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Transition State Theory and the Eyring Equation

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Potential Energy Surfaces and Reaction CoordinatesStatistical Thermodynamics: Properties from Partition Functions+8 more→→Bimolecular Reaction Dynamics: Collisions, Cross Sections, and ScatteringElectrochemical Kinetics: Butler-Volmer Theory+5 more
transition-state-theory activated-complex Eyring-equation activation-enthalpy activation-entropy transmission-coefficient

Core Idea

Transition state theory (TST) assumes that reactants are in quasi-equilibrium with the activated complex (transition state), and that the rate is proportional to the concentration of transition states multiplied by their rate of crossing the barrier. The Eyring equation k = (k_B T/h)·κ·exp(−ΔG‡/RT) provides the rate constant from the free energy of activation ΔG‡ = ΔH‡ − TΔS‡. Unlike collision theory, TST uses thermodynamic quantities for the transition state, making it straightforward to separate enthalpic (barrier height) and entropic (geometric constraint) contributions. The transmission coefficient κ accounts for recrossing trajectories and quantum tunneling (important for proton transfer reactions).

How It's Best Learned

Analyze Eyring plots (ln(k/T) vs 1/T) for several reactions to extract ΔH‡ and ΔS‡. Interpret negative ΔS‡ as an ordered transition state (bimolecular associations) and positive ΔS‡ as a looser one (unimolecular dissociations).

Common Misconceptions

Explainer

Transition state theory builds directly on potential energy surfaces, which describe how a system's energy changes as bonds break and form during a reaction. The reaction coordinate traces the path of lowest energy from reactants to products, and the highest point along that path — the saddle point — is the transition state (or activated complex). TST asks a precise question: given that the transition state exists, how fast does the reaction proceed?

The key assumption is quasi-equilibrium: the population of transition states is assumed to be in rapid equilibrium with the reactant population, governed by the Boltzmann factor exp(−ΔG‡/RT). The rate constant then equals the frequency at which transition states cross over the barrier multiplied by their equilibrium concentration. This gives the Eyring equation: k = (k_BT/h) · κ · exp(−ΔG‡/RT), where k_BT/h is a universal frequency (≈ 6 × 10¹² s⁻¹ at 298 K) and κ is the transmission coefficient. Because ΔG‡ = ΔH‡ − TΔS‡, the rate depends on both the height of the energy barrier (ΔH‡) and how constrained the geometry of the transition state is (ΔS‡). This is TST's major advantage over the Arrhenius equation, which lumps both effects into a single empirical E_a.

The entropy of activation is particularly informative. A large negative ΔS‡ means the transition state is highly ordered relative to the reactants — two molecules must find each other with precisely the right orientation, severely restricting the number of accessible configurations. This is common in bimolecular association reactions. A positive ΔS‡ indicates the transition state is looser than the reactants — a bond is substantially broken while little new constraint has been imposed — typical of unimolecular dissociations.

The transmission coefficient κ corrects for two effects that classical TST ignores. First, some trajectories that reach the barrier top recross back to reactants without proceeding forward, making κ < 1. Second, quantum tunneling allows light particles (most importantly protons) to pass *through* the barrier rather than over it. For proton transfer reactions, tunneling can make rates far higher than the classical Eyring equation predicts, explaining large kinetic isotope effects when hydrogen is replaced by deuterium.

Eyring plots — graphs of ln(k/T) versus 1/T — let you extract ΔH‡ from the slope (−ΔH‡/R) and ΔS‡ from the intercept. This separates the two thermodynamic contributions to reactivity, giving insight into whether a slow reaction suffers from a high barrier, an unfavorable geometric requirement, or both. The limitation to keep in mind is that TST is an approximation: real reaction dynamics on multidimensional potential energy surfaces do not always obey the no-recrossing assumption, and modern trajectory calculations often find κ significantly less than 1 for complex systems.

