Homogeneous Catalysis of Electrochemical Reactions. Steady-State

Subscriber access provided by Universiteit Utrecht

Homogeneous Catalysis of Electrochemical Reactions. Steady-State and Non Steady-State Statuses of Intermediates. Cyrille Costentin, and Jean-Michel Savéant ACS Catal., Just Accepted Manuscript • DOI: 10.1021/acscatal.8b01195 • Publication Date (Web): 26 Apr 2018 Downloaded from http://pubs.acs.org on April 26, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

Homogeneous Catalysis of Electrochemical Reactions. Steady-State and Non Steady-State Statuses of Intermediates. Cyrille Costentin1,2* and Jean-Michel Savéant1* 1

Université Paris Diderot, Sorbonne Paris Cité, Laboratoire d'Electrochimie Moléculaire, Unité Mixte de Recherche Université -

CNRS N° 7591, Bâtiment Lavoisier, 15 rue Jean de Baïf, 75205 Paris Cedex 13, France. 2 Present address : Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, Massachusetts 02138, United States.

[email protected], [email protected],

ABSTRACT. In the context of modern energy challenges, there is an increasing need to decipher the mechanism of complex, multistep catalytic processes, as a basis of their optimization and improvement. Cyclic voltammetry (CV) is one of the most popular electrochemical techniques in this purpose. Mechanistic complexities often trigger a quest for simplification in the treatment of data, such as the application of the steady-state approximation to intermediates. The validity of such assumptions actually need justification. This is the object of the present work, which examines the question for five homogeneous catalytic reaction schemes of practical interest, which can also serve as tutorial examples for the analysis of further schemes. The analysis is simplified by the consideration of pure kinetic conditions and constancy of substrate concentration. These conditions can be achieved in practice by appropriate manipulation of scan rate and concentrations. The currentpotential responses are consequently S-shaped and independent of the scan rate. The CV responses are then depending upon only two dimensionless parameters that group the experimental intrinsic and operational parameters. Limiting subcases reached for extreme values of these parameters are worth considering in terms of mechanism diagnosis and kinetic characterization. They are conveniently represented by kinetic zone diagrams. ACS Paragon Plus Environment

1

ACS Catalysis

Page 2 of 23

The present work not only provides the tools required to check the correctness of the kinetic analysis but also to 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

gauge the possibility of characterizing transient intermediates by structurally informative techniques (e.g. spectroscopic). Keywords: electrochemical reactions, catalysis, steady-state, intermediates, cyclic voltammetry Introduction Contemporary renewable energy challenges are strong incentives for the development of efficient molecular catalysts, including particularly those aiming at the reduction or oxidation of fundamental substrates such as H2O, H2, O2, H+ and CO2.

1,2,3,4,5,6,7

Although the ultimate goal is the implementation of efficient electrolysis

cells yielding the desired product in the shortest possible time, much can be learn in terms of mechanism and catalysis efficiency from the application of microelectrolytic transient or steady-state techniques.

8

Cyclic

voltammetry (CV) is one of the most popular of these “non-destructive” techniques, in which a negligible amount of reactant is consumed during each run (on the order of 1 part per million), thus allowing for a large number of successive experiments.

9,10

Modeling the association between diffusion and chemical reactions

accompanying the electrode electron transfer has been a continual endeavor in the application of cyclic voltammetry to decipher mechanism. It is remarkable that the first contribution to this domain, more than halfcentury ago,

11

has concerned a catalytic reaction scheme. Even if this was the simplest conceivable catalytic

reaction process (the EE’ reaction scheme in Scheme 1), it formed the basis of future developments in the area, especially those that were and are still required by modern energy challenges evoked earlier. In most practical cases, the catalytic reaction schemes are more complicated than the EE’ scheme, involving additional steps such as further electrode and/or solution electron transfers steps as well as chemical steps of varying nature such as acid base reactions (in the general sense) or other bond forming/ bond breaking reactions. Examples of such reactions schemes are listed in Scheme 1. They all involve intermediates that may or may not obey the steady-state approximation. Analysis of the kinetics of the catalytic process is significantly simplified when the steady-state approximation applies. A particularly important aspect of this question relates to the number of parameters that actually govern the CV responses. Experimental parameters that may interfere comprise those ACS Paragon Plus Environment 2

Page 3 of 23

ACS Catalysis EE’C

EE’

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0 EP/Q

0 EP/Q

P + e-

Q

k1

Q + A

P + e-

(0) (E)

P + e-

Q

(0) (E)

P + B

(1)

k1 Q + A

k-1 k2

B

C

C + e-

(0) (E)

P + B

(1)

k2

B

(E)

Q

k3

Q + C

ECCE’

0 EP/Q

0 EP/Q

Q

B

k-1

product (2) (C)

ECCE P + eQ + A

k1

Q + A

(2) (C)

P + product (3)

C + e-

(E’)

(1) (E’)

k2

EE’CE’ 0 EP/Q

P + e-

P + B

k-1

B

EE’CE 0 EP/Q

(0) (E)

k1

Q + A

P + product (1) (E’)

Q

(0) (E)

P + e-

k-1 k2

B

(1) (C)

Q + A

C + product (2) (C) P

B

(3) (E)

(2) (C)

C

P + product (3) (E’)

(0) (E)

Q

k1

k1 k-1 k2

Q + C

k3

(E’)

(1) (C)

B C + product 2P

(2) (C) (3) (E’)

ECE’dim 0 EP/Q

(0) (E) k1

(1) (C)

k-1 kd

(2) (E’dim)

(E designates an electrode electron transfer, E’ an electron transfer taking place in the solution, C an associated chemical step, E’dim is a dimerizing homogeneous electron transfer reaction) Scheme 1: Important Catalytic Reaction Schemes

that can be varied by the operator - concentrations of catalyst and substrate, scan rate - on the one hand and, on the other, all the pertinent thermodynamic and kinetic constants characterizing the successive steps involved the reaction process. Their total number is large, and therefore a strategy that would try to guess their actual values through successive “simulations” 12 based on sounds a hopeless endeavor (noting that the goal is to ACSguess-values Paragon Plus Environment 3

