Difficulty: Intermediate | Prerequisites: Chemical Kinetics Fundamentals (Part 1 of these notes), basic algebra
Big picture: This is the quantitative side of kinetics. Where the fundamentals doc covered why reactions go at different speeds, this one covers how to measure and calculate those speeds. You will learn to write rate laws, determine reaction orders from experimental data, use integrated rate laws to predict concentrations over time, and read the characteristic graphs for each order. If you are not comfortable with the Arrhenius equation and collision theory, review Part 1 first.
The rate law tells you how concentration affects speed: Rate = k[A]^x[B]^y, where the exponents (reaction orders) must be found experimentally. Each order has its own integrated rate law, its own characteristic graph shape and its own half-life formula. Learn to match the graph to the order, and you can solve most exam problems on this topic.
Rate of reaction
The change in concentration of a reactant or product per unit time, expressed in mol/L per second (M/s). Think of it as how quickly the reaction mixture is changing.
Rate law
An equation of the form Rate = k[A]^x[B]^y that relates the reaction rate to the concentrations of reactants raised to experimentally determined powers. In simple terms, it is the recipe that tells you how concentration controls speed.
Rate constant (k)
The proportionality constant in the rate law. Its value is fixed for a given reaction at a given temperature, but its units change depending on the overall reaction order.
Reaction order
The exponent on a reactant's concentration in the rate law. It tells you how sensitive the rate is to changes in that reactant's concentration. The overall order is the sum of all individual orders.
Zero-order reaction
A reaction whose rate is independent of reactant concentration. Rate = k. In simple terms, the reaction runs at the same speed no matter how much reactant you have.
First-order reaction
A reaction whose rate is directly proportional to the concentration of one reactant. Rate = k[A]. Doubling [A] doubles the rate.
Second-order reaction
A reaction whose rate depends on the concentration of one reactant squared, or on the product of two reactant concentrations. Rate = k[A]² or Rate = k[A][B].
Integrated rate law
A rearranged form of the rate law that relates concentration to time directly, letting you calculate how much reactant remains after a given period.
Half-life (t₁/₂)
The time required for the concentration of a reactant to fall to half its initial value. For first-order reactions, half-life is constant and independent of starting concentration.
Mixed-order / broken-order reaction
A reaction with non-integer (fractional) reaction orders, or one whose effective order changes as concentrations shift during the course of the reaction.
Equilibrium constant (K_eq)
The ratio of the forward rate constant to the reverse rate constant: K_eq = k_forward / k_reverse. Uses concentrations of all species (products and reactants), unlike the rate law which uses only reactants.
Rate is expressed in mol/L per second (M/s).
It is measured by tracking the change in concentration of reactants or products over time.
For a general reaction aA + bB → cC + dD:
Rate = −(1/a)(Δ[A]/Δt) = −(1/b)(Δ[B]/Δt) = (1/c)(Δ[C]/Δt) = (1/d)(Δ[D]/Δt)
Rates are usually measured at or near the beginning of the reaction to minimise the effect of the reverse reaction.
The rate law takes the form Rate = k[A]^x[B]^y, where k is the rate constant and the exponents are reaction orders.
Reaction orders must be determined experimentally. They are not read off the balanced equation (except in two specific cases below).
The overall reaction order is the sum of the individual exponents.
The stoichiometric coefficients match the reaction orders only when:
The reaction is a single elementary step and the balanced equation represents the entire mechanism.
The complete mechanism is given and the rate-determining step is identified, in which case the stoichiometric coefficients of the reactants in the RDS equal the reaction orders.
Experimental determination: look for pairs of trials where only one reactant's concentration changes while all others remain constant. The change in rate then reveals the order with respect to that reactant.
The rate law uses only reactant concentrations. The equilibrium expression (K_eq) uses all species.
k is constant only for a specific reaction at a specific temperature.
K_eq = k_forward / k_reverse.
Zero order
Rate = k. The rate does not depend on concentration at all.
The only ways to change the rate of a zero-order reaction are to change temperature (which changes k) or add a catalyst (which lowers Eₐ and therefore changes k).
