Insight on Tafel slopes from a microkinetic analysis of aqueous electrocatalysis for energy conversion

Microkinetic analyses of aqueous electrochemistry involving gaseous H2 or O2, i.e., hydrogen evolution reaction (HER), hydrogen oxidation reaction (HOR), oxygen reduction reaction (ORR) and oxygen evolution reaction (OER), are revisited. The Tafel slopes used to evaluate the rate determining steps generally assume extreme coverage of the adsorbed species (θ ≈ 0 or ≈1), although, in practice, the slopes are coverage-dependent. We conducted detailed kinetic analyses describing the coverage-dependent Tafel slopes for the aforementioned reactions. Our careful analyses provide a general benchmark for experimentally observed Tafel slopes that can be assigned to specific rate determining steps. The Tafel analysis is a powerful tool for discussing the rate determining steps involved in electrocatalysis, but our study also demonstrated that overly simplified assumptions led to an inaccurate description of the surface electrocatalysis. Additionally, in many studies, Tafel analyses have been performed in conjunction with the Butler-Volmer equation, where its applicability regarding only electron transfer kinetics is often overlooked. Based on the derived kinetic description of the HER/HOR as an example, the limitation of Butler-Volmer expression in electrocatalysis is also discussed in this report.

To compare the electrocatalytic activity and to elucidate the reaction mechanism of electrocatalysts, a Tafel analysis is generally utilized. In this method, the sensitivity of the electric current response to the applied potential (Tafel slope) is analyzed, which provides information associated with the rate determining steps. The experimentally observed Tafel slopes can be compared with the theoretically derived slopes assuming different rate-determining steps based on the microkinetic model. Because its derivation process is generally complicated, the surface coverage of the intermediate species is typically assumed as constant: either θ ≈ 0 or θ ≈ 1. This simplification makes it easier for the electrochemist to consider the surface kinetics, and, in many studies, Tafel slopes derived by this method are used [39][40][41] . As previously stated, the coverage should actually vary with the applied potential: the simplification leads to an incomplete description of the actual surface kinetics that depends on the coverage. Furthermore, this assumption of invariable coverage may be applicable for steady-state conditions at constant potential and current conditions; nevertheless, for the Tafel analysis the applicability of such assumption intrinsically involves questionable accuracy. In some studies, a potential-dependent change in the Tafel slope is considered for each reaction (see refs 8, 42, 43 for the HER 21, for the HOR 44, 45, for the ORR and 46 for the OER), although in these cases, the potential and coverage were either described in insufficient detail or the Tafel slopes were conjugated according to the Butler-Volmer equation, which does not fully account for the coverage terms 5,9,31,[47][48][49][50][51] .
This report addresses the theoretical description of the kinetics of these fundamental reactions (HER, HOR, ORR and OER) simply based on microkinetic analyses. Our aims were (1) to describe the dependence of the Tafel slope on the coverage of the formed surface species, e.g., M-H for the HER/HOR, M-OH, M-O, M-OOH and M-OO − for the ORR/OER, where M is the surface site, and (2) to address the applicability of the Butler-Volmer equation in describing electrocatalytic kinetics. The visualization of the electrocatalytic kinetics, i.e., the Tafel slope dependence on coverage, provides the fundamental understanding of the potential-dependent shift in the Tafel slopes associated with the reaction mechanism changes relevant to water electrolysis and fuel cells.