ACS Catalysis

Page 4 of 23

produce a unique set of the unknowns). A more sensible approach consists in gathering these various 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

parameters into the minimal number of effectively governing parameters – usually chosen so as to be dimensionless. In the framework of this strategy, application of the steady-state approximation to one or more intermediates may lead to considerable simplification of the kinetic analysis. It has to be carefully justified, to avoid improper applications that can lead to inaccurate mechanistic conclusions. This is one purpose of the present work, taking as examples the reaction schemes shown in Scheme 1, which were selected either for their basic character or for their practical importance. There is a second important aspect to the intermediate steady-state vs. non-steady state dichotomy. It deals with the possibility of characterizing transient intermediates by structure-telling techniques (e.g., spectroscopic), which are conditioned by the occurrence of non-steady state behaviors compatible with extreme kinetic instability. 13,14 This is not the very first time these problems have been addressed. It has been tackled in the discussion of heterolytic vs. homolytic pathways in the molecular catalysis of H2 evolution. 15 The present contribution gives a more extended and more systematic view of the problem. Results and Discussion Before addressing the detailed analysis of the multistep reaction schemes in Scheme 1, we may note that essential features of the CV responses may be gathered, whatever the reaction scheme, from the simple EE’ mechanism as appears through the kinetic zone diagram recalled in Figure 1. 16, 9

Figure 1. General kinetic zone diagram for the simple EE’ scheme.

0 CP0 , CA : bulk concentrations of the catalyst and substrate

respectively , v: scan rate.

ACS Paragon Plus Environment 4

Page 5 of 23

ACS Catalysis

Catalytic reactions of practical interest are fast, implying that they belong to the upper-right domain of the 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

zone diagram corresponding to the “pure kinetic conditions” as they are usually called. If, in addition, substrate consumption can be neglected during the CV scan, zone KS is reached, corresponding to a stationary regime in which diffusion of the various species is compensated by the occurrence of the homogeneous reactions in which they are involved. S-shaped CV responses are thus obtained, easily characterized by a plateau current value, ipl, and a half wave potential, E1/2. Among all possible regimes, this is the one that provides the most complete access to the kinetic and mechanistic characterization of the catalytic process. This means that all efforts are made to reach this situation by changing concentrations and scan rate according to the directions and magnitude indicated by the compass in Figure 1. We note that what precedes is, strictly speaking, valid for catalyst couples obeying the Nernst law, which is not a serious constraint in practice because they are designed to be fast in most cases. Another important reason for focusing our analyses on this situation deals with the “secondary phenomena” that affect the catalytic current responses in most cases. They derive, not only from substrate consumption that would push the system toward zone KT, but also from other events such as inhibition by products, catalyst instability etc.…. all sharing the common property to interfere proportionally of the charge passed. 17,18 An additional advantage is that the stationary character of the situation entails that the time derivative of concentrations can be cancelled, thus converting the partial derivative equations (vs. time and space) describing the transport-reaction association into simple differential (vs. space) equations. We consider reduction processes, transposition to oxidations being straightforward. In dimensionless terms, the plateau current, i pl , will be dealt with as:

ξ1/ 2 = −

(

F 0 E1/ 2 − EP/Q RT

)

0 ) , and the half-wave potential, ϕ pl = i pl / ( FS DP CP0 k1CA

E1/ 2 , as

0 (S: electrode surface area, DP : catalyst diffusion coefficient, EP/Q ; standard

0 / (k−1CP0 ) for potential of the catalyst couple. The dimensionless kinetic parameters are defined as κ = k1CA

EE’-type schemes or

0 / k−1 EC-type schemes, κ = k1CA

λ = k2 / (k1CA0 ) , and

0 ); ρ = k3CP0 / (k1CA

0 0 / k−1 and λ = 2kd CP0 / (k1CA ) , for the EE’Cdim scheme. κ = k1CA ACS Paragon Plus Environment 5

ACS Catalysis

Page 6 of 23

The basic catalytic EE’C scheme 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Based on the detailed analysis given in the SI, the main characteristics of the CV responses in terms of plateau currents and half-wave potentials are given in Figure 2. For small values of the parameter κ, provided λ is not too small as discussed later on, the steady-state approximation is applicable to the intermediate B, leading to the expressions of CV responses, plateau current, and half-wave potential listed in Table 1, which depend on a single parameter, κλ = k2 / (k−1CP0 ) , featuring the competition between the backward E’ step and the follow-up C step.

0 ) , for Figure 2. Basic catalytic EE’C mechanism. Variations of the plateau current (a) and half-wave potential (b) with λ = k2 / ( k1CA

0 / (k C 0 ) , logκ = -4 (red), -3 (blue), -2 (green), -1 (magenta), 0 (dark yellow), 1 (cyan), 2 (yellow), 3 successive values of κ = k1CA −1 P (orange). The dots are the result of a Finite Difference Resolution (FDR) of the two-parameter diffusion-reaction equation system (see 0 ) in the limiting steady state (S ) and non-steadySI). Full lines and dotted lines: variations of plateau current with λ = k2 / ( k1CA B state (CB) situations, respectively, with the same code of colors for logκ. c: logλ – logκ kinetic zone diagram based on a 5% accuracy on plateau current measurement, showing, besides the two-parameter (κ, λ) –depending general case, G, two one-parameter– depending limiting behaviors corresponding to: steady-state for B, ( S B , κλ ) and constancy of B concentration in the reactiondiffusion layer ( CB , κ λ E). The zero-parameter zones are indicated by the values taken by the parameter of the adjoining oneparameter zone. For the meaning of the bold zone boundary line, see text. The compass in the upper right corner summarizes, in ACS Paragon Plus Environment direction and magnitude, the effects of the various experimental parameters. 6

Page 7 of 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

The full lines in Figure 2 correspond to this situation, showing good adherence with the fully calculated data points. This is no longer the case at the other end of the logκ range, in line with the non-applicability of the steady-state approximation in these conditions. The concentration of B may then build up due to the increasing irreversibility of the E’ step. Another limiting behavior may however be reached, being characterized by a 0 k2 / (k−1CP0 ) . The corresponding expressions of the CV different parameter, namely, κ λ = k1CA

responses, plateau current and half-wave potential are listed in Table 1. Within each of these limiting behaviors, still more simplified situations may be delineated, which do not depend on any parameter. The whole set of limiting behaviors is best represented by a kinetic zone diagram as shown in Figure 2c, where the boundary between the zones are based on a 5% uncertainty on plateau currents. The expressions of the plateau current and half-wave potential for each zones are summarized in Table 1 (the expressions for the full CV responses appear in the SI).

