The rate law and reaction order

Result

The rate law and reaction order

T-038Home CU-202Threads kinetics

Statement

A reaction's rate typically depends on reactant concentrations through a rate law of the form \(\text{rate}=k[A]^m[B]^n\cdots\), where the exponents \(m,n,\ldots\) (the order with respect to each reactant, summing to the overall reaction order) must be determined experimentally, and are not, in general, equal to the reactant's stoichiometric coefficient in the balanced equation. The rate constant \(k\) and the orders \(m,n\) are found by the method of initial rates: varying one reactant's concentration while holding others fixed, and observing how the initial rate changes.

Why it matters

This is the empirical foundation on which the entire kinetics unit is built, and it establishes a crucial, frequently misunderstood point echoing hess-law's and gibbs-free-energy's own thermodynamics-versus-kinetics distinction: a balanced equation fixes stoichiometry and (via gibbs-equilibrium-constant) equilibrium position, but says nothing directly about rate, since rate depends on the actual reaction mechanism — the specific sequence of molecular-level steps — not merely on the overall, net transformation the balanced equation describes.

Hypotheses

The rate law can be written as a simple product of concentration powers, \(\text{rate}=k[A]^m[B]^n\cdots\).This form fits the overwhelming majority of simple reactions well, but is not universally exact: some more complex reactions (particularly those with an enzyme or catalyst involved, developed further in steady-state-approximation) have rate laws that cannot be written as a simple product of powers at all, instead involving a sum of concentration terms in a denominator. Reaction orders \(m,n\) are determined experimentally, via the method of initial rates, not inferred from the balanced equation's stoichiometric coefficients.Orders reflect the actual reaction mechanism, which for most reactions is not simply "all reactants collide and react together in one step" — a subtlety developed fully in collision-theory and transition-state-theory later in this unit; assuming stoichiometric coefficients directly give reaction orders is one of the single most common errors in introductory kinetics (Fails without).

Proof

\text{rate}_2/\text{rate}_1 = \left(\frac{[A]_2}{[A]_1}\right)^m \quad ([B] \text{ held fixed between trials 1 and 2})

If only \([A]\) is varied between two experimental trials, with \([B]\) held constant, the ratio of measured rates depends only on the ratio of \([A]\) values, raised to the unknown order \(m\). A

m = \frac{\ln(\text{rate}_2/\text{rate}_1)}{\ln([A]_2/[A]_1)}

Taking the logarithm of Step 1 and solving directly gives the order with respect to \(A\); the identical procedure, varying \([B]\) instead while holding \([A]\) fixed, gives \(n\). A

k = \frac{\text{rate}}{[A]^m[B]^n}

Once \(m\) and \(n\) are known (Step 2), substituting any one trial's measured rate and concentrations back into the rate law solves directly for the rate constant \(k\). A

Result

\text{rate}=k[A]^m[B]^n\cdots, \qquad \text{overall order}=m+n+\cdots, \qquad m=\frac{\ln(\text{rate ratio})}{\ln(\text{concentration ratio})}

Reading. A short, purely arithmetic procedure — comparing pairs of experimental trials that isolate one concentration's effect at a time — determines a reaction's complete rate law from measured data alone, without any assumption about mechanism needed in advance.

Scope. Applies to reactions whose rate law does take the simple product-of-powers form (Hypotheses); requires at least as many independent trials as reactants whose order is being determined.

Corollaries & converses

The units of \(k\) depend on the overall reaction order \(p=m+n+\cdots\): for concentration in molarity and time in seconds, \(k\) carries units of \(\text{M}^{1-p}\,\text{s}^{-1}\) — a third-order reaction (\(p=3\)), for instance, has \(k\) in \(\text{M}^{-2}\,\text{s}^{-1}\) (Worked examples).
A reaction can be zero order in a reactant (\(m=0\), meaning rate is entirely independent of that reactant's concentration) even though the reactant is chemically essential to the reaction — a genuinely important and often surprising case, common when a catalyst or a fixed active-site population is saturated (a preview of steady-state-approximation).
Converse: once a rate law is experimentally established, it constrains (though does not by itself uniquely determine) which reaction mechanisms are consistent with the observed kinetics, since a proposed mechanism's derived rate law must match the experimentally measured one exactly (developed further in steady-state-approximation).

