Zero-order reactions
A zero-order reaction has a rate that does not change when the reactant concentration changes. The Zero-Order
Reactions: Rate Law and Integrated Rate Law describes a constant reaction rate and the concentration after
a given time.
Core definitions and essential formulas
For a simple reaction \(A \rightarrow \text{products}\), the differential rate law is
\[
\text{rate} = -\frac{\mathrm{d}[A]}{\mathrm{d}t} = k.
\]
Here \([A]\) is the concentration of \(A\) (often \(\mathrm{mol\cdot L^{-1}}\)), \(t\) is time, and \(k\) is the
zero-order rate constant. The minus sign indicates that \([A]\) decreases with time while the forward rate is
treated as positive.
Integrating from \([A]_0\) at \(t=0\) to \([A]_t\) at time \(t\) gives the integrated law:
\[
[A]_t = [A]_0 - kt.
\]
A plot of \([A]_t\) versus \(t\) is linear with slope \(-k\) and intercept \([A]_0\). Useful time relations follow
directly:
\[
t_{1/2}=\frac{[A]_0}{2k}, \qquad t_{\text{complete}}=\frac{[A]_0}{k}.
\]
Larger \(k\) means faster depletion of \(A\) (steeper negative slope), while larger \([A]_0\) increases both
half-life and completion time. If \([A]\) is in \(\mathrm{mol\cdot L^{-1}}\) and \(t\) is in seconds, then \(k\)
has units \(\mathrm{mol\cdot L^{-1}\cdot s^{-1}}\); changing the time unit changes the numerical value of \(k\).
Common pitfalls
- Using the wrong integrated form (zero order is linear in \([A]\), not in \(\ln [A]\)).
- Mixing time units (minutes vs seconds) without converting \(k\) consistently.
- Allowing computed \([A]_t\) to go below zero; physically, concentration cannot be negative.
- Assuming “zero order” from stoichiometry; the order must be supported by experimental data.
Micro example
If \([A]_0=0.80\ \mathrm{mol\cdot L^{-1}}\) and \(k=0.020\ \mathrm{mol\cdot L^{-1}\cdot s^{-1}}\), then
\[
t_{1/2}=\frac{0.80}{2(0.020)}=20\ \mathrm{s}.
\]
Use this tool when a reaction is known to follow zero-order kinetics over the time window of interest and the goal
is \([A]_t\), half-life, or time to completion. Avoid applying it when the rate changes with concentration or when
side reactions become important; a next-step concept is comparing kinetic models using first-order and
second-order integrated rate laws.
