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Reaction rate constant
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In chemical kinetics, a reaction rate constant or reaction rate coefficient (⁠⁠) is a proportionality constant which quantifies the rate and direction of a chemical reaction by relating it with the concentration of reactants.[1]

For a reaction between reactants A and B to form a product C,

a A + b B → c C

where

A and B are reactants
C is a product
a, b, and c are stoichiometric coefficients,

the reaction rate is often found to have the form:

Here ⁠⁠ is the reaction rate constant that depends on temperature, and [A] and [B] are the molar concentrations of substances A and B in moles per unit volume of solution, assuming the reaction is taking place throughout the volume of the solution. (For a reaction taking place at a boundary, one would use moles of A or B per unit area instead.)

The exponents m and n are called partial orders of reaction and are not generally equal to the stoichiometric coefficients a and b. Instead they depend on the reaction mechanism and can be determined experimentally.

Sum of m and n, that is, (m + n) is called the overall order of reaction.

Elementary steps

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For an elementary step, there is a relationship between stoichiometry and rate law, as determined by the law of mass action. Almost all elementary steps are either unimolecular or bimolecular. For a unimolecular step

A → P

the reaction rate is described by , where is a unimolecular rate constant. Since a reaction requires a change in molecular geometry, unimolecular rate constants cannot be larger than the frequency of a molecular vibration. Thus, in general, a unimolecular rate constant has an upper limit of k1 ≤ ~1013 s−1.

For a bimolecular step

A + B → P

the reaction rate is described by , where is a bimolecular rate constant. Bimolecular rate constants have an upper limit that is determined by how frequently molecules can collide, and the fastest such processes are limited by diffusion. Thus, in general, a bimolecular rate constant has an upper limit of k2 ≤ ~1010 M−1s−1.

For a termolecular step

A + B + C → P

the reaction rate is described by , where is a termolecular rate constant.

There are few examples of elementary steps that are termolecular or higher order, due to the low probability of three or more molecules colliding in their reactive conformations and in the right orientation relative to each other to reach a particular transition state.[2] There are, however, some termolecular examples in the gas phase. Most involve the recombination of two atoms or small radicals or molecules in the presence of an inert third body which carries off excess energy, such as O + O
2 + N
2 → O
3 + N
2. One well-established example is the termolecular step 2 I + H
2 → 2 HI in the hydrogen-iodine reaction.[3][4][5] In cases where a termolecular step might plausibly be proposed, one of the reactants is generally present in high concentration (e.g., as a solvent or diluent gas).[6]

Relationship to other parameters

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For a first-order reaction (including a unimolecular one-step process), there is a direct relationship between the unimolecular rate constant and the half-life of the reaction: . Transition state theory gives a relationship between the rate constant and the Gibbs free energy of activation , a quantity that can be regarded as the free energy change needed to reach the transition state. In particular, this energy barrier incorporates both enthalpic () and entropic () changes that need to be achieved for the reaction to take place:[7][8] The result from transition state theory is , where h is the Planck constant and R the molar gas constant. As useful rules of thumb, a first-order reaction with a rate constant of 10−4 s−1 will have a half-life (t1/2) of approximately 2 hours. For a one-step process taking place at room temperature, the corresponding Gibbs free energy of activation (ΔG‡) is approximately 23 kcal/mol.

Dependence on temperature

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The Arrhenius equation is an elementary treatment that gives the quantitative basis of the relationship between the activation energy and the reaction rate at which a reaction proceeds. The rate constant as a function of thermodynamic temperature is then given by:

The reaction rate is given by:

where Ea is the activation energy, and R is the gas constant, and m and n are experimentally determined partial orders in [A] and [B], respectively. Since at temperature T the molecules have energies according to a Boltzmann distribution, one can expect the proportion of collisions with energy greater than Ea to vary with e−Ea⁄RT. The constant of proportionality A is the pre-exponential factor, or frequency factor (not to be confused here with the reactant A) takes into consideration the frequency at which reactant molecules are colliding and the likelihood that a collision leads to a successful reaction. Here, A has the same dimensions as an (m + n)-order rate constant (see Units below).

Another popular model that is derived using more sophisticated statistical mechanical considerations is the Eyring equation from transition state theory:

where ΔG‡ is the free energy of activation, a parameter that incorporates both the enthalpy and entropy change needed to reach the transition state. The temperature dependence of ΔG‡ is used to compute these parameters, the enthalpy of activation ΔH‡ and the entropy of activation ΔS‡, based on the defining formula ΔG‡ = ΔH‡ − TΔS‡. In effect, the free energy of activation takes into account both the activation energy and the likelihood of successful collision, while the factor kBT/h gives the frequency of molecular collision.