Fails without

Assume reaction order equals stoichiometric coefficient, skipping experimental determination (drop Hypotheses' second postulate): \(2\text{N}_2\text{O}_5(g)\to4\text{NO}_2(g)+\text{O}_2(g)\) is a classic, well-documented counterexample — the stoichiometric coefficient of \(\text{N}_2\text{O}_5\) is \(2\), but the experimentally measured rate law is \(\text{rate}=k[\text{N}_2\text{O}_5]^1\), first order, not second order as the coefficient alone might naively suggest. This mismatch is not an anomaly; it reflects that the reaction proceeds through a multi-step mechanism, not a single, direct collision of all species named in the balanced equation.
Compare two trials that vary more than one concentration simultaneously: the simple ratio method (Steps 1–2) requires isolating one reactant's effect at a time; varying two concentrations between compared trials conflates their separate contributions and gives no way to solve for either order individually.

Common errors

Reading reaction orders directly off stoichiometric coefficients rather than determining them experimentally (Fails without, first bullet, and the \(\text{N}_2\text{O}_5\) counterexample).
Confusing "order with respect to a specific reactant" (\(m\) or \(n\) individually) with "overall reaction order" (their sum).
Using generic \(\text{M}^{-1}\text{s}^{-1}\) units for \(k\) regardless of the reaction's actual overall order, rather than the order-dependent units given in the Corollaries.
Comparing trials that vary more than one concentration at once when attempting to isolate a single order (Fails without, second bullet).

Discussion

The systematic, experiment-driven approach to determining rate laws was established chiefly by Jacobus van 't Hoff in his 1884 Études de Dynamique chimique — the same van 't Hoff already credited in chirality-optical-activity for the tetrahedral carbon model and in gibbs-equilibrium-constant for extending Gibbs's thermodynamic framework to chemical equilibrium, illustrating the genuinely broad reach of his contributions across stereochemistry, equilibrium, and kinetics alike. Wilhelm Ostwald, already noted in gibbs-free-energy's Discussion for championing Gibbs's work in Europe, made substantial contributions of his own to physical chemistry and catalysis; van 't Hoff and Ostwald received the first (1901) and third (1909) Nobel Prizes in Chemistry respectively, reflecting how central their contributions were to establishing physical chemistry as a rigorous, quantitative discipline.

The distinction between reaction order and stoichiometric coefficient is one of the clearest, most concrete illustrations available at the introductory level of the broader theme (already raised in hess-law's Discussion) that thermodynamics and kinetics are genuinely separate questions: a balanced equation and its associated thermodynamics (via gibbs-equilibrium-constant) depend only on the overall, net transformation, while the rate law depends on the actual, generally more complicated mechanistic pathway the reaction follows.

Common misconception: that a reaction's rate law can be predicted from its balanced equation alone, the way its equilibrium constant expression can. Equilibrium constant expressions genuinely do follow directly from stoichiometry (gibbs-equilibrium-constant), but rate laws do not — they must always be established experimentally, precisely because they reflect mechanism, which the balanced equation alone does not specify.

Worked examples

Trial	[A] (M)	[B] (M)	rate (M/s)
1	0.10	0.10	\(2.0\times10^{-3}\)
2	0.20	0.10	\(8.0\times10^{-3}\)
3	0.10	0.20	\(4.0\times10^{-3}\)

Comparing trials 1 and 2 (\([A]\) doubled, \([B]\) fixed): \(m=\dfrac{\ln(8.0/2.0)}{\ln(0.20/0.10)}=\dfrac{\ln4}{\ln2}=2\). Comparing trials 1 and 3 (\([B]\) doubled, \([A]\) fixed): \(n=\dfrac{\ln(4.0/2.0)}{\ln(0.20/0.10)}=\dfrac{\ln2}{\ln2}=1\). A

\text{rate}=k[A]^2[B]^1, \qquad k=\frac{2.0\times10^{-3}}{(0.10)^2(0.10)}=2.0\,\text{M}^{-2}\text{s}^{-1}

