Arrhenius Equation, Activation Energy & Reaction Rate Calculator
Solve the two-temperature Arrhenius equation (ln(k₂/k₁) = −(Ea/R)(1/T₂ − 1/T₁)) for activation energy Ea, rate constant k₂, or target temperature T₂, fit multi-point OLS Arrhenius plots (ln k vs. 1/T) with Eyring transition-state parameters (ΔH‡, ΔS‡, ΔG‡), and calculate 0th-, 1st-, and 2nd-order integrated rate laws (half-life t₁/₂, shelf-life t₉₀) with a 1-click toggle between exact CODATA R = 8.3144626 J/(mol·K) and textbook R = 8.31 J/(mol·K).
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Key Formula & Conditions
Learning modelSolves the two-temperature Arrhenius equation ln(k₂/k₁) = −(Ea/R)(1/T₂ − 1/T₁) for Ea, k₂, or T₂, fits multi-point OLS Arrhenius plots (ln k vs. 1/T) with Eyring parameters (ΔH‡, ΔS‡, ΔG‡), and evaluates 0th-, 1st-, and 2nd-order integrated rate laws (t₁/₂, t₉₀) with a toggle between exact CODATA R = 8.31446 J/(mol·K) and textbook R = 8.31 J/(mol·K).
Two-Temperature Arrhenius Parameters
ln(k₂/k₁) = −(Ea/R)(1/T₂ − 1/T₁)Calculated Activation Energy Ea
R = 8.314552,949 J/mol / 12.66 kcal/mol
Experimental (T, k) Dataset
ln k = ln A − (Ea/R)(1/T)Arrhenius Regression & Eyring Activation Parameters
R² = 0.9999| T [K] | 1000/T [K⁻¹] | ln k (実測) | ln k (回帰) | Residual |
|---|
Reaction Order & Integrated Rate Law Parameters
[A]t = [A]₀·exp(−kt)Remaining Concentration [A]t
1st Order40.66% remaining / 59.34% reacted
Reference Activation Energies (Ea) for Representative Reactions
Click a row to load Ea into 2-Temp Solver| Reaction / Process | Ea [kJ/mol] | 25°C→35°C Ratio (Q₁₀) | Kinetic Significance |
|---|---|---|---|
| 2 H₂O₂ → 2 H₂O + O₂ (Catalase enzyme) | 29.0 | 1.46 × | Enzyme lowers barrier from ~75 kJ/mol (uncatalyzed) |
| Classic Rule-of-Thumb (Rate doubles from 25°C to 35°C) | 52.95 | 2.00 × | Benchmark Ea yielding Q₁₀ = 2.0 between 298.15 K and 308.15 K |
| 2 H₂O₂ → 2 H₂O + O₂ (Uncatalyzed) | 75.3 | 2.68 × | Drops to ~56 kJ/mol with I⁻ catalyst |
| 2 N₂O₅(g) → 4 NO₂(g) + O₂(g) | 103.0 | 3.86 × | Canonical 1st-order gas-phase benchmark in physical chemistry |
| 2 HI(g) → H₂(g) + I₂(g) | 184.0 | 11.15 × | High-barrier 2nd-order gas reaction; strongly temperature-sensitive |
Done
Arrhenius Equation, Activation Energy & Reaction Rate Calculator
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Worked Examples & Verified Values
25°C→35°C rate doubles → Find Ea
N₂O₅ decay (Ea=103 kJ/mol, 25→45°C)
1st-order decay (k=0.015 s⁻¹, 60 s)
Common Pitfalls & Calculation Rules
- •Unit mismatch between Ea in kJ/mol and R = 8.314 J/(mol·K). Always convert Ea to J/mol (× 1,000) before dividing by R·T so the exponent −Ea/(RT) is dimensionless.
- •Substituting Celsius (°C) directly into 1/T₂ − 1/T₁. Arrhenius temperatures must be in absolute Kelvin (T [K] = t [°C] + 273.15).
- •Sign reversal in (1/T₂ − 1/T₁). Since rate constants increase with temperature (k₂ > k₁ for T₂ > T₁ when Ea > 0), verify that ln(k₂/k₁) = +(Ea/R)(T₂ − T₁)/(T₁T₂) is positive.
Verification scope and citation information
Use worked examples within their stated inputs, units and assumptions. Tests cover specific cases at a specific time; they do not certify every input, expert review or standards compliance. Check the references and tool-specific verification notes, and enter your own access date when citing.
@misc{tools_hub_arrhenius_calculator_2026,
author = {{Tools Hub}},
title = {Arrhenius Equation, Activation Energy & Reaction Rate Calculator},
year = {2026},
url = {https://the-tools-hub.net/en/tools/arrhenius-calculator/},
note = {Accessed: YYYY-MM-DD}
}How to use
- 1
Select gas constant R mode & calculation tab
Toggle between Exact SI/CODATA (R = 8.3144626 J/(mol·K)) and High-School Textbook (R = 8.31 J/(mol·K)), then choose 2-Temperature Solver, Multi-Point Arrhenius Plot, or 0th/1st/2nd-Order Kinetics.
