Integrated Rate Law

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Core Concept

Integrated rate laws provide a mathematical relationship between the concentration of reactants and time.

  • Purpose: To determine the concentration of a reactant at any given time or to find the time required for a reaction to reach a specific concentration.

Practice Tips

  • Understand the physical meaning: What does each order tell you about how reactants affect the rate?

  • Memorize key formulas: Focus on integrated rate laws, half-life equations, and graphing criteria.

  • Practice with graphs: Use experimental data to plot and determine the reaction order.

  • Solve varied problems: Ensure you can switch between mathematical, graphical, and conceptual approaches.

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Core Concept

Zero-Order First-Order Second-Order
Rate Law Rate=k Rate=k[A] Rate=k[A]2
Integrated Rate Law [A]t=[A]0kt ln[A]t=ln[A]0kt 1[A]t=1[A]0+kt
Graph for Linearity [A] vs. t ln[A] vs. t 1[A] vs. t
Slope k k k
Half-Life (t1/2) t1/2=[A]02k t1/2=ln2k t1/2=1k[A]0
Units of k M/s s1 M1s1
Memorization Suggestion Zero slope is constant decline; concentration decreases linearly. First follows natural logs; think exponential decay. Second order is reciprocal; graphing 1/[A] makes it linear.

Where the differential rate law expresses rate as a function of reactant concentration(s) at an instant in time (hence instantaneous rate), integrated rates express the reactant concentrations as a function of time.  

To solve integrated rate problems, construct a graph with time on the x-axis and then make 3 plots where the y-axis is

Concentration of A [A] vs. t

Natural log of the concentration of A ln [A] vs. t

Reciprocal of [A]

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