Transition State Theory

Historical Context

Transition state theory (TST) was independently developed by Henry Eyring and by Michael Polanyi and Meredith Evans in 1935. It provided the first theoretical framework for calculating absolute rate constants from molecular properties, bridging the gap between quantum mechanics, statistical mechanics, and chemical kinetics.

Svante Arrhenius had proposed his empirical equation $k = A e^{-E_a/RT}$ in 1889, but TST gave physical meaning to both the pre-exponential factor A and the activation energy $E_a$. The concept of a potential energy surface (PES), introduced by Fritz London and developed by Eyring and Polanyi, provided the topographical picture of chemical reactions that remains central to theoretical chemistry.

Derivation 1: The Arrhenius Equation

Arrhenius observed that rate constants increase exponentially with temperature. He proposed:

Taking logarithms:

An Arrhenius plot of $\ln k$ vs $1/T$ yields a straight line with:

The activation energy can also be defined operationally as:

Derivation 2: The Eyring Equation

TST assumes that reactants are in quasi-equilibrium with an activated complex (transition state) at the saddle point of the potential energy surface:

The equilibrium constant for forming the transition state:

The rate of crossing the barrier is determined by the frequency of the transition state decomposing along the reaction coordinate. Statistical mechanics gives this frequency as $k_BT/h$ (the classical vibration frequency of the loose mode at the saddle point):

At 298 K, $k_BT/h = 6.21 \times 10^{12} \text{ s}^{-1}$. This enormous frequency is modulated by the exponential factor: even at the maximum barrier-crossing rate, a barrier of 80 kJ/mol reduces the rate by a factor of $10^{14}$.

Derivation 3: Connecting Arrhenius and Eyring

The Arrhenius activation energy $E_a$ and the Eyring activation enthalpy$\Delta H^\ddagger$ are related but not identical. From the operational definition:

Differentiating the Eyring equation:

The pre-exponential factor is related to the activation entropy:

A large A (or positive $\Delta S^\ddagger$) indicates a loose transition state with more rotational and vibrational freedom. A small A (negative $\Delta S^\ddagger$) indicates a tight, ordered transition state.

Derivation 4: The Eyring Plot

To extract $\Delta H^\ddagger$ and $\Delta S^\ddagger$ separately, we use the Eyring plot. Rearranging the Eyring equation:

A plot of $\ln(k/T)$ vs $1/T$ gives:

The Eyring plot separates the enthalpic and entropic contributions to the activation barrier. Since $\ln(k_B/h) = 23.76$, the intercept directly yields$\Delta S^\ddagger = R(\text{intercept} - 23.76)$.

Derivation 5: Potential Energy Surfaces

The potential energy surface (PES) is a multidimensional function$V(R_1, R_2, \ldots)$ that gives the electronic energy as a function of nuclear coordinates. For a simple A + BC → AB + C reaction with collinear geometry, the PES is a function of two distances: $r_{AB}$ and $r_{BC}$.

The Born-Oppenheimer approximation allows the PES to be computed by solving the electronic Schrodinger equation at fixed nuclear positions. Modern computational chemistry can calculate PES points with high accuracy using methods such as:

The activation energy in TST corresponds to the energy difference between the saddle point and the reactant minimum, corrected for zero-point vibrational energy:$\Delta E^\ddagger_0 = E_{\text{saddle}} - E_{\text{react}} + \Delta\text{ZPE}$.

Applications

Simulation: Eyring, Arrhenius, and PES

This simulation visualizes a one-dimensional potential energy surface, generates Arrhenius plots for different activation energies, computes Eyring equation rate constants, and demonstrates how the Eyring plot separates enthalpic and entropic contributions.

Collision Theory and Its Limitations

Before TST, collision theory provided the first microscopic picture of reaction rates. From the kinetic theory of gases, the collision rate between A and B is:

where $\sigma_{AB} = \pi(r_A + r_B)^2$ is the collision cross-section and$\mu$ is the reduced mass. The rate constant becomes:

Collision theory explains the Arrhenius equation but has significant limitations:

Beyond TST: RRKM Theory

Rice-Ramsperger-Kassel-Marcus (RRKM) theory extends TST to energy-dependent unimolecular reactions. It gives the microcanonical rate constant:

RRKM theory correctly predicts:

When thermally averaged (integrated over a Boltzmann distribution of energies), RRKM reduces to canonical TST in the high-pressure limit, confirming the consistency of the two theories.

Quantum Tunneling and Corrections to TST

Classical TST assumes that the system must have enough energy to surmount the barrier. Quantum mechanics allows particles to tunnel through the barrier even when their energy is below $\Delta E^\ddagger$. This is particularly important for light particles (H atoms, protons, electrons).

where $\kappa$ is the transmission coefficient. The simplest tunneling correction is the Wigner correction:

Evidence for tunneling includes:

Marcus Theory of Electron Transfer

Rudolph Marcus (Nobel Prize 1992) extended TST to electron transfer reactions, providing a remarkable relationship between the rate constant, the driving force ($\Delta G^\circ$), and the reorganization energy ($\lambda$):

This predicts the remarkable "Marcus inverted region": when$-\Delta G^\circ > \lambda$, the reaction actually slows down as it becomes more thermodynamically favorable. This counterintuitive prediction was experimentally confirmed and has profound implications for photosynthesis, photovoltaics, and molecular electronics.