Table 1.The basic catalytic EE’C scheme. Characteristics of the current-potential responses 0 / (k C 0 ) , λ = k / (k C 0 ) κ = k1C A −1 P 2 1 A Zone (definition parameters) G (κ and λ)

S B (κλ )

CB (κ λ ) S B , κλ → ∞

CB , κ λ → 0

Plateau current: i pl ϕ pl = 0 FS DP CP0 k1CA

Half-wave potential: ξ1/ 2 = −

(

F E1/ 2 − E 0 P/Q RT

)

Finite difference resolution of the dimensionless two-parameter diffusion-reaction equation system (see SI)

  1 + λκ ϕ pl = 2λκ  (1 + λκ ) ln   λκ 

   − 1  

    1 + λκ 1 − (1 + λκ ) ln  1  1 + exp ( −ξ1/2 ) 1 + λκ − 1 + exp −ξ  ( ) 1 1/2   = 4  1 + λκ  (1 + λκ ) ln   −1  λκ 

exp ( −ξ1/2 )

ϕ pl 3 1 − ϕ pl 2

=κ λ

1 + exp ( −ξ1/2 )

1

= 1+

ϕ pl

ϕ pl

−

2 1+

2κ λ

ϕ pl 2κ λ

ξ1/2 = 0

1

(κ λ )

1/3

2 3

(

ξ1/ 2 = −0.257 − ln κ λ

)


ACS Catalysis

Page 8 of 23

In the CB zone, the catalytic current is relatively low as compared to what it is in other zones. It may 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

nevertheless be quite exploitable in practice, noting that the situation in this zone is quite interesting due to fact that the intermediate B builds up and may then be characterized by non-electrochemical means or used in additional valuable reactions. The passage between zone S (κλ) and zone CB (κ λ → 0) , marked by a bold line is worth some comment. Indeed, for small values of κ, as λ decreases (typically below 1) the steady-state approximation ceases to be valid over the entire voltammogram. This leads to a splitting of the catalytic wave that renders problematic the very definition of a half-wave potential as well as a precise boundary between zone S (κλ) and zone

CB (κ λ → 0) . At very low λ, B accumulates even at very positive potentials and hence is not at steady-state over the whole voltammogram and CB (κ λ → 0) zone behavior is recovered. We have skipped the detailed analysis of these phenomena in the above discussion, as they involve situations of modest interest in current practice (small catalytic currents and large overpotentials). More information may however be found in the SI. The EE’CE or E’ schemes 0 0 / (k−1CP0 ) , λ = k2 / (k1CA ) , and The CV responses now depend on three parameters, κ = k1CA 0 ) . It is assumed that the intermediate C is much easier to reduce than the resting state of the ρ = k3CP0 / (k1CA

catalyst as it is mostly the case in practice. This implies that C/P standard potential (Scheme 1) is much more positive than P/Q standard potential. ρ measures the competition between two pathways for the reduction of C, which leads to the product and regenerates the catalytic cycle. If ρ is small, C is reduced at the electrode along the EE’CE scheme, whereas if ρ is large the EE’CE’ predominates. These two possibilities are examined successively in the following. The EE’CE scheme It is easy to see (cf. SI) that, compared to the previous scheme, the current is simply doubled due to the additional contribution to the current from the reduction of C at the electrode surface. Provided this modification, all the characteristics of the EE’C scheme apply in the present case (doubling of current, same half-wave potentials), including the discussion of the limiting cases and the kinetic zone diagram. ACS Paragon Plus Environment 8

Page 9 of 23

ACS Catalysis

The EE’CE’ scheme 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Large values of ρ entails the exclusive reduction of the intermediate C by the active form of the catalyst rather than by the electrode. Figure 3 shows the variation of the plateau current and half wave potential with the parameter λ for a series of values κ. ρ has been pushed to such large values that the CV responses do not depend any longer upon this parameter. The main characteristics of the CV responses in terms of plateau currents and half-wave potentials are given in Figure 3. The limiting behaviors defined in Table 2 are represented by lines in Figure 3a,b, allowing the construction of a kinetic zone diagram in Figure 3c (see SI). As in the EE’C case, the direct passage between zone SBSC (κλ) and zone CB SC (κ λ → 0) , is marked by a bold line. The reason is again that for small values of κ, as λ decreases (typically below 1) the steady-state approximation ceases to be valid over the entire voltammogram. This leads to a splitting of the catalytic wave


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 23

0 ) , for successive Figure 3. The EE’CE’ scheme. Variations of the plateau current (a) and half-wave potential (b) with λ = k2 / (k1CA 0 / (k C 0 ) , logκ = -4 (red), -3 (blue), -2 (green), -1 (magenta), 0 (dark yellow), 1 (cyan), 2 (yellow), 3 (orange), 4 values of κ = k1CA −1 P (grey), 5 (light blue), 6 (dark blue), 7 (wine). The dots are the result of a Finite Difference Resolution of the two-parameter diffusionreaction equation system (see SI). Full lines: predicted variations when both B and C are under steady state (zone SBSC with κλ as parameter). Dashed lines: C at steady-state and not B (zone CBSC with κ λ ). Dotted lines: No steady-state for both B and C (zone CBLC with κλ as parameter). Dotted-dashed lines: No steady-state for both B and C (zone IBSC with λ as parameter) . c: logλ – logκ kinetic zone diagram based on a 5% accuracy on plateau current measurement: G two-parameter–depending (κ, λ) general case, four one-parameter–depending limiting behaviors corresponding to steady state for B and C ( S B S C , κλ ), irreversible formation of B and steady state for C ( I B S C , λ ), constant concentration of B in the reaction-diffusion layer and low values of C concentration ( C B LC ,

κλ

), constant concentration of B in the reaction-diffusion layer and steady state for C ( C B S C , κ λ ).The zero-parameter zones are indicated by the extreme values taken by the parameter of the adjoining one-parameter zone. For the meaning of the bold zone boundaries, see text. The compass in the upper right corner summarizes, in direction and magnitude, the effects of the various experimental parameters.

