2.2 Rate laws and reaction orders

Rate laws are the mathematical backbone of chemical kinetics. They show how reaction rates change with reactant concentrations, helping us predict and control reactions. Understanding rate laws is crucial for unraveling reaction mechanisms and optimizing industrial processes.

Reaction orders tell us how sensitive a reaction is to concentration changes. Zero-order reactions don't care about concentration, while higher orders are increasingly affected. Determining reaction orders and rate constants from experimental data is a key skill in chemical kinetics.

Rate Laws and Reaction Orders

Concept of rate law

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• Mathematical expression relating reaction rate to reactant concentrations
• Determined experimentally by measuring reaction rates at different concentrations
• Provides insight into reaction mechanism by revealing which reactants influence rate and to what extent
• Significance in chemical kinetics
  • Predicts how reaction rate changes with reactant concentrations (doubling concentration of a first-order reactant doubles rate)
  • Allows determination of rate constant and reaction order from experimental data
  • Helps in understanding underlying reaction mechanism by suggesting which elementary steps may be involved

Order of chemical reactions

• Exponent to which reactant concentration is raised in rate law
• Can be zero (rate independent of concentration), first (rate directly proportional to concentration), second (rate proportional to square of concentration), or fractional order
• Determining reaction order
  • Method of initial rates
    • Vary concentration of one reactant while keeping others constant
    • Compare ratio of rate changes to ratio of concentration changes (doubling concentration of a first-order reactant doubles rate)
  • Graphical method
    • Plot log(rate) vs. log(concentration) for each reactant
    • Slope of line gives order with respect to that reactant (slope of 1 indicates first order)
• Overall order is sum of orders with respect to each reactant
  • Example: $\text{Rate} = k[\text{A}]^2[\text{B}]$ has overall order of 3 (second order in A, first order in B)

Rate constant determination

• Proportionality constant $k$ in rate law
• Determined experimentally by measuring reaction rate at known reactant concentrations
• Depends on temperature (typically increases with temperature) but not on reactant concentrations
• Units of rate constant depend on overall order of reaction
  • For rate law: $\text{Rate} = k[\text{A}]^m[\text{B}]^n$
    • Units of $k$ = $\frac{\text{concentration}^{1-(m+n)}}{\text{time}}$
    • Example: For a second-order reaction ($m+n=2$), units of $k$ are $\frac{1}{\text{concentration} \cdot \text{time}}$ (M$^{-1}$s$^{-1}$)

Concentration effects on reaction rates

• Increasing concentration of a reactant increases reaction rate
• Extent of rate increase depends on order with respect to that reactant
  1. Zero-order reactions
     • Rate is independent of reactant concentration
     • $\text{Rate} = k$
     • Example: Enzyme-catalyzed reactions at high substrate concentration
  2. First-order reactions
     • Rate is directly proportional to reactant concentration
     • $\text{Rate} = k[\text{A}]$
     • Doubling concentration doubles rate
     • Example: Radioactive decay, many enzyme-catalyzed reactions
  3. Second-order reactions
     • Rate is proportional to square of reactant concentration
     • $\text{Rate} = k[\text{A}]^2$
     • Doubling concentration quadruples rate
     • Example: Dimerization reactions, some bimolecular elementary steps