Half-Life of a Second-Order Reaction

Half-life is a familiar idea imported from first-order kinetics, where it is genuinely a constant: a first-order half-life is $\ln 2 / k$ and contains no concentration at all, which is why radioactive decay has a single quotable number and why the idea feels so natural. Second-order kinetics breaks that intuition, and the break is the entire content of this page.

Integrating $-dC_A/dt = k C_A^2$ gives $1/C_A - 1/C_{A0} = kt$, and setting $C_A = C_{A0}/2$ leaves $t_{1/2} = 1/(k C_{A0})$. The starting concentration is right there in the answer. Halve the feed strength and the half-life doubles. A second-order reaction is fast while it is concentrated and gets progressively, unavoidably slower as it depletes itself — the rate falls as the square of what is left, while the amount remaining to be consumed falls only linearly.

The consequence for a reactor is a long tail. The second half-life is twice the first, the third is four times it, and the seventh is sixty-four times. Getting from 90% to 99% conversion in a second-order system costs roughly ten times the time it took to reach 90% in the first place. That is the arithmetic behind a familiar plant frustration: the last few percent of a second-order reaction can dominate the vessel size, and it is usually cheaper to stop early and recycle the unreacted feed than to build the reactor that finishes the job.

Two practical notes. This form assumes the rate is second order in a SINGLE reactant, $A + A \to$ products. A reaction that is first order in each of two different species, $A + B$, is second order overall but integrates to something else entirely unless the two are fed in exactly stoichiometric ratio — and if $B$ is in large excess, it collapses back to pseudo-first-order behaviour with a constant half-life again. And the units warning from the rate-law page applies in full here: $k$ belongs in this box as m³/(mol·s), while the literature will hand you L/(mol·s), a thousand times larger. Using the tabulated number unchanged makes the half-life come out a thousand times too short, which looks plausible and is not.