Table 2.The EE’CE’ scheme. Characteristics of the current-potential responses 0 / (k C 0 ) , λ = k / (k C 0 ) κ = k1C A −1 P 2 1 A

Zone (definition of parameters)

G (κ &λ)

SBSC (κλ)

FS

DP CP0

0 k1CA

Half-wave potential: ξ1/ 2 = −

(

F E1/ 2 − E 0 P/Q RT

)

Finite difference resolution of the dimensionless two-parameter diffusion-reaction equation system (see SI)

 1 + κλ  κλ



ϕ pl = 4κλ  (1 + κλ ) ln  

ϕ pl 1 +

CBSC ( κ λ )

I BS C ( λ )

i pl

Plateau current: ϕ pl =

ϕ pl κ 2λ

   − 1  

=1

    1 + λκ 1 − (1 + λκ ) ln  1  1 + exp ( −ξ1/2 ) 1 + λκ − 1 + exp −ξ ( 1/2 )  1  = 4  1 + λκ  (1 + λκ ) ln   −1  λκ  ϕ pl exp ( −ξ1/2 ) 1 = − ϕ pl 1 + exp ( −ξ1/2 ) ϕ pl 1+ 2 1+ 2κ 2λ 2κ 2λ

Numerical resolution of two coupled equations: for: λ < 3 − 2 2 or λ > 3 + 2 2 : 2 2  ϕ pl   (1 + (1 − Λ1 ) b0 ) −    ( 2 − Λ1 ) 1  ϕ pl ( 2 − Λ1 )    Λ1  − (1 − Λ1 )  b0 −  −  = −ϕ pl 2 2  Λ1  Λ1 1 ϕ pl ( 2 − Λ1 )  (1 − Λ1 )  b0 −  − 2 2   Λ1 2 2  ϕ pl   (1 + (1 − Λ 2 ) b0 ) −   Λ1  ( 2 − Λ2 ) 1  ϕ pl ( 2 − Λ 2 )    and: − + (1 − Λ 2 )  b0 −  −  = −ϕ pl ϕ 2 2  Λ 2  Λ2 pl ( 2 − Λ 2 )  1 (1 − Λ 2 )  b0 −  − 2 2  Λ2

with: Λ1,2 =

ξ1/2 = 0

(1 + λ ) ± (1 + λ )2 − 8λ 2

for: 3 − 2 2 < λ < 3 + 2 2 :   1 + (1 − Λ1′ ) b0 −   ϕ pl    ( 2 − Λ1′ ) and: z1 1  ϕ pl ( 2 − Λ1′ )    − (1 − Λ1′ )  b0 −  − 1 + (1 − Λ1′ ) b0 +    = ϕ pl ϕ ′ 2 − Λ ( ) z 1 2 2 z z1   1    1   1 − Λ ′  b −  − pl 1  ( 1) 0  2 2 z1   

ϕ pl


Page 11 of 23

ACS Catalysis



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ϕ pl

    1  ϕ pl ( 2 − Λ ′2 )  ( 2 − Λ′2 ) z2   − (1 − Λ′2 )  b0 −  − 1 + (1 − Λ′2 ) b0 +    = ϕ pl ϕ ′ 2 − Λ z 2 2 z z2 ( ) 1   2   2   1 − Λ ′  b −  − pl 2   ( 2 ) 0    2 2 z2  1 + (1 − Λ′2 ) b0 −

ϕ pl  

′ = with: Λ1,2 CBLC ( κλ )

(1 + λ ) ± j 8λ − (1 + λ )2 2

( j 2 = −1 )

z12 = Λ1′ and z22 = Λ ′2

1 2 

 

ξ1/2 = 0

ϕ pl = 2 / 4 − 1 −  κλ

S B S C (κλ → ∞ ) I B S C (λ → ∞ )

CB SC (κ λ → ∞ )

CBSC( κ λ → 0 )

I B SC (λ → 0) CB LC (κλ → ∞)

ξ1/2 = 0

2

ξ1/2 = 0

1

(

ϕ pl = κ 2λ

(

)

1/3

ξ1/2 = −0.257 − ( 2 / 3) ln κ 2λ

)

ξ1/2 = 0

2/ 3

that renders problematic the very definition of a half-wave potential as well as a precise boundary between zone S (κλ) and zone CB (κ λ → 0) . The transition between steady-state vs. non steady-state for C is also worth discussing. Although 0 ) is large, steady-state approximation on C (i.e., k3CCCQ = k2CB , or equivalently ρ = k3CP0 / (k1CA

ρ CCCQ / CP0 = λCB ) may not hold over the entire diffusion-reaction layer if Q concentration becomes vanishingly small before B does. As shown in the SI, such a situation is obtained when λ > EA/B In redox catalysis, EP/Q .

Consequently, κ >> 1 for usual substrate and catalyst concentrations. In addition follow-up reactions are fast. The system is thus relevant to zone SBSC where the steady-state approximation applies to both the B and C intermediates. The ECCE or E’ schemes ACS Paragon Plus Environment 11

ACS Catalysis

Page 12 of 23

0 / k−1 , As in the preceding cases, the CV responses depend on the three parameters, κ = k1CA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0 ) . It is similarly assumed that the intermediate C is much easier to λ = k2 / (k1CA0 ) , and ρ = k3CP0 / (k1CA

reduce than the resting state of the catalyst. ρ measures the competition between two pathways for the reduction of C, which leads to the product and regenerates the catalytic cycle. If ρ is small, C is reduced at the electrode along the ECCE scheme, whereas if ρ is large the ECCE’ pathway predominates. These two possibilities are examined successively in the following. The ECCE scheme This is a rare case in which the CV responses can be entirely described by closed-form expressions depending 0 0 / (k−1CP0 ) and λ = k2 / (k1CA ) (Table 3). The variations of the plateau on the two parameters κ = k1CA

currents and half wave potentials with the two parameters are represented in Figure 4. The equations of the various limiting cases are given in Table 3.


