Graphing data to determine reaction order
Reaction order can often be identified by transforming concentration–time data until it forms a straight line.
Graphing Data to Determine the Order of a Reaction uses linear plots to estimate the order (zero, first,
or second) and to compute a best-fit rate constant from the slope.
Core definitions and essential formulas
For a single-reactant reaction \(A \rightarrow \text{products}\), three common models are tested through their
integrated forms:
\[
[A]_t = [A]_0 - kt \quad (\text{zero order})
\]
\[
\ln\!\left(\frac{[A]_t}{[A]_0}\right) = -kt \quad (\text{first order})
\]
\[
\frac{1}{[A]_t} = kt + \frac{1}{[A]_0} \quad (\text{second order})
\]
Here \([A]_0\) is the initial concentration, \([A]_t\) is the concentration at time \(t\), and \(k\) is the rate
constant. The straight-line plot and slope sign depend on the model: \([A]\) vs \(t\) has slope \(-k\) (zero
order), \(\ln[A]\) vs \(t\) has slope \(-k\) (first order), and \(1/[A]\) vs \(t\) has slope \(+k\) (second
order).
Larger \(|k|\) means faster concentration change. Typical units: zero order \(k\) in \(\mathrm{mol\cdot
L^{-1}\cdot time^{-1}}\), first order \(k\) in \(\mathrm{time^{-1}}\), and second order \(k\) in \(\mathrm{L\cdot
mol^{-1}\cdot time^{-1}}\). When the calculator reports linearity (often via \(R^2\)) for each transformation, the
model with the highest linearity is the best match to the data and its slope gives \(k\).
Common pitfalls
- Using \(\ln[A]\) with \([A]\le 0\) or with concentrations near the detection limit.
- Mixing time units between data points (seconds and minutes in the same dataset).
- Including early/late points affected by mixing delay, induction, or side reactions.
- Assuming the highest \(R^2\) always proves the mechanism; it only supports the best kinetic model over the
measured window.
Micro example
If a plot of \(\ln[A]\) versus \(t\) is linear and the fitted slope is \(-0.30\ \mathrm{min^{-1}}\), then
\[
k = 0.30\ \mathrm{min^{-1}}.
\]
Use this tool when concentration–time data are available for a single reactant and the goal is to identify whether
kinetics are zero, first, or second order and to estimate \(k\). Avoid using it when multiple reactants change
simultaneously or when temperature varies during the run; a next-step concept is the method of initial rates or
fitting more general rate laws with regression and error analysis.
