Eyring Equation

Why the Eyring equation matters

• Universal prefactor. k_BT/h = (1.381×10^-23 × 298)/6.626×10^-34 = 6.21×10¹² s^-1 at room temperature. This is the upper bound for any unimolecular rate constant in solution at 298 K — no reaction can be faster than the thermal flux through the dividing surface, regardless of how favorable ΔG is.
• Separates enthalpy from entropy. Arrhenius lumps everything into A and E_a. Eyring gives ΔH‡ from the slope of ln(k/T) vs 1/T and ΔS‡ from the intercept. ΔS‡ alone often distinguishes mechanisms — Diels-Alder ΔS‡ ≈ −150 J K^-1 mol^-1 proves an ordered, concerted transition state.
• Backbone of physical organic chemistry. Hammett plots, kinetic isotope effects, linear free energy relationships, and Marcus theory are all built on the Eyring framework: rate constants are converted to ΔG‡, and substituent or solvent perturbations are read as ΔΔG‡.
• Henry Eyring, 1935. Published as "The Activated Complex in Chemical Reactions" in J. Chem. Phys. The American Chemical Society's Eyring lecture series and the Henry Eyring Award honor his synthesis of statistical mechanics and chemical kinetics. Eyring shared the priority with Evans and Polanyi who arrived at the same expression independently the same year.
• Underpins enzyme catalysis analysis. Enzymologists routinely report k_cat as ΔG‡ via Eyring; lysozyme has ΔG‡ ≈ 56 kJ/mol versus the uncatalyzed reaction's ≈ 119 kJ/mol — a 63 kJ/mol "catalytic proficiency" that translates to a 10¹¹-fold rate enhancement.
• Predicts kinetic isotope effects. Replacing H with D in a breaking bond shifts the zero-point energy and therefore ΔH‡ by ~5 kJ/mol; predicted k_H/k_D ≈ 7 at 298 K matches observation for primary KIEs. Tunneling pushes this past 10 and breaks the linear Eyring plot — a diagnostic for proton tunneling in many enzymes.
• Universal in computational chemistry. DFT and CCSD(T) calculations report ΔG‡ at the optimized transition state; Eyring converts that single number into a rate constant comparable to experiment. The match is the test of the electronic structure method.

Common misconceptions

• "E_a equals ΔH‡." Not exactly. From d ln k / dT one finds E_a = ΔH‡ + RT for unimolecular and condensed-phase reactions and E_a = ΔH‡ + 2RT for bimolecular gas-phase. At 298 K that is a 2.5 to 5 kJ/mol correction — small but enough to change reported activation energies if you ignore it.
• "Pre-exponential A equals k_BT/h." A also absorbs ΔS‡: A = (e k_BT/h) exp(ΔS‡/R) for unimolecular reactions. A bimolecular A of 10¹¹ M^-1 s^-1 implies a strongly negative ΔS‡, not that the prefactor is universal.
• "κ = 1 always." Convenient default, often wrong. Recrossing reduces κ for many gas-phase abstraction reactions to 0.5 or below, and tunneling makes κ exceed 1 by a factor of 2–10 for H-transfer below room temperature. Variational and quantum TST exist precisely to compute κ.
• "ΔG‡ is a thermodynamic state function." It is a free energy of the transition state relative to reactants, but the transition state is not in true equilibrium — it is the saddle point on the potential energy surface, occupied transiently. The "quasi-equilibrium" assumption underlies the derivation and breaks for very fast reactions.
• "Eyring plot is always linear." Curvature signals tunneling, multiple parallel paths, or a temperature-dependent mechanism. Methylmalonyl-CoA mutase has a curved Eyring plot above 25 °C — the breakpoint diagnoses the proton-tunneling regime change.
• "You can fit Eyring with three temperatures." Statistically yes; reliably no. ΔS‡ from a 30 K range has uncertainty of ±10 J K^-1 mol^-1 which is comparable to the mechanistic differences you are trying to detect. Span ≥40 K with five or more points.

Derivation

Start by treating the transition state X‡ as in quasi-equilibrium with reactants A + B: A + B ⇌ X‡ with equilibrium constant K‡ = [X‡]/[A][B]. Decompose the X‡ partition function by separating the reaction-coordinate vibration (with imaginary frequency along the saddle) from all other modes: q‡ = q‡_⊥ · q_rxn. The reaction-coordinate mode is treated classically; its partition function in the high-temperature limit is q_rxn = k_BT/(hν^‡), where ν^‡ is the imaginary frequency. The rate of barrier crossing is the population of activated complexes times the frequency at which they cross: rate = ν^‡ · [X‡]. Substituting K‡ from the partition function and identifying K‡_⊥ = (k‡_⊥/k_Ak_B) exp(−ΔE^‡/RT), the ν^‡ factor cancels and what survives is k = (k_BT/h) K‡_⊥.