The factor (c⊖)1-M ensures the dimensional correctness of the rate constant when the transition state in question is bimolecular or higher. Here, c⊖ is the standard concentration, generally chosen based on the unit of concentration used (usually c⊖ = 1 mol L−1 = 1 M), and M is the molecularity of the transition state. Lastly, κ, usually set to unity, is known as the transmission coefficient, a parameter which essentially serves as a "fudge factor" for transition state theory.

The biggest difference between the two theories is that Arrhenius theory attempts to model the reaction (single- or multi-step) as a whole, while transition state theory models the individual elementary steps involved. Thus, they are not directly comparable, unless the reaction in question involves only a single elementary step.

Finally, in the past, collision theory, in which reactants are viewed as hard spheres with a particular cross-section, provided yet another common way to rationalize and model the temperature dependence of the rate constant, although this approach has gradually fallen into disuse. The equation for the rate constant is similar in functional form to both the Arrhenius and Eyring equations:

where P is the steric (or probability) factor and Z is the collision frequency, and ΔE is energy input required to overcome the activation barrier. Of note, , making the temperature dependence of k different from both the Arrhenius and Eyring models.

Comparison of models

[edit]

All three theories model the temperature dependence of k using an equation of the form

for some constant C, where α = 0, 1⁄2, and 1 give Arrhenius theory, collision theory, and transition state theory, respectively, although the imprecise notion of ΔE, the energy needed to overcome the activation barrier, has a slightly different meaning in each theory. In practice, experimental data does not generally allow a determination to be made as to which is "correct" in terms of best fit. Hence, all three are conceptual frameworks that make numerous assumptions, both realistic and unrealistic, in their derivations. As a result, they are capable of providing different insights into a system.[9]

Units

[edit]

The units of the rate constant depend on the overall order of reaction.[10]

If concentration is measured in units of mol·L−1 (sometimes abbreviated as M), then

  • For order (m + n), the rate constant has units of mol1−(m+n)·L(m+n)−1·s−1 (or M1−(m+n)·s−1)
  • For order zero, the rate constant has units of mol·L−1·s−1 (or M·s−1)
  • For order one, the rate constant has units of s−1
  • For order two, the rate constant has units of L·mol−1·s−1 (or M−1·s−1)
  • For order three, the rate constant has units of L2·mol−2·s−1 (or M−2·s−1)
  • For order four, the rate constant has units of L3·mol−3·s−1 (or M−3·s−1)

Plasma and gases

[edit]

Calculation of rate constants of the processes of generation and relaxation of electronically and vibrationally excited particles are of significant importance. It is used, for example, in the computer simulation of processes in plasma chemistry or microelectronics. First-principle based models should be used for such calculation. It can be done with the help of computer simulation software.

Rate constant calculations

[edit]

Rate constant can be calculated for elementary reactions by molecular dynamics simulations. One possible approach is to calculate the mean residence time of the molecule in the reactant state. Although this is feasible for small systems with short residence times, this approach is not widely applicable as reactions are often rare events on molecular scale. One simple approach to overcome this problem is Divided Saddle Theory.[11] Such other methods as the Bennett Chandler procedure,[12][13] and Milestoning[14] have also been developed for rate constant calculations.

Divided saddle theory

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The theory is based on the assumption that the reaction can be described by a reaction coordinate, and that we can apply Boltzmann distribution at least in the reactant state. A new, especially reactive segment of the reactant, called the saddle domain, is introduced, and the rate constant is factored:

where αSD
RS is the conversion factor between the reactant state and saddle domain, while kSD is the rate constant from the saddle domain. The first can be simply calculated from the free energy surface, the latter is easily accessible from short molecular dynamics simulations [11]

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Reaction rate constant

View on Grokipedia
from Grokipedia
The reaction rate constant, denoted as k, is a proportionality factor in the rate law of that relates the rate of a to the concentrations of its reactants. For a general reaction aA + bB → products, the rate law is expressed as rate = [A]^m [B]^n, where m and n are the reaction orders with respect to reactants A and B, respectively, and the rate is typically measured as the change in concentration per unit time (e.g., mol L⁻¹ s⁻¹). The value of is determined experimentally and remains constant for a given reaction under specified conditions, reflecting the intrinsic speed of the reaction independent of concentration. The magnitude of the reaction rate constant is highly sensitive to temperature, as described by the : k = A e^(-E_a / RT), where A is the representing the frequency of collisions with proper orientation, E_a is the (the minimum energy barrier for the reaction), R is the (8.314 J mol⁻¹ K⁻¹), and T is the absolute temperature in . This exponential dependence means that even small temperature increases can dramatically accelerate reactions by exponentially raising k. The units of k depend on the overall reaction order: for zero-order reactions, k has units of concentration/time (e.g., mol L⁻¹ s⁻¹); for , time⁻¹ (s⁻¹); and for second-order, concentration⁻¹ time⁻¹ (L mol⁻¹ s⁻¹). Catalysts influence the reaction rate constant by providing an alternative reaction pathway with a lower activation energy, thereby increasing k without being consumed in the process. This effect is particularly notable in industrial processes, where catalysts enable reactions to proceed at milder conditions and higher rates. Overall, the reaction rate constant encapsulates the kinetic behavior of a reaction, guiding predictions in fields from laboratory synthesis to environmental modeling.

