13.3 Transition state theory

Transition state theory (TST) explains how chemical reactions proceed through a high-energy intermediate configuration, connecting the concept of activation energy to measurable rate constants. Rather than just asking "do molecules collide hard enough?" (as collision theory does), TST asks "what does the molecule look like at the top of the energy barrier?" This distinction makes it one of the most powerful frameworks in chemical kinetics for predicting and controlling reaction rates.

Transition state concept

Definition and characteristics

The transition state is the highest-energy molecular configuration along the reaction coordinate. At this point, old bonds are partially broken and new bonds are partially formed. The molecule is neither reactant nor product; it exists at the saddle point on the potential energy surface.

• Denoted by the double-dagger symbol (‡), as in $A^\ddagger$
• It is not a true intermediate you can isolate. It has a lifetime on the order of a single molecular vibration (~$10^{-13}$ s)
• The geometry at the transition state is specific: bond lengths, angles, and dihedral angles all take on values between those of reactants and products

Role in chemical reactions

TST assumes that reactants and the transition state complex are in quasi-equilibrium. This means you can treat the transition state concentration using equilibrium thermodynamics, even though the species itself is fleeting.

• The rate of product formation is proportional to the concentration of the transition state complex
• The formation of the transition state is the rate-determining step in many reactions
• Because the transition state is in quasi-equilibrium with reactants, you can write an equilibrium constant $K^\ddagger$ connecting them, which is the key move that makes TST quantitative

Activation energy and transition state

Relationship between activation energy and transition state

The activation energy ($E_a$) is the energy difference between the reactants and the transition state along the reaction coordinate. It represents the minimum energy input needed for reactants to reach the transition state configuration.

• A higher $E_a$ means a taller energy barrier, so fewer molecules can reach the transition state at a given temperature
• A lower $E_a$ means more molecules have enough energy to cross the barrier, resulting in a faster reaction
• On a potential energy diagram, $E_a$ is measured from the average energy of the reactants up to the peak (the transition state)

Factors affecting activation energy

Catalysts lower $E_a$ by providing an alternative reaction pathway. They stabilize the transition state relative to the reactants, reducing the energy gap without changing the overall thermodynamics of the reaction ($\Delta G_{rxn}$ stays the same).

Temperature does not change $E_a$ itself, but it changes how many molecules have enough kinetic energy to overcome the barrier. At higher temperatures, the Boltzmann distribution shifts so that a larger fraction of molecules exceed $E_a$.

Arrhenius equation

The Arrhenius equation connects $E_a$ to the experimentally measured rate constant:

$k = A \, e^{-E_a / RT}$

• $A$ is the pre-exponential factor (units match those of $k$). It accounts for the frequency of collisions and the fraction with correct orientation.
• $R$ is the gas constant ($8.314 \, \text{J mol}^{-1} \text{K}^{-1}$)
• $T$ is the absolute temperature in Kelvin
• The exponential term $e^{-E_a/RT}$ gives the fraction of molecules with sufficient energy to reach the transition state

Taking the natural log of both sides gives the linearized form:

$\ln k = \ln A - \frac{E_a}{RT}$

A plot of $\ln k$ vs. $1/T$ yields a straight line with slope $-E_a/R$, which is how $E_a$ is determined experimentally.

Predicting reaction rates

The Eyring equation

TST's central quantitative result is the Eyring equation, which expresses the rate constant in terms of thermodynamic activation parameters:

$k = \frac{k_B T}{h} \, K^\ddagger$

where $k_B$ is Boltzmann's constant ($1.381 \times 10^{-23} \, \text{J K}^{-1}$), $h$ is Planck's constant ($6.626 \times 10^{-34} \, \text{J s}$), and $K^\ddagger$ is the quasi-equilibrium constant for forming the transition state from reactants.

The factor $k_B T / h$ has units of $\text{s}^{-1}$ and represents the universal frequency at which the transition state complex crosses the barrier. At 298 K, this equals approximately $6.2 \times 10^{12} \, \text{s}^{-1}$.