Overall order \(=2+1=3\) (third order); the units of \(k\) (\(\text{M}^{-2}\text{s}^{-1}\)) exactly match the Corollaries' prediction \(\text{M}^{1-p}\text{s}^{-1}\) for \(p=3\): \(\text{M}^{1-3}=\text{M}^{-2}\). A

\text{rate}=(2.0\,\text{M}^{-2}\text{s}^{-1})[A]^2[B]^1 \quad (\text{overall order } 3)

Reading. Three experimental trials, each varying only one concentration at a time, fully determine the rate law's exponents and rate constant with no ambiguity.

Scope. The identical procedure extends to reactions with any number of reactants, given a sufficient number of appropriately designed trials.

Problems

Using the data \(\text{Trial 1: }[X]=0.050\,\text{M},[Y]=0.050\,\text{M},\text{rate}=5.0\times10^{-4}\,\text{M/s}\); \(\text{Trial 2: }[X]=0.100\,\text{M},[Y]=0.050\,\text{M},\text{rate}=2.0\times10^{-3}\,\text{M/s}\); \(\text{Trial 3: }[X]=0.050\,\text{M},[Y]=0.100\,\text{M},\text{rate}=5.0\times10^{-4}\,\text{M/s}\), determine the rate law and the value and units of \(k\).
Solution
Trials 1–2 (\([X]\) doubled): \(a=\dfrac{\ln(2.0\times10^{-3}/5.0\times10^{-4})}{\ln(0.100/0.050)}=\dfrac{\ln4}{\ln2}=2\). Trials 1–3 (\([Y]\) doubled, rate unchanged): \(b=\dfrac{\ln(1)}{\ln2}=0\), zero order in \(Y\). Rate law: \(\text{rate}=k[X]^2\) (overall order \(2\)). \(k=\dfrac{5.0\times10^{-4}}{(0.050)^2}=0.20\,\text{M}^{-1}\text{s}^{-1}\), consistent with the Corollaries' predicted units for a second-order reaction (\(\text{M}^{1-2}=\text{M}^{-1}\)).
A reaction has overall order \(4\). State the units of its rate constant \(k\), using concentration in molarity and time in seconds.
Solution
By the Corollaries, \(k\) has units \(\text{M}^{1-p}\text{s}^{-1}\) with \(p=4\): \(\text{M}^{1-4}\text{s}^{-1}=\text{M}^{-3}\text{s}^{-1}\).
Explain why the experimentally observed first-order rate law for \(2\text{N}_2\text{O}_5(g)\to4\text{NO}_2(g)+\text{O}_2(g)\) (rate \(=k[\text{N}_2\text{O}_5]^1\), not second order) is not a contradiction of the balanced equation, but rather evidence about the reaction's mechanism.
Solution
The balanced equation specifies only the overall stoichiometric relationship between reactants consumed and products formed — it makes no claim about how many molecules must collide in a single step, or how many distinct steps the reaction actually proceeds through. A first-order rate law for a reaction with stoichiometric coefficient \(2\) indicates that the reaction does not proceed via a single step in which two \(\text{N}_2\text{O}_5\) molecules collide together; instead, it proceeds through a multi-step mechanism where only one \(\text{N}_2\text{O}_5\) molecule is involved in the rate-determining (slowest) step, consistent with the observed first-order kinetics. This is entirely compatible with, not contradictory to, the balanced equation's overall stoichiometry.
A catalysed reaction is found experimentally to be zero order in its substrate at high substrate concentration. Explain how this is possible, given that the substrate is chemically necessary for the reaction to occur at all.
Solution
Zero order in a reactant (Corollaries, second bullet) means the rate does not change further as that reactant's concentration increases, not that the reactant is unnecessary. In catalysed reactions, this typically occurs once the catalyst's available active sites become fully occupied (saturated) — adding more substrate beyond this point cannot increase the rate further, since the catalyst, not the substrate, has become the limiting factor, even though every individual reaction event still absolutely requires the substrate to be present. This is a preview of the more detailed treatment in steady-state-approximation.