Hubbry Logo
Rate-determining stepRate-determining stepMain
Open search
Rate-determining step
Community hub
Rate-determining step
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Rate-determining step
Rate-determining step
Rate-determining step
View on Wikipedia
from Wikipedia

In chemical kinetics, the overall rate of a reaction is often approximately determined by the slowest step, known as the rate-determining step (RDS or RD-step[1] or r/d step[2][3]) or rate-limiting step. For a given reaction mechanism, the prediction of the corresponding rate equation (for comparison with the experimental rate law) is often simplified by using this approximation of the rate-determining step.

In principle, the time evolution of the reactant and product concentrations can be determined from the set of simultaneous rate equations for the individual steps of the mechanism, one for each step. However, the analytical solution of these differential equations is not always easy, and in some cases numerical integration may even be required.[4] The hypothesis of a single rate-determining step can greatly simplify the mathematics. In the simplest case the initial step is the slowest, and the overall rate is just the rate of the first step.

Also, the rate equations for mechanisms with a single rate-determining step are usually in a simple mathematical form, whose relation to the mechanism and choice of rate-determining step is clear. The correct rate-determining step can be identified by predicting the rate law for each possible choice and comparing the different predictions with the experimental law, as for the example of NO2 and CO below.

The concept of the rate-determining step is very important to the optimization and understanding of many chemical processes such as catalysis and combustion.

Example reaction: NO2 + CO

[edit]

As an example, consider the gas-phase reaction NO2 + CO → NO + CO2. If this reaction occurred in a single step, its reaction rate (r) would be proportional to the rate of collisions between NO2 and CO molecules: r = k[NO2][CO], where k is the reaction rate constant, and square brackets indicate a molar concentration. Another typical example is the Zel'dovich mechanism.

First step rate-determining

[edit]

In fact, however, the observed reaction rate is second-order in NO2 and zero-order in CO,[5] with rate equation r = k[NO2]2. This suggests that the rate is determined by a step in which two NO2 molecules react, with the CO molecule entering at another, faster, step. A possible mechanism in two elementary steps that explains the rate equation is:

  1. NO2 + NO2 → NO + NO3 (slow step, rate-determining)
  2. NO3 + CO → NO2 + CO2 (fast step)

In this mechanism the reactive intermediate species NO3 is formed in the first step with rate r1 and reacts with CO in the second step with rate r2. However, NO3 can also react with NO if the first step occurs in the reverse direction (NO + NO3 → 2 NO2) with rate r−1, where the minus sign indicates the rate of a reverse reaction.

The concentration of a reactive intermediate such as [NO3] remains low and almost constant. It may therefore be estimated by the steady-state approximation, which specifies that the rate at which it is formed equals the (total) rate at which it is consumed. In this example NO3 is formed in one step and reacts in two, so that

The statement that the first step is the slow step actually means that the first step in the reverse direction is slower than the second step in the forward direction, so that almost all NO3 is consumed by reaction with CO and not with NO. That is, r−1 ≪ r2, so that r1 − r2 ≈ 0. But the overall rate of reaction is the rate of formation of final product (here CO2), so that r = r2 ≈ r1. That is, the overall rate is determined by the rate of the first step, and (almost) all molecules that react at the first step continue to the fast second step.

Pre-equilibrium: if the second step were rate-determining

[edit]

The other possible case would be that the second step is slow and rate-determining, meaning that it is slower than the first step in the reverse direction: r2 ≪ r−1. In this hypothesis, r1 − r−1 ≈ 0, so that the first step is (almost) at equilibrium. The overall rate is determined by the second step: r = r2 ≪ r1, as very few molecules that react at the first step continue to the second step, which is much slower. Such a situation in which an intermediate (here NO3) forms an equilibrium with reactants prior to the rate-determining step is described as a pre-equilibrium[6] For the reaction of NO2 and CO, this hypothesis can be rejected, since it implies a rate equation that disagrees with experiment.

  1. NO2 + NO2 → NO + NO3 (fast step)
  2. NO3 + CO → NO2 + CO2 (slow step, rate-determining)

If the first step were at equilibrium, then its equilibrium constant expression permits calculation of the concentration of the intermediate NO3 in terms of more stable (and more easily measured) reactant and product species:

The overall reaction rate would then be

which disagrees with the experimental rate law given above, and so disproves the hypothesis that the second step is rate-determining for this reaction. However, some other reactions are believed to involve rapid pre-equilibria prior to the rate-determining step, as shown below.

Nucleophilic substitution

[edit]

Another example is the unimolecular nucleophilic substitution (SN1) reaction in organic chemistry, where it is the first, rate-determining step that is unimolecular. A specific case is the basic hydrolysis of tert-butyl bromide (t-C
4H
9Br) by aqueous sodium hydroxide. The mechanism has two steps (where R denotes the tert-butyl radical t-C
4H
9):

  1. Formation of a carbocation R−Br → R+
     + Br−
    .
  2. Nucleophilic attack by hydroxide ion R+
     + OH−
     → ROH.

