Arrhenius Equation Calculator
Solve the classic Arrhenius relation k = A · e−Eₐ/(R·T) for any of its four unknowns — rate constant, pre-exponential factor, activation energy, or temperature — with step-by-step math, a live energy-profile schematic, and companion kinetics metrics including the Arrhenius slope and Q₁₀.
A is the collision frequency factor
The pre-exponential factor A represents how often reactant molecules collide with the correct orientation, extrapolated to infinite temperature. For simple gas-phase reactions A typically falls between 10¹⁰ and 10¹³ s⁻¹ or L/(mol·s), close to the molecular collision frequency. For reactions in solution A is usually 10¹⁰–10¹³ s⁻¹, while for surface reactions it can be much smaller because only a fraction of collisions strike an active site. Reaction rate theories — collision theory and transition-state theory — attempt to predict A from first principles.
Eₐ sets the temperature sensitivity
The activation energy is the energy barrier that separates reactants from products, and it appears in the exponential as Eₐ/(R·T). This makes the rate extraordinarily sensitive to Eₐ: at room temperature, a change of 10 kJ/mol in Eₐ changes k by a factor of about 50, while a change of 40 kJ/mol changes it by more than 10⁶. The exponential factor e−Eₐ/RT is the fraction of collisions with energy greater than Eₐ, given by the Boltzmann distribution.
Temperature accelerates reactions exponentially
Because T appears in the denominator of the exponent, k grows exponentially with temperature, not linearly. A reaction with Eₐ = 75 kJ/mol runs about 16 times faster at 50 °C than at 25 °C, and about 250 times faster at 100 °C. This is why a modest temperature rise has a dramatic effect on reaction rate — and why refrigeration preserves food, why autoclaves use steam at 121 °C, and why industrial reactors are often run as hot as the chemistry and equipment allow.
The Arrhenius plot linearises the law
Taking the natural logarithm of both sides gives ln(k) = ln(A) − Eₐ/(R·T). This is a straight line when ln(k) is plotted against 1/T, with slope −Eₐ/R and intercept ln(A). This linearisation is the standard experimental method for measuring activation energies: measure k at several temperatures, plot ln(k) vs 1/T, and extract Eₐ from the slope. A curved Arrhenius plot signals that the mechanism is changing with temperature or that quantum tunnelling is significant.
How to Use This Calculator
The Arrhenius equation links four quantities: k, A, Eₐ and T. Pick which one you want the tool to solve, then fill in the other three.
1. Choose what to solve for
Pick k, A, Eₐ or T from the toggle. The calculator rearranges k = A·e−Eₐ/(RT) automatically for whichever quantity you select, greying out that field.
2. Use realistic presets
k presets span 10⁻¹⁰ to 10⁴ s⁻¹, A presets 10⁶ to 10¹⁵, Eₐ presets 15 to 200 kJ/mol, and T presets from −20 °C to 700 °C — covering cold storage through industrial and combustion chemistry.
3. Match units carefully
k and A are in s⁻¹ for first-order reactions (adjust for other orders), Eₐ in kJ/mol, and T in kelvin. R = 8.314 J/(mol·K) is applied automatically. The hint under each field shows the value in alternative units.
4. Read the result and verdict
The result card shows the solved value plus equivalents — half-life for k, log₁₀ A for A, kcal/mol for Eₐ, and °C/°F for T. A colour-coded verdict classifies k from very slow to diffusion-limited.
5. Check the schematic and charts
The schematic shows the energy profile with reactants, transition state and products, plus the exponential k vs T curve. The main chart plots k against T for three values of Eₐ.
6. Switch to Advanced for depth
Advanced mode adds the classical Arrhenius plot (ln k vs 1/T), a sensitivity check on a +1 % input change, and companion metrics including half-life, Q₁₀, transition-state ΔG‡, and the Boltzmann exponent.
What This Tool Offers
Built around the Arrhenius equation, the fundamental link between reaction rate, activation energy, and temperature.
Four-way solving
Solve for rate constant, pre-exponential factor, activation energy, or temperature. Each rearrangement uses logarithms where needed.
Live energy-profile schematic
A reaction-coordinate diagram with reactants, transition state and products, the Eₐ barrier marked, and an exponential k vs T curve alongside.
Rate-class verdict
Each result is classified from very slow (geological timescales) through moderate laboratory kinetics up to diffusion-limited and barrierless, with a coloured badge.
Logarithmic & linear sliders
k and A use logarithmic sliders spanning 25 and 10 orders of magnitude; Eₐ (0–300 kJ/mol) and T (100–1500 K) use linear sliders for smooth dialling.
Dual comparison charts
Rate constant vs. temperature for three Eₐ values, plus an advanced Arrhenius plot of ln(k) vs 1/T showing the classic linear behaviour.