Fundamentals

Definition and Rate Laws

The reaction rate constant, denoted as kk, serves as the proportionality factor in the rate law of a chemical reaction, relating the reaction rate to the concentrations of the reactants raised to their respective orders. The general form of the rate law is expressed as rate=k[\ceA]n[\ceB]m⋯ ,\text{rate} = k [\ce{A}]^n [\ce{B}]^m \cdots, where nn, mm, etc., represent the reaction orders with respect to each reactant, and for elementary reactions, these orders equal the stoichiometric coefficients in the balanced equation. This formulation arises from the law of mass action, which assumes that the rate is proportional to the product of reactant concentrations, each to the power of its stoichiometric coefficient in single-step processes. The rate constant kk fundamentally quantifies the frequency of effective molecular interactions that result in product formation, encapsulating the probability of successful collisions between reactant molecules. In , kk incorporates both the overall factor—dependent on factors like molecular size and —and the fraction of those collisions that possess sufficient and proper orientation to overcome the barrier. Thus, kk provides a measure of how efficiently a reaction proceeds under given conditions, independent of reactant concentrations. In reversible reactions, distinct forward rate constants (kfk_f) and reverse rate constants (krk_r) are defined to describe the opposing processes, with the equilibrium constant given by K=kf/krK = k_f / k_r. For example, in a simple unimolecular such as \ceA−>products\ce{A -> products}, the rate law simplifies to rate=k[\ceA],\text{rate} = k [\ce{A}], where the first-order dependence reflects the single-molecule process. The value of kk is -dependent, generally increasing with rising to accelerate the .

Elementary Reactions

Elementary reactions represent the fundamental building blocks of mechanisms, defined as single-step processes that occur without intermediates and involve a single . In these reactions, the rate law is directly determined by the , which is the number of reactant participating in the step—typically one, two, or rarely three molecules. Unimolecular reactions involve a single molecule decomposing or rearranging, such as the of to propene, with a rate law of the form rate = k [A]. Bimolecular reactions, the most common type, entail two colliding, yielding rate = k [A][B] for distinct reactants or rate = 2k [A]^2 for identical ones, as seen in the reaction between and : \ce{NO + O3 -> NO2 + O2}. The rate constant k for an elementary step is mathematically expressed as k = rate / ∏ [reactant concentrations]^{stoichiometric coefficients}, ensuring the rate law matches the reaction's stoichiometry. Termolecular reactions, involving three species, follow rate = k [A][B][C] but are exceedingly rare due to the low probability of three particles colliding simultaneously with sufficient energy and proper orientation, as exemplified by the reaction 2NO + O₂ → 2NO₂ (rate = k [NO]^2 [O₂]). Such events are improbable under typical conditions, as the collision frequency for three bodies is orders of magnitude lower than for two. In bimolecular elementary reactions, such as A + B → products, the rate constant encapsulates the collision frequency factor between A and B molecules and the orientation factor, which accounts for the fraction of collisions where the reactants are properly aligned to overcome the energy barrier. This perspective originates from , where the effective rate depends on both the number of encounters per unit time and the steric requirements for reaction. Thus, serves as a quantitative measure of the reaction's intrinsic speed for that specific elementary step, distinct from composite reactions that require mechanistic analysis.

Theoretical Relationships

Activation Energy and Enthalpy

The , denoted as EaE_a, represents the minimum energy barrier that reactant molecules must overcome to form the , enabling the reaction to proceed. This energetic threshold arises from the need for reactants to achieve a specific configuration and sufficient during collisions, as described in and . Without surmounting EaE_a, collisions between molecules are ineffective, resulting in no net reaction progress. The rate constant kk for a reaction is directly influenced by this energy barrier, exhibiting an exponential dependence such that k∝e−Ea/RTk \propto e^{-E_a / RT}, where RR is the and TT is the absolute temperature. This relationship highlights how higher temperatures provide more molecules with energy exceeding EaE_a, thereby accelerating the . In empirical observations, reactions with lower EaE_a values proceed more readily at ambient conditions, underscoring the barrier's role in controlling kinetic behavior. In advanced theoretical frameworks, such as transition state theory, the activation energy connects to thermodynamic quantities like the enthalpy of activation ΔH‡\Delta H^\ddagger, which quantifies the enthalpic change to reach the transition state. For bimolecular reactions, this manifests approximately as Ea≈ΔH‡+2RTE_a \approx \Delta H^\ddagger + 2RT, linking macroscopic kinetic parameters to microscopic enthalpy differences. This approximation accounts for the work associated with forming the activated complex in typical conditions. Catalysts enhance reaction rates by providing an alternative pathway with a reduced EaE_a, allowing more frequent successful collisions without being consumed in the process. Importantly, this lowering of the barrier affects both forward and reverse reactions equally, preserving the and thus the position of . For instance, enzymes in biological systems exemplify this by dramatically increasing kk for specific reactions while maintaining thermodynamic balance.

