Requiem for the Rate-Determining Step in Complex Heterogeneous Catalytic Reactions?

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Article Requiem for the Rate-Determining Step in Complex Heterogeneous Catalytic Reactions?

Dmitry Yu. Murzin Process Chemistry Centre, Åbo Akademi University, 20500 Turku/Åbo, Finland; dmitry.murzin@abo.ﬁ

Received: 1 September 2020; Accepted: 9 September 2020; Published: 10 September 2020

Abstract: The concept of the rate determining step, i.e., the step having the strongest influence on the reaction rate or even being the only one present in the rate equation, is often used in heterogeneous catalytic reactions. The utilization of this concept mainly stems from a need to reduce complexity in deriving explicit rate equations or searching for a better catalyst based on the theoretical insight. When the aim is to derive a rate equation with eventual kinetic modelling for single-route mechanisms with linear sequences, the analytical rate expressions can be obtained based on the theory of complex reactions. For such mechanisms, a single rate limiting step might not be present at all and the common practice of introducing such steps is due mainly to the convenience of using simpler expressions. For mechanisms with a combination of linear and nonlinear steps or those just comprising non-linear steps, the reaction rates are influenced by several steps depending on reaction conditions, thus a reduction in complexity to a single rate limiting step can lead to misinterpretations. More widespread utilization of a microkinetic approach when the reaction rate constants can be computed with reasonable accuracy based on the theoretical insight, and availability of software for kinetic modelling, when a system of differential equations for reactants and products will be solved together with differential equations for catalytic species and the algebraic conservation equation for the latter, will eventually make the concept of the rate limiting step obsolete.

Keywords: rate determining step; linear steps; nonlinear steps

1. Introduction The kinetics of heterogeneous catalytic reactions has been the focus of numerous studies because of its theoretical and practical importance [1–10]. Kinetic analysis was applied for decades to assist reactor design and scaling up of various reactions relevant for oil refining and synthesis of basis and specialty chemicals. More recently, because of a widespread application of so-called flow chemistry to streamline pharmaceutical research and products, kinetic analysis also started to be more widely utilized in the field of fine chemicals [11–13]. It was argued [11] that the framework of a mechanistically based rate equation combined with in situ experimental data is useful for the interpretation of complex reaction networks. Increasing complexity of the catalytic reactions, either relevant for oil refining or pharmaceutical synthesis, poses some restrictions regarding incorporation of such complex mechanisms into tractable rate equations. Thus, another trend is to simplify complex reaction networks and assume that there is a limiting step, the rate of which is a good approximation of the reaction rate of the whole network [14,15]. This approach of a rate-determining step (RDS) became very popular in chemical reaction engineering, allowing the derivation of kinetic equations for reaction mechanisms of differing complexity. Such an assumption of a rate-determining step makes the rate expressions more tractable, facilitating the kinetic modelling of mainly steady-state kinetics even at the expense of losing important mechanistic information. Moreover, in several instances, without a substantial reduction in complexity, kinetic modelling can be computationally very demanding, especially when coupled with transport phenomena [16].

