Transition State Theory

Activated Complex Theory & Reaction Dynamics

Core Concepts

Transition State (Activated Complex)

A high-energy, unstable arrangement of atoms at the peak of the reaction coordinate, denoted ‡

Reactants → [Transition State]‡ → Products

Activation Energy (Ea or ΔG‡)

The minimum energy required to reach the transition state

Arrhenius: Ea (energy barrier)

TST: ΔG‡ (free energy of activation)

Eyring Equation

The Fundamental Equation of TST

k = (κ kBT / h) × e-ΔG‡/RT

k = rate constant (s⁻¹ or M⁻¹s⁻¹)

κ = transmission coefficient (usually ≈ 1)

kB = Boltzmann constant = 1.381 × 10⁻²³ J/K

h = Planck's constant = 6.626 × 10⁻³⁴ J·s

T = temperature (K)

ΔG‡ = Gibbs free energy of activation (J/mol)

R = gas constant = 8.314 J/(mol·K)

Alternative Forms

Using entropy and enthalpy:

k = (κ kBT / h) × eΔS‡/R × e-ΔH‡/RT

Since ΔG‡ = ΔH‡ - TΔS‡

Logarithmic form:

ln(k/T) = ln(κkB/h) + ΔS‡/R - ΔH‡/RT

Useful for Eyring plots

Activation Parameters

ΔH‡ (Enthalpy of Activation)

ΔH‡ = Ea - RT

Energy difference between reactants and transition state

ΔS‡ (Entropy of Activation)

Change in disorder when forming transition state

ΔS‡ > 0: Transition state is more disordered (rare)

ΔS‡ < 0: Transition state is more ordered (common - molecules come together)

ΔS‡ ≈ 0: Little change in disorder

ΔG‡ (Gibbs Free Energy of Activation)

ΔG‡ = ΔH‡ - TΔS‡

Overall barrier to reaction; determines rate constant

Eyring Plot

Determining Activation Parameters Experimentally

ln(k/T) vs (1/T)

Slope = -ΔH‡/R

Intercept = ln(κkB/h) + ΔS‡/R

Method:

1. Measure rate constant k at various temperatures

2. Plot ln(k/T) vs 1/T

3. Extract ΔH‡ from slope

4. Extract ΔS‡ from intercept

5. Calculate ΔG‡ = ΔH‡ - TΔS‡

Reaction Coordinate Diagram

Energy Profile

    Energy
      ↑
      |         [‡]  ← Transition State
      |        /  \
      |       /    \
      |      /      \
      |  R  /        \  P
      |    /          \
      |___/_______ΔG___\______→
           Reaction Coordinate

Key Features:

• R: Reactants (initial state)

• [‡]: Transition state (highest energy)

• P: Products (final state)

• ΔG‡: Activation barrier (R to ‡)

• ΔG: Overall reaction free energy (R to P)

Worked Examples

Example 1: Calculate Rate Constant from ΔG‡

Problem: Calculate k at 298 K if ΔG‡ = 85 kJ/mol (assume κ = 1).

Solution:

k = (kBT/h) × e-ΔG‡/RT

kB/h = (1.381×10⁻²³)/(6.626×10⁻³⁴) = 2.084×10¹⁰ s⁻¹K⁻¹

ΔG‡ = 85,000 J/mol

k = (2.084×10¹⁰)(298) × e-85000/(8.314×298)

k = 6.21×10¹² × e-34.31

k = 6.21×10¹² × 1.08×10⁻¹⁵

k = 6.7×10⁻³ s⁻¹

Example 2: Calculate ΔG‡ Components

Problem: A reaction has ΔH‡ = 75 kJ/mol and ΔS‡ = -45 J/(mol·K). Calculate ΔG‡ at 300 K.

Solution:

ΔG‡ = ΔH‡ - TΔS‡

ΔG‡ = 75,000 J/mol - (300 K)(-45 J/(mol·K))

ΔG‡ = 75,000 + 13,500

ΔG‡ = 88.5 kJ/mol

Note: Negative ΔS‡ indicates a more ordered transition state (common for bimolecular reactions).

Example 3: Relationship to Arrhenius

Problem: If Ea = 90 kJ/mol, what is ΔH‡ at 298 K?

Solution:

ΔH‡ = Ea - RT

ΔH‡ = 90,000 - (8.314)(298)

ΔH‡ = 90,000 - 2,478

ΔH‡ = 87.5 kJ/mol

Note: ΔH‡ and Ea differ by ~2.5 kJ/mol at room temperature.

TST vs Arrhenius

AspectArrheniusTST (Eyring)
Equationk = Ae-Ea/RTk = (kBT/h)e-ΔG‡/RT
BarrierEa (energy)ΔG‡ (free energy)
Pre-exponentialA (empirical)kBT/h (theoretical)
EntropyNot explicitΔS‡ (explicit)
Plotln k vs 1/Tln(k/T) vs 1/T

Common Mistakes

⚠️

Confusing Ea and ΔG‡

Ea is energy; ΔG‡ is free energy (includes entropy!)

⚠️

Wrong Units

ΔH‡ in J/mol, ΔS‡ in J/(mol·K) - watch conversion!

⚠️

Forgetting Temperature Dependence

kBT/h term makes TST temperature-dependent even without exponential!

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