The rate constant k is the proportionality factor in a rate law. For a reaction with rate = k[A]ᵐ[B]ⁿ, the exponents m and n are the orders with respect to each reactant and their sum is the overall order. Orders are found by experiment – they need not equal the coefficients of the balanced equation – except for elementary steps, where the order equals the molecularity: a unimolecular step is first order, a bimolecular step second order.
Because the rate has units of concentration per time, the units of k change with the overall order: M·s⁻¹ for zero order, s⁻¹ for first order, M⁻¹·s⁻¹ for second order and so on. k depends on temperature (through the Arrhenius equation) but not on concentration.
This calculator finds k four ways: from a half-life, using the half-life formula for the chosen order; from one measured rate and the concentrations in the rate law; from the integrated rate law, using the initial concentration, a later concentration and the time between them; or for an elementary step, where the molecularity fixes the rate law. It also reports the overall order, the units, the half-life, the time to 90% conversion and the initial rate.
How to use the Rate Constant Calculator
- Find k from
- Reaction order
- Initial concentration [A]₀
- Half-life t½
- Concentration [A] after time t
- Elapsed time t
- Elementary step
- Measured reaction rate
- Concentration [A]
- Order with respect to A
- Concentration [B]
- Order with respect to B
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Pick the method, enter the data and the order. The calculator rearranges the relevant formula for k – half-life formula, rate law or integrated rate law – using concentrations in mol/L and times in seconds, then converts k to per minute or per hour if you choose.
Worked example
From a half-life: a first-order reaction has t½ = 30 min = 1800 s, so k = ln 2 ÷ 1800 = 3.85 × 10⁻⁴ s⁻¹ (0.0231 min⁻¹). 90% conversion takes ln 10 ÷ k ≈ 5980 s, about 99.7 min.
From the rate law: rate = k[A][B]² with a measured rate of 2.4 × 10⁻⁵ M/s at [A] = 0.10 M and [B] = 0.20 M gives k = 2.4 × 10⁻⁵ ÷ (0.10 × 0.20²) = 6.0 × 10⁻³ M⁻²·s⁻¹; the overall order is 3.
Integrated law: [A] falls from 0.50 M to 0.20 M in 600 s. First order: k = ln 2.5 ÷ 600 = 1.53 × 10⁻³ s⁻¹; second order: k = (1/0.20 − 1/0.50) ÷ 600 = 5.0 × 10⁻³ M⁻¹·s⁻¹.
Rate laws by order
| Order | Rate law | Integrated form | Half-life | Units of k | Straight-line plot |
|---|---|---|---|---|---|
| 0 | k | [A] = [A]₀ − kt | [A]₀ / 2k | M·s⁻¹ | [A] vs t |
| 1 | k[A] | ln[A] = ln[A]₀ − kt | ln 2 / k | s⁻¹ | ln[A] vs t |
| 2 | k[A]² | 1/[A] = 1/[A]₀ + kt | 1 / (k[A]₀) | M⁻¹·s⁻¹ | 1/[A] vs t |
Inputs explained
| Field | What to enter |
|---|---|
| Find k from | Choose one: Half-life (single reactant), Measured rate and the rate law, Concentration change over time (integrated rate law), Elementary step (molecularity sets the order) |
| Reaction order | Choose one: Zero order, First order, Second order |
| Initial concentration [A]₀ | Units: mol/L (M), mmol/L (mM), µmol/L (µM), nmol/L (nM), mol/m³ |
| Half-life t½ | Units: s, min, h, days, years, ms |
| Concentration [A] after time t | Units: mol/L (M), mmol/L (mM), µmol/L (µM), nmol/L (nM), mol/m³ |
| Elapsed time t | Units: s, min, h, days, years, ms |
| Elementary step | Choose one: Unimolecular: A → products (rate = k[A]), Bimolecular: A + B → products (rate = k[A][B]), Bimolecular: 2A → products (rate = k[A]²), Termolecular: 2A + B → products (rate = k[A]²[B]) |
| Measured reaction rate | Units: M/s, M/min, M/h, mM/s, µM/s |
| Concentration [A] | Units: mol/L (M), mmol/L (mM), µmol/L (µM), nmol/L (nM), mol/m³ |
| Order with respect to A | Choose one: 0, 1, 2, 3 |
| Concentration [B] | Units: mol/L (M), mmol/L (mM), µmol/L (µM), nmol/L (nM), mol/m³ |
| Order with respect to B | Choose one: 0 (no second reactant), 1, 2 |
What each result means
- Units of k = [Overall reaction order]
- Initial rate = [Rate constant k]×[Initial concentration [A]₀]^[Reaction order]
- Rate if [A] is doubled = [Measured reaction rate]×2^[Order in A used]
Formula
rate = k[A]ᵐ[B]ⁿ · t½: zero order [A]₀/(2k), first order ln 2/k, second order 1/(k[A]₀) · integrated: [A]₀ − [A] = kt, ln([A]₀/[A]) = kt, 1/[A] − 1/[A]₀ = kt
Worked example
Inputs — Find k from: Half-life (single reactant); Reaction order: First order; Initial concentration [A]₀: 0.5 mol/L (M); Half-life t½: 1,800 s.
Results — Overall reaction order = 1; Rate constant k = 0.000385 per second (M¹⁻ⁿ·s⁻¹); Units of k = s⁻¹; Time to 90% conversion = 5,979.4706 s; Initial rate = 0.000193 M/s.
Key terms
- Rate law
- Equation relating the reaction rate to concentrations, rate = k[A]ᵐ[B]ⁿ.
- Reaction order
- The exponent of a concentration in the rate law; the overall order is their sum.
- Half-life (t½)
- Time for the reactant concentration to fall to half its starting value.
- Molecularity
- The number of particles colliding in an elementary step (uni-, bi- or termolecular).
- Integrated rate law
- The rate law solved for concentration as a function of time.
Frequently asked questions
What is the rate constant?
The proportionality constant k in a rate law such as rate = k[A][B]. It is fixed for a given reaction at a given temperature and does not depend on concentration.
What are the units of the rate constant?
M¹⁻ⁿ·s⁻¹ for overall order n: M·s⁻¹ (zero order), s⁻¹ (first), M⁻¹·s⁻¹ (second), M⁻²·s⁻¹ (third).
How do I get k from a half-life?
First order: k = ln 2 ÷ t½ = 0.693 ÷ t½. A half-life of 30 min gives k = 0.0231 min⁻¹ = 3.85 × 10⁻⁴ s⁻¹. Zero order: k = [A]₀ ÷ 2t½; second order: k = 1 ÷ ([A]₀·t½).
How do I find k from concentration data?
Use the integrated rate law. If a first-order reactant falls from 0.50 M to 0.20 M in 600 s, k = ln(0.50/0.20) ÷ 600 = 1.53 × 10⁻³ s⁻¹.
Does the rate constant change with temperature?
Yes – according to the Arrhenius equation k = A·e^(−Ea/RT). Many reactions roughly double their k for a 10 °C rise near room temperature.
Is the order the same as the coefficient?
Only for elementary steps. For overall reactions the order must be measured; for example 2 N₂O₅ → 4 NO₂ + O₂ is first order in N₂O₅.
Why does the half-life of a first-order reaction not depend on concentration?
Because t½ = ln 2/k contains no concentration term – radioactive decay is the classic example.