Reactions 2020, 1, 37–46; doi:10.3390/reactions1010004 www.mdpi.com/journal/reactions Reactions 2020, 3, x FOR PEER REVIEW 2 of 11 Reactions 2020, 1 38 modelling can be computationally very demanding, especially when coupled with transport phenomena [16]. OnOn the the other other hand, hand, the the concept concept of of RDS RDS was was suggested to to give give fruitful fruitful ideas ideas for for modifying the the reactionreaction conditions conditions or or design design the the catalyst. Subseq Subsequently,uently, there there have have been been many many theoretical papers addressingaddressing how how to to determine determine the the step step which which is isth thee rate rate determining determining [17,18]. [17,18 The]. The most most popular popular is probablyis probably the the degree degree of ofrate rate control control proposed proposed by by Campbell, Campbell, based based on on the the general general principles principles of of differentialdifferential sensitivity analysis analysis to to determine determine sensitivity sensitivity of of the the output output to to one one input input parameter parameter [19–25]. [19–25]. ItIt was was argued argued [16] [16] that that an approach analogous to to electrical networks networks is is even even better better suited suited for for the the identificationidentification of of the the key key steps steps determining determining the the reacti reactionon rate rate allowing the the derivation of of accurate rate rate expressionsexpressions even even for for mechanisms mechanisms with with nonlinear nonlinear step steps.s. Such Such nonlinear nonlinear steps steps are are present present not not only only in in heterogeneousheterogeneous catalytic reactions, such as, for exam example,ple, a a water water gas gas shift shift reaction reaction [16], [16], decomposition decomposition ofof ammonia [26[26,27],27] or or methanol methanol [28 [28],], but but also also in organometallic in organometallic catalysis. catalysis. In the In latter the case, latter the case, reaction the reactionorder in order catalysts in catalysts deviates deviates from unity from [5]. unity The formation[5]. The formation of dimeric of catalyticdimeric catalytic species, onspecies, the contrary, on the contrary,results in results the reaction in the order reaction being order below being one, below which one, is ratherwhich rare is rather [29–31 rare]. [29–31]. ContraryContrary to to the the case case of of linear (Christiansen) se sequencesquences of of a a single-route single-route reaction reaction [5,32,33] [5,32,33] for for single-routesingle-route mechanisms withwith nonlinear nonlinear steps, steps, a generala general expression expression for for the the reaction reaction rate rate in steady-state in steady- stateand quasi-steady-stateand quasi-steady-state conditions conditions cannot cannot be derived be derived [34]. [34]. When When such such derivation derivation is possible, is possible, the ratethe rateexpressions expressions become become rather rather complicated complicated to handle to handle [5,35 [5,35–37].–37]. ToTo overcomeovercome di ffidifficultiesculties in derivingin deriving explicit explicit rate expressions, rate expressions, a semi-empirical a semi-empirical kinetic polynomial kinetic polynomialwas proposed was [38 proposed] as a substitute [38] as a of substi analyticaltute of rate analytical equations. rate equations. However,However, inin catalysis catalysis by by solids solids or organometallicor organometallic complexes, complexes, quite oftenquite either often there either are there nonlinear are nonlinearsteps involved steps involved or the rate or equationsthe rate equations for even for single-route even single-route mechanisms mechan andisms obviously and obviously multi multi route routereactions reactions do not do have not ratehave limiting rate limiting steps [steps34]. [34]. ItIt was was recognized aa longlong timetime ago ago that that for for multi-route multi-route (or (or composite) composite) reactions, reactions, the the concept concept of theof therate-controlling rate-controlling steps steps is not is not required required [34, 39[34,39],], while while in other in other cases cases it should it should be considered be considered with with care, care,as misinterpretation as misinterpretation is easily is easily possible. possible. InIn the the current current work, work, several several cases cases of complex single- and multi-route heterogeneous catalytic reactionsreactions will will be be presented, presented, highlighting highlighting an an apparent apparent danger danger of of falling falling into into a a trap trap of of using using the the rate rate determiningdetermining step, step, which which leads to confusion rather than clarification clarification of reaction mechanisms. 2. Three Step Sequence 2. Three Step Sequence Let us consider first a three-step linear sequence (Figure1) Let us consider first a three-step linear sequence (Figure 1)

FigureFigure 1. PotentialPotential energy energy diagram diagram using using Gibbs Gibbs energy energy an andd Gibbs Gibbs activation activation energy energy for for the the reaction reaction

S->P.S->P. T T11–T–T33 representrepresent transition transition states, states, and and I 1-I–I2 2intermediates,intermediates, I I0 0isis the the free free form form of of catalyst. catalyst. Adapted Adapted fromfrom [40]. [40].

TheThe general general equation equation for for this this three-step three-step sequence sequence in the the case case of of all all reversible reversible steps steps can can be be written written inin a a form form of of frequencies frequencies of of steps steps [6] [6]

ω+1ω+2ω+3 ω 1ω 2ω 3 r = − − − − Ccat (1) ω+2ω+3+ω 3ω+2+ω 3ω 2+ω+3ω+1 + ω 1ω+3+ω 1ω 3+ω+1ω+2+ ω 2ω+1+ω 2ω 1 − − − − − − − − − Reactions 2020, 3, x FOR PEER REVIEW 3 of 11

ωωω− ωωω = +++123 −−− 123 r Ccat (1) Reactions 2020ωω1 +++ ωω ωω ωω + ωω +++ ωω ωω ωω + ωω +2+3, −−− 3+2 3 2 +3+1 −−− 1+3 1 3 +1+2 −−− 2+1 2 1 39

