Chemical kinetics is the branch of chemistry that deals with reaction rates. We observe reaction rates by measuring how much time it takes for reactants to be converted into products.
A rate law relates the concentration of the reactants to the reaction rate in a mathematical expression. It is written in the form rate = k[reactant1][reactant2], where k is a rate constant specific to the reaction. The concentrations of the reactants may be raised to an exponent (typically first or second power).
Most reactions, summarized on paper as a single step, are actually the sum of multiple steps. The reaction rate depends on the slowest of these intermediate steps, or the rate-determining step.
Writing a Rate Law
Find the rate-determining step. Typically, if you are given rate data for an overall reaction, the data includes an indication of which intermediate step is the slowest, or the rate-determining step.
The reactants of the rate-determining step become part of the rate law. For example, if two molecules of O2 gas collide in the slow step, the rate law, at this point, becomes rate=k[O2][O2].
Determine exponents for each reactant in the rate law by observing experimental data given to you. Data should show results of the slow step having been performed several different times, each time changing the concentration of one of the reactants. If, from the baseline, the rate of the reaction doubles when the concentration of the reactant doubles, the reaction is said to be first order in that reactant, and the exponent given that reactant is 1. If doubling the concentration of the reactant quadruples the rate of the reaction, the reaction is said to be second order in that reactant, and the exponent given that reactant is 2.
TL;DR (Too Long; Didn't Read)
Since the rate-determining step may be an intermediate step to the overall reaction, your final rate law may look different from your initial reaction.