Topic 5: Kinetics

Thermodynamics tells you whether a reaction can happen, but kinetics tells you how quickly it actually does. This unit builds the tools to measure reaction rates, express them mathematically with rate laws, and explain them through molecular collisions and step-by-step mechanisms.

Reaction Rate and Concentration

The rate of a reaction describes how fast reactant concentrations fall or product concentrations rise over time, usually reported in units of molarity per second (M/s). For a general reaction aA + bB -> cC + dD, the rate can be written using any species, but each must be divided by its stoichiometric coefficient so that one number describes the whole reaction: rate = -(1/a)(d[A]/dt) = -(1/b)(d[B]/dt) = +(1/c)(d[C]/dt) = +(1/d)(d[D]/dt). Reactant terms carry a negative sign because their concentrations decrease. The instantaneous rate is the slope of the tangent line to a concentration-versus-time curve at a single moment, while the average rate is the slope of a line connecting two points over an interval. Rates generally slow down as a reaction proceeds because reactant concentrations drop.

Rate Law and Reaction Order

A rate law expresses how the reaction rate depends on reactant concentrations: rate = k[A]^m[B]^n. Here k is the rate constant, and the exponents m and n are the reaction orders with respect to each reactant. These orders are determined experimentally and are NOT taken from the balanced equation coefficients. The overall reaction order is the sum of the individual exponents (m + n). A zero-order reactant does not affect the rate; doubling a first-order reactant doubles the rate; doubling a second-order reactant quadruples the rate. The rate constant k is temperature dependent, and its units change with overall order: M/s for zero order, 1/s for first order, and 1/(M*s) for second order.

Method of Initial Rates (Worked Example)

To find a rate law from data, compare experiments in which only one concentration changes at a time and watch how the initial rate responds. Consider the reaction 2NO + O2 -> 2NO2 with these trials: Exp 1: [NO]=0.10, [O2]=0.10, rate=0.0025 M/s. Exp 2: [NO]=0.20, [O2]=0.10, rate=0.010 M/s. Exp 3: [NO]=0.10, [O2]=0.20, rate=0.0050 M/s. From Exp 1 to 2, [NO] doubles while [O2] is fixed, and the rate increases by a factor of 4 (0.010/0.0025 = 4 = 2^2), so the order in NO is 2. From Exp 1 to 3, [O2] doubles while [NO] is fixed, and the rate doubles (0.0050/0.0025 = 2 = 2^1), so the order in O2 is 1. The rate law is rate = k[NO]^2[O2], overall third order. Solve for k using Exp 1: k = rate / ([NO]^2[O2]) = 0.0025 / ((0.10)^2 * 0.10) = 0.0025 / 0.001 = 2.5 M^-2 s^-1.

Integrated Rate Laws and Straight-Line Plots

Integrated rate laws relate concentration directly to time and provide a powerful graphical test for reaction order. For zero order, [A] = -kt + [A]0, so a plot of [A] versus t is linear with slope -k. For first order, ln[A] = -kt + ln[A]0, so a plot of ln[A] versus t is linear with slope -k. For second order, 1/[A] = kt + 1/[A]0, so a plot of 1/[A] versus t is linear with slope +k. The strategy on the exam is simple: take a single set of concentration-time data, make all three plots, and whichever one gives a straight line reveals the order. Remember that only the first-order plot uses the natural log, and only the second-order plot uses the reciprocal with a positive slope.

Half-Life

Half-life (t1/2) is the time required for a reactant concentration to fall to half its starting value. For a first-order reaction, the half-life is constant and independent of concentration: t1/2 = 0.693/k. This constant half-life is the signature of first-order kinetics, which is why radioactive decay (always first order) is described by half-lives. After n half-lives, the fraction of reactant remaining is (1/2)^n. For zero-order and second-order reactions the half-life is not constant: zero-order half-lives shorten as the reaction proceeds, while second-order half-lives lengthen. Recognizing a constant half-life in data is a fast way to identify a first-order process.

Collision Model and the Arrhenius Equation

The collision model states that reactions occur only when particles collide with sufficient energy and the correct orientation. The minimum energy needed for a successful collision is the activation energy (Ea). Raising temperature increases both the frequency of collisions and, more importantly, the fraction of molecules with energy at or above Ea, which is why rates climb sharply with temperature. The Arrhenius equation captures this: k = A * e^(-Ea/RT), where A is the frequency (pre-exponential) factor incorporating collision frequency and orientation, R is the gas constant, and T is absolute temperature. Taking the natural log gives a linear form, ln k = -(Ea/R)(1/T) + ln A, so a plot of ln k versus 1/T is a straight line with slope -Ea/R. This lets you extract activation energy from experimental rate constants at different temperatures.

Energy Profiles and Activation Energy

A reaction energy profile (reaction coordinate diagram) plots potential energy against the progress of the reaction. Reactants start at one energy level and products end at another; the difference between them is the enthalpy change (delta H), which is negative for exothermic and positive for endothermic reactions. Between reactants and products lies an energy hump whose peak corresponds to the transition state, an unstable, high-energy arrangement of atoms. The height of the hump measured from reactants is the forward activation energy, and from products it is the reverse activation energy. A catalyst lowers the activation energy by providing a new pathway with a lower hump, but it does not change delta H, the relative energies of reactants and products, or the equilibrium position.

Reaction Mechanisms, Rate-Determining Step, and Intermediates

Most reactions occur through a sequence of simple steps called elementary steps, and this sequence is the reaction mechanism. The sum of all elementary steps must equal the overall balanced equation. Unlike the overall reaction, an elementary step's rate law CAN be written directly from its molecularity (the number of particles colliding in that step): a unimolecular step is first order, a bimolecular step is second order. The slowest elementary step is the rate-determining step, and it controls the overall rate; the experimentally observed rate law must be consistent with this step. A species produced in one step and consumed in a later step is an intermediate. Intermediates appear in the mechanism but not in the overall equation, and they must not appear in the final rate law (substitute them out using a fast equilibrium step if needed).

Catalysis

A catalyst speeds up a reaction by providing an alternative mechanism with a lower activation energy, and it is regenerated by the end so it is not consumed overall. Because it lowers Ea for both the forward and reverse directions equally, a catalyst increases the rate of both without shifting equilibrium or changing delta H. Homogeneous catalysts exist in the same phase as the reactants (for example, an aqueous catalyst in an aqueous reaction), while heterogeneous catalysts are in a different phase, such as a solid metal surface on which gaseous reactants adsorb and react. A catalyst typically appears as a reactant in an early elementary step and is reformed as a product in a later step, the mirror image of how an intermediate behaves.