The concept of an "activation-energy " barrier to reaction is illustrated with the mountain analogy opposite. The boulder cannot roll off the edge of the mountain without first surmounting the activation barrier crowned by a double dagger symbol (the conventional indication of an activated intermediate state).
A catalyst makes a chemical reaction go faster by providing an alternate path with a lower activation-energy barrier. This is symbolized by the winding path down the side of the mountain.
A rate law is an equation that relates the rate of disappearance of reactants, or appearance of products, to the reactant concentrations. The simplest type of reaction is the spontaneous decomposition of isolated molecules or atoms, and it is encountered most commonly in radioactive decay of unstable nuclei. The rate law for the breakdown of carbon-14 nuclei is
should be read as "the rate of change of carbon-14 concentration with time." (For any quantity, x, whose value changes with time, the expression dx/dt means "the rate of change of x with time.")