Transition state theory: the Eyring equation

Arrhenius fits the temperature dependence of a rate constant and reports two numbers: A (the pre-exponential factor) and E_a (the activation energy). Both are empirical. Transition state theory (TST) — Henry Eyring’s 1935 upgrade — reframes those two numbers into molecular properties of a specific chemical species: the activated complex sitting at the top of the reaction barrier. Eyring’s equation gives ΔH‡ and ΔS‡, activation enthalpy and activation entropy, and separates “why is this reaction slow?” into two independent questions with different remedies.

Assume familiarity with Arrhenius and integrated rate laws from rate laws from mechanism.

The Eyring equation

Eyring’s derivation (McQuarrie §29) treats the activated complex — the arrangement of atoms at the top of the reaction-coordinate barrier — as being in quasi-equilibrium with the reactants. Applying statistical mechanics to that equilibrium gives the rate constant:

$k = \kappa \cdot \frac{k_B T}{h} \exp\left(\frac{\Delta S^{\ddagger}}{R}\right) \exp\left(-\frac{\Delta H^{\ddagger}}{RT}\right)$

with:

• k_B = Boltzmann constant = 1.380 649 × 10⁻²³ J K⁻¹ (CODATA 2018, exact).
• h = Planck constant = 6.626 070 15 × 10⁻³⁴ J s (CODATA 2018, exact).
• T = temperature in K.
• R = gas constant = 8.314 462 618 J K⁻¹ mol⁻¹.
• κ = transmission coefficient (fraction of activated complexes that actually make it over the barrier to products; usually taken as 1 for simple reactions).
• ΔH‡ = activation enthalpy (J/mol) — the “height” of the barrier in energy.
• ΔS‡ = activation entropy (J K⁻¹ mol⁻¹) — the “narrowness” of the barrier in configurational space.

The prefactor k_B T/h ≈ 6.21 × 10¹² s⁻¹ at 298 K — sometimes called the “universal frequency factor.” Every rate constant, in Eyring’s framework, is that number times two exponentials.

Compare Arrhenius:

$k = A \exp(-E_a / RT)$

Correspondence: E_a ≈ ΔH‡ + RT (usually within 2-5 kJ/mol at room T), and A ≈ (k_B T / h) exp(ΔS‡/R). Arrhenius A absorbs the entropy factor; TST separates them.

What ΔH‡ tells you

ΔH‡ is the enthalpic cost of reaching the transition state — bond stretches, angle bends, partial bond breakings that the reactants must undergo. Larger ΔH‡ means the reaction is intrinsically hard, requires more energy input per event, and slows dramatically as T decreases.

Diagnostic values:

• Diffusion-controlled reactions in water: ΔH‡ ≈ 15-20 kJ/mol (activation energy is essentially just the energy to break the solvent cage).
• Fast enzymatic reactions: ΔH‡ = 25-40 kJ/mol.
• Typical bench organic reactions at 25 °C: ΔH‡ = 50-90 kJ/mol.
• Slow reactions (days at 25 °C): ΔH‡ = 90-120 kJ/mol.
• Reactions requiring reflux (80 °C): ΔH‡ = 90-110 kJ/mol.

Reducing ΔH‡ is the traditional goal of catalysis. Enzymes achieve rate enhancements of 10⁶-10¹⁰ almost entirely by lowering ΔH‡; heterogeneous catalysts (Pd/C, Pt/C) do the same for hydrogenation and coupling reactions.

What ΔS‡ tells you

ΔS‡ is the entropic cost of assembling the transition state from reactants. If forming the TS requires two independent molecules to line up in a specific relative orientation and freeze several rotational degrees of freedom, ΔS‡ is strongly negative (order-of-magnitude −100 J K⁻¹ mol⁻¹). If the TS is looser than the reactant — for example, a unimolecular dissociation that partially breaks a bond — ΔS‡ is positive.

