Rate Law Calculator
Solve the classic rate-law relation r = k · [A]ⁿ for any of its four unknowns — reaction rate, rate constant, reactant concentration, or reaction order — with step-by-step math, a live reaction vessel schematic, and companion kinetics metrics including half-life and characteristic time.
k carries the temperature dependence
The rate constant k is not actually constant — it changes strongly with temperature following the Arrhenius law k = A·exp(−Eₐ/RT). For a typical activation energy of 50 kJ/mol, k increases by roughly a factor of 7 between 25 °C and 50 °C. This is why reactions are heated to speed them up, why food spoils faster in the heat, and why pharmaceutical shelf lives are specified with a temperature. The rate law's linear dependence on k means the rate itself follows the same exponential temperature dependence.
Order n controls the concentration sensitivity
The exponent n determines how strongly the rate responds to changes in concentration. For n = 0 the rate is completely independent of concentration — doubling [A] changes nothing. For n = 1 the rate doubles when [A] doubles. For n = 2 it quadruples. This is why reaction order is such a powerful diagnostic: measuring the rate at several concentrations reveals n, which in turn reveals the molecular mechanism. A first-order reaction proceeds via a unimolecular step; a second-order reaction involves a bimolecular collision.
Half-life depends on order
The half-life of a reaction — the time for half the reactant to be consumed — is a direct consequence of the rate law. For a first-order reaction, t½ = ln(2)/k, independent of concentration, which is why radioactive decay has a fixed half-life. For a second-order reaction, t½ = 1/(k·[A]₀), inversely proportional to the initial concentration, so a second-order reaction slows down as it proceeds. For zero-order, t½ = [A]₀/(2k), directly proportional to the initial concentration, as in saturated enzyme kinetics.
Rate laws are empirical, not stoichiometric
A common mistake is to read the reaction order directly off the balanced equation. The stoichiometric coefficients tell you the mole ratios, not the rate law. The decomposition 2 N₂O₅ → 4 NO₂ + O₂ is first-order, not second. The reaction H₂ + Br₂ → 2 HBr has a complex rate law with a [HBr] inhibition term. The order is determined experimentally by the method of initial rates, by integrated rate-law plots, or by half-life analysis, and only then should the mechanism be inferred from it.
How to Use This Calculator
The rate law links four quantities: r, k, [A] and n. Pick which one you want the tool to solve, then fill in the other three.
1. Choose what to solve for
Pick r, k, [A] or n from the toggle. The calculator rearranges r = k·[A]ⁿ for whichever quantity you select, using logarithms when solving for n.
2. Use realistic presets
Rate presets span 10⁻⁷ to 0.1 mol/(L·s), k presets span 10⁻⁴ to 100, concentration presets 10⁻⁴ to 10 mol/L, and order presets cover 0, 0.5, 1, 1.5, 2, 3.
3. Match units carefully
r is in mol/(L·s), k in units that depend on n, [A] in mol/L, and n is dimensionless. Sliders for r, k and [A] are logarithmic for order-of-magnitude control; the slider for n is linear across 0–4.
4. Read the result and verdict
The result card shows the solved value plus equivalents — mol/(L·h) for rate, half-life for k, mol/m³ for concentration, and a classification of the reaction order for n. A verdict badge classifies the rate magnitude.
5. Check the schematic and charts
The schematic shows the reactant in the vessel, the rate law box, the flux of rate out, and a concentration–time decay curve. The chart plots rate against concentration for three values of k.
6. Switch to Advanced for depth
Advanced mode adds a chart of rate against concentration for three reaction orders, a sensitivity check on a +1 % input change, and companion metrics including half-life, characteristic time, k units and reactant consumed per hour.
What This Tool Offers
Built around the classic rate law r = k · [A]ⁿ, the fundamental link between a reaction's rate constant, its concentration sensitivity, and the rate at which it proceeds.
Four-way solving
Solve for reaction rate, rate constant, concentration, or reaction order. Solving for n uses logarithms; the other three are direct algebra.
Live reaction schematic
A reaction vessel with A molecules, a rate-law box showing k and n, an emerging rate arrow whose thickness scales with rate, and a sketch of the concentration decay curve.
Rate-class verdict
Each result is classified from very slow (geological or shelf-life timescales) through slow, moderate and fast, up to very fast or explosive, with a coloured badge.
Logarithmic & linear sliders
r, k and [A] use logarithmic sliders spanning many orders of magnitude; n uses a linear slider from 0 to 4 so fractional orders such as 1.5 are easy to dial in.
Dual comparison charts
Rate against concentration for three values of k, plus an advanced chart of rate against concentration for three reaction orders, showing how the curve shape changes.
Reference tables
Typical rate constants by reaction order, a dynamic rate lookup grid at the current k and n, and a full variable reference table with unit notes.
