Back in the day, scientists knew that reaction rate depended on temperature. It took a little while before they figured out HOW it depended on temperature, but eventually they found a formula that worked ... it depended on the Activation Energy (how much energy was needed to get the reaction going) and the actual temperature (how much energy was available at the moment). The formula is called the Arrhenius Equation...
where A is the "Arrhenius constant" for the reaction, Ea is the Activation Energy, R is the gas constant, and T is the temperature.
This form of the reaction isn't used much, and you'll only use it is you're given A, or asked to solve for A. More often, you'll use the equation indirectly to compare a reaction's rate constant at two different temperatures:
The derivation of this second equation is shown at right, but it's not really necessary for your purposes. It's more important that you're able to use the equation to solve questions like this one...
Q: A particular reaction has a rate constant of k=2.3 x 10^-3 at 20ºC, and 4.3 x 10^-2 at 90ºC. What is the activation energy of the reaction?
A; Substitute the known values into the formula and solve for the activation energy. See the calculation at right. In the end, we find that the activation energy is about 37 kJ/mol.
NOTE: BE CAREFUL of the units! If you are writing a multiple choice test, your teacher will definitely have the answer "36887 kJ/mol" and "37 J/mol", both of which could have been the correct answers IF they had the right units. Most reactions have an activation energy between 1 and 100 kJ/mol.