Pre-exponential Factor

In the , k=Ae−Ea/RTk = A e^{-E_a / RT}, the AA (also known as the frequency factor) quantifies the rate of molecular collisions that possess the correct orientation for reaction, serving as the baseline rate before accounting for the energy barrier. This factor arises from , where AA approximates the product of the ZZ between reactant molecules and the probability that such collisions lead to a reactive encounter. Specifically, A=PZA = P Z, with ZZ depending on and molecular sizes, while PP adjusts for non-ideal collision outcomes. The primary factors influencing AA stem from molecular geometry and environmental conditions. The steric factor PP, which is typically much less than 1 (often ranging from 10−610^{-6} to 0.1 for complex molecules), reflects the of collisions with the precise orientation required for bond breaking and formation, reducing AA below the maximum collision rate. In solution-phase kinetics, solvents further modulate AA by increasing , which lowers the diffusion-controlled , and through effects that alter reactant mobility or stabilize transition states, often resulting in AA values an smaller than in the gas phase. These influences highlight AA's role in capturing probabilistic aspects of reactivity beyond energetic thresholds. Empirically, AA is obtained from Arrhenius plots, where the natural logarithm of the rate constant ln⁡k\ln k is graphed against the inverse 1/T1/T; the resulting linear fit has a y-intercept of ln⁡A\ln A and a slope of −Ea/R-E_a / R. For gas-phase bimolecular reactions, typical AA values span 10910^9 to 101310^{13} L mol−1^{-1} s−1^{-1} , reflecting variations in collision cross-sections and steric hindrances across different molecular systems.

Temperature Dependence

Arrhenius Equation

The provides the foundational empirical model for the temperature dependence of the reaction rate constant in . Developed by Swedish chemist in 1889, it emerged from his analysis of reaction rates in the acid-catalyzed inversion of cane sugar, where he observed that rates increase exponentially with temperature. Arrhenius built upon Jacobus Henricus van't Hoff's earlier investigations into the temperature effects on chemical equilibria, extending those principles to kinetic processes by interpreting the temperature sensitivity in terms of an energy barrier that molecules must overcome. The equation is expressed as k=A e−Ea/RTk = A \, e^{-E_a / RT} where kk is the rate constant, AA is the pre-exponential factor representing the frequency of successful collisions, EaE_a is the activation energy (the minimum energy required for the reaction), RR is the universal gas constant, and TT is the absolute temperature in Kelvin. This exponential form arises from the Boltzmann distribution of molecular energies in a system at thermal equilibrium. The probability that a molecule has energy exceeding EaE_a is proportional to the integral of the Boltzmann factor e−E/RTe^{-E / RT} from EaE_a to infinity, which approximates to e−Ea/RTe^{-E_a / RT} for high activation barriers relative to thermal energy. Thus, the rate constant reflects the fraction of molecules energetic enough to surmount the barrier, multiplied by an attempt frequency captured in AA. For experimental determination of parameters, the is rearranged into its linearized form: ln⁡k=ln⁡A−EaR⋅1T\ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T} A plot of ln⁡k\ln k versus 1/T1/T produces a straight line, with the slope equal to −Ea/R-E_a / R and the equal to ln⁡A\ln A. This graphical method allows extraction of activation energies from measured rate constants at different temperatures, typically yielding reliable values for many reactions. The model assumes a constant and temperature-independent , holding well over moderate temperature ranges (e.g., to a few hundred ) for simple reactions. At extreme temperatures, such as very low cryogenic conditions or high thermal environments, deviations arise due to changes in molecular partitioning or non-ideal behaviors, necessitating more sophisticated theoretical frameworks.