Reference Tables & Conversions
A rate constant unit note, a dynamic rate lookup grid, and typical rate constants by order and reaction.
Rate constant k — units by reaction order
Because k must make both sides of r = k·[A]ⁿ carry the same units of mol/(L·s), the units of k depend on n. Values below are for the same numeric k = 0.1.
| Order n | Units of k | k = 0.1 means | Half-life relation |
|---|---|---|---|
| 0 | mol/(L·s) | 0.1 mol/(L·s) consumed regardless of [A] | t½ = [A]₀ / (2k) |
| 0.5 | mol^0.5 / (L^0.5 · s) | Fractional order — usually empirical | t½ = (√2 − 1)·[A]₀^0.5 / k |
| 1 | s⁻¹ | 0.1 s⁻¹ — 10 % of A reacts per second | t½ = ln(2) / k |
| 1.5 | L^0.5 / (mol^0.5 · s) | Fractional order — often surface-driven | No simple closed form |
| 2 | L/(mol·s) | 0.1 L/(mol·s) — bimolecular rate constant | t½ = 1 / (k·[A]₀) |
| 3 | L²/(mol²·s) | 0.1 L²/(mol²·s) — termolecular (rare) | t½ = 3 / (2·k·[A]₀²) |
Reaction rate r (mol/(L·s)) at k = 0.1000 and n = 1.00
Computed from r = k · [A]ⁿ. Rows are concentration [A], columns are rate constant k multipliers.
| [A] (mol/L) | k × 0.1 | k × 0.5 | k × 1 | k × 2 | k × 5 | k × 10 |
|---|
Typical Rate Constants & Orders
Representative values at 25 °C unless noted. Rate constants depend strongly on temperature, solvent, and catalyst.
| Reaction | Order (n) | k (typical) | Notes |
|---|---|---|---|
| N₂O₅ → NO₂ + O₂ (gas) | 1 | ~3×10⁻⁵ s⁻¹ | Classic first-order gas decomposition |
| H₂O₂ → H₂O + ½O₂ (cat.) | 1 | ~10⁻³–10⁻¹ s⁻¹ | Catalase / MnO₂ catalysed |
| Sucrose + H₂O → glucose + fructose | 1 (in [sucrose]) | ~10⁻⁴ s⁻¹ (25 °C, pH 3) | Acid-catalysed inversion |
| CH₃Br + OH⁻ → CH₃OH + Br⁻ | 2 | ~2×10⁻⁴ L/(mol·s) | SN2 bimolecular |
| H₂ + I₂ → 2 HI (gas) | 2 | ~2×10⁻² L/(mol·s) (400 °C) | Classic second-order gas reaction |
| 2 NO₂ → 2 NO + O₂ | 2 | ~3×10⁻² L/(mol·s) (300 °C) | Second-order decomposition |
| NH₃ on hot W (decomp.) | 0 | ~10⁻⁶ mol/(L·s) | Zero-order — surface saturated |
| Enzyme (Michaelis–Menten) at saturation | 0 | Varies (Vmax) | Rate = Vmax when [S] >> Km |
| H₂ + Br₂ → 2 HBr | Complex (1.5 / −1) | — | Chain reaction with HBr inhibition |
| CH₃CHO → CH₄ + CO | 1.5 | ~10⁻³ L^0.5/(mol^0.5·s) | Free-radical chain — fractional order |
| Radioactive decay (¹⁴C) | 1 | 3.83×10⁻¹² s⁻¹ | t½ = 5730 years |
| Radioactive decay (²³⁸U) | 1 | 4.92×10⁻¹⁸ s⁻¹ | t½ = 4.47 billion years |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| r | Reaction rate | 10⁻⁸ – 10 | mol/(L·s) |
| k | Rate constant | 10⁻⁶ – 10² (depends on n) | (mol/L)^(1−n)·s⁻¹ |
| [A] | Reactant concentration | 10⁻⁶ – 10 | mol/L |
| n | Reaction order | 0 – 3 (sometimes fractional) | dimensionless |
| A | Arrhenius pre-exponential factor | 10⁶ – 10¹³ | same as k |
| Eₐ | Activation energy | 10 – 200 | kJ/mol |
| t½ | Half-life | depends on n, k, [A]₀ | s (or min, h, y) |
| τ | Characteristic time 1/k | depends on k | s |
What Does r = k · [A]ⁿ Measure?
The rate law connects four quantities that any chemist or chemical engineer works with constantly: the reaction rate (r), the rate constant (k), the concentration of a reacting species ([A]), and the reaction order (n). Together they describe how fast a reaction proceeds at any given concentration, and how sensitively that rate responds to changes in concentration. The rate law is the empirical summary of a reaction's kinetics, and it is the starting point for reactor design, shelf-life prediction, enzyme kinetics and atmospheric chemistry.