ω cat where r is the reaction rate, +1 is the frequency of step 1, etc., and C is the catalyst concentration. whereTher frequenciesis the reaction of steps rate, ωare+1 isobtained the frequency by dividing of step the 1, etc.,rate andexpressionsCcat is the of catalyst a particular concentration. step by the Theconcentration frequencies ofof stepscatalytic are obtainedspecies in by that dividing expres thesion. rate For expressions example, ofin a the particular case of step the by reaction the concentrationmechanism of presented catalytic speciesin Figure in that2, the expression. general form For of example, the Equation in the case(1) gives of the [6] reaction mechanism presented in Figure2, the general form of the Equation (1) gives [6] ()kC++ kC k +−−−− kkkC C = 12HCH224 3123 CHcat 26 r (k C k C k k k k C )C (2) +++++++1 H2 +2 C2H4 +3 1 2 3 C2H6 cat ++ = kk+− kC k kC − k −+ kk C kk −+−−− − kkC− − kC kC kkC − kk −− r k +2k 3+k C 3CH26k +2+k C 3 CH 26k 2+k 3k +1 C H 2+ k 1k 3+k k 1 3C CH 26+k +1C H 2k +2C CH 24+ k 2k +1C H 2+k 2k 1 (2) +2 +3 3 C2H6 +2 3 C2H6 2 +3 +1 H2 1 +3 1 3 C2H6 +1 H2 +2 C2H4 2 +1 H2 2 1 − − − − − − − − −

Figure 2. Three-step direct ethylene hydrogenation mechanisms. Figure 2. Three-step direct ethylene hydrogenation mechanisms.

Thus, the frequency of the ﬁrst step in the forward direction is k+1CH2 , while for the reverse step it wouldThus, be just thek frequency. When all of steps the first are irreversible,step in the forward Equation direction (2) can be is easilykC+ simpliﬁed,, while for giving the reverse 1 1 H2 − step it would be just k− . When all steps arek irreversible,C k C k EquationC (2) cank beC easilyC C simplified, giving ω+1ω+2ω+13 +1 H2 +2 C2H4 +3 cat +1 H2 C2H4 cat r = Ccat = = (3) ω+2ω+3+ω+3ω+1 +ω+1ω+2 k+2k+3+k+3k+1 CH +k+1CH k+2CC H k+1 k+1 2 2 2 4 1+ CH + CH CC H k+2 2 k+3 2 2 4 ωωω kC++ kC kC + kC + C C ==+++123 12H224 C H 3 cat = 1 H 224 C H cat rCcat Itωω follows++ ωω from Equation ωω (3) thatkk depending++ + kk C on + the kCkC values of partialkk pressures+1 +1 of, for example,(3) +2 +3 +3 +1 +1 +2 +2 3 3 +1 HHCH2224 +1 +2 1++CCC HHCH2224 hydrogen, the reaction order can change from unity at low pressures, to zero atkk++23 high hydrogen pressures.

The relative contribution of the last term in the denominator of Equation (3) k+1CH2 CC2H4 /k+3 in comparisonIt follows with twofrom other Equation terms (3) defines that depending the reaction on order the towardsvalues of ethylene. partial pressures of, for example, hydrogen,This example the illustratesreaction order an apparent can change danger from of defining unity at the low rate pressures, and even moreto zero turnover at high frequency hydrogen throughpressures. just one The rate relative limiting stepcontribution without taking of the into la accountst term kinetic in the regularities. denominator To highlight of Equation the need (3) kC C/ k+ in comparison with two other terms defines the reaction order towards ethylene. to consider+1HCH224 all steps in 3 a linear catalytic sequence even further, let us consider the apparent activation energy forThis the example three-step illustrates mechanism an (Equationapparent (3)).danger In what of defining follows, thethe termrate “apparent”and even more corresponds turnover to thefrequency observed through activation just energy one rate of alimiting complex step multistep without reaction taking in into the account kinetic regime, kinetic whichregularities. should To nothighlight be confused the withneed theto consider case when all thesteps observed in a linear kinetics catalytic is influenced sequence byeven the further, transport let phenomena.us consider the apparentA more activation simple case energy of the for two-step the three-step sequence mechanism was addressed (Equation in the(3)). literature In what follows, [41] using the the term definition“apparent” of the corresponds apparent activation to the observed energy activation energy of a complex multistep reaction in the kinetic regime, which should not be confused with the case when the observed kinetics is influenced ∂ ln r by the transport phenomena. Ea,app = R (4) − ∂(1/T) A more simple case of the two-step sequence was addressed in the literature [41] using the anddefinition giving of the apparent activation energy ω+2 ω+1 Ea,app = Ea,1 + Ea,2 (5) ω+1 + ω+2 ω+1 + ω+2 Reactions 2020, 1 40

As follows from Equation (5), the apparent activation energy depends not only on the step with the highest activation energy, but also on the contribution of the other step with a lower activation energy. For the three-step sequence using Equations (3) and (4), one gets