Transition State Theory

Transition state theory (TST) provides a fundamental statistical mechanical framework for deriving reaction rate constants from the molecular properties of reactants and the transition state. The core concept is that a bimolecular reaction proceeds through the formation of an activated complex, a transient high-energy species at the saddle point of the potential energy surface (PES), which maps the potential energy as a function of nuclear coordinates. This saddle point is a first-order stationary point on the PES, characterized by a single imaginary vibrational frequency along the reaction coordinate, distinguishing it from minima (reactants or products) that have all real frequencies. The activated complex exists in a shallow potential well perpendicular to the reaction path but is unstable along the path to products. The theory was developed independently in 1935 by Henry Eyring at and by Meredith Gwynne Evans and in . Eyring's formulation emphasized the of the , while Evans and Polanyi focused on surfaces derived from . This approach shifted reaction kinetics from empirical models to a microscopic understanding based on quantum and statistical principles. Within TST, the rate constant kk for a reaction is given by the : k=κkBThexp⁡(ΔS‡R)exp⁡(−ΔH‡RT)k = \kappa \frac{k_B T}{h} \exp\left( \frac{\Delta S^\ddagger}{R} \right) \exp\left( -\frac{\Delta H^\ddagger}{RT} \right) where κ\kappa is the transmission coefficient (approximated as 1 in classical TST), kBk_B is Boltzmann's constant, TT is the absolute temperature, hh is Planck's constant, RR is the gas constant, ΔS‡\Delta S^\ddagger is the standard molar activation entropy, and ΔH‡\Delta H^\ddagger is the standard molar activation enthalpy. This equation expresses the rate constant in terms of thermodynamic activation parameters, with the exponential terms capturing the entropic and enthalpic contributions to the free energy barrier ΔG‡=ΔH‡−TΔS‡\Delta G^\ddagger = \Delta H^\ddagger - T\Delta S^\ddagger. The derivation begins with the postulate of a quasi-equilibrium between the reactants and the , justified when the lifetime of the complex is short compared to the reaction timescale and the activation energy exceeds several kBTk_B T. The equilibrium constant K‡K^\ddagger for activated complex formation is computed using partition functions: K‡=Q‡QAQBexp⁡(−ΔE0/RT)K^\ddagger = \frac{Q^\ddagger}{Q_A Q_B} \exp(-\Delta E_0 / RT), where QQ denotes molecular partition functions (with the transition state partition function treating the reaction coordinate as a rather than ), ΔE0\Delta E_0 is the difference, and standard-state corrections apply for concentrations. The forward rate is then the equilibrium concentration of the complex times the unimolecular frequency kBTh\frac{k_B T}{h} across the dividing surface at the , yielding k=κkBThK‡k = \kappa \frac{k_B T}{h} K^\ddagger; the κ≈1\kappa \approx 1 assumes all complexes crossing the surface react without return. A key limitation of classical TST is its assumption of no recrossing, meaning trajectories reaching the canonical dividing surface at the saddle point proceed irreversibly to products, which overestimates rates for reactions with variational effects or corner-cutting dynamics. This is improved in variational transition state theory (VTST), which locates an optimal dividing surface along the reaction path to minimize the flux and thus recrossing, often yielding rate constants accurate to within 10-20% for gas-phase reactions when combined with accurate PES. The Eyring parameters provide a theoretical underpinning for Arrhenius behavior, where the activation energy approximates ΔH‡+RT\Delta H^\ddagger + RT and the pre-exponential factor incorporates exp⁡(ΔS‡/R)\exp(\Delta S^\ddagger / R).

Comparison of Models

The empirical Arrhenius model, introduced by in 1889, provides a foundational description of temperature dependence through the equation k=Ae−Ea/RTk = A e^{-E_a / RT}, where AA is the and EaE_a is the . This model excels in simplicity and is widely used for fitting experimental data across a range of temperatures, but it lacks a theoretical basis for the pre-exponential factor and does not incorporate entropic contributions. Transition state theory (TST), developed by Henry Eyring in 1935, advances beyond the empirical approach by grounding rate constants in and the concept of a , incorporating both enthalpic and entropic effects via the (detailed in the section). TST offers greater predictive power for computational simulations and mechanistic insights, particularly for complex reactions, though it assumes equilibrium at the transition state, which may not hold under non-equilibrium conditions. The Polanyi-Semenov relation, formulated in the 1930s by and Nikolai Semenov, specifically addresses gas-phase atom-transfer reactions by linking to the reaction through a Ea=E0+αΔHrE_a = E_0 + \alpha \Delta H_r, where α\alpha (typically 0.25–0.5 for exothermic processes) reflects differences. This model is particularly useful for estimating rates in radical or reactions without full quantum calculations, but it is limited to series of related exothermic gas-phase processes and overlooks steric or . Historically, reaction rate modeling evolved from Arrhenius's empirical law, which fit observed exponential temperature dependence, to in the early , and then to TST in , which integrated for a more unified framework. Modern extensions incorporate quantum effects, leading to variational and quantum TST formulations that refine predictions for barrier crossing.
ModelBasisStrengthsLimitationsApplications
ArrheniusEmpirical, exponential fitSimple; excellent for data fitting over moderate temperaturesNo theoretical explanation for AA or ; ignores quantum effectsExperimental rate constant analysis in solution or gas phase
TSTTheoretical, Includes and ; predictive for mechanismsAssumes transition state equilibrium; less accurate at extremes without correctionsComputational predictions for organic and biochemical reactions
Polanyi-SemenovSemi-empirical, bond energiesRapid estimation using ; suited for related reaction seriesLimited to gas-phase exothermic processes; parameter α\alpha variesRadical reactions in or
Arrhenius is preferred for straightforward experimental fitting where mechanistic details are secondary, while TST suits computational predictions requiring thermodynamic consistency, and Polanyi-Semenov applies to extreme conditions like high-temperature gas reactions. Deviations from classical models arise in low-temperature regimes, where quantum tunneling allows reactants to penetrate energy barriers, enhancing rates beyond Arrhenius or standard TST predictions; modified TST incorporates these via semiclassical tunneling corrections, such as in variational formulations, improving accuracy for hydrogen-transfer reactions by factors up to 10-20 at 100-200 K.