Reference tables
Typical A and Eₐ values by reaction class, a dynamic k vs T lookup grid at the current parameters, and a full variable reference table.
Reference Tables & Conversions
An activation-energy unit converter, a dynamic k vs T lookup grid, and typical Arrhenius parameters by reaction class.
Activation energy unit converter
Rate constant k (s⁻¹) at A = 1.000×10¹² s⁻¹ and Eₐ = 75.0 kJ/mol
Computed from k = A · e−Eₐ/(R·T). Rows are temperatures, columns are Eₐ multipliers.
| T (K) | Eₐ × 0.5 | Eₐ × 0.75 | Eₐ × 1.0 | Eₐ × 1.25 | Eₐ × 1.5 | Eₐ × 2 |
|---|
Typical Arrhenius Parameters by Reaction Class
Representative values. Both A and Eₐ vary widely within each class; these are order-of-magnitude guides.
| Reaction class | A (s⁻¹ or L/(mol·s)) | Eₐ (kJ/mol) | k at 298 K (approx) | Notes |
|---|---|---|---|---|
| Diffusion-controlled (in water) | 10¹⁰–10¹¹ | 10–20 | 10⁹–10¹⁰ | Rate limited by solvent viscosity |
| Enzyme-catalysed | 10⁶–10¹³ | 20–60 | 10²–10⁶ | Turnover number 10²–10⁶ s⁻¹ |
| Simple SN2 (CH₃Br + OH⁻) | ~10¹⁰ | ~85 | ~10⁻⁴ | Bimolecular, L/(mol·s) |
| Sucrose hydrolysis (acid) | ~10¹⁴ | ~110 | ~10⁻⁵ | First-order in sucrose |
| N₂O₅ → NO₂ + O₂ (gas) | ~10¹³ | ~103 | ~10⁻⁵ | Classic first-order gas decomp. |
| H₂ + I₂ → 2 HI (gas) | ~10¹¹ | ~165 | ~10⁻¹⁰ | Bimolecular, L/(mol·s) |
| CH₄ + O₂ combustion | ~10¹² | ~200 | ~10⁻²³ | Very high barrier at low T |
| Polymerisation (styrene) | ~10⁹ | ~90 | ~10⁻⁷ | Initiation step |
| Protein denaturation | ~10¹⁵ | ~250 | ~10⁻³⁰ | Very sharp thermal transition |
| Solid-state diffusion (C in Fe) | ~10⁻² (m²/s) | ~80 | ~10⁻²³ | Not a rate constant, but Arrhenius |
| Radioactive decay (¹⁴C) | — | — | 3.83×10⁻¹² | Temperature-independent (nuclear) |
| Bimolecular gas (upper limit) | ~10¹¹ | 0 | ~10¹¹ | Barrierless — every collision reacts |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| k | Rate constant | 10⁻¹⁵ – 10¹¹ | s⁻¹ (order-dependent) |
| A | Pre-exponential factor | 10⁶ – 10¹⁵ | same as k |
| Eₐ | Activation energy | 10 – 300 | kJ/mol |
| T | Absolute temperature | 100 – 1500 | K |
| R | Gas constant | 8.314 | J/(mol·K) |
| R·T | Thermal energy per mole | 0.83 – 12.5 at 100–1500 K | kJ/mol |
| Eₐ/R | Characteristic temperature | 1,200 – 36,000 | K |
| t½ | Half-life (first order) | ln(2)/k | s |
| Q₁₀ | Rate ratio for +10 °C | 1.5 – 4 typical | dimensionless |
What Does k = A · e−Eₐ/(R·T) Measure?
The Arrhenius equation connects four quantities that any chemist, chemical engineer, or materials scientist works with constantly: the rate constant (k), the pre-exponential factor (A), the activation energy (Eₐ), and the absolute temperature (T). Together they describe how fast a reaction proceeds at any temperature, and how sensitively that rate responds to a change in temperature. The equation is one of the most widely used relationships in physical chemistry, and its parameters are routinely extracted from experimental data for reactor design, shelf-life prediction, food preservation, and materials processing.
Arrhenius Equation Formula
The defining relationship is:
Taking the natural logarithm gives the linear form used for parameter extraction:
Rearranged to solve for the other three unknowns:
Eₐ = −R·T·ln(k / A)
T = −Eₐ / (R·ln(k / A)) = Eₐ / (R·ln(A / k))
The Boltzmann factor e−Eₐ/(R·T) is the fraction of molecular collisions that have enough energy to overcome the barrier. For k to be a real positive number less than A, we must have Eₐ > 0 and T > 0, and the argument of the logarithm must satisfy k/A < 1 for positive Eₐ.
