Reaction Order & Rate Law Calculator
Determine the order of a reaction and calculate the rate constant from experimental concentration-time data or initial rate experiments.
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Reaction Kinetics Guide
What should I know about Orders of Reaction?
The order of a reaction describes how the rate depends on reactant concentrations. Rate = k[A]ⁿ. Zero order (n=0): rate is constant, independent of concentration. First order (n=1): rate is directly proportional to [A] — doubling [A] doubles the rate. Second order (n=2): rate proportional to [A]² — doubling [A] quadruples the rate. The overall order = sum of all individual orders. Order must be determined experimentally — it cannot be deduced from the balanced equation (except for elementary steps, where the order does match the stoichiometric coefficients by definition). For any multi-step mechanism, the order must be found from experimental data — typically by the initial rates method, varying one reactant's concentration at a time and measuring how the initial rate responds.
What do I need to know about Initial Rates Method?
Compare two experiments where only [A] changes. Order = log(rate₂/rate₁) / log([A]₂/[A]₁). Example: [A] doubles (×2) and rate quadruples (×4). log(4)/log(2) = 2 → second order. If rate doubles when [A] doubles: log(2)/log(2) = 1 → first order. Rate constant from rate law: k = rate / [A]ⁿ. Units of k depend on order: zero order k in mol/L/s, first order in s⁻¹, second order in L/mol/s.
What's the key thing to understand about Integrated Rate Laws?
Zero order: [A] = [A]₀ − kt. Plot [A] vs t → straight line. t½ = [A]₀/2k. First order: ln[A] = ln[A]₀ − kt. Plot ln[A] vs t → straight line. t½ = 0.693/k (constant, independent of concentration). Second order: 1/[A] = 1/[A]₀ + kt. Plot 1/[A] vs t → straight line. t½ = 1/(k[A]₀). The linearisation technique: try each plot — the one that gives a straight line reveals the order. First order is most common for elementary reactions.
What's the key thing to understand about Mechanism and Rate-Determining Step?
The rate law reflects the slow (rate-determining) step of the mechanism. If mechanism is A → B (slow), then B → C (fast): rate = k[A] (first order in A). If mechanism is A + A → B (slow), then B → products (fast): rate = k[A]² (second order). Intermediates (species produced then consumed) do not appear in the overall rate law. This connection between mechanism and rate law is how chemists propose and test reaction mechanisms experimentally.
What should I know about Catalysts and Reaction Order?
A catalyst speeds up a reaction by providing an alternative, lower-activation-energy pathway, but it does not necessarily change the overall order with respect to the original reactants — it can, however, introduce the catalyst's own concentration into the rate law if the catalysed step is rate-determining. In acid-catalysed reactions, the rate law often includes [H+] as a separate term, even though H+ is regenerated at the end of the mechanism and does not appear in the overall balanced equation; such species appearing in the rate law but absent from the stoichiometric equation is evidence they participate in the rate-determining step. Enzyme-catalysed reactions (Michaelis-Menten kinetics) show an apparent order that changes with substrate concentration: first order in substrate at low concentration (unsaturated enzyme) but zero order at high concentration (enzyme saturated) — a single reaction can therefore show different orders depending on conditions, which is why reaction order is always an experimentally determined quantity rather than something read off the balanced equation.
What's the key thing to understand about Pseudo-Order Reactions?
A pseudo-order reaction is one where the true rate law depends on multiple reactants, but the experiment is designed so only one concentration changes significantly, making the reaction appear to follow a simpler order. The classic example is acid hydrolysis of an ester in a large excess of water: rate = k[ester][H2O], but because water is both the solvent and in such vast excess that its concentration barely changes, this simplifies to rate = k'[ester], where k' = k[H2O] is the pseudo-first-order rate constant. This 'flooding' technique — using a large excess of all reactants except the one being studied — is a standard experimental strategy for determining the order with respect to a single species without several concentrations changing simultaneously. It is widely used in pharmacokinetics (drug degradation where enzyme or cofactor concentrations are effectively constant) and in aqueous solution reactions generally, since water's concentration (about 55.5 mol/L) swamps that of almost any dissolved reactant.
What should I know about Half-Life and Reaction Order?
A reaction's half-life behaves very differently depending on its order. For a first-order reaction, t₁⁄₂ = ln2/k is a true constant, completely independent of starting concentration — which is exactly why radioactive decay (a first-order process) always has the same, fixed half-life no matter how much material you start with. For a second-order reaction, t₁⁄₂ = 1/(k[A]₀) depends directly on initial concentration: more concentrated starting material takes less time to halve, so successive half-lives of the same reaction actually get longer and longer as the reaction proceeds and concentration drops.