Page 13 of 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

0 ) , for Figure 4. ECCE reaction scheme. Variations of the plateau current (a) and half-wave potential (b) with λ = k2 / (k1CA 0 / k−1 , logκ =: -4 (red), -3 (blue), -2 (green), -1 (magenta), 0 (dark yellow), 1 (cyan), 2 (yellow), 5 successive values of , κ = k1CA (light blue). The lines in (a) and (b) correspond to the limiting situation denoted as “IB” in the kinetic zone diagram . The colored full lines in a correspond to zone SB . c: logλ – logκ kinetic zone diagram based on a 5% accuracy on plateau current measurement: G twoparameter–depending (κ, λ) general case, four one-parameter–depending limiting behaviors corresponding to steady state for B ( S B , κλ ), pre equilibrium formation of B ( EB , κ , κ ), constant concentration of B in the reaction-diffusion layer ( C B , κλ ), irreversible formation of B ( I B , λ ). The zero-parameter zones are indicated by the extreme values taken by the parameter of the adjoining one-parameter zone. The wind rose in the upper right corner summarizes, in direction and magnitude, the effects of the various experimental parameters.

The IB situation is particularly interesting as the summum of the non-steady state character of the intermediate B (it corresponds to the black dashed lines in Figures 4a and 4b). These are the best conditions for characterizing the intermediate B by non-electrochemical means or to devise its possible use in additional reactions leading to new products. ACS Paragon Plus Environment 13

ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 23

Table 3.The ECCE scheme. Characteristics of the current-potential responses 0 / k , λ = k / (k C 0 ) κ = k1CA −1 2 1 A Plateau current: i pl ϕ pl = 0 FS DP CP0 k1CA

Zone (definition of parameters) ϕ pl =

G (κ & λ )

SB , (κλ ) CB (κ λ )

IB ( λ )

Half-wave potential: F ξ1/ 2 = − E1/ 2 − E 0 P/Q RT

(

)

2

  (1 − Λ 2 ) Λ 2 − (1 − Λ1 ) Λ1       ( Λ1 − Λ 2 ) Λ 2 Λ1   1   +  Λ Λ Λ + Λ 2 1 2 1  

(



ξ1/ 2 = − ln 1 +  

 Λ1 − Λ 2  Λ 2 − (1 − Λ1 ) Λ1 

(1 − Λ 2 )

)

2 2  1   1   1   1  1 + + λ  +  1 + + λ  − 4λ 1 + + λ  − 1 + + λ  − 4λ  κ   κ   κ   κ  with : Λ1 = and Λ 2 = 2 2 κλ ξ1/ 2 = 0 ϕ pl = 2 1 + κλ   κ ϕ pl = 2 λ ξ1/ 2 = − ln   1+ κ λ  2   ϕ pl = 1 1 +  1 ξ = − ln 1/ 2 1+   1 λ λ + λ λ +1  

(

EB , (κ )

ϕ pl = 2

)

κλ 1+ κ

(

)

ξ1/ 2 = − ln (1 + κ )

SB (κλ → 0) EB (κ → 0)

ϕ pl = 2 κλ

ξ1/ 2 = 0

CB (κλ → ∞) EB (κ → 0)

ϕ pl = 2 λ

ξ1/ 2 = − ln (κ )

SB (λ → ∞) ) I B (λ → ∞)

ϕ pl = 2

ξ1/ 2 = 0

I B (λ → 0) CB (κ λ → ∞)

ϕ pl = 2 λ

ξ1/2 = ln

( λ)

We note that in the present situation, unlike the case of EE’C, EE’CE and EE’CE’ reaction schemes, the steady-state vs. non steady-state status of B transition for small values of κ (i.e. boundary between SB and SBEB zones) is no more an issue and is independent on the potential because the two competing chemical steps ( k−1 and

k2 ) are both first order. The above analysis was applied to H2 evolution from the reduction of NEt3H+ catalyzed by the FeI/0

tetraphenyl porphyrin couple, where the application of the plateau current and E1/2 diagnostic criteria indicate that the system lies in zone

I B (λ → 0) ; CB (κ λ → ∞) , implying that the FeII H• intermediate is formed

irreversibly and is not in steady state. 15 ACS Paragon Plus Environment 14

Page 15 of 23

ACS Catalysis

The ECCE’ scheme 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The parameter ρ is assumed to be so large that the CV responses do not depend any longer upon its value, meaning that the reduction of C into P exclusively takes place in the solution by reaction with Q, thus closing the catalytic loop. The main characteristics of the CV responses in terms of plateau currents and half-wave potentials are given in Figure 5. The limiting behaviors defined in Table 4 are represented by lines in Figure 5a,b, allowing the construction of the kinetic zone diagram in Figure 5c. For the same reason already mentioned in the ECCE case, unlike the case of EE’C, EE’CE and EE’CE’ reaction schemes, in the present situation the steady-state vs. non steady-state status of B transition for small values of κ is not an issue and is independent on the potential leading to a classical boundary between SBSC and SBSC ( κλ → 0 ) / EBSC ( κ → 0 ) zones. Conversely, unlike the ECCE case but alike to the EE’CE’ case, the steady-state vs. non-steady state status of C is worth mentioning because C is reduced by Q through a solution 0 ) being large, steady-state approximation on C may electron transfer step. Therefore, despite ρ = k3CP0 / (k1CA

not hold over the entire diffusion-reaction layer if Q concentration becomes vanishingly small before B does. As shown in the SI, such a situation is obtained when λ < 1 and κλ > 1 . A transition between C being at steadystate and not being at steady-state is thus found for κλ = 1 for small values of λ . It is represented by a bold line in Figure 5c. To better visualize this sharp transition, the plateau current are represented as ϕ pl / λ in Figure 5a’ showing characteristic angular points defining the transition between the CB LC zone and the horizontal line

ϕ pl / λ = 2 (corresponding to the zone CB SC (κ λ → ∞) ). A smoother transition between C being or being not at steady-state is obtained when λ = 1 at large values of κ . It is still an unconventional boundary since it bounds two zones characterized to the same single parameter. It is therefore represented in Figure 5c by a dashed line. Both zones differ from the boundary conditions associated with the diffusion-reaction equations as detailed in the SI.


