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In chemical kinetics, the pre-exponential factor or A factor is the pre-exponential constant in the Arrhenius equation (equation shown below), an empirical relationship between temperature and rate coefficient. It is usually designated by A when determined from experiment, while Z is usually left for collision frequency. The pre-exponential factor can be thought of as a measure of the frequency of properly oriented collisions. It is typically determined experimentally by measuring the rate constant $k$ at a particular temperature and fitting the data to the Arrhenius equation. The pre-exponential factor is generally not exactly constant, but rather depends on the specific reaction being studied and the temperature at which the reaction is occurring.^[1]

$A={\frac {k}{e^{-{\frac {E_{a}}{RT}}}}}=ke^{\frac {E_{a}}{RT}}$

The units of the pre-exponential factor A are identical to those of the rate constant and will vary depending on the order of the reaction. For a first-order reaction, it has units of s⁻¹. For that reason, it is often called frequency factor.

According to collision theory, the frequency factor, A, depends on how often molecules collide when all concentrations are 1 mol/L and on whether the molecules are properly oriented when they collide. Values of A for some reactions can be found at Collision theory.

According to transition state theory, A can be expressed in terms of the entropy of activation of the reaction.

References

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The pre-exponential factor, denoted as

A

, is a key constant in the Arrhenius equation,

k = A e^{-E_a / RT}

, which empirically models the temperature dependence of the rate constant

k

for chemical reactions, where

E_a

is the activation energy,

R

is the gas constant, and

T

is the absolute temperature.^[1] Introduced by Swedish chemist Svante Arrhenius in his 1889 paper on the inversion of cane sugar by acids, this factor captures the intrinsic reaction rate independent of the energy barrier posed by activation energy.^[1] It is also termed the frequency factor because it approximates the frequency of molecular collisions that could potentially lead to reaction, adjusted for molecular orientation and geometry.^[2] In collision theory, the pre-exponential factor is expressed as

A = pZ

, where

Z

is the collision frequency between reactant molecules—proportional to the square root of temperature for gas-phase reactions—and

p

is the steric factor accounting for the fraction of collisions with favorable orientations to surmount the activation barrier.^[2] This interpretation highlights

A

as the maximum possible rate constant at infinite temperature, where all collisions are energetically sufficient, limited only by encounter frequency and geometry.^[3] For example, in bimolecular gas-phase reactions,

Z

derives from kinetic theory as

Z = \sigma \sqrt{\frac{8\pi RT}{\mu}}

Z=σμ8πRT

, with $\sigma$ as the collision cross-section and $\mu$ the reduced mass, underscoring $A$ 's dependence on molecular properties like size and mass.^[2] From the perspective of transition state theory, developed later by Henry Eyring and others, the pre-exponential factor relates to the entropy of activation, $\Delta S^\ddagger$ , via $A = \frac{k_B T}{h} e^{\Delta S^\ddagger / R}$ , where $k_B$ is Boltzmann's constant and $h$ is Planck's constant; this formulation emphasizes $A$ 's role in quantifying the probability of forming the activated complex through entropic contributions to the reaction pathway.^[4] Unlike the exponential term, which dominates temperature sensitivity, $A$ is often treated as temperature-independent for simplicity, though theoretical models predict weak dependencies (e.g., $\propto T^{1/2}$ from collision theory or $\propto T$ from transition state theory).^[2] The units of $A$ match those of the rate constant $k$ , varying with reaction order—for instance, s⁻¹ for first-order reactions or M⁻¹ s⁻¹ for second-order reactions—and it is typically determined experimentally by plotting $\ln k$ versus $1/T$ (an Arrhenius plot), where the y-intercept yields $\ln A$ .^[3] Values of $A$ span wide ranges, around 10^{13} s⁻¹ for unimolecular gas-phase reactions and 10^{10} to 10^{11} M⁻¹ s⁻¹ for bimolecular gas-phase reactions, reflecting differences in collision efficiencies and the factor's utility in predicting reaction rates across diverse systems in physical chemistry and chemical engineering.^[5]

Fundamentals

Definition

The pre-exponential factor, denoted as

A

, is the constant term in empirical rate laws for chemical reactions, representing the maximum possible rate constant in the absence of activation energy barriers.^[6] It quantifies the inherent frequency of successful encounters between reactant molecules under conditions where energy barriers do not impede the reaction.^[7] This parameter is also referred to as the frequency factor, reflecting its association with collision frequencies. In collision theory, it includes a steric factor that adjusts for the probabilities of favorable molecular orientations.^[6] The units of