This reaction is found to be first-order with r = k[R−Br], which indicates that the first step is slow and determines the rate. The second step with OH− is much faster, so the overall rate is independent of the concentration of OH−.

In contrast, the alkaline hydrolysis of methyl bromide (CH
3Br) is a bimolecular nucleophilic substitution (SN2) reaction in a single bimolecular step. Its rate law is second-order: r = k[R−Br][OH−
].

Composition of the transition state

[edit]

A useful rule in the determination of mechanism is that the concentration factors in the rate law indicate the composition and charge of the activated complex or transition state.[7] For the NO2–CO reaction above, the rate depends on [NO2]2, so that the activated complex has composition N
2O
4, with 2 NO2 entering the reaction before the transition state, and CO reacting after the transition state.

A multistep example is the reaction between oxalic acid and chlorine in aqueous solution: H
2C
2O
4 + Cl
2 → 2 CO2 + 2 H+
 + 2 Cl−
.[7] The observed rate law is

which implies an activated complex in which the reactants lose 2H+
 + Cl−
 before the rate-determining step. The formula of the activated complex is Cl
2 + H
2C
2O
4 − 2 H+
 − Cl−
 + x H2O, or C
2O
4Cl(H
2O)–
x (an unknown number of water molecules are added because the possible dependence of the reaction rate on H2O was not studied, since the data were obtained in water solvent at a large and essentially unvarying concentration).

One possible mechanism in which the preliminary steps are assumed to be rapid pre-equilibria occurring prior to the transition state is[7]

Cl
2 + H2O ⇌ HOCl + Cl−
 + H+
H
2C
2O
4 ⇌ H+
 + HC
2O−
4
HOCl + HC
2O−
4 → H2O + Cl−
 + 2 CO2

Reaction coordinate diagram

[edit]

In a multistep reaction, the rate-determining step does not necessarily correspond to the highest Gibbs energy on the reaction coordinate diagram.[8][6] If there is a reaction intermediate whose energy is lower than the initial reactants, then the activation energy needed to pass through any subsequent transition state depends on the Gibbs energy of that state relative to the lower-energy intermediate. The rate-determining step is then the step with the largest Gibbs energy difference relative either to the starting material or to any previous intermediate on the diagram.[8][9]

Also, for reaction steps that are not first-order, concentration terms must be considered in choosing the rate-determining step.[8][6]

Chain reactions

[edit]

Not all reactions have a single rate-determining step. In particular, the rate of a chain reaction is usually not controlled by any single step.[8]

Diffusion control

[edit]

In the previous examples the rate determining step was one of the sequential chemical reactions leading to a product. The rate-determining step can also be the transport of reactants to where they can interact and form the product. This case is referred to as diffusion control and, in general, occurs when the formation of product from the activated complex is very rapid and thus the provision of the supply of reactants is rate-determining.

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Rate-determining step

View on Grokipedia
from Grokipedia
In , the rate-determining step (RDS), also known as the rate-limiting step, is the slowest elementary step within a multi-step that governs the overall speed of the reaction, acting as a bottleneck similar to the narrow neck of a . This step typically possesses the highest among the sequence, ensuring that faster preceding or subsequent steps do not accelerate the process beyond its capacity. The concept applies primarily to mechanisms where one step is significantly slower than the others, allowing the overall to approximate the rate of this critical stage. The RDS plays a pivotal role in deriving the rate law for complex reactions, as the concentrations of reactants and intermediates influencing this step dictate the observed kinetics. If the RDS occurs as the first step, the overall rate law directly reflects its and rate constant; however, when it follows rapid equilibrium steps, the rate law incorporates equilibrium constants from prior stages, often involving intermediate . For instance, in the reaction of with (NO₂ + F₂ → products), the RDS is the initial bimolecular collision, yielding a rate law of rate = k[NO₂][F₂]. In contrast, for the formation of (H₂ + Br₂ → 2HBr), the RDS involves an intermediate bromine atom reacting with , resulting in a more complex rate expression: rate = k[H₂][Br₂]^{1/2}. Identification of the RDS often relies on experimental techniques, such as analyzing kinetic isotope effects, dependence, or computational methods like the degree of rate control (DRC), where a step with DRC ≈ 1 exerts dominant influence. The RDS can shift under varying conditions, such as changes in , , or , which may alter activation barriers and thus the rate-controlling process in heterogeneous or enzymatic reactions. This dynamic nature underscores its importance in fields like design, where optimizing the RDS can enhance reaction efficiency, as seen in electrocatalytic processes or solid-state reactions.

