⚗️Theoretical Chemistry Unit 11 – 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 change in concentration of reactants or products per unit time, typically expressed in units of molarity per second (M/s)
• Rate law is a mathematical expression that relates the reaction rate to the concentrations of reactants, often written as $r = k[A]^m[B]^n$ where $k$ is the rate constant and $m$ and $n$ are the reaction orders with respect to reactants A and B
• Elementary reactions are single-step processes with no intermediate steps, while complex reactions involve multiple elementary steps
• Molecularity refers to the number of reactant molecules that participate in an elementary reaction step
  • Unimolecular reactions involve a single reactant molecule (first-order reactions)
  • Bimolecular reactions involve two reactant molecules (second-order reactions)
• Reaction order is the power to which the concentration of a reactant is raised in the rate law expression
  • Zero-order reactions have rates independent of reactant concentrations
  • First-order reactions have rates proportional to the concentration of a single reactant
  • Second-order reactions have rates proportional to the product of two reactant concentrations or the square of a single reactant concentration
• Half-life ($t_{1/2}$) is the time required for the concentration of a reactant to decrease by half in a first-order reaction, calculated as $t_{1/2} = \frac{\ln 2}{k}$

Reaction Rate Laws and Order

• Rate laws are determined experimentally by measuring the reaction rate at different initial concentrations of reactants
• Method of initial rates involves conducting experiments with varying initial concentrations of one reactant while keeping others constant, then comparing the initial rates to determine the order with respect to each reactant
• Integrated rate laws are obtained by integrating the differential rate law expressions and allow the calculation of reactant concentrations at any given time
  • Zero-order integrated rate law: $[A]_t = [A]_0 - kt$
  • First-order integrated rate law: $\ln [A]_t = \ln [A]_0 - kt$
  • Second-order integrated rate law: $\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt$
• Pseudo-first-order reactions occur when one reactant is present in large excess, making its concentration effectively constant and simplifying the rate law to first-order kinetics
• Reaction order can be determined by plotting the integrated rate law equations and analyzing the linearity of the resulting graphs (concentration vs. time for zero-order, ln(concentration) vs. time for first-order, and 1/concentration vs. time for second-order)
• The rate constant $k$ is specific to a particular reaction at a given temperature and can be determined from the slope of the integrated rate law plots

Collision Theory and Activation Energy

• Collision theory explains the kinetics of chemical reactions based on the idea that reactant molecules must collide with sufficient energy and proper orientation to form products
• Activation energy ($E_a$) is the minimum energy required for reactants to overcome the energy barrier and form the activated complex (transition state) before proceeding to products
  • Higher activation energies result in slower reaction rates, as fewer collisions have enough energy to overcome the barrier
• Effective collisions are those with sufficient energy and proper orientation to lead to a successful reaction
• Collision frequency is the number of collisions per unit time and volume, which depends on the concentrations of reactants, temperature, and molecular size
• Steric factor (probability factor) accounts for the fraction of collisions with the proper orientation for a successful reaction
• Maxwell-Boltzmann distribution describes the distribution of molecular energies in a gas at a given temperature, with the fraction of molecules having energy greater than $E_a$ increasing with temperature
• Increasing temperature raises the average kinetic energy of molecules, leading to more effective collisions and faster reaction rates

Transition State Theory

• Transition state theory (TST) is a more advanced model that describes the formation and decay of the activated complex (transition state) during a chemical reaction
• Activated complex is a high-energy, unstable intermediate formed when reactants collide with sufficient energy and proper orientation
  • It represents the highest energy point along the reaction coordinate (potential energy surface)
• Transition state is the configuration of the activated complex at the saddle point of the potential energy surface, where the energy is highest and the complex is most unstable
• Reaction coordinate is a parameter that describes the progress of a reaction from reactants to products, typically representing the change in potential energy as the reaction proceeds
• TST assumes that the activated complex is in equilibrium with the reactants and that its concentration determines the reaction rate
• The rate constant $k$ is related to the standard Gibbs free energy of activation ($\Delta G^{\ddagger}$) by the Eyring equation: $k = \frac{k_BT}{h}e^{-\Delta G^{\ddagger}/RT}$, where $k_B$ is the Boltzmann constant, $h$ is Planck's constant, and $R$ is the gas constant
• Entropy of activation ($\Delta S^{\ddagger}$) reflects the change in disorder as the reactants form the activated complex, with a more positive value indicating a looser transition state and a faster reaction rate

Temperature Dependence and Arrhenius Equation

• Temperature has a significant effect on reaction rates, with higher temperatures generally leading to faster reactions
• Arrhenius equation relates the rate constant $k$ to the activation energy $E_a$ and temperature $T$: $k = Ae^{-E_a/RT}$, where $A$ is the pre-exponential factor (frequency factor) and $R$ is the gas constant
• The pre-exponential factor $A$ represents the frequency of collisions with proper orientation and is related to the steric factor and collision frequency
• Activation energy $E_a$ can be determined from the slope of an Arrhenius plot (ln $k$ vs. $1/T$), while the pre-exponential factor $A$ is obtained from the y-intercept
• Q10 (temperature coefficient) is a measure of how much the reaction rate increases for every 10°C rise in temperature, typically ranging from 2 to 4 for most chemical reactions
• Enzymes are biological catalysts that lower the activation energy of biochemical reactions, allowing them to proceed at physiological temperatures
  • Enzyme activity is sensitive to temperature changes, with optimal activity at a specific temperature range and denaturation at high temperatures

