# Arrhenius equation The **Arrhenius equation** is the formula that describes how the rate constant of a [[Chemical_reaction|chemical reaction]] depends on [[Temperature|temperature]]: `k = A·exp(−Ea/(R·T))`, where k is the rate constant, A is the pre-exponential or frequency factor, Ea is the [[Activation_energy|activation energy]], R = 8.314 J mol⁻¹ K⁻¹ is the gas constant and T is the absolute temperature.[^af-arrhenius] Named for Svante Arrhenius, who received the 1903 Nobel Prize in Chemistry for his theory of electrolytic dissociation,[^nobel-1903] it is the working law of [[Chemical_kinetics|chemical kinetics]] wherever a rate must be carried from one temperature to another, from the shelf life of a drug to the [[Creep_(deformation)|creep]] of a turbine blade. In the microsim below the reader slides the temperature from 500 to 800 K along the Portal Book's five measured rate constants for the decomposition of hydrogen iodide, and a marker rides the straight line of `ln k` against `1/T` while a readout answers how many times faster the reaction runs than at 555 K.[^af-hi] On the Chemistry flagship's spine this is the main article for Part VI — Reaction, section *Arrhenius and the activation energy*, the sibling that follows Chemical kinetics and its half-life test; [[Catalysis|Catalysis]] takes the barrier idea one step further by lowering Ea without touching the [[Equilibrium_constant|equilibrium constant]]. The equation is empirical in origin and remarkably general. It fits gas reactions, reactions in [[Solution_(chemistry)|solution]], [[Diffusion|diffusion]] in solids, [[Enzyme|enzyme]] turnover below the temperature at which the protein unfolds, and electrode reactions, where electrochemical models write the exchange current in the same semi-empirical form.[^hav-arrhenius] Its two interpretations, [[Collision_theory|collision theory]] and [[Transition_state|transition-state]] theory, each give the exponential a physical meaning and each predict a weak extra temperature dependence in A that the plain form ignores. ## Formulation In the form `k = A·exp(−Ea/(R·T))` the exponential factor is dimensionless and A carries the units of k, which depend on the order of the [[Rate_equation|rate law]]: s⁻¹ for a first-order reaction, L mol⁻¹ s⁻¹ for second order.[^af-units] The same law can be written per molecule with the [[Boltzmann_constant|Boltzmann constant]], `k = A·exp(−Ea/(kB·T))`, with Ea then in joules per molecule rather than per mole. Taking the natural logarithm gives the linear form on which measurements are analysed, `ln k = ln A − (Ea/R)·(1/T)`: a plot of ln k against 1/T is a straight line of slope −Ea/R and intercept ln A.[^af-arrhenius] Eliminating A between two temperatures gives the two-point form `ln(k1/k2) = (Ea/R)·(1/T2 − 1/T1)`, which is the quickest way to find Ea from two rate constants or to predict a rate constant at a third temperature.[^af-twopoint] The exponential is a steep function. Because Ea/R is typically thousands or tens of thousands of kelvin, a modest change in T moves the argument of the exponential by a large fraction, and the rate constant answers by orders of magnitude. The Portal Book's data for 2 HI → H₂ + I₂ run from k = 3.52 × 10⁻⁷ L mol⁻¹ s⁻¹ at 555 K, through 1.22 × 10⁻⁶ at 575 K, 8.59 × 10⁻⁵ at 645 K and 1.16 × 10⁻³ at 700 K, to 3.95 × 10⁻² L mol⁻¹ s⁻¹ at 781 K, a factor of more than 10⁵ over 226 K, and give an activation energy the book reports as 1.8 × 10⁵ J/mol.[^af-hi] The book's second example, N₂O₅ with k = 1.66 L mol⁻¹ s⁻¹ at 650 K and 7.39 at 700 K, gives 1.1 × 10⁵ J/mol from the two-point form.[^af-n2o5] The familiar rule that a rate roughly doubles for each 10 °C rise is a special case, not a law: near room temperature it corresponds to an activation energy of about 50 kJ/mol (a derived figure), and it fails for barriers or temperatures far from those.[^af-rule] ## Derivation Arrhenius did not derive the equation from mechanics; he proposed it by analogy with the temperature dependence of the [[Equilibrium_constant|equilibrium constant]]. The [[Van_'t_Hoff_equation|van 't Hoff equation]], `d ln K/dT = ΔH°/(R·T²)`, follows from the relation between the standard free energy change and K, `ΔG° = −R·T·ln K`, that the Portal Book's thermodynamics chapter establishes.[^af-freeenergy] Since a reversible reaction at [[Chemical_equilibrium|equilibrium]] has `K = kf/kr`,[^af-keq] the logarithm of K is the difference of the logarithms of two rate constants, and Arrhenius's step was to suppose that each rate constant obeys a law of the same shape on its own, `d ln k/dT = Ea/(R·T²)`, with a constant Ea in place of ΔH°. Integrating with Ea independent of temperature gives `ln k = −Ea/(R·T) + constant`, which is the linear form above with the constant identified as ln A. The argument makes a prediction that can be checked: the difference of the two activation energies, `Ea,forward − Ea,reverse`, must equal the reaction enthalpy. On an energy diagram this is the geometry of a barrier between two wells, with Ea the height of the transition state above the reactants and ΔH the height of the products above the reactants,[^af-ea] and it is why a catalyst, which lowers the single barrier seen from both sides, must speed the forward and reverse reactions by the same factor.