Fundamentals

Definition and identification

The rate-determining step (RDS), also referred to as the rate-limiting step, is the slowest elementary step within a multi-step that governs the overall . This step acts as a bottleneck because the concentrations of intermediates produced by preceding steps build up, while subsequent steps cannot consume them faster than the RDS generates them, thereby dictating the pace of the entire process. In mechanisms where multiple steps have comparable rates, the RDS may be a composite of the slowest segments, but typically, it is the single step with the highest , as this correlates with the smallest rate constant under given conditions. The concept of the RDS originated in the early development of during the 1920s, as researchers analyzed complex reaction mechanisms, and was more rigorously formalized in 1935 through Henry Eyring's , which provided a theoretical basis for calculating rate constants of elementary steps and identifying the slowest one in multi-step sequences. Eyring's work emphasized how the free energy of for each step determines its rate, allowing chemists to predict which step would limit the overall kinetics in processes like or industrial syntheses. Identifying the RDS involves experimental and computational approaches to compare the rates of individual steps. Isolation experiments, where intermediates are generated separately and their decay rates measured, directly reveal the slowest step by quantifying rate constants. The steady-state approximation is commonly applied to derive rate laws from proposed mechanisms, with the RDS identified as the step whose rate expression matches the experimentally observed overall rate law./Kinetics/04%3A_Reaction_Mechanisms/4.12%3A_Steady-State_Approximation) Kinetic isotope effects, probed via isotopic labeling of reactants, indicate if bond breaking or forming in a particular step is rate-limiting, as a significant isotope effect (e.g., k_H/k_D > 1.5) signals involvement in the RDS. Additionally, the step with the highest , often determined from Arrhenius plots or computational modeling, serves as a key criterion, though temperature-dependent studies confirm its dominance across conditions. For a simple multi-step mechanism with rate constants k1,k2,…,knk_1, k_2, \dots, k_n, if the RDS is the first step and irreversible, the overall rate approximates the rate of that step: rate≈kRDS[reactants]\text{rate} \approx k_\text{RDS} [\text{reactants}] This holds when subsequent steps are fast relative to the RDS, ensuring no significant backlog of intermediates. If the RDS occurs later, pre-equilibrium assumptions adjust the effective concentration of precursors, but the core principle remains that the slowest kk controls the observed kinetics./Kinetics/04%3A_Reaction_Mechanisms/4.12%3A_Steady-State_Approximation)

Relation to overall rate law

The rate-determining step (RDS) directly influences the form of the overall rate law for a multi-step reaction, as it is the slowest elementary step and thus limits the overall . When the RDS is the first step in the mechanism, the overall rate law is simply the rate law of that elementary step, expressed in terms of the reactant concentrations involved. For instance, in a bimolecular RDS such as A + B → products, the rate is given by rate = k₁[A][B], where k₁ is the rate constant for that step. This direct correspondence arises because prior steps do not exist to build up intermediates, and subsequent steps are faster, allowing the reaction to proceed at the pace set by the initial slow step. If the RDS occurs after one or more fast initial steps, the concentrations of any intermediates must be expressed in terms of the stable reactants to obtain the overall rate law, typically using approximation methods. The steady-state approximation, introduced by Max Bodenstein in the early , assumes that the concentration of a reactive intermediate remains nearly constant over time because its rate of formation equals its rate of consumption (d[I]/dt ≈ 0). Consider a simple mechanism where A → I (fast, rate constant k₁), I → P (slow RDS, k₂), and I → A (reverse, k₋₁): the steady-state condition yields [I] ≈ (k₁[A]) / (k₂ + k₋₁), leading to an overall rate = k₂[I] ≈ (k₁ k₂ / (k₂ + k₋₁)) [A]. This approximation is valid when the intermediate's lifetime is short compared to the overall reaction time, enabling derivation of rate laws that match experimental observations without solving complex differential equations. For mechanisms involving fast reversible pre-steps followed by a slow RDS, the pre-equilibrium approximation is often applied, assuming the initial steps reach equilibrium rapidly relative to the RDS. In such cases, the K_eq for the pre-step (e.g., A ⇌ I, K_eq = [I]/[A] = k₁/k₋₁) allows substitution into the RDS rate law: rate = k_RDS [I] = k_RDS K_eq [A]. This yields a composite rate constant k = k_RDS K_eq, simplifying the overall rate law to rate = k [A], which reflects first-order dependence despite the involvement of intermediates. The approximation holds when the forward and reverse rates of the pre-equilibrium are much faster than the RDS, ensuring the equilibrium is maintained throughout the reaction. Experimental verification of the RDS position involves comparing the predicted rate law from the proposed mechanism to observed kinetic orders and dependencies. By measuring initial rates at varying reactant concentrations, researchers determine the experimental rate law; if it matches the derivation from assuming a particular step as RDS (e.g., second-order if the RDS involves two reactants), this supports the mechanism. labeling or spectroscopic detection of intermediates can further confirm the rate law's consistency with the RDS. A common pitfall in mechanistic analysis is assuming a particular step is the RDS without deriving and testing the corresponding rate law against experimental , which can lead to incorrect mechanisms or overlooked contributions from multiple steps. Not all reactions feature a single dominant RDS, especially in complex systems where rates may be controlled by a combination of steps, underscoring the need for rigorous kinetic analysis.

