second-order reaction
Imagine a dance where progress requires two people to find each other and pair up. When the floor is packed, partners meet constantly and pairings form fast; as people leave, finding a partner gets disproportionately harder, and the pairing rate plummets. A second-order reaction has this flavour: its speed depends on the square of a concentration, or on the product of two concentrations.
Formally, a second-order reaction has rate = k[A]² (or rate = k[A][B]), so it is second order overall. Integrating the simplest case shows that a plot of 1/[A] against time is straight, with slope equal to k — a different telltale plot from the first-order case. Crucially, its half-life is not constant: as the reactant grows scarce, the half-life lengthens, because the two-particle encounters needed to react become rarer.
Second-order kinetics matter because they are typical of reactions where two molecules must actually meet and combine — many bimolecular steps, dimerizations, and ion recombinations. The honest caveat is that the simple 1/[A] formula assumes a single reactant or equal starting concentrations; with two different reactants at different amounts the integrated form is more involved, and you must be careful which case you are in.
Two molecules of nitrogen dioxide collide and combine, following rate = k[NO₂]². Plot 1/[NO₂] against time and the points fall on a straight line, confirming second order — and unlike a first-order case, the time to use up the next half keeps growing as the gas thins out.
A straight 1/[A]-versus-time plot and a lengthening half-life mark second order.
First order plots ln[A] straight; second order plots 1/[A] straight. Picking the wrong plot is a frequent source of misidentifying the order.