13.3 Rate Laws and Reaction Order

This section pins down, precisely, what the last two sections only described in words: what a reaction's rate actually is, why a rate exists at all, and how that rate answers each reactant's concentration through a single algebraic expression — the rate law — whose exponents you read from data, never from the balanced equation.

Why Reactions Have a Rate at All

For a balanced reaction \(a\text{A} + b\text{B} \rightarrow c\text{C} + d\text{D}\), the rate is a change in concentration per unit time — divided by each species' stoichiometric coefficient so every species reports the same number:

Reactants disappear, so \(\Delta[\text{A}] < 0\); the leading minus sign is exactly what makes the reported rate a positive number. And because concentration keeps changing throughout a reaction, "the rate" really means an instantaneous rate — the slope of the tangent to the concentration–time curve at one particular moment, not an average over the whole run.

Zoom in far enough and a rate is just a count of successful collisions per second. Molecules react only when they collide hard enough — and lined up right — to muster the activation energy \(E_a\) needed to reach the transition state. Only the fraction of collisions carrying at least \(E_a\) can cross that barrier, and warming the sample enlarges that fraction sharply, which is why rate climbs so fast with temperature. Picture the reacting system paying \(E_a\) to climb to the transition state, then dropping to a lower-energy product — an exothermic step, \(\Delta E < 0\).

Figure — The reacting system must pay \(E_a\) to reach the transition state, then falls to a lower-energy product — an exothermic step, \(\Delta E < 0\).

The Differential Rate Law

Collisions explain why a reaction has a rate at all; the rate law is the compact bookkeeping that says exactly how fast, and how that speed answers each reactant's concentration. For the general reaction \(a\text{A} + b\text{B} \rightarrow \text{products}\), the rate law takes the form

Reading Order From Data

The order in a reactant tells you exactly how the rate answers a change in its concentration:

So where do \(m\) and \(n\) actually come from? Never from \(a\) and \(b\) — inspect a real mechanism and you'll often find the orders don't match the coefficients at all. Chemists instead run the method of initial rates: run several trials, each changing a single reactant's starting concentration, and compare the initial rates. Between two trials that differ only in \([\text{A}]\),

and solving for the exponent \(m\) gives the order in A. Holding A fixed and varying B the same way gives \(n\). Holding every other concentration constant is what makes this work — it isolates one reactant, so the rate ratio depends on that reactant's exponent alone.

From Rate to Concentration in Time

Once every exponent is nailed down, the overall order does more than label the reaction. Integrating the differential law gives concentration as a function of time — and, for each order, a different quantity that plots as a straight line against \(t\):

The trick: whichever of \([\text{A}]\), \(\ln[\text{A}]\), or \(1/[\text{A}]\) plots straight against time tells you the order directly.

Release the same \([\text{A}]_0\) three ways and the concentration falls in three unmistakable shapes: zero order falls in a straight line to zero; first order is an exponential with a constant half-life; second order is a hyperbola — steep at first, then a long tail. Recognizing the shape of a concentration–time curve is often faster than fitting any equation at all.

Figure — Concentration–time decay for the three common orders, all starting from the same \([\text{A}]_0\).

A straight \(1/[\text{A}]\) vs. \(t\) line is the signature of a second-order reaction, matching \(\dfrac{1}{[\text{A}]}=\dfrac{1}{[\text{A}]_0}+kt\), and its slope equals \(k\) itself, in units of \(\text{M}^{-1}\text{s}^{-1}\). For reference, a straight \([\text{A}]\) plot signals zero order, and a straight \(\ln[\text{A}]\) plot signals first order.

Figure — Equal-width half-lives each halve \([\text{A}]\): \([\text{A}]_0 \to [\text{A}]_0/2 \to [\text{A}]_0/4 \to [\text{A}]_0/8\).

Pulling every order together, the overall order fixes both the units \(k\) must carry and how the half-life behaves as the reaction proceeds:

Read it: only the first-order half-life is independent of \([\text{A}]_0\) — a constant \(t_{1/2}\) is the signature of first-order decay.

Recap

The whole section on one page:

• Rate = \(k[\text{A}]^{m}[\text{B}]^{n}\). The orders \(m,n\) are measured, not read from coefficients; \(m+n\) is the overall order and fixes the units of \(k\).
• Get order from data. Vary one concentration at a time and take the rate ratio \(([\text{A}]_2/[\text{A}]_1)^{m}\); or see which of \([\text{A}]\), \(\ln[\text{A}]\), \(1/[\text{A}]\) plots straight in time.
• Shape and half-life reveal order. A constant half-life \(t_{1/2}=\ln 2/k\) — and an exponential decay curve — are the fingerprints of a first-order reaction.

A rate law is measured, not derived: find each order from how the rate answers a change in concentration, and the units, half-life, and decay curve all follow. Next, this rate law turns outward — toward reaction mechanisms, where the slow step in a multi-step pathway is exactly the one this section just taught you to spot from data alone.