Basic examples

Bimolecular reaction: NO2 + CO

The reaction between nitrogen dioxide and carbon monoxide serves as a classic example of a bimolecular reaction in which the first elementary step is the rate-determining step. The overall balanced equation is \ceNO2+CO−>NO+CO2\ce{NO2 + CO -> NO + CO2}. The accepted mechanism involves two steps: the slow, rate-determining bimolecular step \ceNO2+NO2−>NO3+NO\ce{NO2 + NO2 -> NO3 + NO}, followed by the fast step \ceNO3+CO−>NO2+CO2\ce{NO3 + CO -> NO2 + CO2}. The experimental rate law, determined at low temperatures (below approximately 500 K), is \rate=k[\NO2]2\rate = k [\NO_2]^2, which directly corresponds to the rate law of the slow first step and shows no dependence on [\CO][\CO]. This independence from [\CO][\CO] arises because the second step occurs rapidly after the formation of the intermediate \ceNO3\ce{NO3}, ensuring that the overall rate is governed solely by the production of \ceNO3\ce{NO3}. Supporting evidence for this mechanism and rate law comes from early experimental studies, including temperature-dependent measurements in the that confirmed the second-order dependence on \ceNO2\ce{NO2}. The first step is rate-determining due to its higher relative to the second step, which is highly exothermic and features a low energy barrier, allowing it to proceed quickly once \ceNO3\ce{NO3} is available. The rate expression for the rate-determining step is thus \rate=k1[\NO2]2\rate = k_1 [\NO_2]^2

Pre-equilibrium approximation

In reaction mechanisms featuring a fast initial reversible step followed by a slower rate-determining step (RDS), the pre-equilibrium approximation simplifies the analysis by assuming the initial step achieves rapid dynamic equilibrium, enabling the intermediate's concentration to be related directly to reactant concentrations. This approach is particularly useful when the RDS involves the intermediate formed in the pre-equilibrium, allowing derivation of an effective rate law without solving complex differential equations for transient intermediates. A representative example is a hypothetical acid-catalyzed reaction of substrate A, where protonation occurs rapidly to form AH^+, followed by slow decomposition: A+H+⇌k1k−1AH+(fast equilibrium)\text{A} + \text{H}^+ \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} \text{AH}^+ \quad (\text{fast equilibrium}) AH+→k2products(slow RDS)\text{AH}^+ \stackrel{k_2}{\rightarrow} \text{products} \quad (\text{slow RDS}) The equilibrium constant is K=k1k−1=[AH+][A][H+]K = \frac{k_1}{k_{-1}} = \frac{[\text{AH}^+]}{[\text{A}][\text{H}^+]}, so [AH+]=K[A][H+][\text{AH}^+] = K [\text{A}][\text{H}^+]. The rate of product formation, governed by the RDS, is rate=k2[AH+]=k2K[A][H+]\text{rate} = k_2 [\text{AH}^+] = k_2 K [\text{A}][\text{H}^+], resulting in an observed second-order rate law dependent on both [A] and [H^+]. This derivation highlights how the pre-equilibrium effectively increases the concentration of the reactive species AH^+, altering the apparent kinetics from the intrinsic RDS. The RDS can be identified in such mechanisms if the observed rate depends on [H^+] (from the pre-equilibrium) but remains independent of concentrations involved in subsequent steps, confirming the equilibrium precedes the slow . Early 20th-century studies on reactions, such as the 1913 work by and on inversion, applied this concept to derive rate laws exhibiting saturation-like behavior akin to enzyme-like kinetics. This approximation holds under the condition that the pre-equilibrium is fully established prior to the RDS, requiring k−1≫k2k_{-1} \gg k_2 so the reverse step rapidly replenishes the intermediate; it fails if the reverse rate approaches the RDS forward rate, necessitating alternatives like the steady-state approximation for accuracy.