2 ∂ Ea,app = RT (ln k+ + ln CH + ln k+ + ln C + ln k+ + ln Ccat ∂T 1 2 2 C2H4 3 − (6) ( + + )) ln k+2k+3 k+3k+1 CH2 k+1CH2 k+2CC2H4

Some terms in Equation (6) can be easily diﬀerentiated

∂ 0 ∂ 0 E+1 (ln k + exp( E+ /RT) = (ln k + E+ /RT) = (7) ∂T 1 − 1 ∂T 1 − 1 RT2 Taking Equation (7) into account and also using a standard approach for the diﬀerentiation of a complex function ∂ ln U ∂ ln U ∂U 1 ∂U = = (8) ∂T ∂U ∂T T ∂T one gets the ﬁnal expression for the activation energy of a three step sequence

+ + ω+2ω+3Ea,1 ω+1ω+3Ea,2 ω+1ω+2Ea,3 Ea,app = (9) ω+2ω+3 + ω+3ω+1 + ω+1ω+2

It can be shown that in only exceptional cases the apparent activation energy of the overall reaction is determined by the activation energy of only one rate-limiting step in a steady-state sequence.

3. Four Step Sequence As an example of such a sequence, the liquid-phase hydrogenation of aromatic compounds will be considered. The reaction kinetics was extensively investigated previously in [42–46]. The following generic scheme for hydrogenation of an aromatic compound A to the product B was proposed

1.*A + H *AH 2 ⇔ 2 2.*AH + H *AH 2 2 ⇔ 4 3.*AH *Y 4 ⇒ (10) 3 .*Y + H *B (fast) 0 2 ⇔ 4.*B + A Ξ *A + B A + 3H2 = B where Y is cyclohexene or its derivative, AH2 and AH4 are intermediate complexes, step 30 is fast because hydrogenation of cycloalkenes is much faster than aromatic compounds and step 4 is at quasi-equilibrium. A general equation for the four step sequence is [6]

ω+1ω+2ω+3ω+4 ω 1ω 2ω 3ω 4 r = − − − − − C (11) D cat where D = ω+2ω+3ω+4 + ω 1ω+3ω+4 + ω 1ω 2ω+4 + ω 1ω 2ω 3+ − − − − − − +ω+1ω+2ω+3 + ω 4ω+2ω+3 + ω 4ω 1ω+3 + ω 4ω 1ω 2+ − − − − − − (12) +ω+4ω+1ω+2 + ω 3ω+1ω+2 + ω 3ω 4ω+2 + ω 3ω 4ω 1+ − − − − − − +ω+3ω+4ω+1 + ω 2ω+4ω+1 + ω 2ω 3ω+1 + ω 2ω 3ω 4 − − − − − − Because of irreversibility of step (3), Equations (11) and (12) can be simpliﬁed

ω+ ω+2ω+3ω+ r = 1 4 C (13) D cat Reactions 2020, 1 41

D = ω+2ω+3ω+4 + ω 1ω+3ω+4 + ω 1ω 2ω+4 + ω+1ω+2ω+3 + ω 4ω+2ω+3 + ω 4ω 1ω+3 + ω 4ω 1ω 2+ − − − − − − − − − (14) +ω+4ω+1ω+2 + ω+3ω+4ω+1 + ω 2ω+4ω+1 − Step 4 is at quasi-equilibrium, and therefore the rates are fast in both directions. Further simpliﬁcations of Equation (13) are thus possible by dividing the numerator and the denominator by ω+4, neglecting subsequently the terms ω+1ω+2ω+3/ω+4 and ω 3ω+1ω+2/ω+4. Then the following holds − ω+ ω+2ω+3 r = 1 C (15) ω 4 cat (ω+2ω+3 + ω 1ω+3 + ω 1ω 2)(1 + − ) + ω+1ω+2 + ω+3ω+1 + ω 2ω+1 − − − ω+4 − Replacing frequencies in Equation (15) with the explicit expressions, one gets

k+1CH2 k+2CH2 k+3 r = Ccat (16) ( + + )( + CB ) + + + k+2CH2 k+3 k 1k+3 k 1k 2 1 K C k+1CH2 k+2CH2 k+3k+1CH2 k 2k+1CH2 − − − 4 A − where K4 = k+4/k 4. For a binary mixture of the reactant and substrate, it is possible to use mole − fractions instead of concentrations. Moreover, quite often partial pressures of hydrogen are applied instead of concentrations, resulting in another form of the rate equation