Results and Discussion
Conventionally, the Tafel analysis leads to two important physical parameters: the Tafel slope and the exchange current density. Empirically, the following Tafel relation has been well confirmed: where η defines the overpotential, which is the difference between the electrode and standard potentials (η = E − E 0 ), j denotes the current density, and b is the Tafel slope. Theoretically, simple electrochemical redox reactions can be described by the Butler-Volmer equation 52 : where α is the transfer coefficient, f denotes F/RT (F: the Faraday's constant, R: the universal gas constant, T: the absolute temperature), and j 0 is the exchange current density. The equation represents the total currents from both reduction and oxidation reactions (opposite signs). First, we consider only forward (or backward) rates that are sufficiently larger than the corresponding backward (or forward) reaction rate. From the above equation, the following equation can be derived: The first term in Equation 3 corresponds to a in Equation 1, indicating that the intercept obtained from the plot of η vs. log j can be converted into the exchange current density. The Tafel slope provides insight into the reaction mechanism, and the exchange current density is known as a descriptor of the catalytic activity 1, 53,54 . Thus, for analyzing electrochemical performances, the Tafel analysis is conjugated with the Butler-Volmer equation in many studies. As described in the Introduction, the Tafel slope can be used to address the elementary steps and the rate determining steps. In the following four sections, the Tafel slope is discussed based purely on theoretical microkinetic analyses for the hydrogen evolution reaction (HER), the hydrogen oxidation reaction (HOR), the oxygen reduction reaction (ORR) and the oxygen evolution reaction (OER). Then, the fifth section discusses the rate determining steps of the opposite directions in an oxidation-reduction couple, e.g., HOR and HER. Due to the different rate-determining steps for the respective reactions, a good HOR or OER catalyst may not be a good HER or ORR catalyst, respectively. Therefore, the Butler-Volmer equation is limited to describing chemically reversible electrocatalytic reactions. The constants include Faraday's constant, F = 96500 C mol −1 , the gas constant, R = 8.314 J mol −1 K −1 , temperature T, the electron transfer coefficient, α = 0.5, and the surface area of the electrode, A.
Hydrogen evolution reaction (HER). The HER is generally described in two ways. The first is hydronium ion reduction, and the other is water reduction, A theoretical kinetic description for each reaction is discussed in the following sections.
Hydronium ion reduction. Hydronium ion reduction consists of three steps: the Volmer, Heyrovsky and Tafel steps, as follows [5][6][7][8]42,55,56 . and θ denote the hydronium ion activities and the surface coverage by the hydrogen atom, respectively. Because this step is an electron transfer step, the kinetic rate constant depends on the applied potential, as follows: where k 0 defines the standard rate constant for k, α is the electron transfer coefficient, f denotes F/RT, and η i defines the electrode and standard potential differences (overpotential, E − E 0 ) for ith Equation.
The assumption that the Volmer step completely determines the overall rate leads to faster consumption of adsorbed hydrogen, indicating that the surface coverage should be close to 0. Therefore, using Equations 9 and 10, the reaction rate can be described by the following equation: In the case of the Heyrovsky step determining the rate, the Volmer step can be pre-equilibrated, so the reverse reaction rate for Equation 20, 20 20 OH is the same as Equation 22, resulting in the following coverage description: The forward reaction rate in the Heyrovsky step is given as follows: = 1/1/1 are shown in Fig. 1a,b,c for Equation 13 (or 23), Equation 17 (or 27) and Equation 19 (or 28), respectively. The objective of this study was to elucidate the Tafel slope dependence on the coverage, where hydronium ion reduction and water reduction cannot be differentiated from one another. As widely accepted, Tafel slopes of 120, 40 and 30 mV dec −1 were observed for the Volmer, Heyrovsky and Tafel determining rate steps, respectively, confirming the validity of our kinetic model. Additionally, for the Heyrovsky rate determining step, a Tafel slope of 120 mV dec −1 was observed in the higher coverage region (θ H > 0.6). Therefore, a Tafel slope of 120 mV dec −1 cannot be due only to the Volmer step; it must originate from either the Volmer rate determining step or the Heyrovsky rate determining step with high adsorbed hydrogen atom coverage.