Mechanistic applications

Nucleophilic substitution reactions

In reactions, the rate-determining step (RDS) plays a crucial role in distinguishing between the two primary mechanisms: SN2 and SN1. The SN2 mechanism is a concerted, bimolecular process where the RDS involves the simultaneous attack of the on the carbon atom bearing the and the departure of the , leading to a rate law of rate = k [RX][Nu⁻], where RX is the alkyl halide and Nu⁻ is the . This backside attack results in complete inversion of stereochemical configuration at the reaction . In contrast, the SN1 mechanism proceeds in two steps, with the RDS being the unimolecular dissociation of the to form a intermediate, governed by the rate law rate = k [RX]; the subsequent nucleophilic attack on the planar is rapid and occurs from either side, often leading to or partial inversion depending on ion-pairing effects. These mechanisms were first systematically elucidated in through kinetic studies on alkyl halide reactions by Edward Hughes and Christopher Ingold. Kinetic evidence clearly differentiates the RDS in each mechanism. For SN2 reactions, the second-order dependence on both substrate and nucleophile concentrations confirms the bimolecular RDS, as observed in the hydrolysis of primary alkyl bromides. In SN1 reactions, the first-order kinetics, independent of nucleophile concentration, indicate that carbocation formation is rate-limiting, as demonstrated in the solvolysis of tertiary alkyl halides. Kinetic isotope effects (KIEs) further support these distinctions: SN1 reactions exhibit large secondary α-deuterium KIEs (~1.15–1.23) due to the loose transition state in the RDS, with small α-carbon KIEs (~1.00–1.02) and small leaving-group KIEs (~1.005–1.01). SN2 reactions show smaller secondary α-deuterium KIEs (~1.00–1.10), moderate α-carbon KIEs (~1.03–1.09), small leaving-group KIEs (~1.00–1.01), and small nucleophile KIEs (~1.00–1.03), reflecting partial bond breaking and forming in the concerted RDS. Solvent effects significantly influence the RDS and mechanism preference. Polar protic solvents, such as water or alcohols, stabilize the ionic intermediate and leaving group anion through hydrogen bonding, thereby lowering the for the dissociation step and favoring the SN1 mechanism with its unimolecular RDS. In contrast, these solvents solvate and reduce the nucleophilicity of anionic nucleophiles, disfavoring the bimolecular SN2 RDS. Hughes and Ingold's studies on alkyl in varying solvents highlighted this shift, showing accelerated SN1 rates in protic media due to enhanced stabilization. For instance, the of 2-bromo-2-methylpropane () proceeds via SN1 in aqueous , with the RDS being bromide departure, yielding a racemic product and kinetics.

Chain reactions

Chain reactions, such as free radical processes, differ from linear mechanisms in that they involve a sequence of , , and termination steps, where the overall kinetics are influenced by the interplay among these phases rather than a single slow step. typically generates reactive radicals from an initiator, often requiring high (Ea) and thus proceeding slowly, while involves rapid, low-Ea additions or abstractions that extend the chain, and termination occurs via second-order radical recombination. In such reactions, the rate-determining step (RDS) is frequently the phase, particularly when lengths are long, as each initiating event produces many cycles before termination, making the radical generation rate the primary control on overall reactivity. The steady-state approximation for radical concentrations links the rate constant (k_p) to the length, emphasizing how limits the supply of propagating . The resulting rate law for the overall reaction, such as in free radical polymerization, reflects this by approximating the polymerization rate (R_p) as: Rp≈kp[monomer](ki[initiator]kt)1/2R_p \approx k_p [\text{monomer}] \left( \frac{k_i [\text{initiator}]}{k_t} \right)^{1/2} where k_i is the effective initiation rate constant, k_t the termination rate constant, and the square-root dependence on initiator concentration underscores the effective RDS role of initiation. A classic example is free radical polymerization of vinyl monomers, where Hermann Staudinger's 1920s work established the chain growth mechanism through repeated additions initiated by radicals, demonstrating how slow initiator decomposition governs the process. Another illustrative case is the peroxide-catalyzed addition of HBr to alkenes, known as the peroxide effect, where of the peroxide initiates Br• radicals that propagate via anti-Markovnikov addition, with the overall rate controlled by the initiation efficiency despite fast propagation steps. The (Φ) in chain reactions often exceeds 1—sometimes greatly so—indicating that is not rate-limiting, as a single or initiating event can yield multiple product molecules through the chain cycle, contrasting with the single-step RDS behavior in non-chain processes.

