⚗️Chemical Kinetics Unit 4 – Integrated Rate Laws

Key Concepts

• Integrated rate laws describe the concentration of reactants or products as a function of time
• Derived by integrating the differential rate law and solving for the concentration
• The order of a reaction determines the specific form of the integrated rate law
• Integrated rate laws allow for the calculation of reaction rates, half-lives, and initial concentrations
• The rate constant (k) is a key parameter in integrated rate laws and depends on temperature and the nature of the reactants
• Graphical analysis of concentration-time data can be used to determine the order of a reaction and the rate constant
• Integrated rate laws have practical applications in fields such as chemical engineering, pharmacology, and environmental science

Types of Integrated Rate Laws

• First-order integrated rate law: $\ln[A] = -kt + \ln[A]_0$
  • Concentration of reactant decreases exponentially with time
• Second-order integrated rate law: $\frac{1}{[A]} = kt + \frac{1}{[A]_0}$
  • Inverse of reactant concentration increases linearly with time
• Zero-order integrated rate law: $[A] = -kt + [A]_0$
  • Concentration of reactant decreases linearly with time
• Pseudo-first-order integrated rate law: Applies when one reactant is in large excess, simplifying the rate law to first-order kinetics
• Integrated rate laws for reversible reactions: Account for the presence of both forward and reverse reactions
• Integrated rate laws for complex reactions: Describe the kinetics of multi-step or coupled reactions

First-Order Reactions

• Concentration of reactant decreases exponentially with time according to $[A] = [A]_0e^{-kt}$
• Half-life is independent of initial concentration and is given by $t_{1/2} = \frac{\ln 2}{k}$
• Plotting $\ln[A]$ vs. time yields a straight line with slope $-k$ and y-intercept $\ln[A]_0$
• Examples of first-order reactions include:
  • Radioactive decay
  • Hydrolysis of esters
  • Decomposition of N2O5
• Characteristic doubling of half-life for each successive half-life period
• First-order reactions are common in nature and have important applications in chemical kinetics

Second-Order Reactions

• Inverse of reactant concentration increases linearly with time according to $\frac{1}{[A]} = kt + \frac{1}{[A]_0}$
• Half-life depends on initial concentration and is given by $t_{1/2} = \frac{1}{k[A]_0}$
• Plotting $\frac{1}{[A]}$ vs. time yields a straight line with slope $k$ and y-intercept $\frac{1}{[A]_0}$
• Examples of second-order reactions include:
  • Dimerization of cyclopentadiene
  • Decomposition of NO2
  • Saponification of esters
• Second-order reactions can involve one reactant (A + A) or two different reactants (A + B)
• The units of the rate constant for second-order reactions are $\text{M}^{-1}\text{s}^{-1}$

Zero-Order Reactions

• Concentration of reactant decreases linearly with time according to $[A] = -kt + [A]_0$
• Reaction rate is constant and independent of reactant concentration
• Half-life depends on initial concentration and is given by $t_{1/2} = \frac{[A]_0}{2k}$
• Plotting $[A]$ vs. time yields a straight line with slope $-k$ and y-intercept $[A]_0$
• Examples of zero-order reactions include:
  • Catalytic decomposition of ammonia on platinum
  • Enzymatic reactions at high substrate concentrations (Michaelis-Menten kinetics)
  • Photochemical reactions with high light intensity
• Zero-order reactions are relatively rare but have important applications in catalysis and biochemistry

Half-Life Calculations

• Half-life ($t_{1/2}$) is the time required for the reactant concentration to decrease by half
• For first-order reactions, $t_{1/2} = \frac{\ln 2}{k}$
  • Independent of initial concentration
• For second-order reactions, $t_{1/2} = \frac{1}{k[A]_0}$
  • Depends on initial concentration
• For zero-order reactions, $t_{1/2} = \frac{[A]_0}{2k}$
  • Directly proportional to initial concentration
• Half-life can be used to determine the order of a reaction by comparing the half-lives at different initial concentrations
• The concept of half-life is also applied in fields such as pharmacology (drug elimination) and nuclear physics (radioactive decay)

Graphical Analysis

• Plotting concentration-time data can help determine the order of a reaction and the rate constant
• For first-order reactions, plot $\ln[A]$ vs. time
  • Straight line with slope $-k$ and y-intercept $\ln[A]_0$
• For second-order reactions, plot $\frac{1}{[A]}$ vs. time
  • Straight line with slope $k$ and y-intercept $\frac{1}{[A]_0}$
• For zero-order reactions, plot $[A]$ vs. time
  • Straight line with slope $-k$ and y-intercept $[A]_0$
• Deviations from linearity may indicate a change in reaction order or the presence of multiple reaction steps
• Graphical analysis is a powerful tool for understanding reaction kinetics and determining rate laws from experimental data

Applications and Examples

• Atmospheric chemistry: Integrated rate laws are used to model the formation and depletion of ozone in the stratosphere
• Pharmacokinetics: First-order kinetics describe the elimination of many drugs from the body
  • Half-life is a key parameter in determining dosing intervals
• Enzyme kinetics: Michaelis-Menten kinetics, which follows zero-order behavior at high substrate concentrations, is used to study enzyme-catalyzed reactions
• Radiocarbon dating: The first-order decay of carbon-14 is used to determine the age of organic materials
• Chemical engineering: Integrated rate laws are used to design and optimize chemical reactors
  • Batch reactors
  • Plug flow reactors
  • Continuous stirred-tank reactors (CSTRs)
• Environmental science: Integrated rate laws describe the degradation of pollutants and the persistence of chemicals in the environment