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 23

0 ) , for Figure 5. ECCE’ reaction scheme. Variations of the plateau current (a, a’) and half-wave potential (b) with λ = k2 / (k1CA 0 / k−1 , logκ =: -4 (red), -3 (blue), -2 (green), -1 (magenta), 0 (dark yellow), 1 (cyan), 2 (yellow), 3 successive values of κ = k1CA (orange), 4 (grey), 5 (light blue). The dots are the result of a Finite Difference Resolution (FDR) of the two-parameter diffusionreaction equation system (see SI). Full lines in a: limiting behavior in zone SBSC. Dotted lines in a and a’: complete irreversibility of B formation. Dotted lines: No steady-state for both B and C (zone CBLC with κλ as parameter). In b: black dotted line: complete irreversibility of B formation; colored horizontal dotted lines: limiting behavior in zone EBSC. c: logλ – logκ kinetic zone diagram based on a 5% accuracy on plateau current measurement: G two-parameter–depending (κ, λ) general case, five one-parameter– depending limiting behaviors corresponding to steady state for B and C ( S B S C , κλ ), irreversible formation of B ( I B S C , λ and

I B LC , λ depending on the status steady-state or non-steady-state status of C), pre-equilibrium formation of B and steady state for C (

E B S C , κ ), constant concentration of B in the reaction-diffusion layer and low values of C concentration ( C B LC , κλ ), constant

concentration of B in the reaction-diffusion layer and steady state for C ( C B S C , κ λ ). The zero-parameter zones are indicated by the extreme values taken by the parameter of the adjoining one-parameter zone. For the meaning of the bold zone boundary line, see text. The wind rose in the upper right corner summarizes, in direction and magnitude, the effects of the various experimental parameters.


Page 17 of 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

Table 4 0 / k , λ = k / (k C 0 ) κ = k1CA −1 2 1 A


Zone (definition of parameters) G (κ &λ) CBSC ( κ

Half-wave potential: F ξ1/ 2 = − E1/ 2 − E 0 P/Q RT

(

)

Finite difference resolution of the dimensionless two-parameter diffusion-reaction equation system (see SI) ϕ pl =

λ)

SBSC ( κλ )

 κ  1+ κ λ  

ξ1/ 2 = − ln 

2λ

2κλ 1 + κλ

ϕ pl =

ξ1/ 2 = 0

If 3 − 2 2 > λ or λ > 3 + 2 2  Λ2  1 −    Λ1 − Λ 2 Λ1   ϕ pl = Λ 2  ξ1/ 2 = − ln 1 +  (1 − Λ ) Λ − (1 − Λ ) Λ  Λ Λ2 2 2 1 1  1− 2 Λ1 Λ1

with: Λ1 =

IBSC ( λ ) 20,21

(1 + λ ) + (1 + λ )2 − 8λ 2

and Λ2 =

(1 + λ ) − (1 + λ )2 − 8λ 2

If 3 − 2 2 < λ < 3 + 2 2

ϕ pl =

(a12 + b12 )

8λ − (1 + λ )2

 2 b1 (1 + λ ) + a1 8λ − (1 + λ )    with : a1 =

ϕ pl =

I BL C ( λ )

2 λ ϕ pl = 1+ λ

CBLC ( κλ ) S BS C ( κ → 0 ) EBSC ( κλ → 0 ) E BS C ( κ → ∞ ) C BS C ( κ λ → 0 ) SBSC ( κλ → ∞ ) I BS C ( λ → ∞ )

ϕ pl =

1 + λ + 8λ and b1 = 2

2κλ 1+κ

E BS C ( κ )

2λ κ 1 + κλ

 

ξ1/2 = ln 1 −  

   2 b1 (1 + λ ) + a1 8λ − (1 + λ )  2b1

−1 − λ + 8λ 2

ξ1/ 2 = − ln (1 + κ )    2+ λ λ  ξ1/2 = ln  2   1+ λ   

(

)

(

)

 2λ κ    1 + κλ 

ξ1/2 = ln 

ϕ pl =

2κλ

ξ1/ 2 = 0

ϕ pl =

2λ

ξ1/ 2 = − ln (κ )

ϕ pl = 2

ξ1/ 2 = 0

C BS C ( κ λ → ∞ )

ϕ pl =

2λ

I BL C ( λ → 0 ) CBLC ( κλ → ∞ )

ξ1/2 = ln 2λ

ϕ pl = 2 λ

ξ1/2 = ln 2 λ

(

)


ACS Catalysis

Page 18 of 23

Catalysis of the electrochemical CO2-to-CO conversion by iron 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

“I/0”

tetraphenylporphyrins (PFe) in DMF may

serve as illustrative example of the kinetic analysis depicted in this section (Scheme 2). 22

Scheme 2

0 The fact that E1/ 2 = EP/Q and that the apparent rate constant is first order in CO2 (i.e. the plateau current

would be proportional to

[CO2 ] if canonical conditions were to be met) points to the system lying in zone, -

SBSC ( κ → 0 )/EBSC ( κλ → 0 ), i.e. the intermediates PFe”I”CO2• and PFeIICO are at steady state, the first one being formed form through an unfavorable pre-equilibrium. This situation is quite favorable for a detailed kinetic analysis as demonstrated before, 22,23,24 but this is not the case for a non-electrochemical identification of the intermediates.

Catalytic dimerization schemes. The ECE’dim scheme. Variants of the preceding reaction schemes may involve dimerization reaction. We take here the ECE’dim scheme as tutorial example (Scheme 1). This reaction scheme has been previously analyzed in the particular case of a pre-equilibrium formation reaction of B intermediate

21

and in the particular case of an irreversible

formation reaction of B intermediate at the occasion of a discussion of the mechanisms of molecular catalysis of H2 evolution.

15

The present analysis provides a general description linking both previous analyses which are

subcases of this general description. The main characteristics of the CV responses in terms of plateau currents and half-wave potentials are given in Figures 6a & b together with the zone diagram shown in Figure 6c. The limiting behaviors corresponding to each zone of the zone diagram are summarized in Table As in the case of ECCE and ECCE’ mechanism and ACS Paragon Plus5.Environment 18

Page 19 of 23

ACS Catalysis

unlike EE’C and EE’CE or EE’CE’ mechanisms, there is no bold line for B intermediate in the zone diagram. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Indeed, because P is regenerated through an irreversible step, the transition between steady-state and nonsteady-state status of B is not dependent on the potential.