A

vary with the reaction order, determined by dimensional consistency with the rate constant; for first-order reactions, they are

\mathrm{s^{-1}}

, while for second-order reactions, they are

\mathrm{M^{-1} s^{-1}}

, linking directly to the frequency of molecular collisions.^[8] For example, values of A for unimolecular processes can range from 10^6 s^{-1} to higher magnitudes, while for bimolecular encounters, up to 10^{13} M^{-1} s^{-1}, reflecting differences in collision efficiencies.^[6] In the Arrhenius equation,

A

functions as the pre-exponential term multiplying the exponential factor.^[6]

Arrhenius Equation

The Arrhenius equation provides the foundational context for understanding the pre-exponential factor in chemical kinetics, expressing the temperature dependence of a reaction's rate constant. It is formulated as

k = A \, e^{-E_a / RT},

where

k

is the rate constant,

A

is the pre-exponential factor,

E_a

is the activation energy,

R

is the gas constant, and

T

is the absolute temperature in Kelvin.^[9] This exponential form captures how reaction rates increase dramatically with temperature, as higher thermal energy allows more reactant molecules to overcome the activation barrier.^[9] The equation was proposed empirically by Swedish chemist Svante Arrhenius in 1889, based on his analysis of published experimental data for the acid-catalyzed inversion of cane sugar and other reactions.^[10] Arrhenius observed that rate constants across multiple datasets followed a consistent exponential relationship with inverse temperature, leading him to generalize this pattern for describing temperature effects on reaction velocities.^[10] His work built on earlier observations by Jacobus van 't Hoff regarding temperature influences on equilibria, but Arrhenius was the first to apply the exponential model broadly to kinetic rate constants.^[10] To isolate the pre-exponential factor

A

from experimental data, the equation can be rearranged as

A = k \, e^{E_a / RT}

, which is commonly used in parameter extraction once

k

and

E_a

are determined at a given temperature.^[9] This form highlights

A

as the extrapolated rate constant at infinite temperature, where the exponential term approaches unity.^[9] The Arrhenius equation holds particularly well for elementary reaction steps, where the activation energy

E_a

represents the barrier height from reactants to the transition state.^[9] However, for complex mechanisms involving multiple steps, the overall rate constant may deviate from a simple exponential dependence, as the effective

E_a

becomes a composite of individual step energies and can even vary with temperature or yield negative values in some cases.^[9]

Theoretical Foundations

Collision Theory Perspective

Collision theory posits that the rate of a bimolecular gas-phase reaction depends on the frequency of collisions between reactant molecules and the fraction of those collisions that are effective in producing products. The pre-exponential factor

A

in the Arrhenius equation arises as the product of the collision frequency

Z

and the steric factor

P

, such that

A = P Z

. This framework assumes molecules behave as hard spheres, with reactions occurring only upon collisions that surpass the activation energy barrier and possess the correct orientation.^[11] The collision frequency

Z

represents the number of collisions per unit volume per unit time between reactant molecules under standard conditions, scaled for unit concentrations. For bimolecular reactions in gases,

Z

is proportional to the square root of temperature, reflecting the increased relative speeds of molecules at higher temperatures:

Z \propto \sqrt{T}

Z∝T

. The explicit form for the collision frequency factor in the rate constant is given by $Z = N_A \pi \sigma^2 \left( \frac{8 k_B T}{\pi \mu} \right)^{1/2},$ where $k_B$ is the Boltzmann constant, $\mu$ is the reduced mass of the reactants, $N_A$ is Avogadro's number, and $\sigma$ is the average collision diameter (sum of molecular radii). This expression derives from kinetic theory, treating molecules as hard spheres with a collision cross-section $\pi \sigma^2$ .^[12]^[13] The steric factor $P$ (also denoted $\rho$ ) corrects for the requirement that colliding molecules must be properly oriented for the reaction to proceed, as random orientations often fail to align reactive sites. For simple atomic or diatomic species, $P$ approaches unity, indicating nearly all energetic collisions succeed. However, for complex polyatomic molecules, $P$ is significantly smaller, typically ranging from $10^{-1}$ to $10^{-6}$ , due to the reduced probability of favorable geometries. For instance, in the reaction $\ce{H2 + C2H4 -> C2H6}$ , $P \approx 1.7 \times 10^{-6}$ .^[14]^[15] Collision theory was independently developed in the 1910s by Max Trautz and William Lewis to explain empirical rate laws, particularly the temperature dependence observed in gas-phase reactions. Trautz's 1916 work emphasized collision-based derivations of rate constants, while Lewis's 1918 contribution applied similar ideas to catalytic processes. Despite its foundational role, the theory overestimates reaction rates for systems requiring precise molecular geometries, as it initially assumes all sufficiently energetic collisions are effective without fully accounting for orientation constraints—leading to the introduction of $P$ as an empirical adjustment. This limitation highlights the theory's conceptual value over precise prediction for complex reactions.^[14]

