Comparing Arrhenius, Collision and Eyring Parameters

Introduction

Three familiar rate theories use three different sets of symbols. Arrhenius fits temperature data with Ea and A. Collision theory estimates encounter frequency and introduces a steric factor p. Transition-state theory uses activation enthalpy, activation entropy and a transmission correction. These descriptions can agree about the same measured rate while assigning different meanings to their parameters. A productive comparison begins with definitions and units, then derives relationships only under stated assumptions.

Core explanation

The empirical Arrhenius equation is k = A exp(−Ea/RT) over a temperature range where A and Ea can be treated approximately constant. More generally, define a local Arrhenius activation energy by Ea(T) = −R d ln k/d(1/T). This slope definition remains meaningful even if a plot is slightly curved, though a single constant Ea may then be an inadequate summary. Arrhenius A is the fitted extrapolated factor, not automatically a molecular collision frequency. Its units are the same as the fitted k, so they depend on reaction order.

Simple gas-phase collision theory for a bimolecular A+B event estimates k₂ from a collision frequency term, an energetic fraction, and a steric or orientation factor. A schematic expression is k₂ ≈ p Z(T)exp(−E threshold/RT), with Z(T) representing the collision frequency contribution converted to a concentration-based rate constant. In a hard-sphere model Z usually has a √T dependence through mean relative speed. The steric factor p summarises how many energetically allowed collisions react in that simplified picture. It can absorb defects of the assumed cross-section and potential surface; treating it as literally only the geometric fraction of correctly oriented encounters is often too narrow.

For an ideal classical unimolecular TST treatment, write k = κ(k B T/h)exp(ΔS‡°/R)exp(−ΔH‡°/RT). If κ, ΔH‡° and ΔS‡° are approximately independent of temperature over the interval, take the natural logarithm and differentiate with respect to x = 1/T. Since ln T = −ln x, d ln T/dx = −1/x = −T. The enthalpic exponential contributes −ΔH‡°/R. Thus d ln k/d(1/T) = −T − ΔH‡°/R, and Ea = ΔH‡° + RT. At 298 K, RT ≈ 2.48 kJ mol⁻¹. This is a local relation under those assumptions, not an identity for every mechanism, pressure or temperature range.

The corresponding local Arrhenius prefactor is more subtle. Set A local = k exp(Ea/RT). Substituting the ideal TST form and Ea = ΔH‡°+RT gives A local = κe(k B T/h)exp(ΔS‡°/R) at the selected temperature. It contains an explicit T and usually cannot be treated as a temperature-independent universal constant across a broad range. If κ depends on temperature or activation heat capacities matter, additional derivative terms enter Ea. For a bimolecular elementary step, standard-concentration factors enter A as well, and an order-specific relationship between Ea and ΔH‡° must be derived from the exact chosen expression.

Collision and TST prefactors should not be identified term by term. Collision theory counts approaches based on a cross-section and relative speed; TST counts thermal population at a dividing surface and its positive-direction flux. A small p in collision theory may reflect restricted orientations, but an unfavorable activation entropy in TST may include orientation, translational association, solvent organisation and vibrational-state restrictions. The two are related in spirit yet not numerically equal in general. The Eyring transmission factor κ is different again: it corrects the fraction of counted barrier crossings that commit to products under a stated convention. A trajectory may be perfectly oriented to reach a barrier and still recross.

The parameter comparisons are useful when they reveal a mismatch. If an Arrhenius fit gives curvature, one might test whether a √T collision prefactor explains some of it or whether tunnelling, competing pathways or temperature-dependent solvation are needed. If a computed TST rate is too high, lowering κ for recrossing is one possible correction, but an inaccurate free-energy surface may be another. Neither an unusual A nor an unusual p alone proves a particular mechanism. Primary IUPAC discussions of reaction-rate theory and activation-parameter terminology emphasise the assumptions and definitions behind these quantities.

Step-by-step reasoning

1. Identify whether each quoted parameter comes from an empirical fit, a hard-sphere collision model or an Eyring calculation. 2. State the rate law and units, because A and collision prefactors change units with order. 3. Use Ea = −R d ln k/d(1/T) as the temperature-slope definition. 4. Derive any Ea–ΔH‡° relation from the specific Eyring expression and its assumed constant parameters. 5. Compare prefactors only after including standard-state and temperature factors. 6. Interpret p, ΔS‡° and κ as distinct physical summaries, then use independent experiments to test a proposed mechanism.

Visual explanation

Draw three columns labeled Arrhenius, Collision and Eyring. Under Arrhenius put “measured k(T) → Ea, A.” Under Collision put “encounters × energetic fraction × p.” Under Eyring put “barrier population × crossing frequency × κ.” Connect the Arrhenius slope to the Eyring enthalpy with an arrow labeled “unimolecular ideal: Ea = ΔH‡° + RT.” Do not draw a direct equality arrow from p to exp(ΔS‡°/R) or κ; instead place a note that different assumptions underlie those quantities.