Practical Considerations

Units and Dimensions

The units of the reaction rate constant kk depend on the of an , which determines the reaction order nn. For a unimolecular () reaction, kk has units of inverse time, specifically s−1\mathrm{s}^{-1} in the SI system. For a bimolecular (second-order) reaction, the units are inverse concentration times inverse time, commonly expressed as M−1s−1\mathrm{M}^{-1} \mathrm{s}^{-1} or Lmol−1s−1\mathrm{L} \mathrm{mol}^{-1} \mathrm{s}^{-1}, where M\mathrm{M} denotes molarity (mol L−1^{-1}). Termolecular (third-order) reactions are rarer, but their rate constants have units of M−2s−1\mathrm{M}^{-2} \mathrm{s}^{-1} or mol−2L2s−1\mathrm{mol}^{-2} \mathrm{L}^{2} \mathrm{s}^{-1}. In the SI system, concentration is formally in mol m−3^{-3}, leading to units like m3^{3} mol−1^{-1} s−1^{-1} for second-order reactions, though mol dm−3^{-3} (equivalent to M) and seconds remain standard in chemical kinetics for practicality. For gas-phase reactions, where partial pressures are often used instead of concentrations, rate constants may be reported in units such as cm3^{3} molecule−1^{-1} s−1^{-1} for bimolecular processes; these can be converted to concentration-based units via the ideal gas law PV=nRTPV = nRT and Avogadro's constant. Dimensionally, the rate constant kk for a reaction of overall order nn has dimensions [concentration]1−n[time]−1[\mathrm{concentration}]^{1-n} [\mathrm{time}]^{-1}, ensuring the rate law rate=k[reactants]n\mathrm{rate} = k [\mathrm{reactants}]^{n} yields consistent units of concentration per time (e.g., M s−1^{-1}). This analysis aids in verifying the order from experimental data but highlights potential issues with non-integer orders, where fractional powers (e.g., M−0.5^{-0.5} s−1^{-1} for n=1.5n = 1.5) arise and complicate interpretation. A frequent error in applying rate laws is inconsistently mixing concentration units (e.g., molarity) with pressure-based measures without conversion, resulting in erroneous kk values that misrepresent reaction kinetics.

Determination Methods

Experimental methods for determining constants primarily involve monitoring the progress of a reaction under controlled conditions to extract kinetic parameters from concentration-time . The rates method entails measuring the rate of product formation or reactant consumption at the very beginning of the reaction, where concentrations are well-defined and side reactions are minimal, allowing direct determination of the rate law and constant by varying concentrations. This approach is particularly useful for slow reactions, as it avoids complications from product accumulation or equilibrium effects. For multi-reactant systems, the isolation method simplifies the kinetics by using a large excess of all but one reactant, converting the reaction to pseudo-first-order conditions where the rate depends linearly on the isolated reactant's concentration. This enables sequential determination of partial orders and the overall rate constant by repeating experiments with varied concentrations of the isolated species. Relaxation techniques, such as temperature-jump methods, are employed for fast reactions near equilibrium; a sudden perturbation shifts the system away from equilibrium, and the rate constant is derived from the exponential relaxation back to the new equilibrium state. These methods can resolve rate constants on microsecond timescales by tracking spectroscopic changes post-perturbation. A common example for fast reactions is the stopped-flow apparatus, which rapidly mixes reactants and monitors transients via under pseudo- conditions to yield the second-order rate constant from the observed first-order decay. For instance, in enzyme-substrate kinetics, this technique has provided precise rate constants for association steps by ensuring mixing times shorter than reaction half-lives. Computational approaches complement experiments by predicting rate constants from quantum mechanical calculations. methods, often within , compute the and partition functions to estimate the rate constant without empirical fitting, particularly for gas-phase reactions. Direct dynamics simulations propagate classical trajectories on ab initio-generated surfaces to directly yield rate constants, capturing dynamic effects beyond static approximations. These techniques are essential for inaccessible experimental conditions, such as high temperatures or exotic species. Error analysis in rate constant determination emphasizes precision through replicate runs, which quantify random variations in measurements like or changes, typically achieving uncertainties of 5-10% for well-controlled experiments. is critical, as even 1°C fluctuations can alter rates by several percent due to exponential dependence, necessitating thermostated setups with stability better than 0.1°C. Systematic errors from impure or incomplete mixing are minimized by and validation against standards. Arrhenius plots from temperature-series further refine constants by extracting energies alongside .

