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Transition state theory

T-042Home CU-202Threads kinetics
Statement

Rate from the activated complex at the barrier top.

Why it matters

collision-theory gives a first, mechanical picture of a reaction rate as a collision frequency multiplied by an energetic and orientational requirement, but it treats molecules as crude hard spheres and can only patch up its systematic failure to predict the pre-exponential factor for complex reactions with an ad hoc "steric factor." Transition state theory (TST) replaces that picture with a proper thermodynamic/statistical treatment of the activated complex sitting at the top of the reaction's energy barrier, giving arrhenius-equation's empirical parameters, \(E_a\) and \(A\), a genuine theoretical foundation.

Because TST expresses the rate constant through thermodynamic-style activation parameters (\(\Delta H^{\ddagger}\), \(\Delta S^{\ddagger}\)) rather than through a purely kinetic collision picture, it also connects naturally to steady-state-approximation: TST supplies the rate constant for each individual elementary step in a proposed mechanism, while the SSA is the separate tool for combining those elementary rate constants into an overall observed rate law.

Hypotheses
An activated complex exists at the potential-energy maximum along the reaction coordinate and is treated as being in quasi-equilibrium with the reactants.The activated complex is not a real, isolable chemical species; treating it as though it were in ordinary equilibrium with the reactants, even though it is not stable enough to isolate, is a formal device that lets ordinary equilibrium thermodynamics be applied to it, exactly as though it were a genuine chemical species. Any activated complex that forms proceeds on to products; it never reverts, or "recrosses," back to reactants.Without this assumption, only some fraction of activated complexes would actually become product, and the simple rate expression built from the quasi-equilibrium concentration alone would overstate the true rate. TST as developed here treats barrier crossing purely classically; it neglects quantum mechanical tunnelling through the barrier, a real, additional contribution to the rate that can be significant for light-particle (proton or hydrogen-atom) transfer reactions, especially at low temperature.
Proof
1
\text{A}+\text{B} \rightleftharpoons \text{X}^{\ddagger} \to \text{products}, \qquad K^{\ddagger}=\frac{[\text{X}^{\ddagger}]}{[\text{A}][\text{B}]}
Postulate a quasi-equilibrium between the reactants and the activated complex \(\text{X}^{\ddagger}\), governed by an equilibrium constant \(K^{\ddagger}\) formally analogous to any ordinary equilibrium constant (Hypotheses). A
2
\text{rate} = \nu[\text{X}^{\ddagger}], \qquad \nu = \frac{k_BT}{h}
The rate is the product of the activated complex's concentration and the universal frequency \(\nu\) at which it crosses over irreversibly to products (Hypotheses' no-recrossing assumption); \(\nu=k_BT/h\) follows from a statistical treatment of the reaction-coordinate motion itself. B
3
k = \frac{k_BT}{h}K^{\ddagger}
Substituting the quasi-equilibrium relation \([\text{X}^{\ddagger}]=K^{\ddagger}[\text{A}][\text{B}]\) (Step 1) into the rate expression (Step 2) gives the Eyring equation, expressing the rate constant directly in terms of the activation quasi-equilibrium constant. A
4
K^{\ddagger}=e^{-\Delta G^{\ddagger}/RT}, \qquad \Delta G^{\ddagger}=\Delta H^{\ddagger}-T\Delta S^{\ddagger}
Ordinary equilibrium thermodynamics relates \(K^{\ddagger}\) to a standard Gibbs energy of activation, which expands into separate enthalpic and entropic activation contributions, exactly as for any other equilibrium constant. A
5
k=\frac{k_BT}{h}e^{\Delta S^{\ddagger}/R}e^{-\Delta H^{\ddagger}/RT} \quad\text{vs Arrhenius: } k=Ae^{-E_a/RT}
Comparing directly against arrhenius-equation's empirical form identifies \(\Delta H^{\ddagger}\) closely with \(E_a\), while the pre-exponential factor \(A\) gains a genuine physical interpretation, \(A\approx(ek_BT/h)e^{\Delta S^{\ddagger}/R}\), an activation-entropy term entirely absent from collision theory's cruder, purely geometric picture. B
Result
k=\frac{k_BT}{h}\,e^{-\Delta G^{\ddagger}/RT}=\frac{k_BT}{h}\,e^{\Delta S^{\ddagger}/R}\,e^{-\Delta H^{\ddagger}/RT}