In the literature, various Tafel slopes have been reported. Pt electrocatalysts supported on carbon (Pt/C), one of the most studied catalysts, exhibits a Tafel slope of 30 mV dec −1 in 0.5 M H 2 SO 4 41,57,58 , 120 mV dec −1 under polymer electrolyte membrane fuel cell (PEMFC) conditions 9 and 125 mV dec −1 in 0.5 M NaOH solution 8 . Although this difference could be assigned to the difference in the pH levels of the solutions, in the reported study, the potential region used to obtain the Tafel slope was different: 30 mV dec −1 is taken at the lower overpotential range, whereas a wider overpotential range is considered for 120 mV dec −1 . This indicates that the Tafel slope is indeed potential dependent and, in turn, coverage dependent. In other reports that focused on Pt electrocatalysts, the Tafel slope of bulk Pt disk electrodes is reported to exhibit potential dependence: 36-68 mV dec −1 followed by 125 mV dec −1 with an increasing overpotential in a 0.5 M H 2 SO 4 electrolyte solution 8 . The electrocatalytic activity of Pt toward the HER is known as structure sensitive, as different facets shows various activities and rate determining steps 59,60 . The Tafel slope for Pt (110) is two-step, starting from 55 mV dec −1 shifting to 150 mV dec −1 , Pt(110) exhibits a slope of 75 mV dec −1 that shifts to 140 mV dec −1 , and Pt(111) is reported to exhibit a Tafel slope of 140-150 mV with no transition in a 0.1 M KOH solution 59 . These examples demonstrate the significance in considering potential-dependent Tafel slopes.
Other electrocatalysts in addition to Pt have also been reported to exhibit different Tafel slopes. Ir/C, Pd/C and Rh/C show slopes of 124 ± 5, 127 ± 8 and 95 ± 3 mV dec −1 , respectively, considering a wide range of overpotentials under PEMFC conditions 9   125 mV dec −142 , and Pt-Ce electrodes exhibit a Tafel slope of 114 mV dec −1 in 1.0 M NaOH 61 . Carbon is generally used as a catalyst support, which also becomes active toward the HER by introducing foreign atoms: graphite, P-graphene, N-graphene and N,P-graphene are reported to exhibit Tafel slopes of 206, 133, 116 and 91 mV dec −1 in 0.5 M H 2 SO 4 and 208, 159, 143 and 145 mV dec −1 , respectively, in 0.1 M KOH 62 . When Pt-Pd supported on reduced graphene oxide is used, the Tafel slope is 10-25 mV dec −1 at an overpotential < 40 mV 63 , which is lower than the afore-derived Tafel slopes. However, as shown in Fig. 1, even the theoretical Tafel slopes at smaller overpotentials do not reach the well-known values of 30, 40 and 120 mV dec −1 in every case. Therefore, if too small an overpotential region is considered, the Tafel analysis leads to a misrepresentation of the rate-determining step and furthermore poorly compares to other reported values.
In addition to pure metal and bimetallic electrocatalysts, sulfide 41 . These studies support the importance of evaluating in detail the theoretical Tafel slope as potential-dependent.
Hydrogen Oxidation Reaction (HOR). The HOR can be described in two ways, which are the reverse expressions of the HER. Hydrogen can be oxidized either by a water molecule or a hydroxide anion. In this section, the HOR is described in both cases.
HOR with water molecule. The HOR with a water molecule is the reverse reaction of hydronium ion reduction, and the elementary steps are described using the same equations as Equations 6-8 [20][21][22] , Each case determining the overall reaction rate is discussed in the following sections. When the Heyrovsky step determines the rate, the adsorbed hydrogen is consumed via the Volmer step. In addition, it is assumed that the Tafel step does not occur or is slow and therefore negligible. The forward reaction rate in Equation 7' is described by the following equation: The subscript has a negative value to show similar applicability to the HER. Assuming that the Heyrovsky step determines the overall rate, the consumption of adsorbed hydrogen is faster than its formation, and the surface coverage is close to zero. Combining Equation 29 with Equations 10 and 12 yields the following expression