Now write K‡_⊥ in thermodynamic form: K‡_⊥ = exp(−ΔG‡/RT) with ΔG‡ = ΔH‡ − T ΔS‡. The full result is k = κ (k_BT/h) exp(−ΔG‡/RT), where κ corrects for recrossing and tunneling. To extract ΔH‡ and ΔS‡ from data, take logs and rearrange: ln(k/T) = −ΔH‡/(RT) + ΔS‡/R + ln(k_B/h). Plotting ln(k/T) on the y-axis against 1/T on the x-axis gives a line with slope −ΔH‡/R (units of K) and y-intercept ΔS‡/R + ln(k_B/h) where ln(k_B/h) = ln(2.084×10¹⁰) = 23.76 (k in s^-1, T in K).

The connection to Arrhenius: differentiate ln k with respect to T. d ln k / dT = 1/T + ΔH‡/(RT²). Comparing to Arrhenius's d ln k / dT = E_a/(RT²) gives E_a = ΔH‡ + RT for unimolecular reactions in solution and E_a = ΔH‡ + 2RT for bimolecular gas-phase. The Arrhenius pre-exponential then satisfies A = (eⁿ k_BT/h) exp(ΔS‡/R), with n = 1 or 2.

Eyring vs Arrhenius — what each parameter buys you

Aspect	Arrhenius (1889)	Eyring (1935)
Form	k = A exp(−Ea/RT)	k = κ (kBT/h) exp(−ΔG‡/RT)
Prefactor	Empirical A, fitted	Universal kBT/h = 6.21×1012 s-1 at 298 K
Activation parameter	Ea (single number)	ΔH‡ + ΔS‡ (separable)
Mechanistic content	None — purely phenomenological	ΔS‡ diagnoses associative/dissociative TS
Plot	ln k vs 1/T, slope −Ea/R	ln(k/T) vs 1/T, slope −ΔH‡/R
Tunneling correction	Hidden in temperature-dependent A	Explicit κ > 1
Bimolecular rate ceiling	Implicit collision frequency	Diffusion limit ~1010 M-1 s-1
Derivation	Empirical fit to data	Statistical mechanics on a dividing surface

Reading ΔS‡ — typical values by reaction class

Reaction class	Typical ΔS‡ (J K-1 mol-1)	Interpretation
Diels-Alder cycloaddition	−130 to −180	Highly ordered bicyclic TS, both rotors frozen
SN2 displacement	−40 to −90	Backside attack tightens TS but linear, less ordered
Bimolecular radical recombination	−5 to +20	Loose, late TS — nearly free encounter
SN1 ionization	+30 to +80	Solvent-organized TS but bond cleaving — looser than reactant
Unimolecular cis/trans isomerization	−5 to +15	Internal rotation, modest entropy change
Cope rearrangement	−40 to −60	Cyclic chair-like TS — restricted internal rotors
Enzyme catalysis (kcat)	−20 to +40	Pre-organized active site — ΔS‡ already paid in binding

Applications

• Enzyme kinetics. Convert k_cat to ΔG‡ and compare with the uncatalyzed reaction. Lysozyme drops ΔG‡ from ~119 kJ/mol to ~56 kJ/mol, an enhancement of 10¹¹. Eyring plots of k_cat versus T over 5–45 °C separate ΔH‡ from ΔS‡ and detect tunneling via curvature.
• Physical organic mechanism assignment. ΔS‡ for the Diels-Alder of cyclopentadiene + maleic anhydride is −150 J K^-1 mol^-1, confirming a concerted ordered TS rather than a stepwise diradical (which would give ΔS‡ ≈ 0).
• Pharmaceutical stability prediction. Drug degradation half-lives at storage temperature (5 °C, 25 °C) are extrapolated from accelerated 40 °C data using Eyring with measured ΔH‡ and ΔS‡; ICH Q1A guidelines accept this if the Eyring plot is linear over the range tested.
• Computational chemistry validation. A DFT-computed ΔG‡ of 80 kJ/mol predicts k(298 K) ≈ 6×10^-2 s^-1; experimental measurement of half-life ≈ 12 s confirms the functional. Disagreements over 10 kJ/mol are diagnosed as missing dispersion or basis-set incompleteness.
• Atmospheric chemistry. OH + CH₄ → CH₃ + H₂O has ΔG‡ ≈ 24 kJ/mol; its Eyring-derived rate constant (6.4×10^-15 cm³/molec/s at 298 K) sets the methane atmospheric lifetime to ≈ 9 years — the central number in IPCC methane forcing.