Practice Questions 3 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → Counting to 20 → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Number Bonds to 10 → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Dividing Integers → Unit Rates → Proportions → Percent Concept → Converting Between Fractions, Decimals, and Percents → Operations with Rational Numbers → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Sine, Cosine, and Tangent Ratios → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Indefinite Integrals → Basic Integration Rules → Riemann Sums → Definite Integral Definition → Double Integrals: Definition and Setup → Iterated Integrals and Fubini's Theorem → Double Integrals over Rectangular Regions → Double Integrals over General Regions → Applications of Double Integrals: Area, Mass, and Moments → Triple Integrals in Cartesian Coordinates → Triple Integrals in Cylindrical and Spherical Coordinates → Change of Variables and the Jacobian Determinant → Applications of Triple Integrals: Volume and Mass → Vector Fields and Their Representations → Line Integrals of Vector Fields → Work and Circulation → Line Integrals of Scalar and Vector Functions → Fundamental Theorem for Line Integrals → Conservative Vector Fields → Conservative Vector Fields and Potential Functions → Curl and Divergence of Vector Fields → Curl and Divergence → Divergence Theorem → Electric Flux and Divergence Theorem → Gauss's Law: Integral Form and Meaning → Solving Problems with Gauss's Law → Conductors in Electrostatic Equilibrium → Capacitance and Capacitors → Dielectrics → Dielectric Constant and Relative Permittivity → Electric Field Inside Dielectric Materials → Dielectric Materials and Polarization → Dielectric Susceptibility and Permittivity → Energy Density in Electric Fields → Electric Current and Current Density → Electrical Resistance and Resistivity → Ohm's Law and Circuit Elements → Electromotive Force (EMF) and Batteries → Kirchhoff's Circuit Laws: Voltage and Current → DC Circuit Network Analysis Methods → Transient Response in RC Circuits → RC Circuits → LC and RLC Circuits → AC Circuits: Fundamentals → Impedance and Reactance → AC Power and Resonance → Electromagnetic Waves → Postulates of Special Relativity → Time Dilation → Length Contraction → Lorentz Transformation → Relativistic Velocity Addition → Relativistic Momentum and Energy → Mass-Energy Equivalence and E=mc² → Photons as Particles with Energy and Momentum → Planck-Einstein Relation: Energy and Frequency → Photoelectric Effect → The Photon: Light as Quanta → Compton Scattering → Wave-Particle Duality → de Broglie Wavelength → The Schrödinger Equation → State Vectors and Wavefunctions → Quantum Superposition → Quantum Entanglement → Bell Theorem and Bell Inequalities → Postulates of Quantum Mechanics → Observables and Quantum Operators → Commutators and Commutation Relations → Quantum Angular Momentum → Quantum Mechanical Treatment of Hydrogen → Solving the Schrödinger Equation for Hydrogen Atom → Quantum Numbers → Electron Configuration → Periodic Trends → Covalent Bonding → Electronegativity and Bond Polarity → Ionic Bonding → Lewis Structures → VSEPR Theory and Molecular Geometry → Molecular Geometry and Electron Pair Geometry → Molecular Polarity and Dipole Moments → Intermolecular Forces → States of Matter and Phase Changes: Melting, Boiling, and Sublimation → Gas Laws and the Ideal Gas Equation → Gas Stoichiometry and Volume-Volume Calculations → Thermochemistry and Enthalpy → Heat Capacity and Calorimetry → Entropy and Molecular Disorder → Spontaneity and ΔG → Entropy and Gibbs Free Energy → Chemical Equilibrium → Statistical Mechanics: Ensembles and the Boltzmann Distribution → Partition Function: Definition and Properties → Molecular Partition Functions → Statistical Thermodynamics: Properties from Partition Functions → The van't Hoff Equation: Temperature Dependence of Equilibrium → Arrhenius Equation and Temperature Dependence → Arrhenius Equation and Temperature Dependence of Rate Constants → Transition State Theory and the Eyring Equation

Longest path: 181 steps · 1100 total prerequisite topics

Prerequisites (10)

Leads To (7)