Chemical Kinetics: Integrated Rate Equations, Temperature and Collision Theory

Measuring instantaneous rate directly means drawing a tangent line on a concentration-time graph and carefully reading its slope, a genuinely inconvenient, imprecise process to repeat for every data point. Integrating the differential rate equation solves this cleanly, converting an equation about instantaneous rate into one relating concentration directly to time and the rate constant, values that can simply be measured and plotted, no tangent-drawing required. This integration works out differently for each reaction order, so this chapter derives it specifically for the two simplest, most instructive cases, zero order and first order, and the payoff is genuinely practical: once integrated, both give an equation of the exact form y = mx + c, meaning plotting the right combination of concentration and time produces a straight line whose slope directly reveals the rate constant k.

For a zero-order reaction, rate stays constant regardless of concentration, and integrating d[R]/dt = -k directly gives [R] = [R]0 - kt, an equation matching the straight-line form y = mx + c exactly, meaning plotting [R] against t gives a genuinely straight line, with slope -k and intercept [R]0. Rearranged, this also gives k = ([R]0 - [R])/t, letting the rate constant be calculated directly from just two concentration measurements at two different times. Zero-order kinetics shows up specifically when a reaction's rate is limited by something other than reactant concentration, such as ammonia decomposing on a hot platinum surface: once that surface is fully saturated with gas molecules, adding more ammonia can't speed the reaction up any further, since the platinum's own limited surface area, not concentration, has become the genuine bottleneck.

For a first-order reaction, integrating d[R]/dt = -k[R] gives ln[R] = ln[R]0 - kt, or equivalently, rearranged into base-10 logarithms, k = (2.303/t)log([R]0/[R]); crucially, plotting raw [R] against t here gives a curve, not a straight line, so it's specifically ln[R] (or log[R]) plotted against t that produces the useful straight line, with slope -k (or -k/2.303) and intercept ln[R]0. Rearranging the integrated equation into [R] = [R]0.e^(-kt) reveals the characteristic exponential decay shape, and this exact mathematics describes some of chemistry's and physics's most important processes: hydrogenation of ethene, the decomposition of N2O5, and, notably, every single case of natural or artificial radioactive decay, all genuinely first-order processes obeying this identical equation.

Half-life, t1/2, is the time needed for a reactant's concentration to fall to exactly half its starting value, and the two integrated rate equations produce genuinely, instructively different behaviour for this single quantity. For a zero-order reaction, substituting [R] = [R]0/2 gives t1/2 = [R]0/(2k), meaning half-life is directly proportional to how much reactant you start with, a more concentrated zero-order reaction genuinely takes longer to reach the halfway point. For a first-order reaction, the same substitution gives t1/2 = 0.693/k (since ln 2 = 0.693), and every single [R]0 term cancels out entirely, meaning half-life is completely independent of starting concentration, a first-order reaction's half-life is a fixed, constant property of that reaction, unchanged whether you start with a little reactant or a lot, which is exactly why radioactive isotopes are always described by one single, unchanging half-life value.

Ethyl acetate's hydrolysis, CH3COOC2H5 + H2O to CH3COOH + C2H5OH, is genuinely a second-order reaction, since it truly depends on the concentration of both the ester and water. But run this hydrolysis with water in large excess, as is typical, and water's own concentration barely changes over the course of the reaction, 10 moles of water dropping to 9.99 moles while 0.01 mole of ester is fully consumed, a genuinely negligible shift. Since [H2O] stays effectively constant throughout, it behaves, for all practical purposes, as if it were absorbed into the rate constant itself, leaving the observed rate dependent on ester concentration alone, exactly mimicking true first-order kinetics. This disguised behaviour is called a pseudo first-order reaction, and cane sugar's acid-catalysed inversion into glucose and fructose is a second classic example, again genuinely higher-order in principle, but observed as first order specifically because water, the second reactant, is present in such overwhelming excess.

N2O5's decomposition takes 12 minutes to reach the halfway point at 50 degrees Celsius, 5 hours at 25 degrees, and a full 10 days at 0 degrees, a genuinely dramatic sensitivity to temperature that Swedish chemist Arrhenius captured precisely: k = A.e^(-Ea/RT), where A is the frequency factor (a constant reflecting how often molecules collide with the right geometry), Ea is activation energy, R is the gas constant, and T is absolute temperature. This single equation captures the well-known rule of thumb that a 10-degree rise in temperature roughly doubles a reaction's rate constant. Taking the natural logarithm of both sides gives ln k = -Ea/(RT) + ln A, an equation of the exact straight-line form y = mx + c, meaning a plot of ln k against 1/T gives a straight line with slope -Ea/R, letting a chemist determine a reaction's own activation energy purely from rate measurements made at two or more different temperatures.

Not every molecular collision leads to a reaction; only those between molecules carrying enough combined energy to reach the activation energy, forming a short-lived, unstable activated complex, can actually proceed to products. Since molecules in any real sample don't all move at the same speed, their kinetic energies spread across a Maxwell-Boltzmann distribution, most molecules clustered near some most-probable energy, with a rapidly shrinking tail extending toward higher energies. Raising temperature doesn't shift this whole curve uniformly; it flattens and broadens it, shifting the peak toward higher energy and, crucially, thickening that high-energy tail specifically at and beyond the activation energy threshold. Since only the area under the curve beyond Ea actually represents molecules capable of reacting, and that specific sliver of area genuinely doubles for a roughly 10-degree temperature rise, the exponential term in the Arrhenius equation, e^(-Ea/RT), is precisely this fraction of successfully energetic molecules, explaining exactly why such a modest temperature change produces such an outsized change in rate.

A catalyst increases a reaction's rate without itself undergoing any permanent chemical change, working through the intermediate complex theory: it forms temporary bonds with reactants, creating a short-lived intermediate that then decomposes to release both the products and the unchanged catalyst, ready to repeat the cycle. The genuine mechanism behind this speedup is providing an entirely alternate reaction pathway with a lower activation energy than the uncatalysed route, and since the Arrhenius equation shows rate rising sharply as Ea falls, even a modest reduction in activation energy can produce a substantial increase in rate. Critically, a catalyst changes only the pathway's activation energy, never the reaction's Gibbs energy, and therefore never its equilibrium constant either; it speeds up the forward and reverse reactions by precisely the same factor, meaning a catalysed reaction reaches the exact same equilibrium position as the uncatalysed one, just considerably sooner.

Collision theory, developed by Trautz and Lewis, treats reacting molecules as simple hard spheres, expressing rate as Rate = Z(AB).e^(-Ea/RT), where Z(AB) is collision frequency (collisions per second per unit volume) and the exponential term is the fraction of those collisions carrying sufficient energy. This equation predicts rate constants reasonably well for simple atomic or small-molecule reactions, but shows genuine, significant deviations for more complex molecules, and the reason is that having enough energy alone isn't sufficient; molecules must also collide with the correct spatial orientation to actually let old bonds break and new ones form. Bromoethane reacting to form methanol, for instance, only proceeds when the two molecules approach with their reactive groups properly aligned; a collision with the wrong orientation simply bounces the molecules apart, unreacted, no matter how much energy they carried. Accounting for this adds a steric factor P (the probability that a collision is correctly oriented), refining the equation to Rate = P.Z(AB).e^(-Ea/RT), so an effective collision genuinely requires both sufficient energy and correct orientation together.