k + P k + P k+ 0 1 H2 0 2 H2 3 r = 1 N Ccat (17) − A (k +2PH k+3+ k 1k+3+k 1k 2)(1+ K N )+k +1PH k +2PH +k+3k +1PH + k 2k +1PH 0 2 − − − 4 A 0 2 0 2 0 2 − 0 2 where NA is the mole fraction of the reactant. Equation (16) contains the modiﬁed rate constants k + ; k + as instead of hydrogen concentration in the liquid phase, the pressure of hydrogen is used. 0 1 0 2 Equation (17) presented in a slightly diﬀerent form

k k k 0+1 0+2 +3 P N C k k +k k +k k H2 A cat +3 0+1 2 0+1 0+2 +3 r = k k +k k ( k k +k k )(1 N ) − k k (1 N ) k k (18) 1 +3 1 2 1 1 +3 1 2 − A 1 0+2 +3 − A 0+1 0+2 − − − NAP− H + − − − )P− H + + PH NA+NA k+3k +1+k 2k +1+k +2k+3 2 (k+3k +1+k 2k +1+k +2k+3)K4 2 (k+3k +1+k 2k +1+k +2k+3)K4 k+3k +1+k 2k +1+k +2k+3 2 0 − 0 0 0 − 0 0 0 − 0 0 0 − 0 0 was derived in [43] using a more tedious procedure, namely applying directly the steady-state conditions

r+1 r 1 = r+2 r 2 = r+3 (19) − − − − as well as the quasi-equilibrium for step 4 and ﬁnally the balance equation (i.e., the sum of all coverages is equal to unity). All steps in Equation (10) are needed to explain the observed kinetic regularities, because it was demonstrated that at low hydrogen partial pressures, the reaction order in hydrogen can exceed unity, which is possible when k 1 , 0, while at high pressures the reaction kinetics obeys frequently observed − zero orders in the substrate and hydrogen. In the latter case, the overall rate is determined by the isomerization of the adsorbed AH4 species. The surface of the catalyst is completely covered with this complex and the reaction rate ceases to depend on either hydrogen pressure or the concentration of the substrate. Apparently, the concept of a single rate-determining step is not able to account for a rich kinetic behavior, observed experimentally in hydrogenation of aromatics.

4. Oxidative Dehydrogenation of Ethanol: A Multi-Route Mechanism with Nonlinear Steps The mechanism and kinetics of oxidative dehydrogenation of ethanol over gold catalysts was recently reported [47]. DFT calculations indicated that the activation of molecular oxygen is facilitated in the presence of ethanol acting as a hydrogen donor. As a result of hydrogen abstraction from the Reactions 2020, 3, x FOR PEER REVIEW 6 of 11 as well as the quasi-equilibrium for step 4 and finally the balance equation (i.e., the sum of all coverages is equal to unity). All steps in Equation (10) are needed to explain the observed kinetic regularities, because it was demonstrated that at low hydrogen partial pressures, the reaction order in hydrogen can exceed unity, which is possible when k−1≠ 0, while at high pressures the reaction kinetics obeys frequently observed zero orders in the substrate and hydrogen. In the latter case, the overall rate is determined by the isomerization of the adsorbed AH4 species. The surface of the catalyst is completely covered with this complex and the reaction rate ceases to depend on either hydrogen pressure or the concentration of the substrate. Apparently, the concept of a single rate-determining step is not able to account for a rich kinetic behavior, observed experimentally in hydrogenation of aromatics.

N(1a) N(1b) N(1c) Ξ 1. O2+* O2* 1 1 1 2. EtOH+*ΞEtOH* 2 2 2 3. EtOH* + O2*↔EtO*+OOH* 2 1 2 4. OOH*+*→O*+OH* 2 1 1 5. O*+EtO*→AcH+*+OH* 2 2 2 (10)(20) 6. 2OH*↔O*+H2O+* 2 2 2 7a. 2O*→O2*+* 1 0 0 7b. EtOH*+O*→EtO*+OH* 0 1 0 7c. O*+OOH*→O2*+OH* 0 0 1 (1) N 2EtOH+O2→2AcH+2H2O

