12.4 Integrated Rate Laws

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

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• 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 $-[k](https://www.fiveableKeyTerm:K)$ 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