A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
Integrated rate laws relate concentration to time, enabling prediction of how much reactant remains after a given period. For a reaction A → products: zero order gives [A] = [A]₀ − kt (linear in [A] vs t); first order gives ln[A] = ln[A]₀ − kt (linear in ln[A] vs t, half-life t₁/₂ = 0.693/k); second order gives 1/[A] = 1/[A]₀ + kt (linear in 1/[A] vs t). The graphical method determines order experimentally: plot [A], ln[A], and 1/[A] against time, and whichever gives a straight line reveals the order. Half-life for first-order reactions is uniquely concentration-independent.
Memorize the three integrated forms and their corresponding straight-line plots. Practice determining order from graphical data — the linear plot identifies the order, the slope gives k (with appropriate sign). Work half-life problems for each order and notice how only first-order half-life is constant (radioactive decay is the classic example).
From chemical kinetics, you know that rate laws express how reaction speed depends on concentration: rate = k[A]ⁿ, where n is the reaction order. But rate laws in that form tell you the instantaneous speed at a given moment — they don't directly answer the practical question "how much reactant is left after 30 minutes?" That's what integrated rate laws answer. They are the mathematical result of integrating the differential rate law over time, converting a statement about speed into a statement about concentration as a function of time.
For a zero-order reaction (rate = k, independent of concentration), integration gives [A] = [A]₀ − kt. Concentration decreases linearly with time, like a faucet draining at a constant rate regardless of how much water remains. The half-life is t₁/₂ = [A]₀/2k — it depends on starting concentration, so each successive half-life is shorter. For a first-order reaction (rate = k[A]), integration gives ln[A] = ln[A]₀ − kt, or equivalently [A] = [A]₀e⁻ᵏᵗ. This is exponential decay — the same mathematics that governs radioactive decay, which is why radioactive half-life is constant: t₁/₂ = 0.693/k, independent of how much material remains. For a second-order reaction (rate = k[A]²), integration gives 1/[A] = 1/[A]₀ + kt. The reaction slows dramatically as concentration drops, and the half-life t₁/₂ = 1/(k[A]₀) increases with each successive halving.
The graphical method is the experimental technique for determining reaction order. You measure concentration at several time points, then make three plots: [A] vs t, ln[A] vs t, and 1/[A] vs t. Whichever plot gives a straight line reveals the order — linear in [A] means zero-order, linear in ln[A] means first-order, linear in 1/[A] means second-order. The slope of the straight-line plot gives you the rate constant k (negative slope for zero and first order, positive for second order). This is why the integrated rate laws are written in y = mx + b form: they are designed to be linearized for graphical analysis.
A practical detail worth internalizing: first-order kinetics are by far the most common in chemistry and biology. Drug metabolism, radioactive decay, and many decomposition reactions follow first-order kinetics. The constant half-life property makes first-order processes especially intuitive — if a drug's half-life is 4 hours, then after 4 hours half remains, after 8 hours a quarter remains, after 12 hours an eighth remains, regardless of the initial dose. When a reaction involves multiple reactants, the pseudo-first-order technique simplifies analysis: flood one reactant in large excess so its concentration barely changes, and the rate law reduces to a first-order dependence on the other reactant.