Pseudo-first-order reactions
Some reactions involve two reactants but behave like a one-reactant first-order process because one reactant is in
large excess. Pseudo–First-Order Reactions use an effective rate constant to compute first-order style
quantities such as concentration vs time and half-life.
Core idea and essential formulas
For a bimolecular reaction \(A + B \rightarrow \text{products}\), the true rate law is
\[
\text{rate} = -\frac{\mathrm{d}[A]}{\mathrm{d}t} = k[A][B].
\]
If \(B\) is present in large excess, its concentration changes negligibly during the experiment, so \([B]\approx
[B]_0\) (constant). Define the effective first-order constant
\[
k' = k[B]_0,
\qquad
\text{so that} \qquad
\text{rate}=k'[A].
\]
Here \([A]\) and \([B]_0\) are concentrations and \(t\) is time; \(k\) has units \(\mathrm{L\cdot mol^{-1}\cdot
time^{-1}}\) while \(k'\) has units \(\mathrm{time^{-1}}\). The minus sign keeps the forward rate positive while
\([A]\) decreases.
Treating the kinetics as first order in \(A\), the integrated form is
\[
\ln\!\left(\frac{[A]_t}{[A]_0}\right) = -k't,
\qquad
[A]_t=[A]_0 e^{-k't}.
\]
Practical outputs such as \([A]_t\), fraction remaining \([A]_t/[A]_0\), fraction consumed \(1-[A]_t/[A]_0\), and
\(k'\) follow directly. A linear plot of \(\ln[A]\) versus \(t\) indicates pseudo-first-order behavior, with slope
\(-k'\).
The half-life has the same form as any first-order process:
\[
t_{1/2}=\frac{\ln 2}{k'}\approx \frac{0.693}{k'}.
\]
A larger \(k'\) means faster decay and a shorter half-life; importantly, \(t_{1/2}\) does not depend on \([A]_0\)
under pseudo-first-order conditions.
Common pitfalls
- Using the approximation when \(B\) is not truly in large excess (then \([B]\) is not constant).
- Mixing units when computing \(k' = k[B]_0\) (ensure consistent concentration units).
- Assuming \(k'\) is universal; it changes if \([B]_0\) changes.
- Fitting late-time data where side reactions or depletion of \(B\) breaks linear \(\ln[A]\) behavior.
Micro example
If \(k=0.80\ \mathrm{L\cdot mol^{-1}\cdot s^{-1}}\) and \([B]_0=0.50\ \mathrm{mol\cdot L^{-1}}\), then
\[
k' = 0.80(0.50)=0.40\ \mathrm{s^{-1}}.
\]
Use this tool when a bimolecular mechanism is known but one reactant is buffered or present as a solvent so
\([B]\) stays effectively constant. Avoid using it when both reactants change significantly; a next-step concept
is treating true second-order kinetics or using integrated forms for \(A+B\rightarrow \text{products}\) without
the excess-reactant approximation.
