October 20, 2026
One Exponential, Two Constants
The Arrhenius equation writes a reaction rate constant as k = A·exp(−E_a/RT). The activation energy E_a sets how strongly the rate responds to temperature, the pre-exponential factor A sets its overall scale, and R is the gas constant. The equation is usually credited to Svante Arrhenius, who proposed it in 1889 while studying how acid speeds up the breakdown of sucrose, building on an earlier suggestion by van 't Hoff.
Arrhenius's physical picture was that only molecules in an activated state can react, and the fraction of molecules in that state follows the Boltzmann distribution, which falls exponentially with energy. Raising the temperature does not change the energy barrier, but it greatly increases the fraction of molecules with enough energy to cross it. Laidler's 1984 history of the equation is a good account of how this interpretation was argued over for decades before being accepted.
Where the Ten-Degree Rule Comes From
The old rule of thumb that reaction rates roughly double for every ten degrees of warming falls straight out of the equation. Taking the ratio of rate constants at 298 K and 308 K, a doubling requires E_a ≈ 53 kJ/mol, which happens to be a typical activation energy for many reactions near room temperature. Reactions with larger activation energies are more sensitive still: at 100 kJ/mol the same ten-degree rise multiplies the rate by about 3.7.
The rule is not universal, though. It depends on both the activation energy and the starting temperature, since the sensitivity falls as T rises. A reaction that doubles per ten degrees near room temperature will speed up noticeably less per ten degrees at 300°C. Using a fixed doubling rule over a wide range of temperatures gives poor estimates.
Getting an Activation Energy From Data
Taking logarithms gives ln k = ln A − E_a/(RT), a straight line when ln k is plotted against 1/T. Measure the rate constant at several temperatures, plot the points, and the slope is −E_a/R while the intercept gives ln A. With only two temperatures, the same relationship becomes ln(k₂/k₁) = (E_a/R)(1/T₁ − 1/T₂), the form most often used in exam problems.
A curved Arrhenius plot is itself informative. It can mean the mechanism changes with temperature, that two parallel reactions with different activation energies are competing, or that transport limitations take over at high temperature. Eyring's transition-state theory, published in 1935, also predicts a weak temperature dependence in the pre-exponential factor, and modified forms like k = A·Tⁿ·exp(−E_a/RT) are used in combustion and atmospheric chemistry where very wide temperature ranges matter.
What It Means for Reactor Design
The exponential is why thermal runaway is a central concern with exothermic reactions. Heat generated by the reaction rises exponentially with temperature, while heat removed through a cooling jacket rises only linearly with the temperature difference. If cooling falls behind even briefly, the reactor warms, the reaction accelerates, it releases heat faster, and the temperature can climb out of control. Many serious process accidents follow exactly this pattern.
Temperature is also a selectivity lever. When a desired reaction and an unwanted side reaction compete, the one with the higher activation energy gains more from heating. If the desired reaction has the higher E_a, run hot; if the side reaction does, run cooler and accept a larger reactor. Catalysts work by offering a path with a lower activation energy, and in porous catalysts, strong diffusion limitations can make the measured activation energy appear to be about half its true value, a trap explored in the next post on the Thiele modulus.
References
- Arrhenius, S. (1889). Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren. Zeitschrift für Physikalische Chemie, 4(1), 226–248. doi:10.1515/zpch-1889-0416
- Laidler, K. J. (1984). The development of the Arrhenius equation. Journal of Chemical Education, 61(6), 494. doi:10.1021/ed061p494
- Eyring, H. (1935). The activated complex in chemical reactions. Journal of Chemical Physics, 3(2), 107–115. doi:10.1063/1.1749604
- Fogler, H. S. (2016). Elements of Chemical Reaction Engineering (5th ed.), Chapter 3. Boston: Prentice Hall.