1.2 Integrated Rate Laws

Integrated rate laws are crucial tools in chemical kinetics. They help predict how reactant concentrations change over time, allowing us to calculate concentrations at any point during a reaction.

These laws are derived from differential rate laws and come in three main types: first-order, second-order, and zero-order. Each type has unique equations and applications in real-world chemical processes.

Integrated Rate Laws for Reactions

Deriving Integrated Rate Laws

• Integrated rate laws are derived by integrating the differential rate law
  • Differential rate law relates the rate of a reaction to the concentrations of the reactants
• First-order reaction integrated rate law: $ln[A]_t = -kt + ln[A]_0$
  • $[A]_t$ concentration of reactant A at time t
  • $[A]_0$ initial concentration of A
  • $k$ rate constant
• Second-order reaction integrated rate law: $1/[A]_t = kt + 1/[A]_0$
  • $[A]_t$ concentration of reactant A at time t
  • $[A]_0$ initial concentration of A
  • $k$ rate constant
• Zero-order reaction integrated rate law: $[A]_t = -kt + [A]_0$
  • $[A]_t$ concentration of reactant A at time t
  • $[A]_0$ initial concentration of A
  • $k$ rate constant

Applying Integrated Rate Laws

• Integrated rate laws can be used to calculate the concentration of a reactant or product at any given time
  • Requires knowledge of the initial concentration and rate constant
• First-order reaction concentration at time t: $[A]_t = [A]_0e^{-kt}$
  • $[A]_0$ initial concentration of A
  • $k$ rate constant
  • $t$ time
  • Example: Radioactive decay of carbon-14
• Second-order reaction concentration at time t: $[A]_t = [A]_0/(1 + [A]_0kt)$
  • $[A]_0$ initial concentration of A
  • $k$ rate constant
  • $t$ time
  • Example: Dimerization of cyclopentadiene
• Zero-order reaction concentration at time t: $[A]_t = [A]_0 - kt$
  • $[A]_0$ initial concentration of A
  • $k$ rate constant
  • $t$ time
  • Example: Catalytic decomposition of hydrogen peroxide

Calculating Half-Life

Half-Life Equations

• Half-life ($t_{1/2}$) time required for the concentration of a reactant to decrease to half of its initial value
• First-order reaction half-life: $t_{1/2} = ln(2)/k$
  • Independent of initial concentration
  • $k$ rate constant
• Second-order reaction half-life: $t_{1/2} = 1/([A]_0k)$
  • Depends on initial concentration
  • $[A]_0$ initial concentration of reactant A
  • $k$ rate constant
• Zero-order reaction half-life: $t_{1/2} = [A]_0/(2k)$
  • Depends on initial concentration
  • $[A]_0$ initial concentration of reactant A
  • $k$ rate constant

Half-Life Examples

• First-order reaction example: Decomposition of N2O5
  • Half-life remains constant regardless of initial concentration
• Second-order reaction example: Hydrolysis of sucrose
  • Half-life decreases as initial concentration increases
• Zero-order reaction example: Enzyme-catalyzed reactions
  • Half-life increases as initial concentration increases

Reaction Order Analysis with Integrated Rate Laws

Graphical Analysis

• Reaction order can be determined by analyzing experimental concentration-time data
  • Compare data to integrated rate laws for different reaction orders
• First-order reaction: plot of $ln[A]$ vs. time yields a straight line
  • Slope equals $-k$, where $k$ is the rate constant
• Second-order reaction: plot of $1/[A]$ vs. time yields a straight line
  • Slope equals $k$, where $k$ is the rate constant
• Zero-order reaction: plot of $[A]$ vs. time yields a straight line
  • Slope equals $-k$, where $k$ is the rate constant

Confirming Reaction Order

• Reaction order can be confirmed by comparing calculated half-life values with expected half-life expressions
  • First-order: half-life is independent of initial concentration
  • Second-order: half-life is inversely proportional to initial concentration
  • Zero-order: half-life is directly proportional to initial concentration
• Example: Decomposition of nitrogen pentoxide (N2O5)
  • Plotting $ln[N2O5]$ vs. time yields a straight line, indicating first-order reaction
  • Half-life remains constant at different initial concentrations, confirming first-order