Transition State Theory & Temperature Dependence

Transition State Theory (TST), also known as activated‑complex theory or absolute rate theory, was developed independently by Henry Eyring, Meredith Gwynne Evans, and Michael Polanyi in 1935. It provides a molecular‑level explanation for reaction rates by considering the formation of a high‑energy activated complex or transition state along the reaction pathway. Unlike the Arrhenius equation (which is purely empirical), TST gives a theoretical framework to calculate absolute rate constants from thermodynamic parameters and potential energy surfaces.

1. The Activated Complex & Reaction Coordinate

According to TST, reactants do not convert directly into products in a single step. Instead, they pass through a transient, unstable species called the transition state (denoted by the double dagger ‡). This species exists at the saddle point of the potential energy surface — the point of highest energy along the lowest energy path from reactants to products. Once formed, the activated complex decomposes to give products. The theory postulates a quasi‑equilibrium between reactants and the activated complex:

2. The Eyring Equation (Absolute Rate Theory)

From TST, the rate constant k can be expressed in terms of the Gibbs free energy of activation (ΔG^‡):

where:

• κ = transmission coefficient (usually ≈1)
• k_B = Boltzmann constant (1.381 × 10⁻²³ J·K⁻¹)
• h = Planck’s constant (6.626 × 10⁻³⁴ J·s)
• T = absolute temperature (K)
• R = gas constant (8.314 J·mol⁻¹·K⁻¹)

Since ΔG^‡ = ΔH^‡ – TΔS^‡, the Eyring equation can be rewritten as:

3. The Arrhenius Equation & Temperature Dependence

The empirical Arrhenius equation (1889) describes the temperature dependence of the rate constant remarkably well for many reactions:

where A is the pre‑exponential factor (frequency of collisions) and E_a is the activation energy. Taking natural logarithms:

A plot of ln k vs 1/T (Arrhenius plot) yields a straight line with slope = −E_a/R. From two temperatures:

4. Maxwell–Boltzmann Distribution and Activation Energy

Only molecules with kinetic energy ≥ E_a can overcome the activation barrier. The Maxwell–Boltzmann distribution shows the fraction of molecules with sufficient energy. As temperature increases, the distribution shifts to the right, increasing this fraction dramatically, which explains the exponential rate increase with temperature. Higher temperature → broader distribution → larger fraction of molecules exceeding activation energy → faster reaction rate.

5. Comparison: TST vs. Arrhenius

6. Catalysis and Lowered Activation Energy

A catalyst provides an alternative reaction pathway with a lower activation energy (E_cat). This increases the fraction of molecules with sufficient energy, thereby accelerating both forward and reverse reactions without altering the equilibrium constant.