InIn (20), (20), on on the the right right hand hand side side of of the the equations equations for for the the steps steps (i.e., (i.e., 1 1 to to 7c), 7c), the the stoichiometric stoichiometric (1a) (1c) numbersnumbers (i.e., (i.e., 0, 0, 1, 1, 2) 2) along alternative routes NN(1a)--N N (1c)areare given given [32]. [32]. These These numbers numbers can can be be equal equal toto zero, zero, if if a a particular particular step step is is not not involved involved in in th thee reaction reaction mechanism. mechanism. In In Equation Equation (20), (20), EtOH EtOH stands stands forfor ethanol, ethanol, while while AcH AcH represents represents acetaldehyde acetaldehyde and and * * denotes denotes a a surface surface site. site. AccordingAccording to to the the DFT DFT calculations calculations [47] [47 ]illustrate illustratedd in in Figure Figure 33,, the largestlargest activationactivation barrier correspondscorresponds to to step step 3 3 in in Equation Equation (20), (20), and and thus thus supe superﬁciallyrficially it it can can be be considered considered in in the the conventional conventional approachesapproaches as as the the rate rate limiting limiting step. step. However, However, the the activation activation energy energy of of the the same same step step in in the the reverse reverse direction is very low and step 3 can even be viewed to be close to the quasi-equilibrium. directionReactions 2020 is, very3, x FOR low PEER and REVIEW step 3 can even be viewed to be close to the quasi-equilibrium. 7 of 11 From the considerations above on the apparent activation energy of the three-step sequence with linear steps, it is clear that the apparent activation energy and the reaction rate depend on the contribution of all steps in the mechanism, which could be treated using the steady-state approximation. This approach is possible because of the specific features of the reaction mechanism comprising only nonlinear steps and thus allowing derivation of an analytic rate expression. Such derivation was done in [47] based on the DFT computed reaction energies assuming quasi-equilibria for steps 1 and 2, irreversibility for steps 4 and 5 and reversibility for steps 3 and 6.

Figure 3. The minimum minimum energy path for ethanol (EtOH) de dehydrogenation.hydrogenation. The black and orange bars correspond toto minimaminima andand transitiontransition states, states, respectively. respectively. * denotes* denotes adsorbed adsorbed species, species, while while (g) (g) refers refers to gasto gas phase phase species. species. Square Square brackets brackets enclose enclose a transition a transition state state structure structure where where ‘ - -’ marks‘ - -’ marks the bond the beingbond broken.being broken. Reproduced Reproduced with permission with permission from [from47]. [47].

From the viewpoint considerations of the above global on reaction the apparent kinetics, activation not only steps energy 3-6 of are the important three-step but sequence also the withfinal step linear in steps,the mechanism, it is clear thatstep the7, reflecting apparent the activation fate of atomically energy and adsorbed the reaction oxygen. rate Therefore, depend on three the contributionoptions are possibly of all steps influencing in the mechanism, the stoichiometric which could numbers be treated of other using steps. the steady-state Step 7a in approximation.the route N(1a) Thiscorresponds approach to is recombination possible because of atomic of the specific oxygen, features while such of the recombination reaction mechanism in step comprising 7c (route N only(1c) ) nonlinearinvolves asteps reaction and with thus a allowing peroxy species derivation OOH. of anIn analyticthe route rate N(1b) expression., atomically Such adso derivationrbed oxygen was assists done in the [47 ]abstraction based on the of hydrogen DFT computed from the reaction substrate. energies assuming quasi-equilibria for steps 1 and 2, irreversibilityDerivation for of steps the rate 4 and expression 5 and reversibility (per mole for of stepsoxygen) 3 and for 6.the route N(1b) was described in [47], givingFrom the viewpoint of the global reaction kinetics, not only steps 3–6 are important but also the final step in the mechanism, step 7, reflecting the fate of atomically adsorbed oxygen. Therefore, kkKKCC++3412OEtOH three options are possibly influencing the stoichiometric2 numbers of other steps. Step 7a in the route + k+7b ()kk+−43 CEtOH (1b ) k+ r N = 5 (11) 2kKC kkKC++2 kkKKCC ++ kkkKCC −++ kKKCC+ (1+++KC KC +72bEtOH +341OOEtOHOHO2222 + ( 3412 + 6341 ))+ 312OEtOH2 2 21EtOH O2 kkkk kk++56++++++777bbb+ +7b kkk++74b() − 3 C EtOH kk + 4 − 3 C EtOH kkk ++ 74 b ( − 3 C EtOHEtOH) kk+−43 C kkk+++555k+5

Just for illustration purposes, let us consider a case when step 3 is at quasi-equilibrium and step 6 is irreversible. Equation (21) is then transformed to

(1b ) Kk++ k KKC C/ k r N = 34512OEtOHb2 7 2kKC KkkKC++2 kkKKKC ++ KKKkCC + (1+++KC KC +72bEtOH +34 51OOOEtOH222 + ( 4 5123 ) + 312 5 ) 2 (12) 21EtOH O2 2 kkCkk++57bEtOH ++ 67 b k + 7 b

