =

(36)

Corresponding Author

q Je (s) n DBs

f (t ) = je (s)

(35)

Equation 35 then transforms into eq 17, which is given in the main manuscript.

(31)

m Je (s) BA (s) = − n DA s

CA0 (s) =

q je (τ ) ⎞ ⊖ m − q ⎟⎟ (c ) dτ t−τ⎠

Additionally, we substitute

(30)

The boundary condition 29 then transforms into

∧ BB(s) = +

∫0

t

je (t ) = ncA* πDA a χ (at )

γ=

s Di

= Bi (s)

1 πDB

m je (τ ) ⎞ ⎟⎟ dτ t−τ⎠

t → at τ → az

and in the case of x = 0 to be ∂Ci(x , s) ∂x

∫0

t

As the expression at is dimensionless and proportial to the electrode potential E, it is practical to define the flux of electrons as a function of at. Hence, the following substitutions are made:

(29)

⎛ ∂Ci(x , s) s s = −Bi (s) exp⎜⎜ − ∂x Di Di ⎝

1 πDA

(34)

Substitution of this result in the Nernst equation (eq 16) gives an expression for je as a function of time: 23209


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The Journal of Physical Chemistry C (12) Lin, C.; Jiao, X.; Tschulik, K.; Batchelor-McAuley, C.; Compton, R. G. Influence of Adsorption Kinetics upon the Electrochemically Reversible Hydrogen Oxidation Reaction. J. Phys. Chem. C 2015, 119, 16121−16130. (13) Guidelli, R.; Compton, R. G.; Feliu, J. M.; Gileadi, E.; Lipkowski, J.; Schmickler, W.; Trasatti, S. Definition of the transfer coefficient in electrochemistry (IUPAC Recommendations 2014). Pure Appl. Chem. 2014, 86, 259−262. (14) Guidelli, R.; Compton, R. G.; Feliu, J. M.; Gileadi, E.; Lipkowski, J.; Schmickler, W.; Trasatti, S. Defining the transfer coefficient in electrochemistry: An assessment (IUPAC Technical Report). Pure Appl. Chem. 2014, 86, 245−258. (15) Kätelhön, E.; Compton, R. G. Testing and validating electroanalytical simulations. Analyst 2015, 140, 2592−2598. (16) Compton, R. G.; Laborda, E.; Ward, K. R. Understanding Voltammetry: Simulation of Electrode Processes; Imperial College Press: 2014. (17) Macpherson, J. V.; Unwin, P. R. Determination of the diffusion coefficient of hydrogen in aqueous solution using single and double potential step chronoamperometry at a disk ultramicroelectrode. Anal. Chem. 1997, 69, 2063−2069. (18) Ngamchuea, K.; Eloul, S.; Tschulik, K.; Compton, R. G. Planar diffusion to macro disc electrodes - What electrode size is required for the Cottrell and Randles−Sevcik equations to apply quantitatively? J. Solid State Electrochem. 2014, 18, 3251−3257. (19) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications, 2nd ed.; Wiley: New York, 2001. (20) Bronstein, I. N.; Semendjajew, K. A.; Musiol, G.; Mühlig, H. Taschenbuch der Mathematik, 5., überarbeitete und erweiterte Auflage; Verlag Harri Deutsch: 2001.

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Voltammetric Peak Heights of the ProtonâHydrogen Redox Couple: A

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