Arrhenius Equation Calculator

The Arrhenius equation splits a rate constant into two factors: a pre-exponential factor A that counts how often reactant molecules approach one another in a suitable geometry, and the Boltzmann factor exp(-Ea/RT) giving the fraction of those encounters that carry enough energy to cross the barrier. The exponential dominates. At 298 K the thermal energy RT is only 2.48 kJ/mol, so a barrier of 75 kJ/mol produces an exponent of -30.3 and a Boltzmann factor near 7.3 times 10 to the minus 14. Combined with a typical unimolecular A of 10 to the 13th per second, that still yields a rate constant of about 0.73 per second. A carries the units of k: per second for unimolecular steps, usually between 10 to the 12th and 10 to the 15th, and per molar per second for bimolecular steps, usually 10 to the 9th to 10 to the 11th. Two input conventions cause most errors: temperature must be absolute, and an activation energy entered in kilojoules per mole must be converted to joules before dividing by RT. The equation treats A as temperature independent, which is an approximation; combustion and atmospheric databases therefore use the modified form k = A x T^n x exp(-Ea/RT), with n commonly between 0 and 2. The half-life reported here follows ln(2)/k and is meaningful only for first-order kinetics. A predicted bimolecular constant should also be checked against the diffusion limit in solution, roughly 10 to the 9th to 10 to the 10th per molar per second in water at 25 degrees C.

Quantitative kinetics began in 1850, when Ludwig Wilhelmy tracked sugar inversion with a polarimeter and expressed the result as a differential equation. In the 1860s Augustus Vernon Harcourt and the mathematician William Esson at Oxford analysed the reaction of hydrogen peroxide with hydrogen iodide and produced integrated rate expressions. Temperature dependence was attacked empirically by J. J. Hood in the 1880s and thermodynamically by Jacobus Henricus van't Hoff in 1884. Svante Arrhenius drew these threads together in his 1889 paper, giving the two-parameter exponential form and the activated-molecule interpretation that carry his name. During the twentieth century the equation became a reporting format as much as a theory. Evaluated compilations such as the NIST Chemical Kinetics Database, and the periodic gas-phase evaluations issued by IUPAC and by the NASA Jet Propulsion Laboratory panel, tabulate reactions as A and Ea pairs. Detailed combustion mechanisms such as GRI-Mech list every elementary step in modified-Arrhenius form so that solvers can reconstruct rate constants at any temperature in a simulation. IUPAC responded to the curvature seen in careful measurements by defining the activation energy differentially rather than as a fitted constant, which preserves a usable definition for reactions whose plots bend. The same two-parameter form is now applied to dopant diffusion in silicon, polymer ageing, corrosion and battery degradation.