for current: When the Tafel step is the rate determining step, similar to the previous case, it is assumed that the Heyrovsky step is slow and that the coverage is close to zero. Thus, the rate expression for the Tafel step, In this case of the Volmer step determining the rate, the reactant is provided by two reactions: the Heyrovsky step and the Tafel step. We separately address the two types of adsorbed hydrogen atoms Scientific RepoRts | 5:13801 | DOi: 10.1038/srep13801 as those formed via the Heyrovsky step (θ 1 ) and those formed via the Tafel step (θ 2 ). The coverage of θ 1 is given by considering the pre-equilibrium of Equation 7' . The reverse reaction rate is described by Equation 16, which is equilibrated with Equation 29 to yield the following coverage expression: Notably, K i defines the ratio of k i /k −i (not k −i /k i ) to be consistent with the HER section. The other coverage expression can be obtained by considering the equilibrium of Equations 18 and 31, as follows: is introduced to describe the difference in the practical reactivity of adsorbed hydrogen. If the ratio of ε 1 /ε 2 is unity, there is no difference between the two types of adsorbed hydrogen atoms. The conditions ε 1 « ε 2 and ε 2 « ε 1 correspond to the Tafel-Volmer and Heyrovsky-Volmer steps, respectively. The forward reaction rate in Equation 6' is described by Equation 11, and, by combining Equations 10, 12, 14, 33 and 34, we obtain the following electric current expression, HOR with hydroxide anions. The elementary steps are given by the following equations: When the Heyrovsky step is rate determining, the surface coverage is close to zero, and the Volmer step is negligibly slow. The forward reaction rate for Equation 21' is described by the following equation: The assumptions made here and in Equations 10, 12 and 36 give us the following current expression: In the case of the Tafel step determining the rate, the Volmer step description for the HOR with hydroxide anion is equivalent to that for the HOR with water. Therefore, Equation 32 also gives the electric current for this case.
When the Volmer step is the rate determining step, similar to the previous cases for the HOR with water, we separately address two types of adsorbed hydrogen atoms. The hydrogen adsorbed via the Volmer step is described by Equation 33. The other coverage can be expressed by considering the pre-equilibrium phase of Equation 21' . The reverse reaction rate is given by Equation 26, which is equilibrated with the forward reaction rate of Equation 36. Thus, the following coverage expression, θ 3 , for adsorbed hydrogen via Equation 18 is obtained: Using ε i , the electric current for this case is described by the following equation:  Fig. 2b. The Tafel slope is 40 mV dec −1 with a surface coverage approaching zero (θ H < 0.4), and 120 mV dec −1 is obtained with a high surface coverage (θ H > 0.6). For the Tafel-Volmer step, when θ H is varied from 0 to 1 by tuning K 5 , the Tafel slope is always 120 mV dec −1 . As previously mentioned, 120 mV dec −1 can be observed in many cases, suggesting that the Tafel slope of 120 mV dec −1 cannot be solely used to identify the rate determining step.
In the literature, Tafel slopes using noble metals and their alloys are used to evaluate the HOR kinetics. Tafel slopes of 106, 88, 229, 154 and 784 mV dec −1 were reported for bulk Pt, sputtered Pt, Pt-Ni, Pt-Ti and Ni-Ti, respectively, in 1 M KOH 69 . Tafel slopes of 124 ± 15, 124 ± 5, 258 ± 23 and 180 ± 8 were reported for Pt/C, Ir/C, Pd/C and Rh/C, respectively, where the Tafel step is mentioned as the rate-determining step 9 . The HOR is also a structure sensitive reaction: Pt(110) has a Tafel slope of 28 mV dec −1 (Tafel-Volmer), Pt(100) has a Tafel slope of 37 that increases to 112 mV dec −1 (Heyrovsky-Volmer), and Pt(111) has a Tafel slope of 74 mV dec −1 (Tafel-Volmer, Heyrovsky-Volmer) in 0.05 M H 2 SO 4 70 . These examples corroborate the significance of this study: a Tafel slope of around 120 mV dec −1 is frequently reported but is not evidenced in any rate-determining step.