Transition State Theory Perspective

Transition state theory (TST), formulated by Henry Eyring in 1935, offers a statistical mechanical foundation for understanding the pre-exponential factor in the Arrhenius equation by treating the reaction as proceeding through an activated complex in quasi-equilibrium with the reactants.^[16]^[17] Within this framework, the pre-exponential factor

A

for a bimolecular reaction is given by

A = \frac{k_B T}{h} e^{\Delta S^\ddagger / R},

where

k_B

is the Boltzmann constant,

T

is the temperature,

h

is Planck's constant,

\Delta S^\ddagger

is the activation entropy, and

R

is the gas constant.^[16] This expression arises from the rate constant

k = \frac{k_B T}{h} e^{-\Delta G^\ddagger / RT}

, where

\Delta G^\ddagger

is the Gibbs free energy of activation, linking

A

directly to thermodynamic properties of the transition state.^[17] The term

e^{\Delta S^\ddagger / R}

interprets the pre-exponential factor as a measure of the entropy change associated with forming the transition state from the reactants, reflecting the probability of molecules achieving the necessary configuration for reaction.^[16] A positive

\Delta S^\ddagger

indicates increased disorder in the activated complex, such as loss of rotational or translational freedom, which enhances

A

by increasing the entropy-driven accessibility of the transition state; conversely, a negative

\Delta S^\ddagger

, often due to tighter solvation or orientation constraints, reduces

A

.^[17] This entropic contribution provides a more nuanced view than the collision theory's steric factor, emphasizing equilibrium concentrations of the activated complex rather than mere collision frequencies.^[17] In TST, the activation entropy

\Delta S^\ddagger

and thus

A

are derived from ratios of molecular partition functions for the reactants and the activated complex, capturing contributions from translational, rotational, vibrational, and electronic degrees of freedom.^[17] The rotational partition function in the transition state, approximated as

Q_{rot}^\ddagger = \frac{8 \pi^2 I k_B T}{\sigma h^2}

for a linear complex (with

I

as the moment of inertia and

\sigma

as the symmetry number), influences

A

by accounting for the complex's orientational entropy, often leading to higher values for looser transition states.^[18] Vibrational contributions, via

Q_{vib}^\ddagger = \prod_i \frac{1}{1 - e^{-h \nu_i / k_B T}}

(excluding the imaginary frequency along the reaction coordinate), reflect the stiffness of the activated complex; reduced vibrational frequencies in the transition state compared to reactants typically yield a negative

\Delta S_{vib}^\ddagger

, lowering

A

, as seen in barrier-top vibrations that are softer due to partial bond breaking.^[18] These partition function details enable quantitative predictions of

A

for gas-phase reactions, with typical values around

10^{13}

10^{15}

^{-1}

for unimolecular processes.^[17] Compared to collision theory, TST provides advantages by incorporating solvent effects through modifications to the partition functions or free energy terms, allowing for solvation stabilization of the transition state that alters

A

in solution-phase reactions.^[19] Additionally, extensions of TST, such as variational and semiclassical formulations, account for quantum tunneling by including transmission coefficients that enhance rates below the classical barrier, particularly for reactions with thin barriers like hydrogen transfers, where tunneling can increase effective

A

by factors of 10 to 100.^[20] These features make TST more versatile for complex environments beyond ideal gas collisions.^[17]

Determination Methods

Experimental Techniques

The pre-exponential factor is commonly determined experimentally through the construction of an Arrhenius plot, in which the natural logarithm of the rate constant,

\ln k

, is graphed against the reciprocal of the absolute temperature,

1/T

. This approach assumes the Arrhenius equation holds over the studied range, producing a linear relationship where the slope equals