For two other options also considering the quasi-equilibria of the step 3, the rates are for the routes N(1b)

2kK+73c kk++/2 k + KKCC 45 7cOEtOH 12 2 (1c ) k+ r N = 5 (13) k+4 22kK++++73cc kk 45 kK 73 0.25 2 (1+++++KC KC (1 k++ / 2 k ) KKC C + ( KKC C ) ) 2EtOH 1 O2 5 7 c 12 O2 EtOH 12 O2 EtOH kkkkk+++++75675c c and N(1a), respectively

22/32kk++74a kk++/( KKKCC ) 57aOEtOH2 1232 (1c ) k r N = +5 2kk kk k22 k (14) (1+++KC KC (++74a KKKC C )1/3 (1 ++ + 5+4 ( + 5 ) 1/6 ) + ( + 5 ) 1/3 ( KKKC C ) 2/3 ) 2 21EtOH O222 123 O EtOH 123OEtOH2 kkkkkkk+++++++576747422aa 2 a illustrating that the rate expression and even the kinetic regularities which follow from the corresponding rate expressions are very sensitive to the chemistry of step 7 and the rate constant of this step. The mechanism N(1a) is less probable compared to other alternatives considering a high activation barrier anticipated for this step [47], while other options in principle are possible. For Reactions 2020, 1 43

N(1a) corresponds to recombination of atomic oxygen, while such recombination in step 7c (route N(1c)) involves a reaction with a peroxy species OOH. In the route N(1b), atomically adsorbed oxygen assists in the abstraction of hydrogen from the substrate. Derivation of the rate expression (per mole of oxygen) for the route N(1b) was described in [47], giving

k k K K C C +3 +4 1 2 O2 EtOH k+ (k +k 7b C ) (1b) +4 3 k EtOH N − +5 r = v 2 (21) k k K C tu k k K K C C k k k K C C k K K C C 2k+7bK2CEtOH +3 +4 1 O2 2 +3 +4 1 2 O2 EtOH 6 +3 +4 1 O2 H2O +3 1 2 O2 EtOH (1+K2CEtOH+K1CO + + + ( + − )+ ) 2 k+ k+ k+ k+ k+ k+ 5 k (k +k 7b C ) 6 k +k 7b C k (k +k 7b C ) k +k 7b C +7b +4 3 k EtOH +4 3 k EtOH +7b +4 3 k EtOH +4 3 k EtOH − +5 − +5 − +5 − +5

Just for illustration purposes, let us consider a case when step 3 is at quasi-equilibrium and step 6 is irreversible. Equation (21) is then transformed to

( ) K k k K K C C /k N 1b 3 +4 +5 1 2 O2 EtOH 7b r = r 2 (22) 2k K C K3k+4k+5K1CO k+4k+5K1K2K3CO K3K1K2k+5CO CEtOH (1+K C +K C + +7b 2 EtOH + 2 + 2 ( 2 )+ 2 ) 2 EtOH 1 O2 k 2 k k k +5 k +7bCEtOH +6 +7b +7b

For two other options also considering the quasi-equilibria of the step 3, the rates are for the routes N(1b)

r 2k+7cK3 k+4k+5/2k+7c K1K2CO CEtOH N(1c) k+5 2 r = 2 (23) r r 0.25 k+4 2k+7cK3 k+4k+5 2k+7cK3 (1+K2CEtOH+K1CO + +(1+k+5/2k+7c) K1K2CO CEtOH+ ( K1K2CO CEtOH) ) 2 k+7c k+5 2 k+6k+7c k+5 2 and N(1a), respectively

2/3 2 2k+7ak+4 k + /k+ a( K K K C C ) (1c) 5 7 k2 1 2 3 O2 EtOH rN = +5 r 1/6 1/3 2 (24) 2k k 1/3 k k k2 k2 ( + + +( +7a +4 ) ( + +5 + +4 ( +5 ) )+( +5 ) ( )2/3) 1 K2CEtOH K1CO 2 K1K2K3CO CEtOH 1 2k k 2k k 2k k K1K2K3CO CEtOH 2 k +5 2 +7a +6 +7a +4 +7a +4 2 illustrating that the rate expression and even the kinetic regularities which follow from the corresponding rate expressions are very sensitive to the chemistry of step 7 and the rate constant of this step. The mechanism N(1a) is less probable compared to other alternatives considering a high activation barrier anticipated for this step [47], while other options in principle are possible. For kinetic modelling in [47], a feasible channel of atomically adsorbed oxygen consumption with a low activation energy barrier was considered to be step 7b (Equation (20)).

