Reaction rates and kinetics

Lesson 6 showed how to tell whether a reaction can happen. This lesson asks how fast it happens, which is often the more pressing engineering question. A steel bridge is thermodynamically doomed to rust, but whether that takes five years or a hundred depends on rates. Concrete gains strength over days and weeks at a pace set by temperature; a catalytic converter must clean exhaust within milliseconds; disinfectant chlorine must survive for days in a water main; a sealant must last twenty years on a building, and an engineer must predict that from tests that take a few months. Chemical kinetics, the study of reaction rates, provides the tools: rate laws, integrated rate laws and half-lives, the Arrhenius equation for temperature, reaction mechanisms and catalysts. It also explains some of the most serious industrial hazards, including thermal runaway and dust explosions.

What reaction rate means

The rate of a reaction is how fast reactants are used up or products appear, usually expressed as a change in concentration per unit time, in mol/(L·s). For a reaction aA → bB, the rate is −(1/a) d[A]/dt = (1/b) d[B]/dt; dividing by the coefficients gives one rate for the whole reaction regardless of which substance is measured. Rates generally slow as reactants are consumed.

Four factors control the rate:

• Concentration (or partial pressure): more reactant molecules means more collisions.
• Temperature: a small rise often speeds reactions dramatically.
• Surface area: for reactions involving solids, finely divided material reacts much faster. A log burns slowly; sawdust suspended in air can explode.
• Catalysts: substances that provide a faster pathway without being consumed.

Rate laws

Experiment shows that the rate usually depends on reactant concentrations raised to powers:

Here k is the rate constant, which depends on temperature but not on concentration, and m and n are the reaction orders with respect to A and B; their sum is the overall order. Orders must be measured; they are not, in general, the coefficients in the balanced equation. A common way to find them is the method of initial rates: run the reaction several times, changing one starting concentration at a time, and see how the initial rate responds. If doubling [A] doubles the rate, the reaction is first order in A; if it quadruples the rate, second order; if the rate does not change, zero order.

Integrated rate laws and half-life

A rate law gives the rate at a given moment. Integrating it gives concentration as a function of time, which is usually what an engineer needs. For a first-order reaction, rate = k[A], and:

A plot of ln[A] against time is a straight line of slope −k, the usual test for first-order behavior. The half-life, the time for the concentration to fall to half, is constant for a first-order reaction, whatever the starting concentration:

After one half-life 50 % remains, after two 25 %, after ten about 0.1 %. Radioactive decay is first order: cesium-137, released by nuclear accidents, has a half-life of 30 years, so it takes about three centuries to fall to a thousandth. Many decomposition, decay and drug-elimination processes are first order, or close to it.

For a second-order reaction, rate = k[A]², the integrated law is 1/[A] = 1/[A]₀ + kt, and the half-life, 1/(k[A]₀), grows as the concentration falls. For a zero-order reaction the rate is constant until the reactant runs out, as happens when a catalyst surface is saturated or when the body eliminates alcohol at a fixed rate.

Collision theory and activation energy

For molecules to react, they must collide with enough energy and the right orientation to break existing bonds and form new ones. The minimum energy is the activation energy, E_a. An energy profile plots the energy of the reacting system along the reaction path: reactants climb over an energy barrier, through a short-lived arrangement called the transition state, and descend to products. The height of the barrier above the reactants is E_a; the overall drop from reactants to products is ΔH, which, as Lesson 6 showed, has nothing to do with the rate.

At any temperature, molecular energies are spread over a range, and only the small fraction in the high-energy tail can cross the barrier. Raising the temperature enlarges that fraction sharply, which is why rates are so sensitive to temperature.

The Arrhenius equation

The Swedish chemist Svante Arrhenius found that rate constants follow:

where A, the pre-exponential factor, reflects collision frequency and orientation. Taking logarithms, ln k = ln A − E_a/(RT), so a plot of ln k against 1/T is a straight line of slope −E_a/R. Comparing two temperatures gives the two-point form:

A useful rule of thumb follows: for an activation energy of about 50 kJ/mol, typical of many reactions, the rate roughly doubles for every 10 °C rise near room temperature. This is why food lasts longer in the refrigerator, why batteries age faster in a hot car, and why a fever is dangerous.

