Rate of Reaction and Rate Laws — Notes

Chemical kinetics studies reaction rates, the factors that affect them and reaction mechanisms. Ionic reactions (e.g. Ag⁺ + Cl⁻ → AgCl) are very fast because no bonds need breaking; molecular reactions are slower because covalent bonds must break.

Average rate $= -\Delta[\ce{R}]/\Delta t = \Delta[\ce{P}]/\Delta t$; instantaneous rate $= -d[\ce{R}]/dt$, found as minus the slope of the tangent to the concentration–time curve. For $\ce{aA + bB -> cC + dD}$: rate $= -\frac{1}{a}\frac{d[\ce{A}]}{dt} = -\frac{1}{b}\frac{d[\ce{B}]}{dt} = \frac{1}{c}\frac{d[\ce{C}]}{dt} = \frac{1}{d}\frac{d[\ce{D}]}{dt}$. Units: mol L⁻¹ s⁻¹.

Factors: nature of reactants (ionic fast, covalent slower); surface area (powders faster); concentration or pressure (more collisions); temperature (rate roughly doubles per 10 K rise); catalyst (lower activation energy); radiation (photochemical reactions such as H₂ + Cl₂ in sunlight, photosynthesis, fading of dyes).

Law of mass action (Guldberg and Waage): at constant temperature, the rate of a reaction is proportional to the product of the active masses of the reactants, each raised to its stoichiometric coefficient. Active mass = molar concentration; for pure solids and liquids it is taken as 1. Strictly valid for elementary reactions.

Rate law: rate $= k[\ce{A}]^x[\ce{B}]^y$, with $x$, $y$ found experimentally (not from coefficients). Rate constant $k$: rate at unit concentrations; depends on temperature and catalyst, not on concentration. Order $= x + y$: may be 0, a whole number or a fraction.

Method of initial rates: compare runs in which only one concentration changes; rate ratio = (concentration ratio)ⁿ.

Units of k: $(\text{mol L}^{-1})^{1-n}\ \text{s}^{-1}$ — zero order mol L⁻¹ s⁻¹; first order s⁻¹; second order L mol⁻¹ s⁻¹; third order L² mol⁻² s⁻¹.

Rate $= k$. Integrated: $[\ce{R}] = [\ce{R}]_0 - kt$; $k = ([\ce{R}]_0 - [\ce{R}])/t$. Plot of [R] against t: straight line, slope $-k$. Half-life $t_{1/2} = [\ce{R}]_0/2k$ (proportional to initial concentration). Examples: decomposition of NH₃ on hot Pt at high pressure; enzyme reactions at saturation.

Derivation: $-d[\ce{R}]/dt = k[\ce{R}]$ → $d[\ce{R}]/[\ce{R}] = -k\,dt$ → $\ln([\ce{R}]_0/[\ce{R}]) = kt$, so $k = \dfrac{2.303}{t}\log\dfrac{[\ce{R}]_0}{[\ce{R}]}$ and $[\ce{R}] = [\ce{R}]_0e^{-kt}$. Plot of log[R] against t: straight line, slope $-k/2.303$.

Half-life: $t_{1/2} = 0.693/k$, independent of $[\ce{R}]_0$; $k$ is also independent of initial concentration. After $n$ half-lives, $[\ce{R}] = [\ce{R}]_0/2^n$. Time for 99.9% reaction ≈ $10\,t_{1/2}$.

Examples: radioactive decay, decomposition of N₂O₅, elimination of many drugs. Pseudo-first-order: one reactant in large excess (acid hydrolysis of an ester in water; inversion of cane sugar).

Elementary reaction: one step; complex reaction: several steps (mechanism). Molecularity: number of reacting species colliding in an elementary step — 1, 2 or (rarely) 3, never 0 or fractional; defined only for elementary steps. Order: experimental, for the overall reaction; may be 0 or fractional.

The slowest step is the rate-determining step; the rate law follows from it. Intermediates are eliminated using a fast pre-equilibrium. Example: NO₂ + CO → NO + CO₂ at low T: slow step 2NO₂ → NO₃ + NO gives rate = k[NO₂]².