⚛Molecular Physics Unit 14 – Chemical Kinetics and Reaction Dynamics

Key Concepts and Definitions

• Chemical kinetics studies the rates of chemical reactions and the factors that influence them
• Reaction rate measures the speed at which reactants are consumed or products are formed per unit time
• Rate law expresses the relationship between the reaction rate and the concentrations of reactants
  • Determined experimentally by measuring the change in concentration over time
• Order of reaction refers to the exponent of the concentration term in the rate law
  • Zero-order, first-order, second-order, or mixed-order reactions
• Rate constant ($k$) is a proportionality constant that relates the reaction rate to the concentrations of reactants
• Activation energy ($E_a$) represents the minimum energy required for reactants to overcome and form products
• Catalyst lowers the activation energy of a reaction without being consumed in the process
• Molecularity refers to the number of reactant molecules that participate in an elementary step

Reaction Rate Laws

• Rate law expresses the relationship between the reaction rate and the concentrations of reactants
• General form of a rate law: $\text{Rate} = k[A]^m[B]^n$, where $k$ is the rate constant, $[A]$ and $[B]$ are reactant concentrations, and $m$ and $n$ are the orders of reaction with respect to each reactant
• Integrated rate laws describe the concentration of reactants or products as a function of time
  • Zero-order: $[A]_t = [A]_0 - kt$
  • First-order: $\ln[A]_t = \ln[A]_0 - kt$
  • Second-order: $\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt$
• Half-life ($t_{1/2}$) is the time required for the reactant concentration to decrease by half
  • For a first-order reaction: $t_{1/2} = \frac{\ln 2}{k}$
• Pseudo-first-order reactions occur when one reactant is in large excess, simplifying the rate law
• Determination of rate law and order of reaction involves measuring the initial rates at different reactant concentrations

Factors Affecting Reaction Rates

• Temperature increases the average kinetic energy of molecules, leading to more frequent and energetic collisions
  • Arrhenius equation: $k = Ae^{-E_a/RT}$, where $A$ is the pre-exponential factor, $E_a$ is the activation energy, $R$ is the gas constant, and $T$ is the absolute temperature
• Concentration of reactants affects the frequency of collisions between molecules
  • Higher concentrations result in faster reaction rates
• Surface area of solid reactants influences the rate by determining the number of available reaction sites
  • Smaller particle sizes or porous materials have higher surface areas and faster rates
• Presence of a catalyst lowers the activation energy, allowing more molecules to overcome the energy barrier
  • Catalysts can be homogeneous (same phase as reactants) or heterogeneous (different phase)
• Pressure affects the reaction rate in gaseous systems by changing the concentration of reactants
• Light can initiate or accelerate photochemical reactions by providing the necessary energy for electronic transitions

Collision Theory and Transition State Theory

• Collision theory explains the reaction rate based on the frequency and energy of molecular collisions
  • Effective collisions require sufficient energy (greater than the activation energy) and proper orientation
• Activation energy ($E_a$) is the minimum energy required for reactants to form the activated complex (transition state)
• Transition state theory (TST) describes the reaction rate in terms of the properties of the activated complex
  • Activated complex is a high-energy, unstable intermediate formed during the reaction
• TST assumes that the activated complex is in equilibrium with the reactants
  • Rate constant is proportional to the concentration of the activated complex
• Gibbs free energy of activation ($\Delta G^‡$) determines the concentration of the activated complex
  • $k = \frac{k_BT}{h}e^{-\Delta G^‡/RT}$, where $k_B$ is the Boltzmann constant, $h$ is Planck's constant, and $T$ is the absolute temperature
• Entropy of activation ($\Delta S^‡$) reflects the change in disorder during the formation of the activated complex
• Enthalpy of activation ($\Delta H^‡$) represents the energy difference between the reactants and the activated complex

