What is a pseudo first order reaction, why does it behave like first-order kinetics, and how are the observed rate constant and plots (ln concentration vs time) interpreted?

A pseudo first order reaction occurs when one reactant is in large excess so its concentration stays nearly constant, making the rate appear first order in the limiting reactant with an observed constant \(k_{\text{obs}} = k[B]_0^n\).

Pseudo First Order Reaction in Chemical Kinetics

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Definition of a pseudo first order reaction

A pseudo first order reaction is a reaction that is not truly first order in its full mechanism, but it behaves like first-order kinetics under experimental conditions where one reactant is present in such a large excess that its concentration changes negligibly during the measurement.

Rate-law reduction to an observed first-order form

Consider a reaction where \(A\) is the limiting reactant and \(B\) is in large excess:

\[ A + B \rightarrow \text{products} \]

A common (illustrative) true rate law is:

\[ \text{rate} = k[A][B] \]

If \([B]\) is maintained approximately constant (large excess), then \([B] \approx [B]_0\) during the time window of interest, so:

\[ \text{rate} = k[A][B]_0 = k_{\text{obs}}[A] \qquad\text{with}\qquad k_{\text{obs}} = k[B]_0 \]

More generally, for \(\text{rate} = k[A]^m[B]^n\) with \(B\) in large excess, \(\text{rate} = k_{\text{obs}}[A]^m\) where \(k_{\text{obs}} = k[B]_0^n\). The reaction is “pseudo” because the observed order in \(A\) is measured while \(B\) is effectively constant.

Integrated law and the diagnostic plot

For the common pseudo-first-order case where the observed law is first order in \(A\), \(\text{rate} = k_{\text{obs}}[A]\), the integrated rate law is:

\[ \ln\!\left(\frac{[A]}{[A]_0}\right) = -k_{\text{obs}}t \qquad\Longleftrightarrow\qquad \ln([A]) = -k_{\text{obs}}t + \ln([A]_0) \]

A straight line in a plot of \(\ln([A])\) versus \(t\) indicates pseudo-first-order behavior for \(A\), with slope \(-k_{\text{obs}}\).

When the approximation is valid

\([B]\) must change very little during the run; a practical condition is that \(B\) is at least an order of magnitude larger than \(A\) for a 1:1 stoichiometry, so the fractional change in \([B]\) remains small.
The same temperature and solvent conditions must be maintained so \(k\) is constant during the run.
The measured signal must track \([A]\) (or a quantity proportional to \([A]\)) reliably over time.

Numerical example (computing \([A]\), \(k_{\text{obs}}\), and half-life)

Suppose the true law is \(\text{rate} = k[A][B]\) with \(k = 0.80\ \text{M}^{-1}\text{s}^{-1}\), and \(B\) is prepared at \([B]_0 = 0.50\ \text{M}\) (large excess). Then:

\[ k_{\text{obs}} = k[B]_0 = (0.80\ \text{M}^{-1}\text{s}^{-1}) \cdot (0.50\ \text{M}) = 0.40\ \text{s}^{-1} \]

If \([A]_0 = 0.020\ \text{M}\), then after \(t = 5.0\ \text{s}\):

\[ [A] = [A]_0 e^{-k_{\text{obs}}t} = 0.020 \cdot e^{-(0.40)(5.0)} = 0.020 \cdot e^{-2.0} \approx 0.020 \cdot 0.1353 \approx 0.00271\ \text{M} \]

The pseudo-first-order half-life for \(A\) is:

\[ t_{1/2} = \frac{\ln(2)}{k_{\text{obs}}} = \frac{0.693}{0.40} \approx 1.73\ \text{s} \]

How to recover the true rate constant and the order in the excess reactant

When the true law is \(\text{rate} = k[A][B]^n\) and \(B\) is held in excess, repeated experiments at different \([B]_0\) produce different \(k_{\text{obs}}\) values:

\[ k_{\text{obs}} = k[B]_0^n \]

Taking natural logs yields a linear relationship:

\[ \ln(k_{\text{obs}}) = \ln(k) + n\ln([B]_0) \]

Run	\([B]_0\) (M)	Measured \(k_{\text{obs}}\) (s\(^{-1}\))	\(\ln([B]_0)\)	\(\ln(k_{\text{obs}})\)
1	0.20	0.16	\(\ln(0.20) \approx -1.609\)	\(\ln(0.16) \approx -1.833\)
2	0.40	0.32	\(\ln(0.40) \approx -0.916\)	\(\ln(0.32) \approx -1.139\)
3	0.80	0.64	\(\ln(0.80) \approx -0.223\)	\(\ln(0.64) \approx -0.446\)

The pattern \(k_{\text{obs}}\) doubling when \([B]_0\) doubles indicates \(n \approx 1\). Then \(k = k_{\text{obs}}/[B]_0\), giving \(k \approx 0.80\ \text{M}^{-1}\text{s}^{-1}\) consistently across runs.

Visualization: concentration–time and ln plot for a pseudo first order reaction

A pseudo first order reaction is indicated when \([B]\) stays nearly constant (left), while a plot of \(\ln([A])\) versus \(t\) is linear (right), yielding \(k_{\text{obs}}\) from the slope.

\[ \ln([A]) = -k_{\text{obs}} t + \ln([A]_0) \]

Summary

A pseudo first order reaction results from placing one reactant in large excess so its concentration is effectively constant.
The observed constant satisfies \(k_{\text{obs}} = k[B]_0^n\), converting a multi-reactant rate law into an observed first-order form in the limiting reactant.
Linearity of \(\ln([A])\) versus time is the primary diagnostic for pseudo-first-order behavior in \(A\).

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