Theoretical foundations

Transition state composition

In the context of a multi-step , the of the rate-determining step (RDS) exhibits structural characteristics that depend on its position relative to the reactants and products, as governed by principles of reactivity and energy profiles. If the RDS occurs early in the mechanism, its tends to resemble the reactants more closely, featuring a looser with partial bond breaking and formation that mirrors the initial . Conversely, a late RDS more closely resembles the products, adopting a tighter configuration with bonds more advanced toward the product-like geometry. This resemblance influences the overall reaction selectivity and kinetics, as the RDS dictates the rate law derived from the mechanism. Hammond's postulate, proposed in 1955, provides a foundational framework for understanding this composition by correlating the position of the along the with the of the elementary step. For exothermic steps, where the products are more stable than the reactants, the is early and reactant-like, minimizing the free energy barrier. In endothermic steps, the is late and product-like, reflecting the higher energy of the products and leading to greater selectivity in competing pathways. When applied to the RDS, explains why rate enhancements often favor pathways with stabilized transition states resembling lower-energy species, impacting applications in and . Kinetic isotope effects (KIEs) serve as experimental probes to elucidate the composition of the RDS by measuring changes in upon isotopic substitution. A primary KIE greater than 1 indicates that bond breaking or formation involving the isotopically labeled atom occurs in the RDS , revealing its involvement in the critical structural changes, such as C-H bond cleavage in hydrogen-transfer reactions. For instance, substitution yielding k_H/k_D values of 2–7 confirms that the features partial bond rupture at that site, distinguishing it from non-rate-determining steps where such effects are negligible or inverse. These effects arise from differences in zero-point energies between s, providing quantitative insight into the 's without direct . The theoretical underpinning of RDS transition state composition stems from transition state theory (TST), developed by Eyring in 1935, which posits that the reaction rate is determined by the free energy of activation, ΔG‡\Delta G^\ddagger, separating reactants from the . In TST, the is the highest-energy configuration along the minimum energy reaction path, and its structure dictates the equilibrium constant for forming the activated complex, K‡=e−ΔG‡/RTK^\ddagger = e^{-\Delta G^\ddagger / RT}, which directly influences the rate constant. Computational models, such as (DFT), further refine this by optimizing geometries, confirming that the RDS structure balances reactant and product features based on the . A representative example is the mechanism, where the RDS involves the departure of the from the substrate, forming a intermediate. The in this step features significant stretching of the C-X bond (where X is the ) and partial charge separation, resembling a tight pair with the developing positive charge on carbon delocalized by adjacent groups. Computational studies show this has an elongated C-X distance of approximately 2.5–3.0 Å and solvent-stabilized charge development, aligning with for the endothermic RDS and explaining the sensitivity to substrate stability.

Reaction coordinate diagrams

Reaction coordinate diagrams provide a graphical representation of the energy changes during a chemical reaction, illustrating the pathway from reactants to products along a . The horizontal axis, known as the , quantifies the progress of the reaction, often parameterized by a collective variable such as or . The vertical axis plots the (G), with reactants appearing at an initial energy level, followed by rises to transition states (energy maxima) and dips to intermediates (energy minima), culminating in the products. This one-dimensional projection simplifies the multidimensional , focusing on the most relevant path. In multi-step reactions, the rate-determining step (RDS) corresponds to the with the highest free energy relative to the reactants, representing the largest energy barrier (ΔG‡) that must be surmounted. The overall rate of the reaction is governed by this barrier, approximated as proportional to exp⁡(−ΔGRDS‡/RT)\exp(-\Delta G^\ddagger_\text{RDS} / RT), where RR is the and TT is the , as derived from . Transition states are short-lived configurations at energy maxima, while intermediates reside in potential wells and do not directly limit the rate unless they precede the RDS. For a typical two-step mechanism, the profile features two barriers separated by an intermediate minimum; if the second lies higher in free energy than the first (measured from the reactants' baseline), the second step becomes the RDS, as the population of the intermediate does not accumulate sufficiently to alter the kinetics. Conversely, a lower second barrier relative to the intermediate shifts the RDS to the first step. Catalysts accelerate reactions by selectively lowering the RDS barrier through stabilization of its , without altering the overall . influences the profile indirectly; barriers with significant entropic components (ΔS‡) respond differently to heating, potentially shifting the RDS in reactions where varies across steps. The conceptualization of these diagrams traces back to early in the 1930s, with Eyring's work introducing quantitative energy plots to predict rates. Today, (DFT) computations generate precise profiles by optimizing geometries and calculating electronic energies along the intrinsic , enabling identification of RDS in complex mechanisms. Software like Gaussian or facilitates these calculations, plotting diagrams that integrate zero-point energies and effects for realistic kinetics predictions.

