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
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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: l n [ A ] t = − k t + l n [ A ] 0 ln[A]_t = -kt + ln[A]_0 l n [ A ] t = − k t + l n [ A ] 0
[ A ] t [A]_t [ A ] t concentration of reactant A at time t
[ A ] 0 [A]_0 [ A ] 0 initial concentration of A
k k k rate constant
Second-order reaction integrated rate law: 1 / [ A ] t = k t + 1 / [ A ] 0 1/[A]_t = kt + 1/[A]_0 1/ [ A ] t = k t + 1/ [ A ] 0
[ A ] t [A]_t [ A ] t concentration of reactant A at time t
[ A ] 0 [A]_0 [ A ] 0 initial concentration of A
k k k rate constant
Zero-order reaction integrated rate law: [ A ] t = − k t + [ A ] 0 [A]_t = -kt + [A]_0 [ A ] t = − k t + [ A ] 0
[ A ] t [A]_t [ A ] t concentration of reactant A at time t
[ A ] 0 [A]_0 [ A ] 0 initial concentration of A
k k 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 ] 0 e − k t [A]_t = [A]_0e^{-kt} [ A ] t = [ A ] 0 e − k t
[ A ] 0 [A]_0 [ A ] 0 initial concentration of A
k k k rate constant
t t t time
Example: Radioactive decay of carbon-14
Second-order reaction concentration at time t: [ A ] t = [ A ] 0 / ( 1 + [ A ] 0 k t ) [A]_t = [A]_0/(1 + [A]_0kt) [ A ] t = [ A ] 0 / ( 1 + [ A ] 0 k t )
[ A ] 0 [A]_0 [ A ] 0 initial concentration of A
k k k rate constant
t t t time
Example: Dimerization of cyclopentadiene
Zero-order reaction concentration at time t: [ A ] t = [ A ] 0 − k t [A]_t = [A]_0 - kt [ A ] t = [ A ] 0 − k t
[ A ] 0 [A]_0 [ A ] 0 initial concentration of A
k k k rate constant
t t t time
Example: Catalytic decomposition of hydrogen peroxide
Calculating Half-Life
Half-Life Equations
Half-life (t 1 / 2 t_{1/2} 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 = l n ( 2 ) / k t_{1/2} = ln(2)/k t 1/2 = l n ( 2 ) / k
Independent of initial concentration
k k k rate constant
Second-order reaction half-life: t 1 / 2 = 1 / ( [ A ] 0 k ) t_{1/2} = 1/([A]_0k) t 1/2 = 1/ ([ A ] 0 k )
Depends on initial concentration
[ A ] 0 [A]_0 [ A ] 0 initial concentration of reactant A
k k k rate constant
Zero-order reaction half-life: t 1 / 2 = [ A ] 0 / ( 2 k ) t_{1/2} = [A]_0/(2k) t 1/2 = [ A ] 0 / ( 2 k )
Depends on initial concentration
[ A ] 0 [A]_0 [ A ] 0 initial concentration of reactant A
k k 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 l n [ A ] ln[A] l n [ A ] vs. time yields a straight line
Slope equals − k -k − k , where k k k is the rate constant
Second-order reaction: plot of 1 / [ A ] 1/[A] 1/ [ A ] vs. time yields a straight line
Slope equals k k k , where k k k is the rate constant
Zero-order reaction: plot of [ A ] [A] [ A ] vs. time yields a straight line
Slope equals − k -k − k , where k k 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 l n [ N 2 O 5 ] ln[N2O5] l n [ N 2 O 5 ] vs. time yields a straight line, indicating first-order reaction
Half-life remains constant at different initial concentrations, confirming first-order