The integrated rate laws

zero first second order kineticsreaction orderhalf-life formulasrate lawkinetics equations

Zero-, first- and second-order concentration-versus-time laws and their half-lives — and how to read the order off a straight line.

Zero-Order Integrated Rate Law

[\mathrm{A}] = [\mathrm{A}]_0 - kt

Gives the concentration remaining in a zero-order reaction, where the rate is constant and concentration falls in a straight line with time.

First-Order Integrated Rate Law

[\mathrm{A}] = [\mathrm{A}]_0\,e^{-kt}

Gives the concentration remaining in a first-order reaction, the exponential decay that governs radioactive decay and most drug clearance.

Second-Order Integrated Rate Law

\frac{1}{[\mathrm{A}]} = \frac{1}{[\mathrm{A}]_0} + kt

Gives the concentration remaining in a second-order reaction, where the reciprocal of concentration rises linearly with time.

Half-Life of a Second-Order Reaction

t_{1/2} = \frac{1}{k\,[\mathrm{A}]_0}

Gives the half-life of a second-order reaction, which unlike a first-order half-life depends on the starting concentration.

Half-Life and Decay Constant

t_{1/2} = \frac{\ln 2}{\lambda}

Converts between a half-life and the exponential decay constant via the factor ln 2 = 0.6931471806.

How they fit together

Each is the differential rate law integrated once, and each one is the order that makes a particular plot come out straight. Plot [A] against time and a straight line means zero order. Plot ln[A] and a straight line means first order. Plot 1/[A] and a straight line means second order. The slope of whichever line is straight gives you k, sign and all, which is why kinetics data is fitted three ways before it is fitted once.

The order is an experimental result, not something you can read off the balanced equation, and assuming otherwise is the standard mistake — a reaction between two molecules of A can perfectly well be first order. Half-life is the sharpest tell of all: for a first-order reaction it is constant, so ln 2 / k holds no matter how much is left, which is why radioactive decay and most drug elimination are described by a single number. For zero order the half-life shrinks as the reaction proceeds; for second order it doubles with every halving. Watch the units of k, too — they are different for every order (mol/(L·s), s⁻¹, L/(mol·s)), so the units of a reported rate constant tell you the order before you read anything else.