An integrated rate law is an equation relating a reactant's concentration directly to time, letting you calculate [A] at any moment and determine reaction order from which plot is linear: [A] vs. t (zeroth order), ln[A] vs. t (first order), or 1/[A] vs. t (second order).
A regular (differential) rate law tells you how fast a reaction goes at one instant based on concentration. The integrated rate law answers a more useful question for problem-solving: if I start with this much reactant, how much is left after 30 seconds? It's the rate law with time built in.
Each reaction order has its own version. For zeroth order, [A] = [A]₀ − kt, so a plot of [A] vs. time is a straight line. For first order, ln[A] = ln[A]₀ − kt, so ln[A] vs. time is linear (EK 5.3.A.2). For second order, 1/[A] = 1/[A]₀ + kt, so 1/[A] vs. time is linear (EK 5.3.A.3). That's the whole game on the AP exam. You're given concentration-time data or graphs, you figure out which transformation makes a straight line, and that tells you the order. The slope of that line hands you the rate constant k (EK 5.3.A.4). Notice each equation is just y = mx + b in disguise, where time is x and the slope is ±k.
Integrated rate laws live in Topic 5.3 (Concentration Changes Over Time) in Unit 5: Kinetics, supporting learning objective 5.3.A, which asks you to identify the rate law of a reaction from concentration-versus-time data. This is the partner skill to the method of initial rates from earlier in Unit 5. Initial rates use multiple experiments; integrated rate laws use one experiment tracked over time. The AP equation sheet gives you the first-order and second-order integrated rate laws, so the exam tests whether you know what they mean and when to use them, not whether you memorized them. Graph interpretation here is one of the most reliably tested skills in all of Unit 5.
Keep studying AP Chemistry Unit 5
Reaction Order (Unit 5)
The integrated rate law is how you diagnose order from time data. Whichever plot is linear ([A], ln[A], or 1/[A] versus time) tells you whether the reaction is zeroth, first, or second order in that reactant (EK 5.3.A.1).
Rate Constant (Unit 5)
Once you've found the linear plot, its slope gives you k. For zeroth and first order the slope is −k; for second order the slope is +k. Same constant as the differential rate law, just extracted from a graph instead of a rate table.
Half-life (Unit 5)
Half-life comes straight out of the first-order integrated rate law. Set [A] = ½[A]₀ and you get t½ = 0.693/k, which is constant no matter the starting concentration. That constant half-life is the fingerprint of a first-order process.
Partial Pressure (Unit 3)
For gas-phase reactions, partial pressure is proportional to concentration, so you can swap P in for [A] in any integrated rate law. A linear ln(P) vs. time plot still means first order.
This term shows up almost entirely as a graph and data skill. Multiple-choice stems give you a description like "ln[A] decreases linearly with time" and ask for the order (that one is first order), or hand you a data table of 1/[A] values increasing by equal steps and ask what conclusion the data supports (second order, and the slope is k). Calculation questions plug numbers into the equations directly, for example finding 1/[A] after 10.0 s given k = 0.25 M⁻¹s⁻¹ and [A]₀ = 0.50 M for a second-order reaction. On FRQs, kinetics questions routinely ask you to identify which plot would be linear for a given order, sketch or interpret that plot, calculate k from a slope, or find the concentration remaining after some time. Know all three linear plots cold and remember the first- and second-order equations are on your reference sheet.
The differential rate law (rate = k[A]ⁿ) relates rate to concentration at one instant and is found using the method of initial rates across several experiments. The integrated rate law relates concentration to time and is found from one experiment monitored as it runs. They describe the same reaction with the same k and the same order; they just answer different questions. If a problem gives you a rate table, think differential. If it gives you concentration-vs-time data or a graph, think integrated.
The integrated rate law connects a reactant's concentration to elapsed time, so you can calculate how much reactant remains at any point in the reaction.
You determine reaction order by finding which plot is linear: [A] vs. t means zeroth order, ln[A] vs. t means first order, and 1/[A] vs. t means second order.
The slope of the linear plot gives you the rate constant k (negative slope for zeroth and first order, positive slope for second order).
The first-order and second-order integrated rate laws are printed on the AP Chem equation sheet, so focus on knowing when and how to apply them.
First-order reactions have a constant half-life (t½ = 0.693/k), which falls directly out of the first-order integrated rate law.
Use the method of initial rates when you have multiple trials, and use integrated rate laws when you have one trial tracked over time.
It's an equation that relates reactant concentration directly to time. Each order has its own form: [A] = [A]₀ − kt for zeroth order, ln[A] = ln[A]₀ − kt for first order, and 1/[A] = 1/[A]₀ + kt for second order.
Mostly no. The first-order and second-order integrated rate laws (plus the first-order half-life equation) are on the AP equation sheet. What you do need to memorize is which plot is linear for each order, because the exam tests that interpretation constantly.
The regular (differential) rate law, rate = k[A]ⁿ, tells you the instantaneous rate from concentration and is found by comparing initial rates across experiments. The integrated rate law tells you concentration as a function of time from a single experiment. Same reaction, same k, different question.
Transform the concentration data. If ln[A] vs. time is a straight line, the reaction is first order (EK 5.3.A.2). If 1/[A] vs. time is a straight line, it's second order (EK 5.3.A.3). If plain [A] vs. time is already linear, it's zeroth order.
Almost. The magnitude of the slope equals k in all three cases, but the sign differs. Zeroth and first order plots slope downward (slope = −k), while the second-order 1/[A] plot slopes upward (slope = +k).