Real-world analogy

One observer reports how rapidly people enter a building as the weather changes; another estimates the number reaching the doorway and the fraction with valid access; a third counts those gathered at a checkpoint and the fraction who pass through. All three can describe the same overall entry rate, but their intermediate counts are defined differently. Likewise, the kinetic models can fit one rate without making their prefactors interchangeable. The analogy does not reproduce molecular energy distributions or quantum effects.

Real-world example

For a gas reaction, experimentalists might fit k(T) to Arrhenius form, while theorists calculate a potential-energy surface and predict k(T) with TST. A collision estimate can provide an order-of-magnitude check. If all agree near room temperature but diverge at high temperature, examine whether the hard-sphere cross-section, activation heat capacity or assumed mechanism changes. Agreement at one temperature is not enough to validate every microscopic interpretation of A, p or ΔS‡°.

Why?

Why is Ea slightly larger than ΔH‡° in the simple unimolecular Eyring treatment? The Eyring rate includes an explicit factor of T. When T rises, that prefactor alone increases k even if the enthalpy-controlled exponential were unchanged. The Arrhenius slope captures the full temperature response, so its local energy parameter includes an extra RT. The result follows from differentiating ln T, not from adding an arbitrary energy correction to a potential-energy diagram.

Common misconception

“A small Arrhenius prefactor proves an unfavorable orientation factor.” It may suggest a restricted transition-state ensemble, but A depends on rate-law units, temperature range, entropy, standard states and possible mechanism changes. Another mistake is setting collision p equal to TST κ. The former modifies an encounter model; the latter concerns commitment after a barrier-surface crossing. Finally, Ea is not always a simple potential-energy barrier height; its definition comes from the temperature derivative of a rate constant.

Worked example

A unimolecular reaction has ΔH‡° = 50.0 kJ mol⁻¹, and over a narrow interval around 298 K its ΔS‡° and κ are approximately constant. Using Ea = ΔH‡° + RT gives Ea ≈ 50.0 + (8.314 × 298)/1000 = 52.5 kJ mol⁻¹. If an Arrhenius fit over that interval reported about 52.5 kJ mol⁻¹, it would be consistent with this ideal TST prediction. It would not establish the actual value of κ or prove the potential-energy surface correct, because different errors or pathways could produce a similar local slope.

Quick check

1. What are the units of Arrhenius A in a first-order rate law? Answer: s⁻¹, the same units as the first-order rate constant k.

Exam focus

Begin with the mathematical definitions. State k = Aexp(−Ea/RT), Ea = −R d ln k/d(1/T), and the specified Eyring form. For the ideal unimolecular case with nearly constant activation parameters and κ, show the derivative that gives Ea = ΔH‡° + RT. Avoid transferring that result to every bimolecular or complex reaction without deriving the appropriate form. Match prefactor units to reaction order.

Advanced insight

If κ varies with temperature, differentiation yields an extra term −R d ln κ/d(1/T) in Ea. Activation heat capacity similarly makes ΔH‡° and ΔS‡° vary; a temperature-dependent ΔH‡° must be handled consistently with thermodynamics. Collision theory's √T prefactor also contributes to an Arrhenius slope, so an energy threshold in that model is not automatically identical to fitted Ea. At high precision, even the definition of an observed activation energy should specify whether pressure, ionic strength or other conditions are held fixed during the temperature series.

Summary

Arrhenius Ea and A summarise measured temperature dependence; collision theory estimates encounter frequency, energetic accessibility and a steric factor; TST combines activation free energy with a crossing frequency and transmission correction. Under ideal unimolecular TST assumptions, local Ea = ΔH‡° + RT. Prefactors and correction factors from the three descriptions have different definitions and often different units or conventions. Use them to compare models carefully, not as interchangeable labels for one physical quantity.

Practice questions

1. Find the local ideal Ea at 300 K if ΔH‡° = 60.0 kJ mol⁻¹ for a unimolecular step. Answer: Ea ≈ 60.0 + (8.314 × 300)/1000 = 62.5 kJ mol⁻¹.

2. Why can an Arrhenius A have units L mol⁻¹ s⁻¹? Answer: In a second-order concentration-based rate law, k has those units and the exponential is dimensionless, so A has the same units.

3. Give one reason a fitted Ea may differ from a computed saddle-point energy difference. Answer: Ea is a temperature-slope parameter and includes entropic, prefactor, dynamical or mechanism effects; the saddle energy difference alone is not the complete thermal rate model.

4. What data would help test whether a low collision-model p truly reflects orientation restrictions? Answer: Orientation- or state-resolved scattering measurements and comparison with an accurate potential-energy surface could test that interpretation.

5. Is p in simple collision theory generally equal to κ in transition-state theory? Answer: No. They refer to different model corrections: encounter reactivity versus successful dividing-surface passage.