Oxygen reduction reaction (ORR). The ORR mechanism is complicated and continues to be discus sed 26,28,40,44,47,48,71,72 . Recently, Adzic and coworkers reported that OO − can be the intermediate species for ORR under alkaline conditions 73 . Therefore, as elementary steps, the following associative mechanism is considered in this study [26][27][28][29][30][31]74,75 : Further details of the elementary steps have been discussed by Koper, where charge transfer and proton-coupled charge transfer reactions are separately considered 76 . Although above steps are in the more simplified form, the resultant Tafel slope results in identical values. Each step of Equations 41-44 is assumed to be rate determining to describe the electric currents in the following sections. Regarding the The current expression is provided as follows using Of note, Equation 42 is not an electron transfer reaction, and therefore that the reaction rate dependence on the applied potential originates from that of the coverage. where θ 3 denotes the surface coverage by MO, and the following relationship is true: ( ) − r r 57 42 42 Among the three adsorbed species, the following relationship is true: the overall reaction rate is described, as follows: . It should be noted that the Tafel slope itself is not affected by these figures, but the potential region where the specific Tafel slope is observed is quantitatively dependent on these values, although this falls outside the scope of our study. The primary differences between sections describing Equation 41 and 44 being rds are the constants, as can be observed in Equations 50 and 63. Therefore, both cases can be described simultaneously (Fig. 3a). Figure 3b,c describe Equations 55 and 61, respectively. The Tafel slope of 120 mV dec −1 is obtained when the rate is determined by the first discharge step or the upon consumption of the MOOH species with high coverage of MOO − (Equation 43). In the other cases, the simulated Tafel slope is lower than 120 mV dec −1 , as Equations 55 and 61 correspond to Tafel slopes of 60 mV dec −1 and 40 mV dec −1 , respectively.
Various Tafel slopes have been reported in the literature. Pt is one of the most active electrocatalysts for the ORR and has been studied under various conditions. The Pt/C catalyst under acidic conditions exhibits a two-step Tafel slope: ca. 60 mV dec −1 shifts to 120 mV dec −1 with increasing potential in 0.5 M H 2 SO 4 , 0.05 M H 2 SO 4 and 0.5 M HClO 4 [77][78][79][80][81] . A single Tafel slope of 50-80 mV dec −1 (higher with decreasing Pt mean diameter from 6 to 1 nm) has been reported in 5 mM HClO 4 , possibly measured at a lower overpotential range (corresponding to the aforementioned 60 mV dec −1 ) 82 . In an alkaline solution of 1 M NaOH, Pt/C also exhibits a two-step Tafel slope of 65-82 mV dec −1 that increases to > 100 mV dec −1 with increasing overpotential 83 . Bare Pt disks exhibit similar behavior: a Tafel slope of 60 mV dec −1 at a low overpotential and 120 mV dec −1 at higher overpotentials in 0.1 M HClO 4 31,44,84 and in HClO 4 or H 2 SO 4 at pH 0.3-4 71,72 . When an additional supporting electrolyte is used, a single Tafel slope of 120 mV dec −1 is observed 85 . In addition, under PEMFC conditions, a single Tafel slope of 109-120 mV dec −1 at 65 °C was reported 86 . Under neutral conditions of 0.05 M K 2 SO 4 , various Tafel slopes were found: 175 mV dec −1 that decreased to 120 mV dec −1 at pH 4 and 120 → 175 → 120 mV dec −1 at pH 6 85  . Under alkaline conditions, the Tafel slope generally increases. A constant Tafel slope of 75 mV dec −1 for Pt (111), a two-step Tafel slope of 86 that increases to 167 mV dec −1 for Pt (100), and a two-step Tafel slope of 89 that increases to 265 mV dec −1 for Pt (110) in 0.1 M KOH were reported 92 .
Alloying Pt with other metals is a common way to improve the ORR activities, among which Pt-Ni alloy is one of the most studied and active electrocatalysts. PtNi mesostructured thin film exhibit a Tafel slope of 40  Regarding other electrocatalysts that do not contain platinum, Ir was reported to exhibit Tafel slopes of 60 and 120 mV dec −1 at low and high overpotentials, respectively, in LiClO 4 solutions at pH values of 2.2, 3.1, and 11.0 97 . In addition to metal catalysts, some oxides (57 mV dec −1 for MnO x /C, 56 mV dec −1 for Ni-MnO x /C, 47 mV dec −1 for Mg-MnO x /C) 98 and carbon species (110 mV dec −1 for pyrolytic graphite) 49 exhibit ORR activities under alkaline conditions. In many cases, a variety of Tafel slopes are reported, and most are potential-dependent. To identify or at least consider the rate determining step, deriving the theoretical Tafel slope dependence on each coverage is of particular importance.