0 ), Figure 6. ECE’dim reaction scheme. Variations of the plateau current (a) and half-wave potential (b) with strucure λ = 2kd CP0 / (k1CA

for successive values of κ = k1C A0 / k − 1 : logκ = -4 (red), -3 (blue), -2 (green), -1 (magenta), 0 (dark yellow), 1 (cyan), 2 (yellow), 3 (orange), 4 (grey), 5 (dark green), 6 (dark blue), 7 (wine). The dots are the result of a Finite Difference Resolution (FDR) of the twoparameter diffusion-reaction equation system (see SI). In full lines are predicted variations for zone SB. The black dots are the results of a FDR resolution for a totally irreversible formation of B and the dashed line the predicted behavior in zone IB. In b, the dashed lines is the predicted behavior in zone IB: and the full lines is the predicted behavior in zone CB . c: logλ – logκ kinetic zone diagram based on a 5% accuracy on plateau current measurement: G two-parameter–depending (κ, λ) general case, four one-parameter– depending limiting behaviors corresponding to steady state for B ( S B , κ λ ), pre-equilibrium formation of B ( E B , κ ), irreversible formation of B ( I B , λ ), constant concentration of B in the reaction-diffusion layer ( CB , κ λ ). The zero-parameter zones are indicated by the extreme values taken by the parameter of the adjoining one-parameter zone. The wind rose in the upper right corner summarizes, in direction and magnitude, the effects of the various experimental parameters.


ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 23

Table 5.The The ECE’dim scheme. Characteristics of the current-potential responses

(

)

0 /k 0 0 κ = k1CA −1 , λ = 2kd CP / ( k1CA ) Zone(definition of parameters)

CB ( κ

Half-wave potential: F E1/ 2 − E 0 ξ1/ 2 = − P/Q RT

(

)

Finite difference resolution of the dimensionless two-parameter diffusion-reaction equation system (see SI) Coupled resolution of: 3 3      2λ − 1  3 1 + 4κ 2λ + 1 1 + 4κ 2 λ q − 1   3 1 + 4κ 2 λ q0,1/ 2 + 1   1 + 4 κ ϕ pl 0,1/2         =    ϕ pl =  2 2 2 2 48 κ λ 48 κ 2λ

G (κ & λ )

SB ( κ


λ )

( )

λ)

ϕ pl =

2 λ 3

( )

q ,1/ 2 1 + exp ( −ξ1/ 2 )  = 1 0

      1/3 4 −1     ξ1/2 = − ln κ  1/3   4  κ λ + 1    6  

IB ( λ )

FDR leading to the black dots and the dashed black curve in Figure 6a

EB ( κ )

ϕ pl =

(

)

κ 2 λ 3 1+ κ

ξ1/2 = − ln  41/3 − 1 (1 + κ )  

(

)

SB (κ λ → 0) EB (κ → 0)

ϕ pl =

2 λκ 3

ξ1/2 = − ln 41/3 − 1

CB (κ λ → 0) EB (κ → ∞)

ϕ pl =

2 λ 3

ξ1/2 = − ln  41/3 − 1 κ   

SB (κ λ → ∞) I B (λ → ∞ )

)

I B (λ → 0) CB (κ λ → ∞)

ϕ pl = 1

ϕ pl =

2 λ 3

(

)

ξ1/2 = 0

(

)

ξ1/2 = ln λ − ln  6 1 − 4 −1/3  ≈ ln λ 

Conclusions Removal of the steady state approximation usually applied to intermediates of multistep catalytic processes is a necessary requirement for a reliable analysis of experimental reactions and for their optimization. As appears in the manuscript and in the supporting information, this may lead to cumbersome physico-mathematical developments even if important simplifying assumptions are made. This is indeed the case here, where it has been assumed that pure kinetic conditions and negligible consumption of the substrate are achieved in the framework of the cyclic voltammetric experiment. These are rather commonly completed in current practice of homogeneous molecular catalysis and are experimentally ascertained by the S-shape of the CV current potential ACS Paragon Plus Environment 20

Page 21 of 23

ACS Catalysis

responses. This behavior is the result of mutual compensation of diffusion of the catalyst and chemical steps 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

involved to produce a time-independent situation resulting in CV responses that can be recognized not only by their peculiar shape but also by their independence from the scan rate. Under these conditions, we have shown that, for a series of practically important reaction schemes, the CV responses can be analyzed by means of only two independent parameters that gather the large number of experimental parameters that come into play. When these parameters become either very large of very small, limiting behaviors are reached that allow a simpler mechanistic and kinetic characterization. For the five reaction schemes considered here, our detailed analysis has allowed the establishment of diagnostic criteria allowing the determination of mechanism and of kinetic and thermodynamic constants or groups of these constants. In spite of the necessity of taking into account situations where steady-state conditions do not applied to intermediates, they actually apply in a number of circumstances. This is generally the case when the intermediate is formed through an up-hill reaction (small κ s). The achievement of such conditions, not only simplifies the physico-mathematical analysis of the problem but open opportunities to kinetically dissect mechanism by manipulating the nature of the rate determining step. A useful precaution, in such cases, is to test the self-consistency of the analysis, checking that the rate constants thus determined do match the applicability of the steady-state approximation Reaction schemes in which the formation of the intermediate is fast and irreversible is at the opposite of these situations. They conversely, provide good opportunities to characterize the intermediate by non-electrochemical means (e.g., spectroscopic), provided the final step is not too fast, thus allowing the building up of the intermediate concentration. Delineation of such conditions is also of help when planning the involvement of the intermediate in additional reactions leading to most preferred alternative products. The present contribution aims at a general analysis of the problem, potentially applicable to a large variety of situations. Systematic illustration by specific systems and experimental conditions is clearly beyond our present scope of this work. Nevertheless, the discussion of the homolytic vs. heterolytic mechanistic dichotomy in the catalysis of hydrogen evolution by low-valent iron tetraphenylporphyrins 15 may serve as an illustrating tutorial example. ACS Paragon Plus Environment 21