Special cases

Diffusion-controlled reactions

In diffusion-controlled reactions, the rate-determining step is the physical transport of reactants through the medium to form a reactive encounter pair, as the subsequent chemical transformation occurs much faster than this diffusion process. The overall reaction rate follows second-order kinetics, expressed as rate = k_diff [A][B], where k_diff is the diffusion-limited rate constant derived from the Smoluchowski equation: kdiff=4π(DA+DB)(rA+rB)NA1000,k_{\text{diff}} = \frac{4\pi (D_A + D_B)(r_A + r_B) N_A}{1000}, with D_A and D_B the diffusion coefficients of the reactants (in cm² s⁻¹), r_A and r_B their radii (in cm), and N_A = 6.022 \times 10^{23} mol^{-1} Avogadro's constant, yielding units of M⁻¹ s⁻¹ (the factor of 1000 converts cm³ to L). This formulation, originally developed by Smoluchowski in 1916 for colloidal coagulation, applies broadly to bimolecular encounters in solution where transport limits the reaction. Such reactions prevail under conditions of high reactant concentrations and low viscosity, which enhance molecular mobility while still making the bottleneck. They are particularly prevalent in radical recombination processes, where highly reactive combine immediately upon encounter, and in certain fast bimolecular associations like enzyme-substrate binding limited by diffusional approach. Experimental evidence supports this regime through bimolecular rate constants in that saturate at approximately 10⁹ to 10¹⁰ M⁻¹ s⁻¹, values that remain largely independent of the for the chemical step since , not the barrier-crossing, governs the kinetics. In partially diffusion-controlled scenarios, where the chemical rate constant k_chem is on the order of k_diff, the observed rate constant reflects both contributions additively: kobs=(1kdiff+1kchem)−1.k_{\text{obs}} = \left( \frac{1}{k_{\text{diff}}} + \frac{1}{k_{\text{chem}}} \right)^{-1}. This relationship arises from conceptualizing the process as two sequential resistances in series, allowing rates to transition smoothly from diffusion-limited to chemically limited behavior. A representative example is the of by molecular oxygen in solution, where Debye's 1942 extends Smoluchowski's framework to account for electrostatic effects in ionic media, predicting diffusion-controlled quenching efficiencies that match observed rates near 10¹⁰ M⁻¹ s⁻¹.

Enzyme kinetics

In enzyme kinetics, the rate-determining step (RDS) is frequently the catalytic conversion of the enzyme-substrate (ES) complex to products, as modeled in the classic Michaelis-Menten framework. The mechanism posits rapid reversible binding of the free enzyme (E) and substrate (S) to form ES, followed by the slower, irreversible catalytic step yielding free enzyme and product (P): \ceE+S⇌ES(fast equilibrium),\ceES−>E+P(slow, RDS).\begin{align*} &\ce{E + S ⇌ ES} \quad (\text{fast equilibrium}), \\ &\ce{ES -> E + P} \quad (\text{slow, RDS}). \end{align*} The overall reaction rate is v=kcat[\ceES]v = k_\text{cat} [\ce{ES}], where kcatk_\text{cat} is the turnover number. Applying the steady-state approximation to [ES] yields the Michaelis-Menten equation: v=Vmax[\ceS]Km+[\ceS],v = \frac{V_\text{max} [\ce{S}]}{K_m + [\ce{S}]}, with maximum velocity Vmax=kcat[\ceE]totalV_\text{max} = k_\text{cat} [\ce{E}]_\text{total} and Michaelis constant KmK_m reflecting substrate affinity. Identification of the RDS relies on comparing kinetic rate constants within the mechanism. When kcat≪k−1k_\text{cat} \ll k_{-1} (the ES dissociation rate), the binding equilibrium is rapidly established, rendering the catalytic step the RDS and approximating Km≈Kd=k−1/k1K_m \approx K_d = k_{-1}/k_1, where k1k_1 is the association rate constant. In contrast, a high KmK_m (typically >1 mM) signals low substrate affinity, implying that binding becomes effectively rate-limiting at subsaturating physiological [S], as the enzyme struggles to form sufficient ES. Allosteric effects can dynamically shift the RDS by modulating transition state stabilization without directly competing at the active site. For instance, V-type allosteric inhibitors bind remotely to alter the catalytic machinery, shifting the RDS from chemistry to a conformational step, as observed in α-isopropylmalate synthase where L-leucine binding perturbs the rate-limiting isomerization. This regulatory mechanism allows fine-tuned control of enzymatic flux in metabolic pathways. The foundational 1913 Michaelis-Menten model assumed steady-state conditions but overlooked transient phases; modern experimental approaches like pre-steady-state kinetics using stopped-flow spectroscopy resolve this by capturing millisecond-scale events to pinpoint the RDS through observation of intermediate accumulation or fluorescence changes. In extensions to multi-substrate enzymes, single-molecule studies from the 2000s revealed that product release often emerges as the RDS, constraining turnover in ordered sequential mechanisms; for example, single-molecule fluorescence on demonstrated slow, heterogeneous product dissociation limiting the cycle. This aligns with the pre-equilibrium approximation for initial binding steps discussed earlier.