- 2
Enter temperatures, rate constants, or concentration data
Input T₁ and T₂ (°C or K) with Ea or k₂/k₁ (or load N₂O₅ / textbook presets), paste experimental (T, k) pairs for linear regression, or select reaction order and initial concentration [A]₀.
- 3
Inspect Ea, Arrhenius plot, Eyring parameters & working
Review activation energy Ea (kJ/mol, J/mol, kcal/mol), pre-exponential factor A, Q₁₀, interactive ln k vs. 1000/T SVG plot, Eyring ΔH‡/ΔS‡/ΔG‡, half-life t₁/₂, and step-by-step substitutions.
Features
- One-click toggle between Exact SI/CODATA (R = 8.3144626 J/(mol·K)) and High-School Textbook (R = 8.31 J/(mol·K)) gas constants
- Three-way two-temperature Arrhenius solver for activation energy Ea, rate constant k₂ (and ratio k₂/k₁), or target temperature T₂
- Simultaneous output in kJ/mol, J/mol, and kcal/mol alongside pre-exponential factor A, Boltzmann factor exp(−Ea/RT), and temperature coefficient Q₁₀
- Multi-point Ordinary Least Squares (OLS) Arrhenius plot linear regression with slope standard error (±SE), R², and residual diagnostic table
- Eyring transition-state theory evaluation of enthalpy (ΔH‡), entropy (ΔS‡), and Gibbs free energy of activation (ΔG‡)
- 0th-, 1st-, and 2nd-order integrated rate law solver ([A]t, t, k) with half-life t₁/₂, 90% shelf-life t₉₀, and 99% completion time t₉₉%
- Interactive SVG Arrhenius plots (ln k vs. 1000/T) for both 2-temperature comparisons and multi-point experimental fits
- Curated reference table of benchmark activation energies (catalase vs. uncatalyzed H₂O₂, N₂O₅, sucrose inversion, ester hydrolysis) and 1-click Markdown report copy
Use cases
High-School & University Chemistry Exam Verification
Verify 2-temperature activation energy problems, Q₁₀ rate multipliers, and 1st/2nd-order half-life questions with complete step-by-step substitutions.
Physical & Analytical Chemistry Lab Regression Reports
Paste multi-temperature (T, k) experimental data to immediately obtain OLS slope −Ea/R, activation energy ± standard error, R², and Eyring ΔH‡, ΔS‡, and ΔG‡.
Pharmaceutical & Food Shelf-Life (t₉₀) Estimation
Extrapolate room-temperature rate constants from accelerated thermal stability studies and compute 90% potency shelf-life (t₉₀) and half-life (t₁/₂).
Details
Proposed by Svante Arrhenius in 1889, the Arrhenius equation k = A · exp(−Ea / RT) is the cornerstone of chemical kinetics, relating the reaction rate constant k to absolute temperature T [K] and activation energy Ea [J/mol]. Here R is the molar gas constant (exact SI value 8.314462618 J/(mol·K); textbook value 8.31 J/(mol·K)), A is the pre-exponential (frequency) factor, and the Boltzmann factor exp(−Ea / RT) represents the fraction of molecular collisions possessing kinetic energy at least equal to the activation barrier Ea.
Taking the natural logarithm yields the linear relation ln k = ln A − (Ea / R)(1 / T). Evaluating this at two temperatures T₁ and T₂ eliminates A, giving the two-point Arrhenius equation: ln(k₂ / k₁) = −(Ea / R)(1 / T₂ − 1 / T₁) = (Ea / R)(T₂ − T₁) / (T₁ T₂). The classic textbook rule of thumb that "a 10°C rise near room temperature (25°C → 35°C) doubles the reaction rate" corresponds specifically to a reaction with Ea ≈ 52.95 kJ/mol. By contrast, a reaction with Ea = 100 kJ/mol accelerates by ~3.70× over the same 10°C interval.
In physical chemistry laboratories and pharmaceutical accelerated stability studies, rate constants measured at multiple temperatures (Tᵢ, kᵢ) are analyzed via an Arrhenius plot of y = ln k versus x = 1/T using Ordinary Least Squares (OLS) linear regression. The slope m = −Ea / R yields the activation energy Ea = −m · R (and its standard error SE from the slope variance), while the y-intercept b = ln A gives A = exp(b). Connecting with Eyring transition-state theory k = (kB T / h) exp(−ΔG‡ / RT) further provides the enthalpy of activation ΔH‡ = Ea − R T_ref, entropy of activation ΔS‡ = R [ln(A h / (kB T_ref)) − 1], and Gibbs energy of activation ΔG‡ = ΔH‡ − T_ref ΔS‡.
