Transition State Theory

The activated complex as a real species

Arrhenius (1889) gave us a phenomenological formula k = A·exp(−Ea/RT), where Ea was an empirical fitting parameter and A was a pre-factor of obscure physical meaning. Forty-six years later, Henry Eyring, Michael Polanyi and Meredith Evans took the next step: they treated the molecular configuration at the top of the barrier as a real, if short-lived, species — the activated complex — and asked how many of them exist at any moment.

The TST argument runs like this. Suppose reactants R and the activated complex R‡ are in quasi-equilibrium (the population of R‡ at any instant equals the equilibrium fraction set by their relative free energies). Then once R‡ exists, it falls apart into products at the universal vibrational frequency k_B·T/h, which is about 6.2 × 10¹² s⁻¹ at 298 K — the speed at which the imaginary-frequency mode at the saddle point pulls the complex apart.

Combining the two ideas:

k = (k_B · T / h) · K‡

where K‡ is the equilibrium constant between reactants and the activated complex. Substituting K‡ = exp(−ΔG‡/RT) gives the Eyring equation:

k = (k_B · T / h) · exp(−ΔG‡ / RT)

Splitting ΔG‡ into ΔH‡ and ΔS‡

Because ΔG‡ = ΔH‡ − T·ΔS‡, the Eyring equation expands into:

k = (k_B · T / h) · exp(ΔS‡ / R) · exp(−ΔH‡ / RT)

Take the logarithm of both sides, divided by T:

ln(k/T) = −ΔH‡/(R·T) + [ln(k_B/h) + ΔS‡/R]

An Eyring plot of ln(k/T) versus 1/T gives a straight line with slope −ΔH‡/R and intercept ln(k_B/h) + ΔS‡/R. From three or more rate constants at three temperatures you extract both ΔH‡ and ΔS‡ — separating the part of the activation barrier that is enthalpic (bond breaking, electrostatic strain) from the part that is entropic (loss of translational and rotational freedom).

Worked example: a Diels–Alder reaction

Cyclopentadiene + maleic anhydride at three temperatures gives:

T (K)    k (M⁻¹ s⁻¹)
278      9.4 × 10⁻³
298      4.5 × 10⁻²
318      1.7 × 10⁻¹

Build the Eyring plot points:

1/T (K⁻¹)         ln(k/T)
3.597 × 10⁻³      ln(9.4×10⁻³/278) = −10.39
3.356 × 10⁻³      ln(4.5×10⁻²/298) = −8.79
3.145 × 10⁻³      ln(1.7×10⁻¹/318) = −7.54

Linear regression: slope ≈ −6,300 K, intercept ≈ +12.3.

ΔH‡ = −R · slope = −8.314 × (−6,300) = 52,400 J/mol = 52 kJ/mol
ΔS‡ = R · (intercept − ln(k_B/h))
    = 8.314 × (12.3 − 23.76)
    = 8.314 × (−11.46)
    = −95 J/(mol·K)

Interpretation: the enthalpic barrier of 52 kJ/mol is moderate (typical organic concerted reaction). The strongly negative ΔS‡ of −95 J/(mol·K) is the signature of a concerted bimolecular reaction — two molecules combining into one constrained transition state, losing about 11 J/(mol·K) of entropy per atom that gets locked down. ΔG‡ at 298 K is ΔH‡ − T·ΔS‡ = 52 − 298·(−0.095) = 52 + 28 = 80 kJ/mol.

TST/Eyring vs Arrhenius

	Arrhenius (1889)	Eyring TST (1935)
Form	k = A·exp(−Ea/RT)	k = (k_B·T/h)·exp(−ΔG‡/RT)
Pre-factor origin	Empirical fit	Universal vibrational frequency
Barrier parameter	Ea (single number)	ΔH‡ and ΔS‡ separately
Provides mechanism insight	Limited	Yes — ΔS‡ flags concerted vs stepwise
Plot	ln(k) vs 1/T	ln(k/T) vs 1/T
Slope gives	−Ea/R	−ΔH‡/R
Connection	Ea = ΔH‡ + n·RT (n=1 or 2)	—
Best for	Engineering correlations	Mechanism, computational chemistry

The two equations are not competing — they are different parameterizations of the same physics. Arrhenius is older, simpler, and adequate when you only need to predict rates. Eyring is more informative when you want to know whether your transition state is loose or tight, dissociative or associative, concerted or stepwise.