-E_a / R

(with

R

as the gas constant) and the y-intercept corresponds to

\ln A

. Linear regression is then applied to the data points to extract

\ln A

and thus

A

, providing a robust estimate when multiple rate constants are measured at precisely controlled temperatures.^[21] Another practical method involves deriving rate constants from integrated rate laws tailored to the reaction order, followed by Arrhenius analysis across temperatures. For a first-order reaction, the integrated rate law

\ln [A]_t = \ln [A]_0 - kt

is used to compute

k

from concentration-time data at varying temperatures; these

k

values are subsequently fitted to obtain

A

. This technique is particularly useful for reactions where direct monitoring of concentrations is feasible, such as spectroscopic measurements in solution or gas-phase systems, ensuring the pre-exponential factor reflects the specific kinetic regime.^[22] To achieve reliable results, experiments are conducted over a temperature span of 10–50 K, which provides adequate variation in rate constants for accurate linear fits without introducing mechanistic changes or non-Arrhenius behavior. Key error sources include catalyst impurities, which can inadvertently lower activation barriers or alter collision frequencies, leading to inflated or inconsistent

A

values if not rigorously controlled through purification protocols.^[23] A historical example is the early 20th-century study of hydrogen iodide decomposition (2 HI → H₂ + I₂) by Max Bodenstein, where rate data over 283–508°C enabled Arrhenius plotting to yield

A \approx 10^{11} \, \mathrm{M^{-1} s^{-1}}

, highlighting the method's application in establishing kinetic parameters for gas-phase reactions.^[24]

Computational Estimation

Computational estimation of the pre-exponential factor relies on theoretical models and simulations that predict its value a priori, often complementing experimental measurements by providing insights into molecular-level mechanisms. Within transition state theory (TST), ab initio quantum chemistry calculations compute the pre-exponential factor

A

as the product of a transmission coefficient and the ratio of partition functions for the transition state and reactants, evaluated at the saddle point. Software like Gaussian facilitates these computations through frequency analysis at optimized geometries, yielding vibrational, rotational, and translational contributions to the partition functions that determine

A

. For example, in molecule-surface reactions, such ab initio TST calculations produce pre-exponential factors corresponding to entropic changes, with values reflecting the degrees of freedom in the system.^[25] Molecular dynamics (MD) simulations offer a dynamical perspective for estimating

A

in complex or condensed-phase systems, where they quantify collision rates or transmission coefficients that influence the frequency factor. By propagating trajectories of reactant ensembles, MD identifies the fraction of collisions that lead to reactive events, directly informing the pre-exponential term in the Arrhenius equation. Reactive MD studies, such as those examining electric field effects on reaction kinetics, use collision frequency analysis to derive

A

, revealing how environmental factors modulate effective collision orientations and steric hindrances.^[26] Empirical correlations via quantitative structure-property relationship (QSPR) models predict

A

by relating it to molecular descriptors like size, polarity, and electronic properties, trained on datasets of Arrhenius parameters from diverse reactions. These models decompose the rate constant into its pre-exponential and activation components, enabling extrapolation to unstudied compounds without full simulations. Conjugated QSPR approaches integrated with the Arrhenius equation accurately forecast

A

for cycloaddition reactions, capturing temperature dependencies and structural influences on entropy changes.^[27] A representative case study involves SN2 nucleophilic substitution reactions, where variational transition state theory (VTST) optimizes the dividing surface along the reaction path to compute

A

. VTST accounts for recrossing trajectories, yielding pre-exponential factors consistent with the entropic bottlenecks in these concerted mechanisms. This method, advanced in seminal reviews, enhances predictive accuracy over conventional TST for barrier reactions like SN2 by minimizing variational free energy.^[28]

Applications

Chemical Reaction Kinetics

In chemical reaction kinetics, the pre-exponential factor serves as a key parameter in the Arrhenius equation, quantifying the frequency of effective collisions between reactant molecules that lead to product formation. It encapsulates the probability of proper molecular orientation during collisions, allowing researchers to predict reaction rates and infer mechanistic details from experimental data. High values of the pre-exponential factor indicate mechanisms dominated by random collisions with minimal steric hindrance, while lower values suggest additional constraints, such as specific geometric requirements for the transition state.^[29] The magnitude of the pre-exponential factor aids in discriminating between reaction mechanisms. For simple gas-phase bimolecular reactions involving atoms or small molecules, A often reaches values around

