Integrated rate laws connect reactant concentration to time, making them essential for predicting how fast a chemical reaction will proceed. They help chemists determine whether a reaction is zero order, first order, or second order in a reactant. This matters in medicine, environmental chemistry, food preservation, and industrial production because concentration changes over time affect safety, yield, and shelf life.
By graphing the right quantity against time, students can turn experimental data into a clear reaction model.
The key idea is that each reaction order has a different integrated rate law and a different graph that becomes a straight line. For zero order reactions, concentration decreases linearly with time, while first order reactions show a linear relationship between ln[A] and time. For second order reactions, 1/[A] increases linearly with time.
Half-life behavior also reveals reaction order because it is constant only for first order reactions.
Key Facts
- Zero order integrated rate law: [A]t = [A]0 - kt
- First order integrated rate law: ln[A]t = ln[A]0 - kt
- Second order integrated rate law: 1/[A]t = 1/[A]0 + kt
- Zero order straight-line plot: [A] vs. t has slope = -k
- First order straight-line plot: ln[A] vs. t has slope = -k
- Half-lives: zero order t1/2 = [A]0/(2k), first order t1/2 = 0.693/k, second order t1/2 = 1/(k[A]0)
Vocabulary
- Integrated rate law
- An equation that relates reactant concentration to time for a specific reaction order.
- Reaction order
- The exponent pattern in a rate law that shows how rate depends on reactant concentration.
- Rate constant
- The proportionality constant k that connects reaction rate to concentration terms for a given reaction at a fixed temperature.
- Half-life
- The time required for the concentration of a reactant to decrease to one half of its initial value.
- Linear plot
- A graph that forms a straight line when the correct concentration expression is plotted against time.
Common Mistakes to Avoid
- Using the differential rate law when the problem asks for concentration after time. Integrated rate laws are needed when time and changing concentration are involved.
- Assuming [A] vs. time is always the correct straight-line plot. Only zero order reactions give a straight line for [A] vs. t.
- Forgetting that first order plots use ln[A], not log[A] unless the equation is adjusted. Using the wrong logarithm changes the slope relationship and the calculated k.
- Treating half-life as constant for every reaction order. Half-life is independent of initial concentration only for first order reactions.
Practice Questions
- 1 A zero order reaction has [A]0 = 0.800 M and k = 0.0200 M/s. What is [A] after 15.0 s?
- 2 A first order reaction has k = 0.0350 s^-1 and [A]0 = 1.20 M. What is [A] after 40.0 s?
- 3 Experimental data give a straight line when 1/[A] is plotted against time, but not when [A] or ln[A] is plotted against time. What is the reaction order, and what does the slope of the line represent?