Arrhenius Variables
| Symbol | Meaning | Notes |
|---|---|---|
| k | Rate constant | Units depend on reaction order; describes rate at a given T |
| A | Pre-exponential factor | Frequency factor; k at T = ∞; reflects collision frequency × steric factor |
| Eₐ | Activation energy | Minimum energy for reaction; strong influence on temperature sensitivity |
| T | Absolute temperature | In kelvin; T(K) = T(°C) + 273.15 |
| R | Gas constant | 8.314 J/(mol·K) = 0.008314 kJ/(mol·K) |
| Eₐ/R | Characteristic temperature | Temperature scale over which k changes by a factor of e |
| Q₁₀ | Temperature coefficient | Rate ratio for a 10 °C rise; derived from Eₐ and T |
How Each Variable Affects the Rate Constant
Because the equation is linear in A but exponential in Eₐ/T, the four variables have very different sensitivities. Doubling A doubles k at any temperature — a modest effect. But increasing Eₐ by 10 kJ/mol at room temperature multiplies k by roughly 1/50, and decreasing it by 10 kJ/mol multiplies k by roughly 50. The rate's temperature sensitivity is governed by the ratio Eₐ/T, which is why reactions with high activation energy are extremely temperature-sensitive, while reactions with low Eₐ (diffusion-limited, barrierless radical recombinations) are nearly temperature-independent.
The quantity Eₐ/(R·T) is the number of "thermal energy units" needed to surmount the barrier. At room temperature, R·T = 2.48 kJ/mol, so an activation energy of 75 kJ/mol corresponds to 75/2.48 ≈ 30 thermal units — meaning roughly e−30 ≈ 10⁻¹³ of collisions have enough energy. This is why A must be enormous (10¹²–10¹³ s⁻¹) for a rate constant of any appreciable size at room temperature. As temperature rises, R·T grows, the exponent shrinks, and k grows exponentially.
Arrhenius Calculation Examples
Eₐ/(R·T) = 75,000 / (8.314 × 373.15) = 24.18
k = 1×10¹² · e−24.18 = 5.00×10⁻² s⁻¹
A = k · eEₐ/(R·T) = 5×10⁻² · e24.18 = 1.00×10¹² s⁻¹
Eₐ = −R·T·ln(k/A) = −8.314 × 373.15 × ln(5×10⁻¹⁴) = 75.0 kJ/mol
T = Eₐ / (R·ln(A/k)) = 75,000 / (8.314 × 29.93) = 373.15 K = 100 °C
Eₐ = R·ln(k₂/k₁) / (1/T₁ − 1/T₂) = 8.314 × ln(10) / (0.00333 − 0.00313) ≈ 95.6 kJ/mol
Frequently Asked Questions
A is the frequency factor — the rate constant extrapolated to infinite temperature, where the exponential factor equals 1. In collision theory, A represents the product of the collision frequency Z and a steric factor P (the fraction of collisions with the correct orientation). For simple gas reactions A is close to the collision frequency, 10¹¹–10¹³ s⁻¹ or L/(mol·s). For reactions in solution A usually falls in a similar range, while for surface reactions A may be much smaller. In transition-state theory, A = (k_B·T/h)·eΔS‡/R, so A is directly related to the entropy of activation ΔS‡.
Because ln(k) = ln(A) − Eₐ/(R·T), which has the form y = c + m·x with y = ln(k), x = 1/T, slope m = −Eₐ/R, and intercept c = ln(A). This is the standard graphical method for extracting activation energy from experimental data: measure the rate constant at several temperatures, plot ln(k) against 1/T, fit a straight line, and read Eₐ = −slope × R and A = eintercept. A curved Arrhenius plot indicates that the reaction mechanism changes with temperature, that there is quantum tunnelling, or that there is a competing parallel reaction.
No. The Arrhenius equation is an empirical fit that works remarkably well for most reactions over moderate temperature ranges, but it fails in several cases. At very low temperatures, quantum-mechanical tunnelling can carry light atoms through the barrier, giving rates higher than the classical Arrhenius prediction and curved Arrhenius plots. At very high temperatures, reactions with no barrier become diffusion-limited, and k becomes temperature-insensitive. In reactions with complex mechanisms, the effective Eₐ can be a weighted combination of several elementary steps and can even be negative. The more general form of the temperature dependence, from transition-state theory, is k = (k_B·T/h)·e−ΔG‡/(R·T), which is not identical to the Arrhenius form.
Measure the rate constant k at several temperatures (at least 4–5) covering as wide a range as feasible. For each temperature, compute 1/T (in K⁻¹) and ln(k). Plot ln(k) on the y-axis against 1/T on the x-axis. Fit a straight line by least squares. The slope is −Eₐ/R, so Eₐ = −slope × R (in J/mol; divide by 1000 for kJ/mol). The intercept is ln(A), so A = eintercept. Alternatively, the two-point form Eₐ = R · ln(k₂/k₁) / (1/T₁ − 1/T₂) can be used with just two temperatures, but the result is less reliable. Always report the temperature range over which the fit was made, because curvature is common.