References

  1. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_%28Physical_and_Theoretical_Chemistry%29/Kinetics/03%253A_Rate_Laws/3.02%253A_Reaction_Mechanisms/3.2.03%253A_Rate_Determining_Step
  2. https://www.pearson.com/channels/general-chemistry/textbook-solutions/mcmurry-8th-edition-9781292336145/ch-13-chemical-kinetics/what-distinguishes-the-rate-determining-step-from-the-other-steps-in-a-reaction-
  3. https://www.sciencedirect.com/topics/chemistry/rate-determining-step
  4. https://www.sciencedirect.com/science/article/abs/pii/S0021951721004097
  5. https://www.sciencedirect.com/science/article/pii/B9780128136430000093
  6. https://pubs.acs.org/doi/10.1021/cr60056a006
  7. http://ch302.cm.utexas.edu/kinetics/mechanisms/submodule.php?name=rate-limit-step
  8. https://www.chem.tamu.edu/class/fyp/keeney/ch16-k.pdf
  9. https://pubs.aip.org/aip/jcp/article/3/2/107/203352/The-Activated-Complex-in-Chemical-Reactions
  10. https://www.chm.uri.edu/weuler/chm112/lectures/lecture6.html
  11. https://www.osti.gov/servlets/purl/1605198
  12. https://www.epfl.ch/labs/lspn/wp-content/uploads/2024/09/The-Rate-Determining-Step.pdf
  13. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Kinetics/03%3A_Rate_Laws/3.02%3A_Reaction_Mechanisms/3.2.03%3A_Rate_Determining_Step
  14. https://sites.engineering.ucsb.edu/~jbraw/chemreacfun/ch5/slides-kinetics.pdf
  15. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Kinetics/04%3A_Reaction_Mechanisms/4.12%3A_Steady-State_Approximation
  16. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Kinetics/03%3A_Rate_Laws/3.02%3A_Reaction_Mechanisms/3.2.02%3A_Pre-equilibrium_Approximation
  17. https://www.jove.com/science-education/v/11382/rate-law-rate-determining-steps-and-reaction-mechanisms
  18. https://link.springer.com/chapter/10.1007/978-3-319-15482-4_5
  19. https://chem.libretexts.org/Courses/Indiana_Tech/EWC%3A_CHEM_1030_-_General_Chemistry_II_(Budhi)/02%3A_Chemical_Kinetics/2.6%3A_Reaction_Mechanisms
  20. https://onlinelibrary.wiley.com/doi/abs/10.1002/kin.550090605
  21. https://pmc.ncbi.nlm.nih.gov/articles/PMC4624389/
  22. https://pubs.acs.org/doi/10.1021/cr60051a004
  23. https://www.dalalinstitute.com/wp-content/uploads/Books/A-Textbook-of-Organic-Chemistry-Volume-1/ATOOCV1-3-9-Isotope-Effects.pdf
  24. https://onlinelibrary.wiley.com/doi/pdf/10.1002/jlcr.1434
  25. https://doi.org/10.1039/jr9350000244
  26. https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Polymer_Chemistry/03%3A_Kinetics_and_Thermodynamics_of_Polymerization/3.03%3A_Kinetics_of_Chain_Polymerization
  27. https://employees.csbsju.edu/cschaller/Advanced/Polymers/KPchain.html
  28. https://ethz.ch/content/dam/ethz/special-interest/chab/icb/morbidelli-dam/documents/Education/PRCE/DOC_2016/Chapter1.pdf
  29. https://onlinelibrary.wiley.com/doi/10.1155/2009/893234
  30. https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Supplemental_Modules_(Organic_Chemistry)/Reactions/Addition_Reactions/Hydrohalogenation/Hydrogen_Bromide_and_Alkenes:_The_Peroxide_Effect
  31. https://pubs.rsc.org/en/content/articlehtml/2015/sc/c5sc02185e
  32. https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Organic_Chemistry_(Morsch_et_al.)/07:_Alkenes-_Structure_and_Reactivity/7.11:_The_Hammond_Postulate
  33. https://pubs.acs.org/doi/10.1021/ja01607a027
  34. https://pubs.acs.org/doi/abs/10.1021/acscatal.9b05637
  35. https://pubs.acs.org/doi/10.1021/ja00052a055
  36. https://ocw.mit.edu/courses/5-12-organic-chemistry-i-spring-2003/105aa1cc1658da51469e91c49084d818_07.pdf
  37. https://ethz.ch/content/dam/ethz/special-interest/chab/organic-chemistry/bode-group-dam/documents/open-source-lecture-notes/key-concepts-in-catalysis-kinetics-thermodynamics-2015.pdf
  38. https://pubs.rsc.org/en/content/articlepdf/2024/cp/d4cp00266k
  39. https://pubs.acs.org/doi/10.1021/acs.jchemed.3c00319
  40. https://chem.libretexts.org/Bookshelves/Biological_Chemistry/Concepts_in_Biophysical_Chemistry_(Tokmakoff)/06%3A_Dynamics_and_Kinetics/21%3A_Binding_and_Association/21.05%3A_Diffusion-Limited_Reactions
  41. https://pmc.ncbi.nlm.nih.gov/articles/PMC10674959/
  42. https://iopscience.iop.org/article/10.1149/1.3071413
  43. https://pubs.acs.org/doi/10.1021/bi201284u
  44. https://www.sciencedirect.com/science/article/pii/S2213020914000627
  45. http://people.reed.edu/~glasfeld/Chem391/notes/ChKinetics_2015.pdf
  46. https://pubs.acs.org/doi/10.1021/jp5014434
Add your contribution
Related Hubs
User Avatar
No comments yet.