Once k is known at a given temperature, integrated rate laws govern reactant depletion over time t. For 0th-order kinetics, [A]t = [A]₀ − kt with half-life t₁/₂ = [A]₀ / (2k); for 1st-order kinetics, [A]t = [A]₀ exp(−kt) with concentration-independent half-life t₁/₂ = ln(2) / k ≈ 0.6931 / k and 90% shelf-life t₉₀ = −ln(0.9) / k ≈ 0.1054 / k; for 2nd-order kinetics, 1/[A]t = 1/[A]₀ + kt with half-life t₁/₂ = 1 / (k[A]₀).
FAQ
What is the most common unit mistake when using the Arrhenius equation?
Activation energy Ea is typically reported in kJ/mol, whereas the molar gas constant R = 8.314 J/(mol·K) is in J/(mol·K). Because the exponent −Ea / (RT) must be dimensionless, you must convert Ea to J/mol (multiply kJ/mol by 1,000) before dividing by R·T. Additionally, temperatures must always be converted from Celsius (°C) to absolute Kelvin (T = t + 273.15).
How does adding a catalyst affect the parameters in the Arrhenius equation?
A catalyst provides an alternative reaction pathway with a lower activation energy barrier Ea without altering the overall thermodynamic enthalpy change ΔH of the reaction. For example, uncatalyzed H₂O₂ decomposition has Ea ≈ 75 kJ/mol, which drops to ~49 kJ/mol on colloidal platinum and ~8 kJ/mol with catalase enzyme—increasing the Boltzmann factor exp(−Ea/RT) at 25°C by more than 11 orders of magnitude (~5.5 × 10¹¹×).
How does Arrhenius activation energy Ea relate to Eyring enthalpy of activation ΔH‡?
In condensed-phase and unimolecular reactions, experimental Arrhenius activation energy Ea relates to the Eyring enthalpy of activation ΔH‡ via Ea = ΔH‡ + RT (or ΔH‡ = Ea − RT). At 25.0°C (298.15 K), the thermal energy term RT equals 2.479 kJ/mol, so ΔH‡ is 2.48 kJ/mol smaller than Ea.
How are half-life (t₁/₂) and 90% shelf-life (t₉₀) calculated for a first-order reaction?
For a first-order reaction [A]t = [A]₀ exp(−kt), setting [A]t / [A]₀ = 0.5 yields t₁/₂ = ln(2) / k ≈ 0.69315 / k. Setting [A]t / [A]₀ = 0.90 (10% degradation) yields the shelf-life t₉₀ = −ln(0.90) / k ≈ 0.10536 / k, which is approximately 15.2% of the half-life (t₉₀ ≈ 0.152 × t₁/₂).
What activation energy Ea corresponds to the textbook rule that "reaction rate doubles for a 10°C rise"?
Near room temperature (25°C = 298.15 K to 35°C = 308.15 K), a 2-fold rate increase corresponds to Ea = R·ln(2) / (1/298.15 − 1/308.15) ≈ 52.95 kJ/mol (~53 kJ/mol). For a reaction with Ea = 100 kJ/mol, the same 10°C rise increases the rate by ~3.70×.
How does half-life t₁/₂ depend on initial concentration [A]₀ across 0th-, 1st-, and 2nd-order reactions?
For a 1st-order reaction, t₁/₂ = ln(2)/k ≈ 0.693/k is strictly independent of initial concentration [A]₀. For a 2nd-order reaction, t₁/₂ = 1/(k[A]₀) is inversely proportional to [A]₀ (slowing down as reactant depletes), whereas for a 0th-order reaction, t₁/₂ = [A]₀/(2k) is directly proportional to [A]₀.
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Verified: Checked November 11, 2026 via independent calculation and Playwright tests in both languages: 2-temperature Ea (25°C→35°C, k₂/k₁=2.00) gives 52.95 kJ/mol (CODATA R) and 52.92 kJ/mol (textbook R=8.31); N₂O₅ at 45°C (Ea=103.0 kJ/mol, k₁=3.46×10⁻⁵ s⁻¹ at 25°C) gives k₂/k₁=13.63× and k₂=4.71×10⁻⁴ s⁻¹; target T₂ for 10× rate (Ea=80.0 kJ/mol from 27°C) gives 50.23°C (323.38 K); 4-point N₂O₅ OLS regression gives Ea=102.68±1.54 kJ/mol, A=3.42×10¹³ s⁻¹, R²=0.9996, and Eyring ΔH‡=+100.20 kJ/mol; 1st-order decay (k=0.0150 s⁻¹, [A]₀=1.00 M, 60 s) gives [A]t=0.4066 M, t₁/₂=46.21 s, and t₉₀=7.024 s. Applies to stated ideal kinetic models.