Worked Example: Finding Reaction Order from Initial Rates Data
The method of initial rates determines reaction order by comparing how the rate changes when a reactant's concentration changes, holding other reactants constant. Consider this data for a reaction A → products:
| Experiment | [A] (M) | Initial Rate (M/s) |
|---|---|---|
| 1 | 0.10 | 0.0020 |
| 2 | 0.20 | 0.0080 |
| 3 | 0.40 | 0.0320 |
Comparing experiments 1 and 2: concentration doubles (0.10 → 0.20) and rate quadruples (0.0020 → 0.0080), a factor of 4. Since 4 = 2^n, solving gives n = 2 — the reaction is second order in A. Comparing experiments 2 and 3 confirms it: concentration doubles again and rate again quadruples (0.0080 → 0.0320). With the order known, the rate law is rate = k[A]². Solving for k using experiment 1: k = 0.0020 / 0.10² = 0.0020 / 0.01 = 0.20 M⁻¹s⁻¹. Checking with experiment 3: k = 0.0320 / 0.40² = 0.0320 / 0.16 = 0.20 M⁻¹s⁻¹ — the same value, confirming the rate law.
Reference Table: Half-Life Formulas by Reaction Order
How a reaction's half-life behaves as the reaction proceeds is itself a diagnostic clue to its order:
| Order | Half-Life Formula | Behavior as Reaction Proceeds |
|---|---|---|
| 0 | t½ = [A]₀ / (2k) | half-life shrinks as initial concentration decreases between trials |
| 1 | t½ = 0.693 / k | constant, independent of concentration |
| 2 | t½ = 1 / (k[A]₀) | half-life grows as the reaction proceeds (each successive half-life is longer) |
| 3 | t½ = 3 / (2k[A]₀²) | half-life grows even more sharply |
The first-order row explains why first-order kinetics (such as radioactive decay or many drug elimination processes) are described with a single constant half-life no matter how much material is present — that property is unique to first order and is often used as a quick experimental test: if measured half-lives stay constant as a reaction proceeds, the reaction is first order; if they visibly lengthen over time, it's second order or higher.
Reference Table: Linear Plots Used to Identify Reaction Order
Beyond comparing initial rates, chemists often identify reaction order graphically, by plotting concentration data in a form that becomes a straight line only for the correct order:
| Order | Integrated Rate Law | Plot That Gives a Straight Line | Slope |
|---|---|---|---|
| 0 | [A] = [A]₀ − kt | [A] vs. t | −k |
| 1 | ln[A] = ln[A]₀ − kt | ln[A] vs. t | −k |
| 2 | 1/[A] = 1/[A]₀ + kt | 1/[A] vs. t | +k |
In practice, a chemist plots concentration-vs-time data all three ways and sees which one produces a straight line; whichever plot is linear reveals the order directly, and the slope of that line gives the rate constant k without needing a separate initial-rates experiment. This graphical method is often preferred over the initial-rates method because it uses an entire data set rather than just two or three points.
Zero-Order Kinetics: When Rate Doesn't Depend on Concentration
A zero-order reaction has a constant rate that doesn't change as concentration drops, which often happens when a catalyst or enzyme is saturated with substrate so that reaction speed is limited by the catalyst, not by how much reactant is available. For a zero-order reaction with rate constant k = 0.50 M/s and starting concentration [A]₀ = 2.0 M, the half-life is t½ = [A]₀ / (2k) = 2.0 / (2 × 0.50) = 2.0 / 1.0 = 2.0 s. Unlike first-order half-life, this value depends directly on the starting concentration — doubling [A]₀ would double the half-life, since the constant rate simply takes proportionally longer to use up a larger starting amount.
Third-Order Reactions: Rare but Real
Third-order reactions, where rate depends on the cube of a single concentration (or the product of three separate concentration terms), are uncommon but do occur, usually in gas-phase reactions involving three-body collisions, such as 2NO + O2 → 2NO2. Because three species must collide simultaneously with the correct orientation and energy, third-order reactions tend to be slower and rarer than first- or second-order ones, and their half-life (t½ = 3/(2k[A]₀²)) shrinks with the square of the starting concentration, making them extremely sensitive to how concentrated the reactants are.