Half-Life of a Reaction

The half-life of a reaction, written $t_{1/2}$, is the time required for the

concentration of a reactant to fall to exactly half of its value at the start of that interval. Because

zero order and first order reactions have very different integrated rate equations, their half-lives

behave in strikingly different ways -- a difference important enough to use as a direct experimental

test of a reaction's order.

Zero order half-life. Starting from the zero order integrated equation, $[R] = [R]_0 - kt$, and

substituting $[R] = [R]_0/2$ at $t = t_{1/2}$:

The zero order half-life is directly proportional to the initial concentration $[R]_0$ -- it is

not a fixed number for the reaction, but changes depending on how much reactant one starts with.

Consequently, each successive half-life of a zero order reaction is only half as long as the one

before it (since the "initial" concentration for the second half-life is itself only half of the true

starting concentration).

First order half-life. Starting from the first order integrated equation,

$k = (2.303/t)\log([R]_0/[R])$, and substituting $[R] = [R]_0/2$ at $t = t_{1/2}$:

Strikingly, the initial concentration $[R]_0$ cancels out completely: the first order half-life

depends only on the rate constant $k$, and not at all on the starting concentration. This means

that for a first order reaction, the time taken for any concentration to fall to half its value --

whether that is the very first half-life or, say, the fifth -- is always exactly the same fixed number, …