Chemical kinetics is the branch of physical chemistry that deals with the rates of chemical reactions and the factors that affect these rates. While thermodynamics tells us whether a reaction can occur and in which direction, kinetics tells us how fast it will occur and by what mechanism. This distinction is crucial: a thermodynamically favourable reaction such as the formation of water from hydrogen and oxygen can proceed so slowly at room temperature that no observable change occurs without a catalyst.
The rate of a reaction is defined as the change in concentration of a reactant or product per unit time. Reaction rates can be expressed either as the average rate over a time interval or as the instantaneous rate at a particular moment. The rate law expresses the dependence of the rate on the concentrations of reactants, and the order of a reaction is determined experimentally from this rate law rather than from the stoichiometric coefficients.
This chapter introduces the concept of rate constants, rate laws, and reaction orders, along with the integrated rate equations for zero, first, and second order reactions. The half-life of a reaction, the Arrhenius equation for temperature dependence, and collision theory with the concept of activation energy complete the picture. These concepts are applied to analyse real reactions, including nuclear decay which follows first-order kinetics.
The rate of a reaction is the change in concentration of a reactant or product per unit time.
For a general reaction: $$aA + bB \rightarrow cC + dD$$ $$\text{Rate} = -\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$$
The units of rate are mol L^-1 s^-1 (or mol/dm^3 per second).
The rate law expresses the relationship between the rate and the concentrations of reactants. $$\text{Rate} = k [A]^x [B]^y$$ where $k$ is the rate constant, and $x$ and $y$ are the orders with respect to A and B.
The overall order of the reaction is the sum of the individual orders: $$\text{Order} = x + y$$
The order of a reaction is determined experimentally, usually by the initial rate method or by the integrated rate equation method. It can be zero, fractional, or a whole number, and it is not necessarily equal to the stoichiometric coefficients.
The units of $k$ depend on the overall order: $$\text{Units of } k = (mol\ L^{-1})^{1-n} s^{-1}$$ where $n$ is the order of the reaction. For a first-order reaction, $k$ has units of s^-1; for a zero-order reaction, mol L^-1 s^-1.
For a zero-order reaction, the rate is independent of the concentration of the reactant: $$[A]t = -k t + [A]_0$$ The half-life of a zero-order reaction is directly proportional to the initial concentration: $$t$$} = \frac{[A]_0}{2k
For a first-order reaction: $$A \rightarrow \text{Products}, \qquad \text{Rate} = k[A]$$ $$\ln \frac{[A]_0}{[A]_t} = k t, \qquad \text{or} \qquad [A]_t = [A]_0 e^{-kt}$$
In terms of the rate constant: $$k = \frac{2.303}{t} \log \frac{[A]_0}{[A]_t}$$
The half-life of a first-order reaction is independent of initial concentration: $$t_{1/2} = \frac{0.693}{k}$$
This is why radioactive decay, which follows first-order kinetics, has a fixed half-life regardless of the amount present.
For the reaction $2A \rightarrow \text{Products}$: $$k = \frac{1}{t} \left( \frac{1}{[A]t} - \frac{1}{[A]_0} \right)$$ The half-life is inversely proportional to the initial concentration: $t = 1/(k[A]_0)$.
The temperature dependence of the rate constant is given by the Arrhenius equation: $$k = A e^{-E_a/RT}$$ where $A$ is the frequency factor, $E_a$ is the activation energy, $R$ is the gas constant, and $T$ is the absolute temperature.
In logarithmic form: $$\ln k = \ln A - \frac{E_a}{RT}, \qquad \text{or} \qquad \log k = \log A - \frac{E_a}{2.303 R T}$$
The activation energy can be calculated from rate constants at two temperatures: $$\log \frac{k_2}{k_1} = \frac{E_a}{2.303 R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)$$
Reactions occur when molecules collide with sufficient energy and correct orientation. Only a small fraction of collisions, called effective collisions, lead to reaction. The activation energy is the minimum energy that colliding molecules must possess for reaction to occur. A catalyst works by lowering the activation energy, thereby increasing the rate constant.
| Order | Integrated Rate Equation | Half-Life | Units of k | Plot for Straight Line |
|---|---|---|---|---|
| Zero | [A]t = -kt + [A]0 | [A]0/2k | mol L^-1 s^-1 | [A] vs t |
| First | log([A]0/[A]t) = kt/2.303 | 0.693/k | s^-1 | log[A] vs t |
| Second | 1/[A]t = kt + 1/[A]0 | 1/(k[A]0) | L mol^-1 s^-1 | 1/[A] vs t |
| Quantity | Formula | Remarks |
|---|---|---|
| Instantaneous rate | -d[A]/dt | Slope of tangent |
| First order k | k = (2.303/t) log([A]0/[A]t) | Time independent constant |
| Half-life first order | t1/2 = 0.693/k | Independent of [A]0 |
| Arrhenius equation | k = A e^(-Ea/RT) | Temperature dependence |
| Activation energy | log(k2/k1) = (Ea/2.303R)(1/T1 - 1/T2) | Two temperatures |
Chemical kinetics reveals how fast reactions proceed and provides the tools to control their speed. The rate law and its constants, determined experimentally, allow chemists to predict how changes in concentration and temperature affect reaction rates. Integrated rate equations and half-life relationships enable quantitative analysis of reaction progress, with first-order kinetics applying to everything from drug metabolism to radioactive decay. The Arrhenius equation and collision theory explain the temperature sensitivity of reactions and the critical role of activation energy. Catalysts, by lowering this energy barrier, are central to industrial chemistry and biochemistry. Together these concepts make kinetics indispensable for understanding and designing chemical processes in research, industry, and medicine.