Integrated rate laws

Result

Integrated rate laws

T-039Home CU-202Threads kinetics

Statement

Concentration-time behaviour for zero, first and second order.

Why it matters

rate-law-order established how the instantaneous rate depends on concentration through a differential rate law, but an instantaneous rate is not directly what a chemist typically measures in the laboratory; concentration as a function of time, tracked over the course of a whole reaction, is. Integrated rate laws are exactly the bridge between the two: solving the differential rate law explicitly gives concentration versus time directly, and — just as importantly — gives a specific, testable linear plot for each reaction order, making the integrated form the standard practical tool for determining a reaction's order from real experimental concentration-time data.

Hypotheses

The reaction proceeds with a single, well-defined, constant order in the reactant of interest throughout the entire time course studied.If the true mechanism changes order partway through (for instance, as a co-reactant is depleted, or as the reaction approaches equilibrium and a significant reverse rate develops), a single integrated rate law derived assuming one fixed order will fit the early data well but systematically deviate later; this is precisely how curvature in a supposedly linear plot is used diagnostically (Corollaries). Temperature, and hence the rate constant \(k\), is held constant throughout the reaction.Since \(k\) itself depends on temperature (arrhenius-equation), any significant temperature drift during a kinetic run — whether from external heating/cooling changes or from the reaction's own heat of reaction in a poorly thermostatted setup — would make \(k\) effectively time-dependent, invalidating the constant-\(k\) integration performed in the Proof.

Proof

\text{Zero order: } -\frac{d[A]}{dt}=k \ \Longrightarrow\ [A]=[A]_0-kt

Direct integration of a rate independent of concentration gives a linear decline in \([A]\) with time; a plot of \([A]\) versus \(t\) is linear with slope \(-k\) for a genuinely zero-order reaction (commonly seen when a reaction is limited by something other than reactant concentration, such as a saturated catalytic surface, langmuir-isotherm). A

\text{First order: } -\frac{d[A]}{dt}=k[A] \ \Longrightarrow\ \ln[A]=\ln[A]_0-kt, \qquad t_{1/2}=\frac{\ln2}{k}

Separating variables and integrating gives an exponential decay in \([A]\); a plot of \(\ln[A]\) versus \(t\) is linear with slope \(-k\), and the resulting half-life is a constant, independent of the starting concentration — a distinctive diagnostic signature unique to first order among the three cases here. A

\text{Second order: } -\frac{d[A]}{dt}=k[A]^2 \ \Longrightarrow\ \frac{1}{[A]}=\frac{1}{[A]_0}+kt, \qquad t_{1/2}=\frac{1}{k[A]_0}

Separating variables and integrating this case gives a reciprocal-concentration form; a plot of \(1/[A]\) versus \(t\) is linear with slope \(+k\), and here the half-life depends explicitly on the starting concentration (unlike the first-order case), a second distinctive diagnostic used to distinguish second order from first order experimentally. A

\text{Plot } [A],\ \ln[A],\ \text{and } 1/[A]\ \text{ vs. } t\text{; whichever plot is linear reveals the true order, and its slope gives } k.

Since each order predicts a distinct linearising transformation of \([A]\), applying all three transformations to the same experimental concentration-time data and checking which gives a genuinely straight line is the standard, direct experimental method for simultaneously determining both the reaction order and the rate constant. A

Result

[A]=[A]_0-kt\ (0);\quad \ln[A]=\ln[A]_0-kt\ (1);\quad \frac{1}{[A]}=\frac{1}{[A]_0}+kt\ (2)

Reading. Each reaction order has a unique concentration-time functional form and a correspondingly unique linear plot; identifying which plot is linear for a given data set identifies the order directly, and the slope of that linear plot gives the rate constant without any further fitting.

Scope. Strictly valid only while the assumed order remains constant and temperature is held fixed throughout the run (Hypotheses); reactions that change mechanism or order partway through (for instance, as they approach equilibrium, or as a co-reactant becomes limiting) will show curvature away from the naive single-order fit at longer times.

Corollaries & converses

The half-life expressions differ characteristically by order: constant for first order (Step 2), proportional to \([A]_0\) for zero order (Step 1, since \(t_{1/2}=[A]_0/2k\)), and inversely proportional to \([A]_0\) for second order (Step 3) — measuring how \(t_{1/2}\) changes across runs at different starting concentrations is itself a standalone method for determining order.
steady-state-approximation often produces observed rate laws that are effectively pseudo-first-order or otherwise simplified relative to the true underlying mechanism's full complexity; the integrated forms derived here are frequently applied to such an experimentally observed, simplified rate law rather than to a genuinely elementary reaction.
Converse: if a concentration-time data set is found to give a linear plot only for, say, the \(\ln[A]\) transformation and not the other two, this is itself direct, suficient evidence that the reaction is first order in that reactant, without needing any separate confirmation.

Fails without

Assume a fixed order (Hypotheses) throughout the entire reaction when the true mechanism actually changes order partway through: the appropriate linear plot (Step 4) will fit the early-time data well but curve away from linearity at later times, as the reaction's true kinetics diverge from the single, fixed-order model assumed in the integration.
Allow temperature to drift during the kinetic run: since \(k\) is temperature-dependent (arrhenius-equation), a drifting temperature makes \(k\) effectively time-dependent, and the constant-\(k\) integration performed in the Proof no longer applies; the resulting plot will show curvature or an apparent (spurious) change in slope unrelated to any real change in reaction order.