Graph signature: [A] vs time is a straight line with slope = −k.
Integrated rate law: [A] = [A]₀ − kt
Half-life: t₁/₂ = [A]₀ / 2k
Units of k: M/s
First order
Rate = k[A]. The rate is directly proportional to the concentration of one reactant.
Doubling [A] doubles the rate.
A first-order reaction with a single reactant typically involves a molecule undergoing a change on its own (e.g. decomposition, radioactive decay) without colliding with another molecule.
Graph signature: [A] vs time is a curved (exponential) decay. ln[A] vs time is a straight line with slope = −k.
Integrated rate law: [A] = [A]₀ × e^(−kt)
Half-life: t₁/₂ = ln(2) / k ≈ 0.693 / k
The half-life is constant, independent of starting concentration.
Units of k: 1/s (s⁻¹)
Second order
Rate = k[A]² (one reactant) or Rate = k[A][B] (two reactants).
A second-order dependence on a single reactant suggests two molecules of that species must collide.
Graph signature: [A] vs time is a steep curve. 1/[A] vs time is a straight line with slope = +k.
Integrated rate law: 1/[A] = 1/[A]₀ + kt
Half-life: t₁/₂ = 1 / (k[A]₀)
The half-life depends on the initial concentration and gets longer as [A]₀ decreases.
Units of k: 1/(M·s) (M⁻¹s⁻¹)
Higher-order reactions
Third-order and above are rare because the simultaneous collision of three or more particles with proper orientation is highly improbable.
Mixed-order / broken-order reactions
These have non-integer (fractional) reaction orders.
The effective order can change during the course of the reaction as concentrations shift.
Example: Rate = k₁[C][A]² / (k₂ + k₃[A]).
At low substrate concentrations, enzyme-catalysed reactions approximate first-order kinetics (rate proportional to [substrate]). At very high substrate concentrations the enzyme is saturated and the reaction approximates zero-order kinetics (rate independent of [substrate]). This is visible on the classic Michaelis-Menten curve where the rate plateaus at V_max.
General rate law
Rate = k[A]^x[B]^y
Definition of rate (for aA + bB → cC + dD)
Rate = −(1/a)(Δ[A]/Δt) = −(1/b)(Δ[B]/Δt) = (1/c)(Δ[C]/Δt) = (1/d)(Δ[D]/Δt)
Radioactive / first-order decay
[A]_t = [A]₀ × e^(kt) (note: some texts write this with −k in the exponent depending on sign convention)
Summary table: reaction orders at a glance
Order | Rate law | Integrated rate law | Half-life | Units of k | Linear plot |
|---|---|---|---|---|---|
0 (zero) | R = k | [A] = [A]₀ − kt | t₁/₂ = [A]₀ / 2k | M/s | [A] vs t, slope = −k |
1 (first) | R = k[A] | [A] = [A]₀ e^(−kt) | t₁/₂ = ln 2 / k | 1/s | ln[A] vs t, slope = −k |
2 (second) | R = k[A]² | 1/[A] = 1/[A]₀ + kt | t₁/₂ = 1 / (k[A]₀) | 1/(M·s) | 1/[A] vs t, slope = +k |
The linear-plot column is how you identify the order experimentally: whichever transformation of concentration gives a straight line against time tells you the order.
Radioactive decay follows first-order kinetics, which is why radioactive isotopes have a constant half-life regardless of how much material remains. Carbon-14 dating relies on this: the half-life is fixed at roughly 5,730 years, so measuring how much C-14 is left tells you how old a sample is.
Enzyme saturation in the body follows the Michaelis-Menten model. At low drug concentrations, the body eliminates a drug at a rate proportional to its concentration (first-order). At high concentrations the metabolic enzymes are saturated and elimination becomes constant (zero-order), which is why overdoses can be disproportionately dangerous: the body cannot clear the drug any faster.
Students often assume the stoichiometric coefficients in the balanced equation are the reaction orders. They are not, unless the reaction is a single elementary step or you are given the mechanism with the rate-determining step identified.