Reading. The rate constant is expressed through the thermodynamic activation parameters of a hypothesised transition state rather than through a purely kinetic collision picture, giving both a barrier height (\(\Delta H^{\ddagger}\)) and an order/probability term (\(\Delta S^{\ddagger}\)) direct physical meaning.

Scope. Requires the quasi-equilibrium and no-recrossing assumptions (Hypotheses); neglects quantum tunnelling, most significant for light-atom transfer at low temperature.

Corollaries & converses
  • A negative \(\Delta S^{\ddagger}\) corresponds to a highly ordered, restrictive transition state — typical of a bimolecular association reaction — exactly the situation collision-theory could only capture crudely with its empirical steric factor.
  • Comparing Steps 4 and 5 shows \(E_a\approx\Delta H^{\ddagger}+RT\) for a reaction in solution, linking arrhenius-equation's empirical activation energy directly to a genuine thermodynamic quantity.
  • steady-state-approximation and transition-state-theory address complementary aspects of the same kinetic problem: SSA extracts an observed rate law from a proposed multi-step mechanism, while TST supplies the theoretical rate constant for each individual elementary step feeding into that mechanism.
Fails without
  • Drop the no-recrossing assumption: if a significant fraction of activated complexes revert back to reactants rather than proceeding on to products, Step 2's rate expression, built from the full quasi-equilibrium concentration of \(\text{X}^{\ddagger}\) alone, systematically overestimates the true rate; a transmission coefficient correcting for recrossing is needed to fix this in a more refined treatment.
  • Ignore quantum tunnelling for a light-particle transfer reaction (Hypotheses' \(t3\)): the classical TST rate can substantially underestimate the true observed rate for hydrogen or proton transfer, especially at low temperature, since tunnelling opens an additional, purely quantum-mechanical pathway through, rather than over, the barrier that the classical treatment cannot capture at all.
Common errors
  • Treating the activated complex as a real, isolable chemical species, rather than a formal, quasi-equilibrium construct at the potential-energy maximum.
  • Confusing \(\Delta H^{\ddagger}\) with the reaction's overall thermodynamic \(\Delta H\) — the activation enthalpy concerns only the barrier from reactants to the transition state, independent of whether the overall reaction is exothermic or endothermic.
  • Assuming a negative \(\Delta S^{\ddagger}\) indicates an error in the analysis, rather than recognising it as the expected, typical sign for essentially any bimolecular association reaction.
  • Neglecting that classical TST, as developed here, entirely omits quantum tunnelling, which can noticeably raise the true rate above the TST prediction for light-particle transfer.
Discussion

Transition-state theory was developed largely by Henry Eyring, along with Meredith Gwynne Evans and Michael Polanyi, in the mid-1930s, which is why the rate expression derived here is commonly called the Eyring equation. It represented a substantial conceptual advance over collision theory, replacing a purely mechanical, hard-sphere collision picture with a genuinely thermodynamic and statistical treatment grounded in the quantum statistical mechanics then being developed.

Quantum mechanical tunnelling allows a reacting system some non-zero probability of passing through, rather than over, the activation barrier, an effect entirely outside classical TST's scope; it is most pronounced for the lightest nuclei (hydrogen and its isotopes) and becomes proportionally more important as temperature is lowered, since the classical over-the-barrier route becomes increasingly disfavoured while the tunnelling probability is comparatively insensitive to temperature.

Common misconception: that a large activation energy inherently reflects a large activation enthalpy alone, with entropy playing no role. Step 5's expansion of \(\Delta G^{\ddagger}\) shows the observed rate is jointly controlled by both an enthalpic barrier and an entropic term, and a slow reaction can have a comparatively modest \(\Delta H^{\ddagger}\) but a strongly unfavourable, very negative \(\Delta S^{\ddagger}\).