Connecting to Gibbs free energy of activation

Since $K^\ddagger$ is an equilibrium constant, it relates to the Gibbs free energy of activation:

$K^\ddagger = e^{-\Delta G^\ddagger / RT}$

Substituting into the Eyring equation:

$k = \frac{k_B T}{h} \, e^{-\Delta G^\ddagger / RT}$

Breaking down $\Delta G^\ddagger$

The Gibbs free energy of activation splits into enthalpic and entropic contributions:

$\Delta G^\ddagger = \Delta H^\ddagger - T \Delta S^\ddagger$

• $\Delta H^\ddagger$ (enthalpy of activation): reflects the energy cost of bond stretching/breaking to reach the transition state. Closely related to $E_a$ (specifically, $E_a = \Delta H^\ddagger + RT$ for a unimolecular reaction in the gas phase).
• $\Delta S^\ddagger$ (entropy of activation): reflects how ordered or disordered the transition state is compared to the reactants.

The sign of $\Delta S^\ddagger$ tells you something physical:

• $\Delta S^\ddagger < 0$: the transition state is more ordered than the reactants (common for bimolecular reactions where two molecules must come together)
• $\Delta S^\ddagger > 0$: the transition state is looser or more disordered than the reactants (common for dissociation reactions)

Transition state stability

Factors influencing transition state stability

Anything that lowers the energy of the transition state relative to the reactants will reduce $\Delta G^\ddagger$ and speed up the reaction. Several factors matter:

• Electronic effects: Electron-donating or electron-withdrawing substituents can stabilize developing charges in the transition state. For example, a carbonyl group adjacent to a reaction center can stabilize a transition state through resonance delocalization of partial charges.
• Steric effects: Bulky groups can destabilize a transition state if they create unfavorable steric strain at the critical geometry.
• Reaction conditions: Temperature, pressure, and solvent all influence the relative stability of the transition state.

Catalysts and transition state stability

Catalysts work by stabilizing the transition state, not by destabilizing the reactants. This is a subtle but important distinction.

• Heterogeneous catalysts (e.g., metal surfaces) bind reactants in orientations that lower the energy of the transition state
• Enzymes are remarkably effective biological catalysts. They achieve rate enhancements of $10^6$ to $10^{17}$ by binding the transition state far more tightly than the substrate. Lysozyme, for instance, distorts its substrate toward the transition state geometry for glycosidic bond hydrolysis, dramatically lowering $\Delta G^\ddagger$.

Solvent effects on transition state stability

The solvent interacts differently with reactants and the transition state, and this difference affects the rate.

• If the transition state is more polar than the reactants, a polar solvent will stabilize it preferentially, lowering $\Delta G^\ddagger$ and speeding up the reaction. Ester hydrolysis, for example, proceeds faster in water than in nonpolar solvents because the transition state carries significant partial charges.
• If the transition state is less polar than the reactants, a polar solvent can actually slow the reaction by stabilizing the reactants more than the transition state.
• Hydrogen bonding, dipole-dipole interactions, and solvent reorganization all contribute to these effects.

Transition state theory vs. other theories

Comparison with collision theory

Collision theory is useful for simple gas-phase reactions but struggles with reactions in solution or those involving complex molecules. TST handles these cases more naturally.

Eyring equation vs. Arrhenius equation

Both relate the rate constant to temperature, but they come from different theoretical foundations:

• Arrhenius: $k = A \, e^{-E_a/RT}$ — empirical in origin, with $A$ and $E_a$ as fitting parameters
• Eyring: $k = \frac{k_B T}{h} \, e^{-\Delta G^\ddagger/RT}$ — derived from statistical mechanics, with $\Delta H^\ddagger$ and $\Delta S^\ddagger$ as physically meaningful parameters

The Eyring equation has a $T$ in the pre-exponential factor, which means it predicts a slight temperature dependence beyond the exponential term. For most practical purposes over moderate temperature ranges, both equations fit experimental data well.

Hammond postulate and Marcus theory

• The Hammond postulate states that the transition state resembles whichever species (reactant or product) it is closer to in energy. For an exothermic reaction, the transition state is early (resembles reactants). For an endothermic reaction, it is late (resembles products). This is useful for predicting transition state geometry without detailed calculations.
• Marcus theory applies specifically to electron transfer reactions. It introduces the concept of reorganization energy ($\lambda$), which accounts for the structural rearrangement needed for the solvent and reactants to reach the transition state. The Marcus equation predicts that the activation energy depends on both $\Delta G_{rxn}$ and $\lambda$:

$\Delta G^\ddagger = \frac{(\lambda + \Delta G^\circ)^2}{4\lambda}$

This predicts the counterintuitive "inverted region" where making a reaction more exergonic actually slows it down, a prediction that was experimentally confirmed and earned Rudolph Marcus the 1992 Nobel Prize in Chemistry.