Oxygen evolution reaction (OER). The oxygen evolution reaction (OER) is known as a 2 or 4 electron step, and the reaction mechanism is complicated 39,56,99 . Based on the literature and considering that the OER with hydroxide anion is the backward reaction of the ORR, we considered the following mechanism under alkaline conditions, assuming a single-site mechanism:  ( ) − r r 78 65 65 The backward reaction rate for Equation 65 can be written as: 65 65 H O 2 2 Then, the following relation for the coverage is obtained: Additionally, the following is true:  which yields the following kinetic rate equation:   Therefore, the kinetic current is given as:    ) determines both reactions, the HER can be described by Equation 13, and the HOR can be described by Equation 35. By introducing the exchange current, I 0 , Equation 13 can be simplified as follows: (Heyrovsky-Volmer step, and θ H is close to 0)} And {(3) if the surface coverage during hydronium ion reduction remains close to zero at any potential} And {(4) if the surface coverage during hydrogen oxidation by water molecules remains close to unity at any potential} then the HOR/HER may be described by the Butler-Volmer equation. Regarding point (1), assuming that the Volmer step determines the rate is difficult in practice. A Tafel slope of 120 mV dec −1 for the HER cannot be used as evidence for the Volmer step being the rate-determining step, as mentioned in the HER section. Additionally, as described in the HOR section, Tafel slopes that differ from 120 mV dec −1 for the HOR can be theoretically obtained only if adsorbed hydrogen species are formed via the Heyrovsky step and if the surface coverage θ H is close to 0. These rationales suggest that there are other scenarios in which the Tafel slope can be 120 mV dec −1 . Hence, an experimentally observed slope of 120 mV dec −1 does not imply that the Volmer step limits the rate. Furthermore, to meet criteria (3) and (4), there should be a certain potential range where coverage changes with potential, which contradicts the criteria themselves. Thus, the assumption that the Volmer step determines the overall rate at any potential is not true in any case. If all of the above criteria are satisfied, then the Butler-Volmer equation may be applied to fit the HOR/HER. Otherwise, the obtained fitting parameter may result in misleading information.
Nevertheless, in some studies, the Butler-Volmer equation 52 : is applied to elucidate the surface kinetics 5,9,31,[47][48][49][50]  Other physical parameters associated with electrocatalytic activity. As described in this report, the electrocatalytic activity is not only potential-dependent but also temperature dependent, which alters the Tafel slope [107][108][109] . As well-known in the field of catalysis, the rate constant for the chemical reaction follows the Arrhenius's equation: where A represents the collision frequency, and E a represents the apparent activation energy. By varying the temperature during electrochemical measurement, the rate constants for each step can be altered, which in turn defines the Tafel slope. In the literature, Equation 3 is typically used to describe the dependence of the Tafel slope on temperature. In this equation, − α RT F corresponds to the Tafel slope. This equation is derived based on the Butler-Volmer equation in conjunction with the Tafel equation, which is doubtful in our opinion, as discussed. Simply speaking, the Tafel slope indeed increases with elevated temperature 50,51,109 , but this increase likely arises from the increased rate constant. When the overall electric current is differentiated by potential, the detailed theoretical Tafel slope description is obtained. Further differentiating it by temperature reveals the dependence of the Tafel slope on temperature. This exercise is highly complicated and beyond the scope of this study.
The rate constant for the electrochemical reaction is generally given in Equation 10, where α is the transfer coefficient. The reference 52 states that "α, the transfer coefficient, can range from zero to unity". Because the transfer coefficient is included in the rate constant description, the Tafel slope is also dependent on the transfer coefficient. Experimentally identifying the transfer coefficient is quite difficult, but some have tried to determine it by directly correlating the transfer coefficient with the Tafel slope given by Equation 3 31,51 which is based on the assumption that the Butler-Volmer equation is applicable. The effort to experimentally elucidate the transfer coefficient should be made with considerable care, especially considering the applicability of the Butler-Volmer equation.