5. Conclusions and Outlook Apparently, the concept of the rate limiting will continue to be utilized in the future. This concept is useful as a part of the education of future specialists in chemical reaction engineering and catalyst development. For simple single-route reaction mechanisms, the application of the rate-determining step along with the quasi-equilibria of other steps results in easily derived rate equations. For more complex cases with several linear steps, the general approach based on the theory of complex reactions of Horiuti–Temkin allows the derivation of the corresponding rate equations. Apparent difficulties in doing this, as illustrated above for the four-step sequence, often result in a reductionistic approach assuming a rate-determining step either based on chemical intuition or a more rigorous theoretical analysis based on quantum chemical calculations. There are cases when the rate equation for linear sequences with a small number of steps was not derived and a system of differential equations was solved even if the rate equation for the steady -state system can be readily derived. This was done, for example in [48], when a solver of differential equations was used for a four step sequence of the Fujiwara–Moritani reaction (Figure4), unfortunately not reporting how the balance equation for catalytic species was taken into account, Reactions 2020, 3, x FOR PEER REVIEW 8 of 11 kinetic modelling in [47], a feasible channel of atomically adsorbed oxygen consumption with a low activation energy barrier was considered to be step 7b (Equation (20)).

5. Conclusions and Outlook Apparently, the concept of the rate limiting will continue to be utilized in the future. This concept is useful as a part of the education of future specialists in chemical reaction engineering and catalyst development. For simple single-route reaction mechanisms, the application of the rate-determining step along with the quasi-equilibria of other steps results in easily derived rate equations. For more complex cases with several linear steps, the general approach based on the theory of complex reactions of Horiuti–Temkin allows the derivation of the corresponding rate equations. Apparent difficulties in doing this, as illustrated above for the four-step sequence, often result in a reductionistic approach assuming a rate-determining step either based on chemical intuition or a more rigorous theoretical analysis based on quantum chemical calculations. There are cases when the rate equation for linear sequences with a small number of steps was not derived and a system of differential equations was solved even if the rate equation for the steady -state system can be readily derived. This was done, for example in [48], when a solver of differential Reactionsequations2020 ,was1 used for a four step sequence of the Fujiwara–Moritani reaction (Figure 4),44 unfortunately not reporting how the balance equation for catalytic species was taken into account,

Figure 4.4. ReactionReaction mechanism mechanism for for the the Fujiwara–Moritani Fujiwara–Moritan reactioni reaction [48]. Published[48]. Published by The by Royal The Society Royal ofSociety Chemistry. of Chemistry.

For a particular case of the mechanism in Figure4 4,, the the raterate equationequation can can bebe easilyeasily derivedderived as as thisthis mechanism isis aa specialspecial casecase of of Equations Equations (11) (11) and and (12) (12) with with irreversible irreversible steps steps 2 and2 and 3 simply3 simply resulting resulting in in ω+1ω+2ω+3ω+4Ccat r = ωωωωC (25) r =ω2ω3ω4 + ω 1ω3ω4 + ω1ω2ω3 + ++++ω12344ω2ω3 + ωcat4ω 1ω3 + +ω4ω1ω2 + ω3ω4ω1 ωωω+++++++ ω− ωω ωωω ω− ωω ω− ω− ω ωωω ωωω (15) Or, after introducing234 the−−−− 134 explicit expressions 123 for 423 the frequencies 4 13 of steps 412 341

Or, after introducing the explicit exprk k+essionsk+ k C forC theC frequenciesC of steps r = +1 2 3 +4 ace tan ilide acrylate BQ cat k+2k+3k+4CacrylateCBQ+k 1k+3k+4CacrylateCBQ+k+1k+2k+3Cace tan ilideCacrylate+ k 4k+2k+3Cacrylate+k 4k 1k+3Cacrylate+k+1k+2k+4Cace tan ilideCBQ+ k+1k+3k+4Cace tan ilideCacrylateCBQ (26) − − − −

In a more general case, when the reaction mechanisms comprise linear and nonlinear steps without any rate-limiting ones, derivation of an explicit rate equation can be too tedious. Instead, a comparison between the experimental and calculations should be done in an implicit form using a system of diﬀerential equations considering also an algebraic balance equation for the catalytic species. Availability of computer programs with a customer written subroutine comprising a set of elementary reactions will pave the way for eventual marginalization of the concept of the rate-determining step in the kinetic modelling of complex heterogeneous catalytic reactions.

Funding: This research received no external funding. Conﬂicts of Interest: The author declares no conﬂict of interest.

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