Special Cases

Gases and Plasmas

In gas-phase kinetics, reaction rate constants for bimolecular processes are conventionally expressed in units of concentration, such as cm³ molecule⁻¹ s⁻¹, but in systems where partial pressures are more convenient—particularly at low pressures or in engineering contexts—the rate laws incorporate partial pressures, yielding units like atm⁻¹ s⁻¹ for second-order reactions. This adjustment accounts for the relationship between concentration and pressure via the ideal gas law, P = cRT, where the rate constant in pressure units, k_p, relates to the concentration-based k_c by k_p = k_c (RT)^{Δn}, with Δn being the change in moles of gas. Such formulations facilitate modeling in environments like dilute gases or vacuum systems. A key feature of gas-phase unimolecular reactions is the fall-off regime at low pressures, explained by the , which posits that reactant molecules must be energized through collisions with a bath gas M before . At high pressures, frequent collisions maintain a steady of energized intermediates, yielding a pressure-independent high-pressure limit rate constant. However, at low pressures, the collision rate drops, limiting energization and causing the effective rate constant to decrease linearly with pressure, transitioning to a second-order dependence on the bath gas concentration. This pressure dependence is critical for accurate predictions in dilute conditions. In plasmas, the rate constants deviate from neutral gas-phase behavior due to the abundance of ions, excited states, and free , which enhance reactivity and elevate effective rate constants beyond typical values. For electron-impact reactions, such as or dissociation, rate constants often exceed 10^{-9} cm³ molecule⁻¹ s⁻¹, reflecting high cross sections and velocities of energetic . High prevalent in plasmas accelerate these rates in line with general Arrhenius temperature dependence, but the non-Maxwellian nature of electron distributions—often featuring overpopulated high-energy tails—necessitates modified Arrhenius expressions or direct integration over the distribution function to compute accurate rate coefficients. These specialized rate constant behaviors underpin applications in modeling, where fall-off effects and dependencies are vital for simulating ignition, speeds, and formation in engines and reactors using detailed kinetic mechanisms. In , gas-phase rate constants, including those for radical reactions and pressure-limited processes, drive models of oxidant cycles, such as OH-initiated degradation of volatile organic compounds, essential for predicting air quality and tropospheric levels.

Surface Reactions

In heterogeneous catalysis, reaction rate constants for surface reactions describe the kinetics of processes where reactants adsorb onto a catalyst surface before reacting, often following mechanisms like Langmuir-Hinshelwood (LH). In the LH mechanism, both reactants adsorb dissociatively or associatively on adjacent active sites, and the surface reaction between adsorbed species determines the rate, given by r=kθAθBr = k \theta_A \theta_B, where θA\theta_A and θB\theta_B are the fractional surface coverages of species A and B, respectively, and kk is the rate constant for the bimolecular surface reaction step. The coverages θA\theta_A and θB\theta_B are derived from the Langmuir adsorption isotherm, θi=KiCi1+∑KjCj\theta_i = \frac{K_i C_i}{1 + \sum K_j C_j}, where KiK_i is the adsorption equilibrium constant and CiC_i the gas-phase concentration, leading to an overall rate law of r=kKAKBCACB(1+KACA+KBCB)2r = \frac{k K_A K_B C_A C_B}{(1 + K_A C_A + K_B C_B)^2} for bimolecular reactions under steady-state conditions. The rate constant kk in surface reactions incorporates the frequency of collisions between adsorbed species on the limited surface area, typically following Arrhenius behavior k=Aexp⁡(−Ea/RT)k = A \exp(-E_a / RT), where AA is the reflecting surface mobility and site . However, the effective rate constant is modified by preceding adsorption and desorption steps, which can lower kk compared to gas-phase analogs due to energy barriers for adsorption and site blocking at high coverages; for instance, adsorption is often rate-limiting at low temperatures, reducing the observed kk. Units for kk in LH mechanisms are commonly s^{-1} for unimolecular surface steps or site^{-1} s^{-1} when normalized per , emphasizing per-site reactivity rather than bulk concentration. Temperature dependence in surface rate constants exhibits a compensation effect, where variations in activation energy EaE_a across related catalysts correlate linearly with ln⁡A\ln A, such that ln⁡A=αEa+β\ln A = \alpha E_a + \beta, resulting in similar effective kk values at typical operating temperatures despite differences in individual parameters. This arises from enthalpy-entropy compensation in adsorption and surface diffusion, with exothermic adsorption weakening at higher temperatures, shifting kinetics from zero-order (coverage-independent) at low TT to first-order at high TT. A representative example is synthesis on iron-based catalysts via the Haber-Bosch process, where the LH mechanism governs dissociation and steps on surface sites, with the rate constant derived from turnover frequency (TOF), typically 0.1–1 s^{-1} site^{-1} at 400–500°C and 100–300 atm, reflecting the slow N₂ adsorption as the rate-determining step. In such systems, the effective kk is tuned by promoters like or Al₂O₃ to enhance adsorption equilibria, achieving industrially viable rates.