Common errors

Fitting a straight line to \([A]\) vs. \(t\) data that is genuinely first or second order, mistaking approximate short-time linearity (any smooth curve looks locally linear over a short enough interval) for true zero-order behaviour.
Using the first-order half-life formula (\(t_{1/2}=\ln2/k\), independent of \([A]_0\)) for a reaction that is not actually first order, where half-life genuinely depends on starting concentration (Step 1 and Step 3).
Confusing the sign or slope convention between the three plots — \([A]\) and \(\ln[A]\) plots have negative slope \((-k)\), while the \(1/[A]\) plot has positive slope \((+k)\), per Steps 1–3.
Applying an integrated rate law derived for a single reactant directly to a multi-reactant reaction without first confirming (e.g. via large excess of the other reactants, giving pseudo-order conditions) that the reaction is genuinely behaving as simple order in the reactant being tracked.

Discussion

The systematic determination of reaction order from concentration-time data, using exactly the linearisation technique of Step 4, has been a cornerstone experimental method of chemical kinetics since the field's earliest quantitative studies in the late 19th century; it remains, alongside the closely related method of initial rates, one of the two standard practical routes to establishing a rate law experimentally before any mechanistic interpretation is attempted.

Radioactive decay, though a nuclear rather than chemical process, follows exactly the same first-order integrated rate law (Step 2) with its own characteristic half-life; the mathematical identity between chemical first-order kinetics and radioactive decay kinetics is a direct consequence of both processes sharing the same underlying differential structure — a constant per-particle (or per-molecule) probability of the "event" occurring per unit time, independent of how many events have already occurred.

Common misconception: that a reaction's order must match its molecularity (the number of molecules colliding in an elementary step) or its overall stoichiometric coefficients. Reaction order is an entirely empirical, experimentally determined quantity (found via exactly the methods developed here), and for any reaction proceeding through a multi-step mechanism, it need not match either the overall balanced equation's coefficients or any single step's molecularity at all.

Worked examples

[A]_0=1.00\,\text{M}; \text{ measured } [A] \text{ vs. } t \text{ (s): } (0,1.00),(50,0.607),(100,0.368),(150,0.223)

Testing the first-order transformation: \(\ln[A]\) at each time gives \(0,\ -0.499,\ -1.00,\ -1.50\), which is exactly linear in \(t\) with slope \(-0.0100\,\text{s}^{-1}\), confirming the reaction is first order with \(k=0.0100\,\text{s}^{-1}\). A

t_{1/2} = \frac{\ln2}{k} = \frac{0.693}{0.0100} = 69.3\,\text{s}

Applying Step 2's first-order half-life formula directly to the fitted rate constant gives the time for the concentration to fall to half its initial value, independent of what that initial value actually was. A

\text{First order confirmed: } k=0.0100\,\text{s}^{-1},\ t_{1/2}=69.3\,\text{s}

Reading. Linearity of the \(\ln[A]\) vs. \(t\) plot, and only that plot, is sufficient evidence to establish first order directly from raw concentration-time data, with the rate constant read straight off the fitted slope.

Scope. The identical fitting procedure, tried against all three linearised forms in turn (Step 4), determines the order of any reaction whose kinetics fall cleanly into one of the three cases treated here.

Problems

A reaction's \(1/[A]\) vs. \(t\) plot is found to be linear with slope \(0.0250\,\text{M}^{-1}\text{s}^{-1}\) and intercept \(2.00\,\text{M}^{-1}\). Identify the reaction order, find \(k\) and \([A]_0\), and compute the half-life.
Solution
Linearity of \(1/[A]\) vs. \(t\) indicates second order (Step 3). Comparing to \(1/[A]=1/[A]_0+kt\): \(k=0.0250\,\text{M}^{-1}\text{s}^{-1}\) (the slope), and \(1/[A]_0=2.00\,\text{M}^{-1}\), so \([A]_0=0.500\,\text{M}\). \(t_{1/2}=1/(k[A]_0)=1/(0.0250\times0.500)=1/0.0125=80.0\,\text{s}\).
For the same reaction as Problem 1, if the initial concentration were instead doubled to \(1.00\,\text{M}\), predict the new half-life, and explain qualitatively why it differs from the first case.
Solution
\(t_{1/2}=1/(k[A]_0)=1/(0.0250\times1.00)=40.0\,\text{s}\), exactly half the previous half-life. This matches Step 3's second-order half-life formula, which is inversely proportional to \([A]_0\): doubling the starting concentration for a second-order reaction halves the half-life, the opposite of first-order behaviour, where half-life is independent of starting concentration altogether (Corollaries).
A student obtains concentration-time data that gives a reasonably straight line on both the \([A]\) vs. \(t\) plot and the \(\ln[A]\) vs. \(t\) plot, over a narrow, short time window. Explain, using the Common errors, why this ambiguity can arise, and suggest how to resolve it.
Solution
Over a sufficiently short time interval, any smooth, continuously decreasing function (including a true exponential decay, \(\ln[A]\) genuinely linear) can appear approximately linear when plotted directly as \([A]\) vs. \(t\), simply because a short enough segment of any smooth curve is well-approximated by its local tangent line (Common errors, first bullet). The ambiguity is resolved by collecting data over a substantially longer time window (ideally spanning at least one or two half-lives, or a large fractional change in \([A]\)): only the genuinely correct order's linearised plot will remain straight across the full, extended range, while the others will visibly curve.