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 \(\text{Rate} = k\) \(\text{Rate} = k[A]\) \(\text{Rate} = k[A]^2\)
Integrated Rate Law \([A]_t = [A]_0 - kt\) \(\ln[A]_t = \ln[A]_0 - kt\) \(\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt\)
Graph for Linearity \([A]\) vs. \(t\) \(\ln[A]\) vs. \(t\) \(\frac{1}{[A]}\) vs. \(t\)
Slope \(-k\) \(-k\) \(k\)
Half-Life (\( t_{1/2} \)) \(t_{1/2} = \frac{[A]_0}{2k}\) \(t_{1/2} = \frac{\ln 2}{k}\) \(t_{1/2} = \frac{1}{k[A]_0}\)
Units of \( k \) \(\text{M/s}\) \(\text{s}^{-1}\) \(\text{M}^{-1}\text{s}^{-1}\)
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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