Why is a first-order half-life independent of concentration?

For a first-order reaction the half-life is ln 2 divided by k, which contains no concentration term. Because the rate is proportional to how much reactant is present, the time to lose half of it is always the same no matter where you start. This is why radioactive decay and many drug-elimination processes, both first order, have a fixed half-life. Zero-order and second-order half-lives do depend on the starting concentration.

How do the half-life formulas differ by order?

Zero order: half-life = [A]0 / 2k, so it gets shorter as the reactant is used up. First order: half-life = ln 2 / k, constant regardless of concentration. Second order: half-life = 1 / (k[A]0), so it doubles with each successive half-life as the concentration falls. Noticing whether successive half-lives are constant, shortening, or lengthening is a quick way to identify the order.

How do I find the reaction order from data?

Plot the data three ways against time: [A], the natural log of [A], and 1/[A]. The plot that comes out as a straight line reveals the order. A linear [A] plot means zero order, a linear ln[A] plot means first order, and a linear 1/[A] plot means second order. The slope of that straight line is related to the rate constant k.

What units does the rate constant k have?

The units of k depend on the overall order so that the rate always comes out in concentration per time. For a zero-order reaction k is in M per second, for first order it is in inverse seconds (per second), and for second order it is in inverse molar per second, 1/(M s). If you ever forget the order, the units of k give it away.

Reaction Half-Life & Rate Law Calculator, Free

Reaction order in the reactant

Rate constant k (1/s)

Initial concentration [A]₀ (M)

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Concentration [A] at this time

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Half-life t_½

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Linear plot

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How reaction order sets the half-life

The half-life is the time for a reactant to fall to half its concentration, and its formula depends entirely on the reaction order. The most important case is first order, where the half-life is ln 2 / k and does not depend on concentration at all, so each successive half-life takes the same time. That is why radioactive decay and many drug-elimination curves have a single fixed half-life. A zero-order half-life shortens as the reactant runs out, and a second-order half-life doubles with each halving of concentration. Watching whether successive half-lives stay constant, shrink, or grow is a fast way to read the order straight off a graph.

The three orders, side by side

Order	Integrated rate law	Half-life	Straight-line plot
Zero	[A] = [A]₀ − kt	[A]₀ / 2k	[A] vs t
First	[A] = [A]₀ e^−kt	0.693 / k (constant)	ln[A] vs t
Second	1/[A] = 1/[A]₀ + kt	1 / (k[A]₀)	1/[A] vs t

Only the first-order half-life is independent of concentration, which is what makes it so useful and so heavily tested.

FAQ

Why is a first-order half-life constant?
t½ = ln 2 / k has no concentration term, so the time to lose half is always the same. Radioactive decay and many drugs are first order for this reason.

How do the formulas differ?
Zero: [A]₀/2k (shortens). First: 0.693/k (constant). Second: 1/(k[A]₀) (doubles each half-life).

What are the integrated rate laws?
Zero: [A] = [A]₀ − kt. First: [A] = [A]₀e^−kt. Second: 1/[A] = 1/[A]₀ + kt.

How do I find the order from data?
Plot [A], ln[A], and 1/[A] vs time. The one that is a straight line gives the order (zero, first, second).

What units does k have?
Zero order M/s, first order 1/s, second order 1/(M·s), chosen so the rate is always M/s.