To accurately describe the kinetic component from overall current, the contribution of mass-transport must be effectively isolated. The following Levich equation has been established for the mass-transport limited current in the configuration of the rotating disk electrode (RDE) 52 : , where i L is the Levich currents (limiting diffusion current), F is the Faraday constant, A represents the electrode surface area, ω denotes the disk electrode rotation speed, ν is the kinematic viscosity and δC represents the difference between surface and bulk reactant concentrations. The relationship between the overall current i with the Levich current i L and the kinetic current i k is described in the Koutecky-Levich (KL) equation: Using these equations, the kinetic current can be determined only when the system satisfies the assumptions required to establish the Levich equation. For example, a system consisting of considerable electrode roughness may lead to large deviations from the theoretical value simply expected from the KL equation. Thus, the KL analysis must be treated with special care based on these requirements. In general, the obtained values can be mostly overestimated, underestimated or even misled due to the multi-step nature of the reaction mechanism 110 . The considerations on mass-transport effects have been studied elsewhere for HER [111][112][113] , HER/HOR 4,112,113 and ORR 114 and are out of the scope of this study. However, it should be emphasized that improper subtraction of mass-transport contribution would cause misinterpretation of the rate determining steps and kinetics due to inaccurate Tafel slopes for elucidating kinetics 112,113 . Microkinetic analyses are powerful for addressing reaction mechanisms and rate-determining steps. In addition, there is natural limitation in that the postulated mechanisms must be correct to describe the Tafel slopes. As observed in this study, the same Tafel slopes can be obtained for different elementary steps with varied coverages. For instance, in the OER section, the OER was described as a single-site Scientific RepoRts | 5:13801 | DOi: 10.1038/srep13801 mechanism, but recent advances in mechanistic studies of PSII in photosynthesis have proposed radical coupling mechanisms in which two metals are involved in the generation of one oxygen molecule 115 . In another example, it was proposed that HER via water reduction is facilitated by Ni hydroxide islands on Pt surfaces via a bifunctional mechanism at the periphery, where two different sites are involved in one reaction 4,116,117 . This study provides an aspect of electrocatalytic kinetics that focus on Tafel analyses and the applicability of the Butler-Volmer equation. The development of improved electrocatalysts should aim to identify catalysts that proceed via unexpected elementary steps, which breaks the volcano plot trend with one activity descriptor, such as metal-adsorbate bond strength 34,36,82,[118][119][120][121] .

Conclusions
Fundamental electrocatalytic reactions of hydrogen evolution reaction (HER), hydrogen oxidation reaction (HOR), oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) were revisited considering conventional microkinetics and focusing on Tafel analyses. Our kinetic model reproduces the well-known Tafel slopes of 30, 40 and 120 mV dec −1 for the Tafel, Heyrovsky and Volmer steps of the HER, respectively, which confirms the validity of our method. Although in the literature, a Tafel slope of 120 mV dec −1 for the HER is generally assigned to the Volmer step, based on our analysis, this slope was also observed when the Heyrovsky step determined the rate with a high coverage of adsorbed hydrogen (> 0.6). Similarly, a Tafel slope of 120 mV dec −1 was also found for the HOR, ORR and OER, but not only in the first discharge step as the rate-determining step. This observation suggests that a Tafel slope of 120 mV dec −1 (often considered as a single-electron transfer rate limiting) does not conclusively identify the rate-determining step. Furthermore, the validity of the Butler-Volmer equation to describe electrocatalytic kinetics in redox reactions was addressed in this study. Theoretical modeling suggests that only in very limited cases, where the electron transfer reaction determines the rate, is the Butler-Volmer equation applicable in describing the electrocatalytic kinetics. Therefore, the kinetics should be considered based on a microkinetic model that includes coverage terms rather than a model conjugated with the Butler-Volmer equation. Although the Tafel analysis is useful in elucidating the rate-determining steps, too simplified discussion, such as determination of the kinetics based only on the Butler-Volmer assumption, fails to accurately describe the surface electrocatalysis. This work provides a more accurate and concrete vision of the kinetics of H 2 and O 2 aqueous electrocatalysis, which is essential for the advancement of electrolysis and fuel cells.