10^{11}

L mol

^{-1}

^{-1}

, consistent with collision theory predictions where most collisions are productive due to low orientation barriers. In contrast, reactions with significant steric or orientational requirements, such as those in enzyme catalysis, exhibit lower A values, typically on the order of

10^{8}

10^{12}

^{-1}

, reflecting the necessity for precise substrate alignment within the enzyme's active site to achieve the transition state.^[30] Unlike the activation energy term, which dominates the temperature dependence through its exponential form, the pre-exponential factor shows only weak variation with temperature, often approximated as constant over typical experimental ranges. This relative stability facilitates reliable extrapolation of rate constants to unstudied temperatures using Arrhenius parameters derived from plots of

\ln k

versus

1/T

. Such predictions are essential for modeling reaction behavior in industrial processes or biological systems where direct measurements at all conditions are impractical.^[29] Isotopic substitution can influence the pre-exponential factor through alterations in molecular mass, which affect collision frequencies in gas-phase reactions according to collision theory. Heavier isotopes reduce the relative velocity and thus the collision rate, leading to modestly lower A values; for instance, the ratio of pre-exponential factors for hydrogen versus deuterium abstraction reactions is approximately 1.38, arising from the square root of the mass ratio in the collision frequency expression.^[31] A representative gas-phase example is the bimolecular reaction

\ce{H2 + I2 -> 2HI}

, for which the pre-exponential factor is on the order of

10^{11}

^{-1}

^{-1}

, closely matching theoretical estimates from collision theory and supporting a direct, elementary mechanism without significant orientation restrictions.^[32]

Diffusion and Activation Processes

The pre-exponential factor extends to diffusion processes described by Fick's laws, where the diffusion coefficient

D

follows an Arrhenius-type dependence:

D = D_0 \exp\left(-\frac{E_d}{RT}\right),

with

D_0

serving as the pre-exponential factor analogous to that in chemical kinetics, representing the frequency of atomic jumps or entropy effects in the diffusion mechanism.^[33] In solids,

D_0

typically ranges from

10^{-3}

1

cm²/s for vacancy-mediated self-diffusion, as seen in face-centered cubic metals like silver, where

D_0 \approx 0.67

cm²/s for lattice self-diffusion.^[34] This value reflects the structural and vibrational contributions to the baseline diffusivity without thermal activation barriers. In activation-controlled processes beyond pure diffusion, such as vacancy-mediated atomic transport in semiconductors or creep deformation in materials, pre-exponential terms appear in the jump frequencies or rate constants following similar Arrhenius forms. For vacancy diffusion in semiconductors like silicon, the atomic jump rate

\Gamma = \Gamma_0 \exp(-E_m / kT)

includes a pre-exponential

\Gamma_0

on the order of

10^{13}

s⁻¹, tied to the Debye frequency of lattice vibrations, enabling dopant redistribution during device fabrication.^[35] In creep, particularly Nabarro-Herring creep dominated by volume diffusion, the strain rate

\dot{\epsilon}

incorporates a pre-exponential

A

\dot{\epsilon} = A \sigma^n \exp(-Q / RT)

, where

A

encapsulates geometric factors, diffusion coefficients, and grain boundary characteristics, influencing high-temperature deformation in alloys.^[36] A key example in materials science is self-diffusion in metals, where the pre-exponential

D_0

is strongly influenced by lattice vibrations, as captured in transition state models; the attempt frequency for atomic jumps arises from phonon modes, modulating

D_0

across elements like copper (

D_0 \approx 0.2

cm²/s) and nickel (

D_0 \approx 0.5

cm²/s).^[37] This vibrational contribution ensures

D_0

scales with the entropy of the saddle-point configuration, distinguishing self-diffusion from impurity pathways. Cross-disciplinarily, the pre-exponential factor manifests in Kramers' theory for escape rates from metastable potentials, applicable to activated processes in noisy environments like polymer dynamics or quantum tunneling approximations; the rate

\Gamma \approx \frac{\omega_a \omega_b}{2\pi \gamma} \exp(-\Delta V / kT)

features a pre-exponential involving well and barrier curvatures (

\omega_a, \omega_b

) and damping (

\gamma

), bridging diffusion analogies to broader stochastic activation.^[38]

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