What ΔS‡ values mean physically

Reaction type	ΔS‡ (J/(mol·K))	Why
Unimolecular dissociation (e.g. peroxide cleavage)	+30 to +60	One molecule → two; freedom increases
SN1 ionization	+10 to +50	C–X bond stretching; some loosening
SN2 substitution	−50 to −90	Trigonal bipyramidal TS; tight
Diels–Alder concerted cycloaddition	−120 to −180	Two molecules locked into ring
Bimolecular radical recombination	−40 to −80	Two radicals → one molecule, but loose TS
Enzymatic catalysis	−10 to −60	Substrate pre-organized; less entropy lost
Solvolysis with solvent in TS	−80 to −150	Solvent molecules lock around TS

The sign and magnitude of ΔS‡ alone often distinguishes mechanisms that have indistinguishable ΔH‡ values. A reaction whose ΔS‡ is +20 J/(mol·K) is unlikely to be SN2; one with ΔS‡ = −150 J/(mol·K) is unlikely to be a radical chain.

Where TST gets used in practice

• Computational reaction discovery. Modern density-functional codes (Gaussian, ORCA, Q-Chem) locate transition state geometries by saddle-point optimization, then compute ΔG‡ from vibrational frequencies. A complete pathway map is built by chaining TS calculations, and the lowest ΔG‡ path is the predicted mechanism. Industrial drug discovery uses this routinely.
• Enzyme kinetics. The 10⁸–10¹⁷ rate accelerations of enzymes are decomposed into ΔΔH‡ (bond-energy effects) and ΔΔS‡ (substrate pre-organization). Transition-state stabilization theory (Pauling, 1948; Wolfenden, 1972) attributes much of the rate gain to TS-shaped binding — the principle behind TS-analog inhibitors.
• Astrochemistry and combustion modeling. The CHEMKIN and Cantera reaction databases store ΔH‡ and ΔS‡ for thousands of elementary reactions. Combustion simulations propagate rate constants over 10²–10⁴ K via Eyring, where Arrhenius extrapolations would diverge.

When TST breaks down

• Tunneling. Light atoms (H, D, occasionally C) can pass through the barrier rather than over it. Below ~150 K, hydrogen tunneling dominates, and pure TST under-predicts the rate by orders of magnitude. Add a tunneling correction (Wigner, Eckart, or instanton).
• Recrossing. Sometimes the system reaches the saddle point but bounces back into the reactant well rather than committing to products. The transmission coefficient κ corrects for this; for floppy or strongly anharmonic systems κ can drop to 0.3–0.5.
• Non-equilibrium (femtosecond) chemistry. If the reaction is faster than vibrational relaxation, the reactants are not Boltzmann-distributed and the quasi-equilibrium assumption fails. Direct dynamics simulations replace TST in this regime.
• Multi-state reactions. Photochemistry, electron transfer near surface crossings, and radical reactions with spin-state changes need a multi-surface treatment (e.g. nonadiabatic TST).

Common pitfalls

• Confusing Ea and ΔH‡. They differ by RT for solution phase, 2RT for gas phase. The numbers are close but not identical; mixing them in derivations introduces ~5 kJ/mol systematic error.
• Using k_B·T/h as if it were the actual rate constant. 6.2 × 10¹² s⁻¹ is the upper bound — it's what you get for ΔG‡ = 0. The exp(−ΔG‡/RT) factor reduces real reactions by 10⁰ to 10⁻²⁵.
• Ignoring the κ in computational papers. Many DFT-based predictions silently assume κ = 1 even when tunneling matters. Hydrogen-atom transfer reactions in particular need explicit tunneling corrections at room temperature.
• Treating ΔS‡ as a thermodynamic property. ΔS‡ is a kinetic quantity — it characterizes a non-stationary point on the surface. It does not appear in equilibrium constants, only in rate expressions.

Variants and refinements

• Variational TST (VTST). Locates the minimum-flux dividing surface rather than the saddle point. Used when the saddle is broad or when there are multiple equivalent paths.
• Marcus theory. A specialized TST for electron transfer; ΔG‡ = (λ + ΔG)²/(4λ) where λ is reorganization energy. Predicts the inverted region.
• Kramers theory. Adds friction from the solvent — for very viscous solvents, the rate is suppressed below the TST value because the activated complex re-equilibrates with solvent.
• RRKM theory. The unimolecular extension that handles internal energy redistribution between vibrational modes; foundation of master-equation modeling for combustion and atmospheric chemistry.