Integrated rate laws are essential tools in chemistry, linking reactant concentration to time in chemical reactions. They enable us to predict reaction progress and kinetics without constant monitoring, providing a mathematical relationship between concentration and time.

These laws come in different forms for zero-, first-, and second-order reactions, each with unique characteristics. Understanding half-life and how to determine reaction order are crucial skills in applying integrated rate laws to real-world chemical processes.

Integrated Rate Laws

Purpose of integrated rate laws

• Relate reactant concentration to time in a chemical reaction enables determination of concentration at any given time or time required to reach a specific concentration
• Derived by integrating the differential rate law provides a mathematical relationship between concentration and time
• Allows prediction of reaction progress and kinetics without continuous monitoring of reactant concentration (spectroscopy or titration)

Calculations with integrated rate laws

• Zero-order reactions:
  • Integrated rate law: $[A]_t = -kt + [A]_0$ concentration decreases linearly with time
  • Rate constant units: concentration/time (M/s) indicates the change in concentration per unit time
  • Graphical representation: [A] vs. t is linear with a slope of $-k$ straight line with negative slope
• First-order reactions:
  • Integrated rate law: $\ln[A]_t = -kt + \ln[A]_0$ natural logarithm of concentration decreases linearly with time
  • Rate constant units: 1/time (s$^{-1}$) indicates the fraction of reactant consumed per unit time
  • Graphical representation: ln[A] vs. t is linear with a slope of $-k$ straight line with negative slope
• Second-order reactions:
  • Integrated rate law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$ reciprocal of concentration increases linearly with time
  • Rate constant units: 1/(concentration × time) (M$^{-1}$s$^{-1}$) indicates the change in reciprocal concentration per unit time
  • Graphical representation: 1/[A] vs. t is linear with a slope of $k$ straight line with positive slope

Half-life in chemical reactions

• Time required for reactant concentration to decrease to half of its initial value represents the speed of the reaction
• Related to rate constant and reaction order allows calculation of half-life from rate constant and vice versa
  1. Zero-order: $t_{1/2} = \frac{[A]_0}{2k}$ half-life increases with initial concentration
  2. First-order: $t_{1/2} = \frac{\ln 2}{k}$ half-life is constant and independent of initial concentration
  3. Second-order: $t_{1/2} = \frac{1}{k[A]_0}$ half-life decreases with increasing initial concentration
• For first-order reactions, half-life is constant and independent of initial concentration simplifies calculations and comparisons
• For zero-order and second-order reactions, half-life depends on initial concentration requires recalculation for different starting concentrations

Determination of reaction order

• Plot concentration-time data using integrated rate law equations for different reaction orders:
  • Zero-order: [A] vs. t
  • First-order: ln[A] vs. t
  • Second-order: 1/[A] vs. t
• The plot that yields a straight line indicates the correct reaction order allows visual determination of reaction order
  • Slope of the line is related to rate constant (k) enables calculation of rate constant from graph
• Compare half-lives at different initial concentrations an alternative method to determine reaction order
  • If half-life is constant, the reaction is first-order
  • If half-life is inversely proportional to initial concentration, the reaction is second-order
  • If half-life is directly proportional to initial concentration, the reaction is zero-order
• The order of reaction describes how the rate of a reaction depends on the concentration of reactants

Reaction Kinetics and Rate Laws

• Differential rate law expresses the rate of reaction as a function of reactant concentrations
• Integrated rate law relates concentration to time and is derived from the differential rate law
• Rate constant (k) is a proportionality factor that relates reaction rate to reactant concentrations
• Reaction kinetics study how fast chemical reactions occur and the factors affecting reaction rates