[^af-catalyst] The derivation also shows what the equation assumes: an Ea that does not change with temperature and a constant pre-exponential factor. Both hold well over the range of a typical study and neither holds exactly, which is the subject of the modified equation below. ## Arrhenius plot An Arrhenius plot is the graph of ln k (or log₁₀ k) against 1/T in K⁻¹. If the data fall on a straight line the reaction obeys the equation over that range, the slope gives −Ea/R and the intercept at 1/T = 0 gives ln A. The reciprocal must be taken of the absolute temperature; a plot against 1/°C has no meaning. Because A is found by extrapolating to infinite temperature, small errors in slope become large errors in A, and the Portal Book's advice for rate laws in general, that a graphical or [[Least_squares|least-squares]] fit is more reliable than a two-point estimate, applies with force here.[^af-graphical] The microsim on this page is the Arrhenius plot for the hydrogen iodide data drawn live. At load the sim fits A and Ea to the five book points by least squares in the (1/T, ln k) plane; the fit gives Ea = 185 kJ/mol and ln A = 25.2 with A in L mol⁻¹ s⁻¹ (both derived here from the book's table), and a two-point fit between 555 K and 781 K gives the same 185 kJ/mol, so the book's 1.8 × 10⁵ J/mol is a rounding of the same number.[^af-hi] The reader's one control is T from 500 to 800 K in 1 K steps, starting at 555 K. As T moves, a marker slides along the fitted line, the five measured points stay fixed for comparison, and a readout on a logarithmic scale shows `k/k(555 K)`. Dragged to 781 K the readout passes 10⁵, the book's own span; near 560 K each 10 K step almost exactly doubles k, because with Ea = 185 kJ/mol the ratio `exp((Ea/R)·(1/560 − 1/570))` comes to 2.0, which is why the doubling rule works there and nowhere in particular else. The HUD shows `k = A·exp(−Ea/(R·T))`. An optional second panel sketches the energy distribution of the molecules with the tail beyond Ea shaded; that curve is ILLUSTRATIVE, since the book's own figure is a schematic rather than a computed distribution.[^af-collision] Two cautions belong to the reading of any Arrhenius plot. The printed 1/T values in a textbook table carry only three significant figures, so the last digit of a slope computed by hand is not meaningful. And a straight line over 555–781 K does not guarantee straightness outside that range; the plot is a test of the equation over the data, not a proof of it beyond them. ## Modified Arrhenius equation Both theoretical interpretations of the equation predict that the pre-exponential factor itself depends weakly on temperature, so the general form used in fitting is `k = A'·T^n·exp(−Ea/(R·T))`. [[Collision_theory|Collision theory]] gives n = 1/2, because the collision frequency of gas molecules grows with the mean speed, which by the [[Kinetic_theory_of_gases|kinetic theory of gases]] rises as the square root of T; transition-state theory gives n = 1 through its factor `kB·T/h`.[^af-collision] Over a narrow range the power of T is invisible, since T^n changes by a few per cent while the exponential changes by orders of magnitude, and the plain form fits as well as the modified one. Over a wide range, or for very accurate data, the curvature of the Arrhenius plot is real, and the exponent n is either fixed by theory or fitted as a third parameter. Combustion modelling commonly uses the three-parameter form for its elementary reactions, and the same expression describes the temperature dependence of [[Diffusion|diffusion]] coefficients in solids and of the viscosity of liquids, any process obeying it being called thermally activated. When a modified form is used, the quantity called the activation energy is no longer simply the slope of the Arrhenius plot. The operational definition that survives every form is `Ea = R·T²·(d ln k/dT)`, the local slope of ln k against 1/T at the temperature of interest; for the plain equation this reproduces the constant Ea, while for the modified form it gives `Ea + n·R·T`, larger by a small temperature-dependent term. ## Theoretical interpretation The Arrhenius equation is a summary of data before it is a theory, and three lines of interpretation give its two parameters a physical meaning. Each reproduces the exponential factor; they differ in what they say about A and in where they predict the simple law to fail. ### Arrhenius's concept of activation energy Arrhenius pictured a reaction as passing through an energetic state that only a fraction of the molecules can reach, the fraction being set by the [[Boltzmann_distribution|Boltzmann distribution]]: the proportion of molecules with energy at least Ea above the mean is proportional to `exp(−Ea/(R·T))`, and the reaction rate is proportional to that proportion. The Portal Book draws the same picture as a distribution of molecular energies with the region beyond the activation energy shaded, and notes that raising the temperature broadens the distribution and enlarges the shaded tail far more than it raises the average energy.