Predicting service life from accelerated tests

Materials such as sealants, coatings, plastics, adhesives and battery electrolytes degrade by slow chemical reactions, typically oxidation. Nobody can wait twenty years to find out whether a product will last twenty years, so manufacturers age samples in ovens at several elevated temperatures, measure the degradation rate at each, fit the Arrhenius equation and extrapolate to service temperature.

import math

R = 8.314
# Accelerated aging of a sealant: first-order rate constants (per day) at oven temperatures
data = [(60, 0.00366), (70, 0.00834), (80, 0.0200), (90, 0.0457)]
x = [1 / (t + 273.15) for t, k in data]          # 1/T in 1/K
y = [math.log(k) for t, k in data]               # ln k
n = len(data)
xm, ym = sum(x) / n, sum(y) / n
slope = sum((a - xm) * (b - ym) for a, b in zip(x, y)) / sum((a - xm) ** 2 for a in x)
lnA = ym - slope * xm
Ea = -slope * R
print(f"Activation energy {Ea / 1000:.1f} kJ/mol, A = {math.exp(lnA):.3g} per day")

for t in [25, 40, 60, 90]:
    T = t + 273.15
    k = math.exp(lnA - Ea / (R * T))
    half = math.log(2) / k                        # days to lose half the property
    print(f"{t:3d} C: k = {k:.2e} per day, half-life {half:8.0f} days = {half / 365:5.1f} years")

Activation energy 84.9 kJ/mol, A = 7.39e+10 per day
 25 C: k = 9.71e-05 per day, half-life     7135 days =  19.5 years
 40 C: k = 5.01e-04 per day, half-life     1382 days =   3.8 years
 60 C: k = 3.55e-03 per day, half-life      195 days =   0.5 years
 90 C: k = 4.48e-02 per day, half-life       15 days =   0.0 years

The fitted activation energy, 85 kJ/mol, predicts a half-life of about 20 years at 25 °C but under 4 years at 40 °C, a reminder that a sealant on a sun-baked south-facing roof can run much hotter than the air. Extrapolation assumes the same reaction controls degradation at every temperature; if a different process takes over at high temperature (a polymer softening, a plasticizer evaporating), the prediction can be badly wrong, so good test programs use the lowest oven temperatures practical and confirm with real-world exposure.

Reaction mechanisms

Most reactions do not happen in a single collision but through a sequence of elementary steps, the mechanism. Each elementary step's rate law does follow its own molecularity: a step in which two molecules collide is second order. The slowest step, the rate-determining step, acts like the narrowest point in a pipe and controls the overall rate, which is why experimental rate laws often differ from the overall balanced equation. Combustion, for example, proceeds through dozens of steps involving highly reactive fragments called free radicals; flame retardants work by capturing those radicals and breaking the chain.

Catalysts

A catalyst speeds a reaction by providing an alternative pathway with lower activation energy, and is regenerated at the end. Because it lowers the barrier for the forward and reverse reactions equally, a catalyst does not change ΔH, ΔG or the equilibrium constant: it gets the reaction to equilibrium faster but does not shift where equilibrium lies (Lesson 8).

• Heterogeneous catalysts are in a different phase, usually a solid on which gas or liquid reactants adsorb. A car's three-way catalytic converter uses platinum, palladium and rhodium spread over a ceramic honeycomb with an enormous surface area to convert carbon monoxide, unburned hydrocarbons and nitrogen oxides into CO₂, water and N₂. It works only above its light-off temperature of about 250 to 300 °C, so most of a modern car's emissions come in the first minute after a cold start.
• Homogeneous catalysts are dissolved in the same phase as the reactants, as in many processes that make plastics.
• Enzymes are biological catalysts, proteins that speed specific reactions in living cells by factors of a million or more, and are used industrially in detergents, biofuels and wastewater treatment.

Catalysts can be poisoned: lead in old gasoline coated converter catalysts and destroyed them, which is one reason leaded gasoline was phased out; sulfur in fuel poisons them too.

Rates and safety: runaway reactions and explosions

The exponential dependence of rate on temperature can turn deadly. In an exothermic reaction, heat raises the temperature, which speeds the reaction, which releases heat faster. If the heat is removed fast enough, the reaction settles at a steady temperature; but heat generation rises exponentially with temperature while cooling rises only roughly linearly, so beyond a critical point the reaction runs away. Thermal runaway has caused chemical reactor explosions, fires in stockpiles of coal and oily rags that self-heat, and lithium-ion battery fires, in which internal decomposition reactions accelerate one another once a cell passes roughly 150 °C.

Surface area matters equally. Grain, sugar, flour, coal, wood and metal dusts suspended in air can burn so fast that they explode. Dust explosions have leveled grain elevators and factories; preventing them requires good housekeeping, dust collection, controlling ignition sources and explosion venting. Gas mixtures burn only between their lower and upper flammability limits (5 % and 15 % for methane in air), which is why gas detectors alarm at a fraction of the lower limit.