Diagnostic values:

• Bimolecular associations (SN2, Diels-Alder, radical recombination): ΔS‡ ≈ −80 to −150 J K⁻¹ mol⁻¹. Very negative.
• Unimolecular dissociations (SN1 ionization, retro-Diels-Alder): ΔS‡ ≈ +10 to +30 J K⁻¹ mol⁻¹. Slightly positive.
• Isomerizations, conformational changes: ΔS‡ ≈ 0 to −20 J K⁻¹ mol⁻¹. Near zero.
• Enzyme reactions with well-positioned substrates: ΔS‡ often less negative than the uncatalyzed case, because the enzyme has pre-organized the reactants — one of enzymes’ entropic tricks.

The sign of ΔS‡ is a mechanism diagnostic. A strongly negative ΔS‡ almost always means an associative mechanism; a positive ΔS‡ often means a dissociative mechanism. Steinfeld Ch. 10 has case studies of TST used this way in organic mechanistic assignment.

Extracting ΔH‡ and ΔS‡ from rate data

Take the log of the Eyring equation:

$\ln\frac{k}{T} = -\frac{\Delta H^{\ddagger}}{R T} + \ln\frac{k_B}{h} + \frac{\Delta S^{\ddagger}}{R}$

A plot of ln(k/T) vs 1/T is linear. Slope = −ΔH‡/R, intercept = ln(k_B/h) + ΔS‡/R. This is the Eyring plot — the same idea as an Arrhenius plot but with the y-axis rescaled to divide by T, and the intercept now returns ΔS‡ rather than ln A.

From two data points at temperatures T₁ and T₂ with rate constants k₁ and k₂:

$\Delta H^{\ddagger} = R \cdot \frac{\ln(k_1 T_2 / k_2 T_1)}{1/T_2 - 1/T_1}$

Then ΔS‡ falls out of Eyring’s equation at either temperature. Modern kinetics software does this automatically; the half-life calculator does not do Eyring plots, but any Arrhenius-based analysis converts to Eyring with the same data.

Worked example — SN2 reaction from two temperatures

Why TST helps where Arrhenius does not

Two mechanisms that give the same Arrhenius fit can be distinguished by TST. Consider two hypothetical reactions:

• Reaction A: ΔH‡ = 60 kJ/mol, ΔS‡ = −20 J K⁻¹ mol⁻¹ (mild association).
• Reaction B: ΔH‡ = 40 kJ/mol, ΔS‡ = −150 J K⁻¹ mol⁻¹ (strong association).

At 298 K both give similar k (≈ 10⁻³ s⁻¹), so an Arrhenius plot at 25 °C alone cannot tell them apart. But the temperature dependence of the temperature dependence differs: Reaction A speeds up more with T (steeper ln k vs 1/T slope). If you have data over a wide T range, the Eyring plot cleanly separates them. Steinfeld Ch. 10 has real examples where mechanistic assignment turned on exactly this discrimination.

TST also handles isotope effects (KIE) more cleanly: the ratio k_H/k_D can be computed from the difference in zero-point vibrational energies of the C-H versus C-D bond, and the ratio’s magnitude tells you whether the H atom is being transferred at the TS. Primary KIE k_H/k_D = 6-8 → H being broken in the RDS. Secondary KIE k_H/k_D = 0.9-1.1 → H not directly involved but the bond hybridization changes.

Practice problems

Where to go next

• The next layer of theory is the Marcus equation for electron-transfer reactions, which further decomposes ΔH‡ into reorganization energy and driving-force terms. Steinfeld Ch. 10 covers Marcus theory.
• For heterogeneous catalysis and enzyme active sites, TST extends via the Bronsted-Evans-Polanyi (BEP) relation: linear correlation between ΔH‡ and reaction ΔH across a series of similar reactions. It is the theoretical foundation of most scaling relations in catalysis research.
• The half-life calculator uses first-order kinetics; once you have k from Eyring, half-life follows immediately.
• The sig figs cheatsheet reminds that activation-parameter uncertainty compounds quickly through the exponential; report ΔH‡ to ±2 kJ/mol and ΔS‡ to ±5 J K⁻¹ mol⁻¹ unless your data are exceptional.

TST’s real value is diagnostic. Arrhenius gives you two numbers that fit; Eyring gives you two numbers that mean something about the transition state. When two mechanisms fit the same Arrhenius plot, ΔS‡ often names which one is running.