ACS Catalysis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 23

Supporting Information Derivation of the equations. References

(1) Hoffert, M. I.; Caldeira, K.; Jain, A. K.; Haites, E. F.; Harvey, L. D. D.; Potter, S. D.; Schlesinger, M. E.; Schneider, S. H.; Watts, R. G.; Wigley, T. M. L.; Wuebbles, D. J.: Energy implications of future stabilization of atmospheric CO2 content. Nature 1998, 395, 881-884. (2) Lewis, N. S.; Nocera, D. G.: Powering the planet: Chemical challenges in solar energy utilization. Proc. Natl. Acad. Sci. U. S. A. 2006, 103, 15729-15735. (3) Gray, H. B.: Powering the planet with solar fuel. Nat Chem 2009, 1, 7-7. (4) Nocera, D. G.: Chemistry of Personalized Solar Energy. Inorg. Chem. 2009, 48, 10001-10017. (5) Abbott, D.: Keeping the Energy Debate Clean: How Do We Supply the World's Energy Needs? Proceedings of the IEEE 2010, 98, 42-66. (6) Chu, S.; Majumdar, A.: Opportunities and challenges for a sustainable energy future. Nature 2012, 488, 294-303. (7) Artero, V.; Fontecave, M.: Solar fuels generation and molecular systems: is it homogeneous or heterogeneous catalysis? Chem. Soc. Rev. 2013, 42, 2338-2356. (8) Bard, A. J. & Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications 2nd ed. (John Wiley & Sons, 2001). (9) Savéant, J.-M. Elements of Molecular and Biomolecular Electrochemistry: An Electrochemical Approach to Electron Transfer Chemistry. (John Wiley & Sons, 2006). (10) Other transient or stationary non-destructive techniques 8,9 could be used instead, transposition of what is said here from CV to this techniques being well feasible. (11) Savéant, J.-M.; Vianello, E.: Recherches sur les courants catalytiques en polarographie oscillographique à balayage linéaire de tension. Étude théorique. Adv. Polarogr. 1959, 1, 367-374. (12). a: The available “simulation” packages enter given values of these various parameters into a finite difference resolution of a set of partial derivative equations featuring the association between transport and reaction, with the appropriate initial and boundary conditions. 12b,c KISSA: Amatore, C.; Klymenko, O.; Svir, I. A new strategy for simulation of electrochemical mechanisms involving acute reaction fronts in solution: Principle. Electrochem. Commun. 2010, 12, 1170-1173. c: DIGIELCH: Rudolph, M.: Digital simulations on unequally spaced grids.: Part 2. Using the box method by discretisation on a transformed equally spaced grid. J. Electroanal. Chem. 2003, 543, 23-39. (13) Chatterjee, S.; Sengupta, K.; Mondal, B.; Dey, S.; Dey, A.: Factors Determining the Rate and Selectivity of 4e–/4H+ Electrocatalytic Reduction of Dioxygen by Iron Porphyrin Complexes. Acc. Chem. Res. 2017, 50, 1744-1753. (14) Mondal, B.; Rana, A.; Sen, P.; Dey, A.: Intermediates Involved in the 2e–/2H+ Reduction of CO2 to CO by Iron(0) Porphyrin. J. Am. Chem. Soc. 2015, 137, 11214-11217. (15) Costentin, C.; Dridi, H.; Savéant, J.-M.: Molecular catalysis of H2 evolution. Diagnosing heterolytic vs. homolytic pathways. J. Am. Chem. Soc. 2014, 136, 13727-13734. (16) Savéant, J.-M.; Su, K.-B.: Homogeneous redox catalysis of electrochemical reaction. Part VI. Zone diagram representation of the kinetics regimes. J. Electroanal. Chem. 1984, 171, 341-349. (17) Costentin, C.; Drouet, S.; Robert, M.; Saveant, J.-M.: Turnover numbers, turnover frequencies and overpotential in molecular catalysis of electrochemical reactions. Cyclic voltammetry and preparative-scale electrolysis. J. Am. Chem. Soc. 2012, 134, 1123511242. ACS Paragon Plus Environment 22

Page 23 of 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Catalysis

(18) Costentin, C.; Passard, G.; Robert, M.; Savéant, J.-M.: Ultraefficient homogeneous catalyst for the CO2-to-CO electrochemical conversion. Proc. Natl. Acad. Sci. U.S.A. 2014, 111, 14990-14994. (19) Andrieux, C. P.; Blocman, C.; Dumas-Bouchiat, J.-M.; M'Halla, F.; Savéant, J.-M.: Determination of the life-times of unstable ion radicals by homogeneous catalysis of electrochemical reactions. Application to the reduction of aromatic halides. J. Am. Chem. Soc. 1980, 102, 3806-3813. (20).This corrects the fact that this second set of equations was skipped in reference 21. We thank Dr Mandfred Rudoph (Leipzig) for drawing our attention on errors that this omission could generate. (21) Costentin, C.; Savéant, J.-M.: Multielectron, multistep molecular catalysis of electrochemical reactions: benchmarking of homogeneous catalysts. ChemElectroChem 2014, 1, 1226-1236. (22) Costentin, C.; Drouet, S.; Passard, G.; Robert, M.; Savéant, J.-M.: Proton-coupled electron transfer cleavage of heavy-atom bonds in electrocatalytic processes. Cleavage of a C–O bond in the catalyzed electrochemical reduction of CO2. J. Am. Chem. Soc. 2013, 135, 9023-9031. (23) Costentin, C.; Robert, M.; Savéant, J.-M.: Current issues in molecular catalysis illustrated by iron porphyrins as catalysts of the CO2-to-CO electrochemical conversion. Acc. Chem. Res. 2015, 48, 2996-3006. (24) Azcarate, I.; Costentin, C.; Robert, M.; Savéant, J.-M.: Dissection of electronic substituent effects in multielectron-multistep molecular catalysis. Electrochemical CO2-to-CO conversion catalyzed by iron porphyrins. J. Phys. Chem. C 2016, 120, 28951-28960.

TOC


Homogeneous Catalysis of Electrochemical Reactions. Steady-State

Recommend Documents