Rate Law Formula
The defining relationship for a single-species rate law is:
Rearranged to solve for the other three unknowns:
[A] = (r / k)1/n (for n > 0)
n = ln(r / k) / ln([A])
For a reaction with multiple reactants, the general form is:
where the exponents a, b, c are the individual orders and their sum is the overall order. This calculator handles the single-reactant form; the same rearrangement applies to each factor in turn.
Rate Law Variables
| Symbol | Meaning | Notes |
|---|---|---|
| r | Reaction rate | Concentration consumed or produced per unit time |
| k | Rate constant | Temperature-dependent via Arrhenius; units depend on n |
| [A] | Concentration of reactant A | Usually molar (mol/L); activities for non-ideal systems |
| n | Reaction order | Experimental, not from stoichiometry; can be fractional or negative |
| A | Arrhenius pre-exponential factor | Frequency factor; related to collision rate |
| Eₐ | Activation energy | Energy barrier; determines temperature sensitivity |
| t½ | Half-life | Time for [A] to fall to half its initial value |
How Each Variable Affects the Reaction Rate
Because the relation is linear in k but a power law in [A], the effect of each variable is different. Doubling k doubles the rate at any concentration. Doubling [A] multiplies the rate by 2ⁿ: ×1 for zero-order, ×2 for first-order, ×4 for second-order. The rate's sensitivity to concentration is therefore governed entirely by n, which is why determining the order experimentally is the central task of a kinetics study.
The rate constant k carries the entire temperature dependence through the Arrhenius equation k = A·exp(−Eₐ/RT). A plot of ln(k) versus 1/T gives a straight line of slope −Eₐ/R, so the activation energy can be extracted from measurements at several temperatures. The pre-exponential factor A reflects the frequency of collisions with the correct orientation, and usually falls between 10⁶ and 10¹³ s⁻¹ for first-order reactions. Together, A and Eₐ give a complete picture of how the reaction rate responds to temperature, while n and the concentration terms give a complete picture of how it responds to composition.
Rate Law Calculation Examples
r = 0.1 × 0.1¹ = 1.000×10⁻³ mol/(L·s)
k = 1×10⁻³ / 0.1 = 1.000×10⁻² s⁻¹
[A] = (1×10⁻³ / 0.1)1/1 = 0.0100 mol/L
n = ln(4×10⁻³ / 0.1) / ln(0.1) = ln(0.04) / ln(0.1) ≈ 1.398
r = 2×10⁻⁴ × 0.5² = 5.000×10⁻⁵ mol/(L·s)
t½ = ln(2) / k = 0.693 / 0.1 = 6.931 s
Frequently Asked Questions
The rate law is the full mathematical expression relating rate to concentrations, e.g. r = k·[A]ⁿ·[B]ᵐ. The rate constant k is just the proportionality factor in that expression. The rate law tells you both the magnitude (through k) and the concentration sensitivity (through the orders n, m). Two different reactions might have the same k but very different rate laws if their orders differ. Conversely, two reactions with the same rate law at one temperature may have very different rates at another, because their activation energies — and hence their k values — differ.
Because both sides of r = k·[A]ⁿ must have the same units. The left side is always mol/(L·s). The right side is k times [A]ⁿ, which has units of k times (mol/L)ⁿ. For the two to match, k must have units of mol/(L·s) divided by (mol/L)ⁿ, which simplifies to (mol/L)^(1−n) · s⁻¹. So for n = 1, k has units s⁻¹ (the concentration units cancel). For n = 2, k has units L/(mol·s). For n = 0, k has units mol/(L·s). This is why you cannot compare k values from reactions of different order directly — you must compare them at the same order or convert to a common basis such as the activation energy.
Yes to both. Fractional orders — 0.5, 1.5, 2.5 — arise when the reaction mechanism involves a chain reaction, a surface-catalysed step, or a pre-equilibrium, so that the overall rate has a non-integer dependence on one of the species. The H₂ + Br₂ → 2 HBr reaction is the classic example, with an order of 1.5 in Br₂ and −1 in HBr. Negative orders mean that increasing the concentration of that species slows the reaction down — usually because the species is an inhibitor or a product that appears in the denominator of the rate expression. Negative and fractional orders are always empirical; they never correspond to a simple molecularity.
Temperature affects the rate only through the rate constant k, via the Arrhenius law k = A · exp(−Eₐ / R·T). The concentrations and the order are unchanged. Because the dependence is exponential, the effect is dramatic: a reaction with Eₐ = 50 kJ/mol runs roughly 7 times faster at 50 °C than at 25 °C, and about 50 times faster at 100 °C. A rough rule of thumb, valid for many reactions near room temperature, is that the rate doubles for every 10 °C rise — but this is only approximate, and reactions with low Eₐ (under 20 kJ/mol) barely respond to temperature at all.
