Last updated: 2026-06-16

A first-order reaction proceeds at a rate proportional to the concentration of a single reactant. As that reactant is used up, the reaction slows in step, tracing out an exponential decay curve. This calculator applies the first-order integrated rate law to find the concentration remaining after any elapsed time and reports the reaction's half-life. First-order kinetics govern radioactive decay, many drug eliminations, and a wide range of decompositions and isomerizations.

The integrated rate law

The differential rate law $-\dfrac{d[A]}{dt} = k[A]$ integrates to a logarithmic straight-line form, which rearranges to an exponential:

$\ln[A] = \ln[A]_0 - kt \qquad\Longrightarrow\qquad [A] = [A]_0\,e^{-kt}$

A plot of $\ln[A]$ against time is therefore linear with slope $-k$, the standard graphical test for first-order behaviour. The rate constant $k$ has units of inverse time, such as $\mathrm{s^{-1}}$.

Constant half-life

Setting $[A] = [A]_0/2$ in the integrated law gives a half-life that contains no concentration term:

$t_{1/2} = \frac{\ln 2}{k}$

The reactant always takes the same time to halve, no matter how much is present — the defining signature of first-order kinetics.

Worked example

Suppose $[A]_0 = 1.0\ \mathrm{M}$ and $k = 0.05\ \mathrm{s^{-1}}$. After 60 seconds:

The half-life is $t_{1/2} = \ln 2 / 0.05 \approx 13.9\ \mathrm{s}$, so 60 seconds is a little over four half-lives, leaving about one-sixteenth of the original amount.

Why the curve is exponential

Because the rate is always a fixed fraction of what remains, equal time steps remove equal fractions, and repeated multiplication by a constant factor is exactly exponential decay. For reactions of other orders the shape changes: see the Second-Order Integrated Rate Law Calculator and the Zero-Order Integrated Rate Law Calculator. To measure an overall rate from two concentration readings, use the Average Reaction Rate Calculator.