Worked examples
1
k=3.2\times10^{-4}\,\text{s}^{-1}\ \text{at}\ T=298\,\text{K (illustrative values)}: \quad K^{\ddagger}=\frac{k}{k_BT/h}
The universal frequency factor \(k_BT/h\) at \(298\,\text{K}\) is \(\dfrac{(1.381\times10^{-23})(298)}{6.626\times10^{-34}}\approx6.21\times10^{12}\,\text{s}^{-1}\); dividing the given rate constant by this factor gives \(K^{\ddagger}=3.2\times10^{-4}/6.21\times10^{12}\approx5.15\times10^{-17}\). A
2
\Delta G^{\ddagger}=-RT\ln K^{\ddagger} = -(8.314)(298)\ln(5.15\times10^{-17})
Substituting \(\ln(5.15\times10^{-17})\approx-37.5\) into Step 4's relation gives \(\Delta G^{\ddagger}\approx-(2478)(-37.5)\approx92{,}900\,\text{J/mol}\approx92.9\,\text{kJ/mol}\), the activation Gibbs energy consistent with the given rate constant and temperature. A
\Delta G^{\ddagger}\approx92.9\,\text{kJ/mol from } k=3.2\times10^{-4}\,\text{s}^{-1}\text{ at }298\,\text{K}

Reading. A single measured rate constant, together with temperature, converts directly through the Eyring equation into an activation Gibbs energy, without needing to separately measure \(\Delta H^{\ddagger}\) and \(\Delta S^{\ddagger}\) individually.

Scope. Separating \(\Delta G^{\ddagger}\) into \(\Delta H^{\ddagger}\) and \(\Delta S^{\ddagger}\) individually requires measuring \(k\) at several temperatures and analysing the resulting Eyring plot, analogous to how an Arrhenius plot separately extracts \(E_a\) and \(A\).

Problems
  1. A reaction has \(k=1.0\times10^{-2}\,\text{s}^{-1}\) at \(T=310\,\text{K}\). Compute \(K^{\ddagger}\) using \(k_BT/h\approx6.46\times10^{12}\,\text{s}^{-1}\) at this temperature.
    Solution\(K^{\ddagger}=k/(k_BT/h)=1.0\times10^{-2}/6.46\times10^{12}\approx1.55\times10^{-15}\).
  2. Two bimolecular reactions have similar \(\Delta H^{\ddagger}\) but very different rates at the same temperature. Explain, in terms of \(\Delta S^{\ddagger}\), how this is possible.
    SolutionBy Step 5's Eyring expression, the rate constant depends on both \(\Delta H^{\ddagger}\) (through the exponential barrier term) and \(\Delta S^{\ddagger}\) (through the pre-exponential entropy term). If the two reactions have similar \(\Delta H^{\ddagger}\) but very different rates, the difference must arise predominantly from \(\Delta S^{\ddagger}\): a reaction requiring a much more restrictive, precisely ordered transition state (more negative \(\Delta S^{\ddagger}\)) will be markedly slower than one with a more loosely constrained transition state, even at an identical activation enthalpy.
  3. Explain qualitatively why a sterically demanding bimolecular reaction (requiring precise mutual orientation of two large reactants) is expected to have a strongly negative \(\Delta S^{\ddagger}\), and relate this to how collision theory would instead describe the same effect.
    SolutionForming a highly specific, precisely oriented transition state from two independently tumbling reactant molecules represents a large loss of rotational and translational freedom, giving a strongly negative \(\Delta S^{\ddagger}\) (Corollaries). Collision theory, lacking any thermodynamic activation-entropy concept, can only capture this same steric demand indirectly, through an empirically fitted "steric factor" reducing the predicted rate below the simple collision-frequency estimate — TST's \(\Delta S^{\ddagger}\) gives this same physical effect a proper theoretical basis instead.