Advanced Theories

Rate Constant Calculations

Classical methods for calculating reaction rate constants often involve integrating the differential rate laws derived from experimental data in batch reactors, allowing the determination of the rate constant kk by fitting concentration-time profiles. For a second-order reaction involving a single reactant AA, the integrated rate law is given by 1[A]=1[A]0+kt\frac{1}{[A]} = \frac{1}{[A]_0} + kt where [A][A] is the concentration at time tt, and [A]0[A]_0 is the initial concentration; plotting 1/[A]1/[A] versus tt yields a straight line with slope kk./12%3A_Kinetics/12.05%3A_Integrated_Rate_Laws) Similar integrations apply to zero- and first-order reactions, enabling kk extraction from linear regressions of transformed concentration data in constant-volume batch systems. Quantum chemistry approaches compute rate constants by scanning potential energy surfaces (PES) to identify transition states, which are then used as input for transition state theory (TST) predictions. These methods involve ab initio calculations to map the PES, locating minima for reactants and products, and saddle points for transition states, with the activation energy EaE_a derived from the energy barrier height. The resulting PES data feed into TST formulations, such as the Eyring equation, to predict thermal rate constants kk over temperature ranges, often incorporating variational TST to refine barrier locations for improved accuracy. Software tools like Gaussian and facilitate these computations by optimizing molecular geometries, calculating EaE_a via or coupled-cluster methods, and applying TST to derive kk. For instance, 's capabilities include accurate barrier height evaluations using domain-based local pair natural orbital methods, directly linking to TST rate predictions. Gaussian similarly supports frequency calculations at transition states to confirm imaginary frequencies and compute partition functions for TST. For unimolecular dissociation rates, extends these tools by evaluating microcanonical rate constants from vibrational frequencies and energies on the PES, accounting for intramolecular energy redistribution. Validation of computed rate constants typically compares predictions to experimental values, with agreements within a factor of 10 often considered reliable given uncertainties in PES accuracy and anharmonic effects. High-level methods achieve this precision for simple systems, though larger molecules may require semi-empirical corrections for broader applicability.

Divided Saddle Theory

Divided Saddle Theory (DST) addresses challenges in modeling multi-dimensional transition states for constants by dividing the region on the free energy surface into discrete Saddle Domains (SDs). This division allows for the construction of variational dividing surfaces that minimize recrossing of reactive trajectories, a common issue in standard (TST) where the dividing surface is fixed at the . By focusing on these domains, DST enables more precise rate calculations for systems where the exhibits complex topology, reducing the overestimation of rates inherent in conventional approaches. The formulation of DST integrates elements of variational TST by optimizing the effective transition state location through postprocessing of data. Auxiliary rate constants (k_SD) are computed within each SD using the average number of transitions per unit time from equilibrium simulations, then reweighted by the fractional concentration of the SD relative to the reactant state (α_SD_RS) to yield the overall rate constant: k_DST = k_SD × α_SD_RS. This approach dynamically selects the best dividing surface along the to account for recrossings without requiring additional specialized simulations beyond standard equilibrium sampling and committor . DST finds applications in reactions involving submerged barriers or , such as biomolecular conformational changes and pericyclic rearrangements. For instance, in the alanine dipeptide (modeled in implicit ), DST yields forward and backward rate constants of 0.257 × 10¹¹ s⁻¹ and 1.564 × 10¹¹ s⁻¹, respectively, closely matching unbiased direct dynamics results. Similarly, for the barbaralane , it computes a rate constant of 1.926 × 10¹² s⁻¹. These applications demonstrate improvements in accuracy over standard TST by factors of 3–5 (corresponding to 200–400% error reduction), particularly beneficial for condensed-phase systems where recrossing is pronounced. DST emerged in 2014 from the work of János Daru and András Stirling, building on earlier variational methods such as the Bennett-Chandler formalism and effective positive flux approaches to enhance rate constant predictions in complex landscapes.

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