Reaction Mechanisms and Elementary Steps

• Reaction mechanism is a sequence of elementary steps that describes the detailed molecular pathway from reactants to products
• Elementary steps are single-step processes that represent the actual molecular events occurring during a reaction
  • Unimolecular (one reactant molecule), bimolecular (two reactant molecules), or termolecular (three reactant molecules) steps
• Rate-determining step (RDS) is the slowest step in a multi-step reaction mechanism
  • Overall reaction rate is determined by the rate of the RDS
• Intermediate is a species formed in one step of the mechanism and consumed in a subsequent step
  • Steady-state approximation assumes that the concentration of intermediates remains constant over time
• Pre-equilibrium approximation assumes that the initial steps of a mechanism are in equilibrium
• Kinetic isotope effect (KIE) compares the reaction rates of isotopically labeled and unlabeled reactants
  • Primary KIE indicates that the labeled atom is involved in the RDS
• Catalytic mechanisms involve the participation of a catalyst in the reaction pathway
  • Enzymes are biological catalysts that lower the activation energy by binding to substrates

Experimental Methods in Kinetics

• Spectroscopic techniques monitor the concentration of reactants or products over time
  • UV-visible spectroscopy measures the absorption of light by species in solution
  • Infrared (IR) spectroscopy detects the vibrational transitions of molecules
  • Nuclear magnetic resonance (NMR) spectroscopy probes the magnetic properties of atomic nuclei
• Stopped-flow methods rapidly mix reactants and measure the concentration changes on a short timescale
• Flash photolysis initiates a reaction with a short pulse of light and monitors the subsequent concentration changes
• Calorimetry measures the heat evolved or absorbed during a reaction
  • Isothermal calorimetry maintains a constant temperature and relates the heat flow to the reaction rate
• Pressure measurements can be used to study gas-phase reactions
  • Manometry measures the change in pressure as the reaction proceeds
• Chromatographic techniques (GC, HPLC) separate and quantify the components of a reaction mixture
• Kinetic isotope labeling uses isotopically labeled reactants to elucidate the reaction mechanism

Mathematical Models and Data Analysis

• Differential rate laws express the reaction rate as a function of the instantaneous concentrations of reactants
  • $\frac{d[A]}{dt} = -k[A]^m[B]^n$, where $[A]$ and $[B]$ are reactant concentrations, $k$ is the rate constant, and $m$ and $n$ are the orders of reaction
• Integrated rate laws describe the concentration of reactants or products as a function of time
  • Obtained by integrating the differential rate law
• Linearization of integrated rate laws allows the determination of rate constants and orders of reaction from experimental data
  • Plot of $\ln[A]$ vs. time for a first-order reaction yields a straight line with slope $-k$
• Arrhenius plot ($\ln k$ vs. $1/T$) determines the activation energy and pre-exponential factor from temperature-dependent rate constants
• Eyring plot ($\ln(k/T)$ vs. $1/T$) relates the rate constant to the thermodynamic parameters of the activated complex
• Nonlinear regression analysis fits experimental data to kinetic models and estimates the rate constants and orders of reaction
• Numerical methods solve complex kinetic equations and simulate reaction profiles
  • Runge-Kutta methods, finite difference methods, and stochastic simulations

Applications in Molecular Physics

• Atmospheric chemistry studies the kinetics of reactions in the Earth's atmosphere
  • Ozone depletion, photochemical smog, and greenhouse gas emissions
• Combustion kinetics investigates the rates and mechanisms of fuel oxidation reactions
  • Design of efficient engines and pollution control strategies
• Catalysis plays a crucial role in industrial processes and renewable energy technologies
  • Heterogeneous catalysts (solid-gas or solid-liquid interfaces) and homogeneous catalysts (organometallic complexes)
• Biochemical kinetics explores the rates of enzyme-catalyzed reactions in living organisms
  • Michaelis-Menten kinetics, allosteric regulation, and metabolic pathways
• Polymer kinetics studies the rates of polymerization reactions and the properties of polymer materials
  • Chain-growth and step-growth polymerization, kinetic control of molecular weight and dispersity
• Photochemistry investigates the kinetics of light-induced reactions
  • Photosynthesis, photocatalysis, and photopolymerization
• Kinetics of charge transfer and energy transfer processes in molecular systems
  • Electron transfer in photovoltaic devices and energy transfer in light-harvesting complexes
• Computational modeling of reaction kinetics predicts the rates and mechanisms of complex chemical systems
  • Quantum chemical calculations, molecular dynamics simulations, and kinetic Monte Carlo methods