Transition-state theory (TST) gives the more fundamental expression k = (k_B·T/h) · K‡, where k_B is Boltzmann's constant, h is Planck's constant, and K‡ is the equilibrium constant between reactants and the transition state. Expanding K‡ = e−ΔG‡/(R·T) and separating ΔG‡ into enthalpy and entropy gives k = (k_B·T/h) · eΔS‡/R · e−ΔH‡/(R·T). Comparing with the Arrhenius form, we identify Eₐ ≈ ΔH‡ + R·T and A ≈ (k_B·T/h) · eΔS‡/R. So the Arrhenius pre-exponential factor is essentially the entropy of activation, and Eₐ is essentially the enthalpy of activation plus a small thermal correction.
Because typical activation energies are in the range 20–200 kJ/mol, and expressing them in J/mol gives numbers like 20,000–200,000 that are harder to read and remember. Using kJ/mol keeps the numbers in the same intuitive range as bond energies (which are also conventionally quoted in kJ/mol), making it easy to compare Eₐ with the energies of the bonds being broken and formed. In the Arrhenius formula you must convert to J/mol to match the gas constant R = 8.314 J/(mol·K); the calculator does this conversion for you automatically.
Q₁₀ is the factor by which the rate increases when the temperature rises by 10 °C. It is widely used in biology and food science because it is more intuitive than Eₐ. From the Arrhenius equation, Q₁₀ = k(T+10)/k(T) = eEₐ·10/(R·T·(T+10)). For a typical Eₐ of 50–80 kJ/mol near room temperature, Q₁₀ falls in the range 2–4 — meaning reactions roughly double to quadruple their rate for each 10 °C rise. This is why refrigeration slows food spoilage, and why fever accelerates metabolism. Note that Q₁₀ is not constant: it depends on T and decreases as temperature increases.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| k | Rate constant | |
| A | Frequency factor | |
| Ea | Activation energy | J/mol |
| R | Gas constant | J/(mol·K) |
| T | Absolute temperature | K |
What it means
The Arrhenius equation is a foundational formula in chemical kinetics that relates the rate constant k to temperature T and activation energy Ea. It is written as k = A exp(−Ea/(R T)), where A is the pre‑exponential factor (frequency factor), R is the universal gas constant, and Ea is the activation energy. This equation arises from the collision theory and transition state theory. It shows that even a small increase in temperature can greatly increase the reaction rate because of the exponential term. The activation energy is the minimum energy that reactant molecules must have to undergo reaction. In practice, the Arrhenius equation is used to extrapolate rate constants to different temperatures, to determine Ea from experimental data (Arrhenius plot), and to design reactors with temperature control. It is also applied in materials science, biology (enzyme kinetics), and environmental chemistry to model degradation rates. The equation highlights the trade‑off between reaction speed and energy input, which is crucial for process economics and safety.
Worked example
Arrhenius Equation – Two Examples
Real‑World| Parameter | Value |
|---|---|
| A | 1×10¹⁰ s⁻¹ |
| Eₐ | 50,000 J/mol |
| R | 8.314 J/mol·K |
| T | 298 K |
| Parameter | Value |
|---|---|
| A | 5×10¹² s⁻¹ |
| Eₐ | 75,000 J/mol |
| T | 310 K |
Common mistakes
- Activation energy Ea: Must be in J/mol (not kJ/mol) when using R=8.314 J/(mol·K) – convert kJ to J.
- Temperature T: Must be in Kelvin – never Celsius or Fahrenheit.
- Frequency factor A: Has the same units as k; check consistency.
- Exponential term: The exponent −Ea/(RT) is dimensionless; ensure R and Ea use consistent units.
- Assumption of constant Ea: This equation assumes Ea is independent of T; for some reactions it varies.
Applications
The Arrhenius equation, k = A·e^(−Ea/RT), is the cornerstone of temperature‑dependent reaction kinetics. It relates the rate constant k to the activation energy Ea, the absolute temperature T, and the pre‑exponential factor A, which accounts for collision frequency and orientation. This equation is vital for predicting how reaction rates change with temperature, enabling engineers to design reactors that operate safely and economically. It is used to determine optimal operating temperatures, to assess catalyst performance, and to calculate the shelf‑life of pharmaceuticals and food products. In materials science, it models diffusion and creep processes. By understanding the Arrhenius relationship, engineers can extrapolate rates from lab data to industrial conditions and can diagnose issues like catalyst deactivation or unwanted side reactions that may become significant at elevated temperatures.
- Reactor design and temperature optimisation
- Kinetic parameter estimation from experimental data
- Stability and shelf‑life prediction for thermally sensitive products
- Catalyst activity and deactivation analysis
- Diffusion and solid‑state kinetics modelling
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