Reaction Mechanisms and Elementary Steps

• Reaction mechanisms describe the sequence of elementary steps that occur during a complex reaction, including the formation and decay of any intermediate species
• Elementary steps are single-step processes that represent the actual molecular events taking place during a reaction, with each step having its own rate law and rate constant
• Intermediates are species formed during the reaction but not present in the overall balanced equation, often highly reactive and short-lived
• Rate-determining step (RDS) is the slowest elementary step in a reaction mechanism, which determines the overall rate of the reaction
  • The rate law for the overall reaction is determined by the rate law of the RDS
• Steady-state approximation assumes that the concentration of intermediates remains constant during the majority of the reaction, as their rates of formation and consumption are equal
• Pre-equilibrium approximation assumes that the initial elementary steps are fast and reversible, establishing an equilibrium between reactants and intermediates before the RDS occurs
• Kinetic isotope effect (KIE) is the change in reaction rate when an atom in a reactant is replaced by its isotope, often used to identify the RDS and provide insight into the reaction mechanism
  • Primary KIE occurs when the isotopic substitution is directly involved in the bond-breaking or bond-forming process of the RDS
  • Secondary KIE occurs when the isotopic substitution affects the reaction rate indirectly, such as through changes in vibrational frequencies or steric effects

Catalysis and Inhibition

• Catalysts are substances that increase the rate of a chemical reaction without being consumed in the process, providing an alternative reaction pathway with a lower activation energy
• Homogeneous catalysts are in the same phase as the reactants (e.g., acids, bases, and metal complexes in solution)
• Heterogeneous catalysts are in a different phase from the reactants, typically solid catalysts with reactants in the gas or liquid phase (e.g., metal surfaces, zeolites)
• Enzymes are highly specific biological catalysts that lower the activation energy of biochemical reactions through substrate binding and stabilization of the transition state
• Michaelis-Menten kinetics describes the rate of enzyme-catalyzed reactions, relating the reaction rate to the substrate concentration: $v = \frac{v_{max}[S]}{K_M + [S]}$, where $v_{max}$ is the maximum rate and $K_M$ is the Michaelis constant
• Inhibitors are substances that decrease the rate of a chemical reaction by interfering with the reaction mechanism or blocking the active sites of a catalyst
  • Competitive inhibitors bind to the same active site as the substrate, preventing substrate binding and reducing the reaction rate
  • Non-competitive inhibitors bind to a different site on the enzyme, altering its conformation and reducing its catalytic activity
• Catalyst poisoning occurs when impurities or reaction products irreversibly bind to the active sites of a catalyst, reducing its effectiveness over time
• Turnover number (TON) and turnover frequency (TOF) are measures of catalyst efficiency, representing the number of substrate molecules converted per catalyst molecule and the TON per unit time, respectively

Experimental Methods and Data Analysis

• Spectroscopic techniques, such as UV-Vis, IR, and NMR, can be used to monitor the concentrations of reactants, products, and intermediates over time, providing kinetic data for reaction rate analysis
• Stopped-flow techniques involve rapidly mixing reactants and measuring the concentrations at very short time intervals, useful for studying fast reactions with half-lives in the millisecond to second range
• Flash photolysis uses a short, intense pulse of light to generate reactive intermediates, which can then be monitored using spectroscopic methods to study their kinetics
• Isotopic labeling involves replacing atoms in reactants with their isotopes (e.g., deuterium, 13C, 15N) to track the fate of specific atoms during a reaction and elucidate the reaction mechanism
• Kinetic isotope effect (KIE) experiments compare the reaction rates of labeled and unlabeled reactants to identify the rate-determining step and provide insight into the reaction mechanism
• Lineweaver-Burk plot (double-reciprocal plot) is a graphical method for analyzing enzyme kinetics data, plotting 1/v vs. 1/[S] to determine $v_{max}$ and $K_M$ from the y-intercept and slope, respectively
• Eyring plot (ln(k/T) vs. 1/T) is used to determine the activation enthalpy ($\Delta H^{\ddagger}$) and activation entropy ($\Delta S^{\ddagger}$) from the slope and y-intercept, respectively, based on the Eyring equation
• Hammett plot (log(k/k0) vs. σ) relates the reaction rates of substituted aromatic compounds to the electronic effects of the substituents, with the slope (ρ) indicating the sensitivity of the reaction to these effects
• Computational methods, such as density functional theory (DFT) and molecular dynamics (MD) simulations, can provide insights into reaction mechanisms, transition state structures, and energy barriers, complementing experimental kinetic data