Students frequently confuse the equilibrium constant (K_eq) with the rate law. K_eq uses concentrations of all species (products and reactants); the rate law uses only reactants.
Students sometimes forget that the units of k change with reaction order. Zero order: M/s. First order: 1/s. Second order: 1/(M·s). Getting the units wrong is a common mark lost on exams.
Students often think half-life is always constant. It is constant only for first-order reactions. For zero-order reactions it depends on [A]₀, and for second-order it depends on [A]₀ as well (and gets longer as the reactant is consumed).
⚠️ Determining reaction order from experimental data (the method of initial rates) is one of the most common exam question types. Practise isolating the effect of one reactant at a time.
⚠️ Know which graph gives a straight line for each order. If the exam gives you a data set, you may need to plot [A], ln[A] and 1/[A] against time and see which is linear.
⚠️ Half-life questions often test whether you know the difference between first-order (constant half-life) and the other orders (half-life depends on [A]₀).
⚠️ Units of k are a reliable check. If your calculated k has the wrong units for the order you determined, something has gone wrong.
⚠️ The connection between the rate law and the mechanism (stoichiometric coefficients of the RDS = reaction orders) is a frequent exam topic. Be clear on when this applies and when it does not.
Fill in the blank: The overall reaction order is the ______ of the individual exponents in the rate law. (sum)
True or False: For a first-order reaction, the half-life depends on the initial concentration. (False. First-order half-life = ln 2 / k, independent of [A]₀.)
Fill in the blank: To identify a second-order reaction graphically, plot ______ vs time and look for a straight line. (1/[A])
True or False: The rate constant k has the same units regardless of reaction order. (False. Units of k change: M/s for zero order, 1/s for first order, 1/(M·s) for second order.)
Fill in the blank: The equilibrium constant K_eq equals k_forward divided by ______. (k_reverse)
Q: In an experiment, doubling [A] while holding [B] constant quadruples the rate. What is the order with respect to A?
A: Second order with respect to A. If doubling [A] multiplies the rate by 4 (which is 2²), the exponent on [A] is 2.
Q: A reaction has the rate law Rate = k[A][B]. What is the overall order, and what are the units of k?
A: The overall order is 1 + 1 = 2 (second order overall). The units of k for a second-order reaction are 1/(M·s) or M⁻¹s⁻¹.
Q: You are given concentration vs time data for a reaction with one reactant. You plot [A] vs t, ln[A] vs t and 1/[A] vs t. The ln[A] vs t plot is linear. What is the reaction order, and what does the slope of that line represent?
A: The reaction is first order. The slope of ln[A] vs t equals −k (the negative of the rate constant).
Q: A first-order reaction has a rate constant of 0.025 s⁻¹. Calculate the half-life.
A: t₁/₂ = ln 2 / k = 0.693 / 0.025 = 27.7 seconds.
Q: For the mechanism below, write the rate law. Step 1 (slow): A₂ + B → A₂B Step 2 (fast): A₂B + B → 2AB
A: The rate-determining step is Step 1. The rate law is written from the reactants of the slow step: Rate = k[A₂][B]. The intermediate A₂B does not appear in the rate law because it is produced and consumed within the mechanism.
Q: Explain why the half-life of a zero-order reaction gets shorter as the reaction proceeds.
A: For zero order, t₁/₂ = [A]₀ / 2k. Each successive "half-life" starts from a lower effective [A]₀, so the time to halve again is shorter. The reaction consumes reactant at a constant rate, so the relative fraction consumed per unit time increases as the total amount shrinks.
This material connects directly to chemical equilibrium: the equilibrium constant is the ratio of the forward and reverse rate constants (K_eq = k_f / k_r), so rate law concepts underpin equilibrium calculations. It also connects back to the fundamentals in Part 1, since the Arrhenius equation explains why k depends on temperature. If you continue to biochemistry, Michaelis-Menten enzyme kinetics is built on the zero-order and first-order behaviour described here, with V_max corresponding to enzyme saturation (zero-order regime) and the initial linear region corresponding to first-order kinetics.
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