12.2 Complex reactions and rate-determining steps

Complex reactions involve multiple steps, unlike elementary reactions that occur in a single collision. The rate-determining step, the slowest in the mechanism, controls the overall reaction rate. Understanding this concept is crucial for deriving rate laws and predicting reaction behavior.

The rate law for complex reactions can't be determined directly from the balanced equation. Instead, it depends on the rate-determining step and involves applying the steady-state approximation to reactive intermediates. This approach helps predict how changes in conditions affect reaction rates.

Elementary vs Complex Reactions

Reaction Mechanisms

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• Elementary reactions involve a single step and occur in a single molecular collision
  • The reaction mechanism and the balanced chemical equation are the same for an elementary reaction
• Complex reactions involve two or more elementary steps and occur through a series of molecular collisions
  • The reaction mechanism for a complex reaction consists of multiple elementary steps
  • The overall balanced equation is the sum of these steps

Rate Laws

• The rate law for an elementary reaction can be determined directly from the balanced chemical equation
  • The order of the reaction with respect to each reactant equals its stoichiometric coefficient
  • The overall order of an elementary reaction is the sum of the orders with respect to each reactant
• The rate law for a complex reaction cannot be determined directly from the balanced overall equation
  • It depends on the relative rates of the elementary steps in the reaction mechanism
  • The slowest step often determines the rate (rate-determining step)

Rate-Determining Step Significance

Characteristics of the Rate-Determining Step

• The rate-determining step (RDS) is the slowest step in a multi-step reaction mechanism
  • It limits the overall rate of the reaction, acting as a bottleneck for the entire process
• The RDS often has the highest activation energy
  • This corresponds to the largest energy barrier that must be overcome for the reaction to proceed

Impact on Overall Rate Law

• The overall rate law for a complex reaction is determined by the rate law of the RDS
  • The concentrations of the reactants involved in the RDS appear in the overall rate law
  • The concentrations of reactants not involved in the RDS do not appear in the overall rate law
• If the RDS changes due to a change in reaction conditions (temperature or reactant concentrations), the overall rate law for the complex reaction may also change

Deriving Rate Laws for Complex Reactions

Steady-State Approximation

• To derive the rate law for a complex reaction, consider the rate laws for each elementary step in the reaction mechanism
• Identify the RDS by comparing the relative rates of the elementary steps
  • The step with the slowest rate will be the RDS
• Apply the steady-state approximation to any reactive intermediates formed in the mechanism
  • This assumes that the concentrations of these intermediates remain constant over time
  • Their rates of formation and consumption are equal

Expressing Concentrations and Substitution

• Express the concentrations of the reactive intermediates in terms of the concentrations of the reactants and the rate constants of the elementary steps
• Substitute these expressions into the rate law for the RDS to obtain the overall rate law for the complex reaction
  • The concentrations of the reactants involved in the RDS appear in the overall rate law
  • The concentrations of reactants not involved in the RDS do not appear in the overall rate law

Rate Effects of Concentration and Conditions

Reactant Concentrations

• Changes in reactant concentrations can affect the rates of individual elementary steps in a complex reaction mechanism, potentially leading to a change in the RDS
• Increasing the concentration of a reactant involved in the RDS increases the overall reaction rate
  • The RDS is the slowest step and limits the overall rate
• Increasing the concentration of a reactant not involved in the RDS has no effect on the overall reaction rate

Temperature and Catalysts

• Changing the temperature of a reaction can affect the rates of individual elementary steps differently, depending on their activation energies
  • If the RDS changes as a result of a temperature change, the overall rate law and the temperature dependence of the reaction rate may also change
• Catalysts lower the activation energy of one or more elementary steps in a complex reaction mechanism
  • If the catalyzed step becomes faster than the previous RDS, the RDS may change, leading to a change in the overall rate law

Activation Energy and Rate Constant Relationship

Arrhenius Equation

• The overall rate constant for a complex reaction depends on the rate constant of the RDS
  • The RDS is the slowest step and limits the overall rate of the reaction
• The rate constant for the RDS, and thus the overall rate constant for the complex reaction, can be expressed using the Arrhenius equation:
  • $k = A * e^(-Ea/RT)$
  • $k$ is the rate constant, $A$ is the pre-exponential factor, $Ea$ is the activation energy, $R$ is the gas constant, and $T$ is the absolute temperature

Temperature Dependence and Pre-Exponential Factor

• The activation energy ($Ea$) of the RDS determines the temperature dependence of the overall rate constant and the reaction rate
  • A higher $Ea$ results in a stronger temperature dependence
  • The reaction rate increases more rapidly with increasing temperature
• The pre-exponential factor ($A$) of the RDS relates to the frequency of collisions between reactant molecules and their orientation
  • It represents the fraction of collisions with sufficient energy and proper orientation to lead to a successful reaction
• Changes in the RDS due to changes in reaction conditions can lead to changes in the activation energy and pre-exponential factor
  • This affects the overall rate constant and temperature dependence of the complex reaction