First-order reactions
A first-order reaction is one where the reaction rate is directly proportional to the concentration of a single reactant. The First-Order Reactions: Rate Law and Integrated Rate Law is used to compute concentration after time, the rate constant, or the half-life from concentration–time data.
Core definitions and essential formulas
For \(A \rightarrow \text{products}\), the differential rate law is
\[
\text{rate} = -\frac{\mathrm{d}[A]}{\mathrm{d}t} = k[A].
\]
Here \([A]\) is the concentration of \(A\), \(t\) is time, and \(k\) is the first-order rate constant. The minus sign reflects reactant consumption while keeping the forward rate positive. Since \([A]\) cancels one concentration unit, \(k\) has units of \(\text{time}^{-1}\) (such as \(\mathrm{s^{-1}}\) or \(\mathrm{min^{-1}}\)).
Separating variables and integrating from \([A]_0\) at \(t=0\) to \([A]_t\) at time \(t\) gives
\[
\ln\!\left(\frac{[A]_t}{[A]_0}\right) = -kt,
\qquad
[A]_t = [A]_0 e^{-kt}.
\]
A larger \(k\) means a faster exponential decay of \([A]\) and a shorter characteristic timescale. For linear plotting, \(\ln[A]\) versus \(t\) is a straight line with slope \(-k\), which is useful when \(k\) is determined from experimental data.
The half-life for a first-order process is
\[
t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k}.
\]
Unlike zero-order reactions, \(t_{1/2}\) is independent of the initial concentration \([A]_0\); the fraction remaining after a given time depends only on \(kt\).
Common pitfalls
- Using base-10 logarithms without converting (natural log \(\ln\) is required in the standard form).
- Mixing time units so that \(k\) and \(t\) are inconsistent (seconds vs minutes).
- Applying the first-order model when \(\ln[A]\) versus \(t\) is not approximately linear.
- Using negative or zero concentrations in logarithms (inputs must be positive).
Micro example
If \(k=0.20\ \mathrm{min^{-1}}\), then
\[
t_{1/2}=\frac{0.693}{0.20}=3.47\ \mathrm{min}.
\]
Use this tool for processes that show exponential decay, such as unimolecular decompositions, radioactive-style kinetics, or pseudo-first-order conditions. Avoid using it when concentration changes are driven by competing pathways or changing reaction order; a next-step concept is testing alternative models with second-order kinetics or using the Arrhenius equation for temperature effects.
