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Rate Law and Arrhenius Equation Calculator
In short: Rate law and Arrhenius calculator: integrated rate laws and half-lives for zero, first and second order, reaction order from initial rates or concentration–time data, and activation energy from the Arrhenius equation. It is free, works in your web browser and needs no download.
Three calculators in one. Solve integrated rate laws for [A], [A]₀, k or t with half-lives. Find reaction orders and k (with units) from initial-rate experiments, or test concentration–time data for zero, first or second order. Use the Arrhenius equation to find Ea, k or T, in °C or K.
Fill in three values and leave the one you want to find blank.
Method of initial rates
Enter 2 to 4 experiments. Leave the [B] column empty for a single reactant. Rate law: rate = k[A]m[B]n.
| Exp. | [A] (M) | [B] (M) | Initial rate (M/time) |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 |
Order from concentration–time data
Two-point Arrhenius equation
ln(k₂/k₁) = (Ea/R)(1/T₁ − 1/T₂). Fill in four values and leave one blank. k₁ and k₂ must have the same units.
k = A·e−Ea/RT (leave one blank)
Integrated rate laws and half-lives
| Order | Rate law | Integrated form | Straight-line plot | Half-life | Units of k |
|---|---|---|---|---|---|
| Zero | rate = k | [A] = [A]₀ − kt | [A] vs t, slope −k | [A]₀ / 2k | M s⁻¹ |
| First | rate = k[A] | ln[A] = ln[A]₀ − kt | ln[A] vs t, slope −k | 0.693 / k | s⁻¹ |
| Second | rate = k[A]² | 1/[A] = 1/[A]₀ + kt | 1/[A] vs t, slope +k | 1 / (k[A]₀) | M⁻¹ s⁻¹ |
For an overall order p, k has units of M1−p time⁻¹. First-order reactions are often written with log₁₀: k = (2.303/t) log([A]₀/[A]).
Method of initial rates
Compare two experiments where only one concentration changes. If doubling [A] multiplies the rate by 4, then 2m = 4 and m = 2. In general m = log(rate ratio) / log(concentration ratio). The calculator also fits all runs at once by least squares on ln(rate) = ln k + m ln[A] + n ln[B], rounds orders that are within 0.1 of a whole or half number, then averages k over all runs.
Arrhenius equation
k = A·e^(−Ea/RT) ln k = ln A − (Ea/R)(1/T) ln(k₂/k₁) = (Ea/R)(1/T₁ − 1/T₂)
R = 8.314 J mol⁻¹ K⁻¹. A plot of ln k against 1/T is a straight line with slope −Ea/R. In log₁₀ form: log(k₂/k₁) = Ea/(2.303R) × (T₂ − T₁)/(T₁T₂).
Worked examples
First-order decay
N₂O₅ decomposes with k = 6.2 × 10⁻⁴ s⁻¹. [A]₀ = 0.050 M. Find [A] after 600 s.
[A] = 0.050 × e^(−6.2 × 10⁻⁴ × 600) = 0.050 × e^(−0.372)
[A] = 0.0345 M
75% complete (NEET style)
A first-order reaction is 75% complete in 40 min. Find t½.
75% complete = 2 half-lives, so t½ = 40/2. Check: k = ln 4 / 40 = 0.0347 min⁻¹.
t½ = 20 min
Second order
2NO₂ → 2NO + O₂ is second order with k = 0.50 M⁻¹ s⁻¹. How long to fall from 0.010 M to 0.0010 M?
t = (1/0.0010 − 1/0.010) / 0.50 = 900 / 0.50
t = 1800 s (30 min)
Initial rates (AP style)
Exp 1: [A] 0.10, [B] 0.10, rate 2.0 × 10⁻³. Exp 2: [A] 0.20, rate 8.0 × 10⁻³. Exp 3: [B] 0.20, rate 4.0 × 10⁻³.
Doubling [A] gives ×4, so m = 2. Doubling [B] gives ×2, so n = 1. k = 2.0 × 10⁻³ / (0.10² × 0.10)
rate = k[A]²[B], k = 2.0 M⁻² s⁻¹
Activation energy (NCERT)
The rate quadruples when T rises from 293 K to 313 K. Find Ea.
Ea = 8.314 × ln 4 / (1/293 − 1/313) = 8.314 × 1.386 / 2.181 × 10⁻⁴
Ea = 52.9 kJ/mol
The "rate doubles every 10 °C" rule
Rate doubles between 25 °C and 35 °C.
Ea = 8.314 × ln 2 / (1/298.15 − 1/308.15)
Ea ≈ 52.9 kJ/mol
Units and common mistakes
- Temperature in kelvin. The Arrhenius equation needs K. Using °C gives nonsense (and division by zero at 0 °C).
- Ea in J vs kJ. R = 8.314 J mol⁻¹ K⁻¹, so convert Ea to J/mol inside the exponent. This calculator takes kJ/mol and converts for you.
- Units of k tell you the order. s⁻¹ means first order. M⁻¹ s⁻¹ means second order. M s⁻¹ means zero order.
- Keep time units consistent. If k is in min⁻¹, time must be in minutes.
- Orders come from experiment. They are not the coefficients of the balanced equation, except for elementary steps.
- Half-life depends on [A]₀ for zero and second order. Only first-order half-lives are constant.
- Pick experiments where one concentration changes. If both change together, you cannot separate the two orders.
Drawing reaction mechanisms? Sketch intermediates, transition states and energy profiles in MolDraw, a free online chemical structure editor.
Open MolDraw editorFrequently asked questions
How do I find the order of a reaction from initial rates?
Choose two experiments where only one concentration changes. Order = log(rate₂/rate₁) / log(conc₂/conc₁). Doubling a concentration that doubles the rate means first order; ×4 means second order; no change means zero order.
How do I tell the order from concentration–time data?
Plot [A], ln[A] and 1/[A] against time. The one that gives a straight line shows the order: zero, first or second. The calculator compares the R² values of all three fits.
What are the units of the rate constant?
For overall order n, k has units M^(1−n) time⁻¹. Zero order: M s⁻¹. First order: s⁻¹. Second order: M⁻¹ s⁻¹. Third order: M⁻² s⁻¹.
How do I calculate activation energy from two temperatures?
Use Ea = R ln(k₂/k₁) / (1/T₁ − 1/T₂) with T in kelvin and R = 8.314 J mol⁻¹ K⁻¹. If the rate quadruples from 293 K to 313 K, Ea = 52.9 kJ/mol.
What is the half-life of a first-order reaction?
t½ = ln 2 / k ≈ 0.693/k. It does not depend on the starting concentration. After n half-lives, (1/2)ⁿ of the reactant remains.
Why does a 10 °C rise roughly double the rate?
For reactions with Ea around 50 kJ/mol near room temperature, the Arrhenius equation predicts about a twofold increase per 10 °C. It is a rule of thumb, not a law.
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