[^af-collision] On this view Ea is the minimum energy the reacting molecules must possess, and the [[Activated_complex|activated complex]], the fleeting arrangement at the top of the barrier, is the state they pass through. Ea is defined relative to the reactants and is independent of the reaction enthalpy: a strongly exothermic reaction may have a high barrier and a mildly endothermic one a low barrier.[^af-ea] ### Collision theory [[Collision_theory|Collision theory]] gives the pre-exponential factor a mechanical origin. For a bimolecular gas reaction the rate is the number of collisions per second between the two kinds of molecule, multiplied by the fraction of collisions with relative kinetic energy along the line of centres at least Ea, multiplied by a steric factor for the fraction of sufficiently energetic collisions that also have the right orientation.[^af-collision] The collision frequency depends on the molecular sizes, the concentrations and the mean relative speed, which by the [[Maxwell–Boltzmann_distribution|Maxwell–Boltzmann distribution]] scales as the square root of temperature, so collision theory writes `k = p·Z·exp(−Ea/(R·T))` with Z proportional to T^1/2. The theory works well for simple gas reactions between small molecules and increasingly poorly as molecules grow, because the steric factor p, which it cannot calculate, then falls far below one. ### Transition state theory Transition-state theory treats the activated complex as a species in quasi-equilibrium with the reactants, located at the saddle point of the potential energy surface along the [[Reaction_coordinate|reaction coordinate]].[^af-ea] The rate constant becomes `k = (kB·T/h)·K‡`, the frequency `kB·T/h` at which complexes cross the saddle multiplied by an equilibrium constant for forming them, and writing K‡ in terms of a free energy of activation gives `k = (kB·T/h)·exp(ΔS‡/R)·exp(−ΔH‡/(R·T))`. Compared with the Arrhenius form, the enthalpy of activation plays the part of Ea, the [[Entropy|entropy]] of activation sets the size of A, and the factor of T is the source of the n = 1 modified form. The theory explains why A can be much smaller than a collision frequency, since a tightly ordered transition state has a negative entropy of activation, and it extends naturally to reactions in solution and on surfaces where collision theory has no footing. ### Limitations of the idea of Arrhenius activation energy The equation describes an enormous range of processes, and its failures are instructive. Some reactions show curved Arrhenius plots because two mechanisms with different barriers compete and trade dominance as the temperature changes; the observed Ea is then a weighted average with no single physical meaning. Reactions with no barrier at all, such as the recombination of two radicals, can have rate constants that fall as the temperature rises, a negative apparent activation energy in the operational definition above. Reactions in solution limited by [[Diffusion|diffusion]] of the partners toward each other show the small activation energy of the solvent's viscosity rather than of the chemistry. At low temperatures, light particles such as hydrogen atoms and electrons can pass through a barrier rather than over it, and the plot flattens as [[Quantum_tunnelling|quantum tunnelling]] takes over. [[Enzyme|Enzyme]]-catalysed reactions obey the equation only below the temperature at which the protein unfolds. In each case the Arrhenius activation energy is still defined, as `R·T²·(d ln k/dT)`, but it is a fitted descriptor rather than the height of a single barrier, and the microsim's straight line is what a single-barrier reaction looks like over a limited range. ## See also - [[Activation_energy]] - [[Collision_theory]] - [[Transition_state]] - [[Activated_complex]] - [[Reaction_coordinate]] - [[Chemical_kinetics]] - [[Catalysis]] - [[Van_'t_Hoff_equation]] ## References [^af-arrhenius]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.5 Collision Theory, pp. 819–820 (the Arrhenius equation; its linearized form; R = 8.314 J mol⁻¹ K⁻¹). https://open.umn.edu/opentextbooks/textbooks/chemistry-atoms-first [^af-twopoint]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.5, pp. 820–821 (two-point form ln(k1/k2) = (Ea/R)(1/T2 − 1/T1)). [^af-hi]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.5, Example 17.13, pp. 819–821 (hydrogen iodide rate constants at 555, 575, 645, 700 and 781 K; Ea reported as 1.8 × 10⁵ J/mol. The 185 kJ/mol fit, ln A = 25.2 and the 10 K doubling near 560 K are Wikitube values derived from the book's table). [^af-n2o5]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.5, p. 821 (N₂O₅ decomposition, k = 1.66 L mol⁻¹ s⁻¹ at 650 K and 7.39 at 700 K; Ea = 1.1 × 10⁵ J/mol). [^af-units]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.3–17.4, pp. 805, 816 (units of the rate constant by order). [^af-rule]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.2 Factors Affecting Reaction Rates (the rule of thumb that rates roughly double per 10 °C; page to pin). The 50 kJ/mol equivalent near 298 K is derived from the two-point form. [^af-collision]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.5 Collision Theory, pp. 818–819 (collision frequency, orientation and energy requirements; the schematic distribution of molecular energies with the fraction beyond Ea). [^af-ea]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.5 and §17.6, pp. 818, 828 (activation energy as the height of the transition state above the reactants; ΔH as the product–reactant difference; reaction diagrams). [^af-catalyst]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.7 Catalysis, p. 828 (a catalyst lowers the activation energy through a different mechanism; reactant and product energies unchanged). [^af-keq]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.7, pp. 834–835 (K = kf/kr; a catalyst does not change K). [^af-graphical]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 17 Kinetics, §17.5, p. 821 (the graphical or regression approach is "more reliable" than two-point estimates). [^af-freeenergy]: Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 12 Thermodynamics, §12.4 Free Energy, pp. 597–622 (ΔG° = −RT ln K and the temperature dependence of K; page to pin). [^hav-arrhenius]: Haverkort, Willem (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Chapter 1 Electrochemistry, p. 29 (the Arrhenius law in semi-empirical form for electrode kinetics). https://open.umn.edu/opentextbooks/textbooks/electrolysers-fuel-cells-and-batteries-analytical-modelling [^nobel-1903]: The Nobel Prize. "The Nobel Prize in Chemistry 1903 — Svante Arrhenius." NobelPrize.org, Nobel Prize Outreach. https://www.nobelprize.org/prizes/chemistry/1903/summary/ ### Cited works - Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Portal Book 051: Chapter 17 Kinetics, pp. 791–850; Chapter 12 Thermodynamics, pp. 597–622. - Haverkort, Willem (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*. Portal Book 053: Chapter 1 Electrochemistry, pp. 20–41. - The Nobel Prize in Chemistry 1903 — Svante Arrhenius. NobelPrize.org. ## General references and further reading - Boyd, W. Christopher (2025). *Exploring Inorganic and Organometallic Chemistry*. Portal Book 052, Chapter 10 Kinetics, pp. 267–282. https://open.umn.edu/opentextbooks/textbooks/exploring-inorganic-and-organometallic-chemistry - Flowers, Paul; Neth, Edward; Robinson, William et al. (2019). *Chemistry: Atoms First 2e*. OpenStax, Chapter 17 Kinetics. https://open.umn.edu/opentextbooks/textbooks/chemistry-atoms-first - Haverkort, Willem (2024). *Electrolysers, Fuel Cells and Batteries: Analytical Modelling*, Chapter 1. https://open.umn.edu/opentextbooks/textbooks/electrolysers-fuel-cells-and-batteries-analytical-modelling ## External links - *Chemistry: Atoms First 2e* (OpenStax, 2019), Portal Book 051 — Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/chemistry-atoms-first - *Electrolysers, Fuel Cells and Batteries: Analytical Modelling* (2024), Portal Book 053 — Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/electrolysers-fuel-cells-and-batteries-analytical-modelling - The Wikipedia pair's External links section lists the pair's own links. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Arrhenius_equation.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Arrhenius equation* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Arrhenius_equation.html" data-title="Arrhenius equation"></div> *Built from `MICROSIM_GUIDE/specs/sims/Arrhenius_equation.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Arrhenius_equation) : [Wikitube](https://en.wikitube.io/wiki/Arrhenius_equation) · pinned revision [1374206680](https://en.wikipedia.org/w/index.php?oldid=1374206680) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Chemistry]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Chemistry row K33 · sim pending (matter/Arrhenius_equation).*