Reaction Rate Calculator

The Arrhenius equation states that the rate constant k equals a pre-exponential factor A multiplied by the Boltzmann term exp(-Ea/RT), with R fixed at 8.314 J per mol per kelvin. The exponential term is the fraction of molecular collisions carrying at least the activation energy Ea, while A absorbs how often collisions occur and how many have the geometry needed to react. At 298 K the quantity RT equals only 2.478 kJ/mol, so every additional 5.7 kJ/mol of activation energy costs a full factor of ten in rate. That steepness is the reason the familiar rule of thumb about rates doubling per 10 degrees C holds near room temperature for reactions with Ea around 50 kJ/mol. Ordinary bond-breaking reactions fall between roughly 50 and 250 kJ/mol, and unimolecular pre-exponential factors typically sit between 1 x 10^9 and 1 x 10^16 per second. The half-life reported alongside k assumes first-order kinetics, where the time to consume half the reactant is ln(2)/k and is independent of starting concentration. For second-order or more complex rate laws the units of A change and that half-life no longer applies. Arrhenius behavior is empirical rather than exact: over wide temperature spans a modified form with a T^n prefactor fits better, and a plot of ln k against 1/T bends whenever a multi-step mechanism changes its rate-determining step, when a reaction becomes diffusion limited, or when quantum tunneling contributes at low temperature. Catalysts do not appear in the equation at all; they act by lowering Ea along a different pathway and leave the equilibrium position untouched.

Jacobus Henricus van t Hoff proposed in 1884, in his Etudes de dynamique chimique, that a rate constant should vary with temperature in a way parallel to the thermodynamic equation for equilibrium constants, producing the exponential form on theoretical grounds. Svante Arrhenius took up the idea in 1889 and supplied the physical interpretation that made it stick: only molecules already raised to an activated state above an energy barrier can react, and the Boltzmann factor gives their fraction. The equation carries Arrhenius name for that interpretation, though van t Hoff arrived at the mathematics first. Arrhenius received the 1903 Nobel Prize in Chemistry for his separate work on electrolytic dissociation. What A actually represents took several more decades. Max Trautz in 1916 and William Cudmore McCullagh Lewis in 1918 independently developed collision theory, which calculated A from kinetic theory as a collision frequency and exposed the discrepancy that had to be patched with a steric factor. In 1935 Henry Eyring at Princeton, and Meredith Gwynne Evans and Michael Polanyi at Manchester, independently formulated transition state theory, recasting the pre-exponential factor in terms of the entropy of activation for a transient activated complex sitting at the saddle point of the potential energy surface. Modern practice retains the Arrhenius plot of ln k against reciprocal temperature as the standard way to extract activation energy from experimental data, and IUPAC defines the Arrhenius activation energy precisely as the negative of R times the slope of that line.