Three standard methods. (1) Method of initial rates: measure the initial rate at several starting concentrations of A, holding all else constant. Then n = log(r₂/r₁) / log([A]₂/[A]₁). This is the most direct and is what the calculator's n-solve mode is reproducing. (2) Integrated rate-law method: fit the concentration–time data to the integrated forms — [A] vs t is linear for n = 0, ln[A] vs t is linear for n = 1, 1/[A] vs t is linear for n = 2 — and see which plot gives the straightest line. (3) Half-life method: measure the half-life at several initial concentrations. If t½ is independent of [A]₀ the reaction is first order; if t½ is proportional to [A]₀ it is zero order; if t½ is inversely proportional to [A]₀ it is second order.
Molecularity is a theoretical concept: the number of molecules that come together in a single elementary step. It can only be 1 (unimolecular), 2 (bimolecular), or rarely 3 (termolecular), and it is always a positive integer. Reaction order is an experimental concept: the exponent on a concentration term in the observed rate law. It can be 0, fractional, or negative, and it applies to the overall reaction, not just one step. For an elementary step, the order and molecularity coincide. For an overall reaction, they usually do not, because the overall rate law reflects the mechanism as a whole — the slowest step, the pre-equilibria, and any chain-propagation steps.
The half-life is derived by integrating the rate law from [A]₀ to [A]₀/2. The result depends on n. For zero-order, t½ = [A]₀ / (2k). For first-order, t½ = ln(2) / k, independent of concentration. For second-order, t½ = 1 / (k · [A]₀), inversely proportional to initial concentration. For third-order, t½ = 3 / (2k·[A]₀²). Half-life is therefore a diagnostic: measuring how it depends on starting concentration is one of the cleanest ways to determine the reaction order. It is also why first-order processes such as radioactive decay have a fixed, characteristic half-life, while second-order processes slow down as they consume reactant.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| r | Reaction rate | mol/(L·s) |
| k | Rate constant | varies |
| [A] | Concentration of reactant A | mol/L |
| n | Order of reaction |
What it means
The rate law expresses the relationship between the reaction rate and the concentrations of reactants. For a general reaction aA + bB → products, the rate is often written as r = k [A]^m [B]^n, where m and n are the reaction orders with respect to A and B, respectively. The overall order is m + n. The rate constant k is temperature-dependent and is determined experimentally. Rate laws are fundamental in chemical kinetics: they allow engineers to design reactors, predict reaction progress, and optimise conditions. For elementary reactions, the orders equal the stoichiometric coefficients, but for complex reactions they must be determined from experimental data. The units of k depend on the overall order. Understanding rate laws is essential for safe and efficient operation of chemical processes, such as in pharmaceuticals, petrochemicals, and environmental engineering. They also underpin the development of kinetic models used in process simulation.
Worked example
Rate Law – Two Examples
Real‑World| Parameter | Value |
|---|---|
| k | 0.05 s⁻¹ |
| [A] | 0.10 M |
| n | 1 |
| Parameter | Value |
|---|---|
| k | 0.030 M⁻¹s⁻¹ |
| [A] | 0.20 M |
| n | 2 |
Common mistakes
- Rate constant k: Its units depend on reaction order – for first order, s⁻¹; for second, M⁻¹·s⁻¹. Using wrong units leads to incorrect rates.
- Concentration units: Ensure [A] is in the same units as those used for k (e.g., mol/L, mol/m³).
- Reaction order n: Must be determined experimentally; do not assume it equals the stoichiometric coefficient.
- Elementary vs. non‑elementary: The rate law form applies only to elementary reactions; for complex mechanisms, use derived rate expressions.
- Temperature dependence: The rate constant changes with temperature; this formula is for a fixed T.
Applications
The rate law for a chemical reaction, r = k[A]ⁿ, quantifies how the reaction rate depends on the concentration of reactants. The rate constant k reflects the intrinsic speed of the reaction, while the order n (often 0, 1, or 2) indicates how the rate changes with concentration. This equation is fundamental to chemical kinetics, enabling engineers and chemists to design reactors, predict reaction progress, and control industrial processes. In practice, rate laws are determined experimentally and are used to scale up reactions from laboratory to production scale. They are essential for optimising conditions such as temperature, pressure, and catalyst loading to achieve desired conversion and selectivity. Understanding the rate law is also critical in environmental chemistry, where it models pollutant degradation, and in biochemistry for enzyme kinetics. By applying this formula, engineers can ensure efficient and safe operation of chemical plants.
- Design of batch, CSTR, and PFR reactors
- Scale‑up of chemical processes
- Optimisation of reaction conditions (temperature, concentration)
- Environmental modelling of pollutant degradation
- Enzyme kinetics and biochemical engineering
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