Activated Rates of Chemical Reactions: the Arrhenius Formula

Chemical reactions happen at different speeds at different temperatures. The reaction
Gasoline + Oxygen -> Carbon Dioxide + Water
happens very slowly at room temperature. That's why you can pour gasoline into your car without it bursting into flames! At high temperatures, it happens quickly. The reaction also releases a lot of heat. That's why, if you light a match while pouring gasoline into your car, it can cause a problem. The gasoline nearest your match heats up, reacts with the oxygen in the air, and gives off enough heat to raise the temperature of the next bit of gasoline. That's why they have NO SMOKING signs at gas stations.

This speedup with temperature is often found to have an activated, or Arrhenius form:

Speed = C exp(-E/T).
Here, T is the temperature, measured from absolute zero, and C is the speed of the reaction if the temperature is very high (often around ten trillion reactions a second, 1013). The exponential function
exp(-x) = e-x = 10-0.43 x
gets very small when x gets big. As T gets smaller than E, the reaction gets slower and slower: when T=E/30, the gasoline molecules would each take about one second to react (Speed ~ 1). Even on a hot day, gasoline vapors in an enclosed space still smell after many hours (DON'T TRY THIS AT HOME!), so we know E is bigger than 30 T for hot days.

What is the physical (or chemical) explanation for this formula? Well, the gasoline and oxygen are happy, tightly bound molecules just the way they are. The atoms would be even happier if they could figure out to rearrange into carbon dioxide and water: this difference in happiness is the energy given off when the gasoline burns. In the picture, we see that the gasoline and air mixture we show in a little bowl or well, where the energy is fairly low. The other well represents the burned-up gasoline, and is even lower: the difference in height is the energy released. The main thing to notice is that there is a mound, or barrier between the two wells! The gasoline and oxygen have to get torn apart and mushed together in unhappy ways before they can reach their best state. The barrier represents the easiest way to get from one state to another. The height of this barrier is E in Arrhenius' formula.

The idea is that at high temperatures, the atoms wiggle fast, and the gasoline and oxygen can occasionally wiggle over the barrier. At low temperatures, they can only vibrate a bit inside their wells, and the reaction doesn't go. The particular formula Speed = C exp(-E/T) comes from the amazing physics truth that at temperature T, every possible configuration of atoms with energy E occurs with probability proportunate to exp(-E/T).

More information:

Last modified: August 13, 1997

James P. Sethna, sethna@lassp.cornell.edu

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).