In fact, radioactive decay is a first-order process and can be described in terms of either the differential rate law () or the integrated rate law: \[N = N_0e^ \] \[\ln \dfrac=-kt \label\] Because radioactive decay is a first-order process, the time required for half of the nuclei in any sample of a radioactive isotope to decay is a constant, called the half-life of the isotope.
The half-life tells us how radioactive an isotope is (the number of decays per unit time); thus it is the most commonly cited property of any radioisotope.
Often students get bogged down in the fact that they don't "understand" how and why radioactive elements decay and miss the whole point of this exercise.
This becomes evident when we rearrange the integrated rate law for a first-order reaction (Equation 14.21) to produce the following equation: Figure \(\Page Index\): The Half-Life of a First-Order Reaction.
This plot shows the concentration of the reactant in a first-order reaction as a function of time and identifies a series of half-lives, intervals in which the reactant concentration decreases by a factor of 2.
If two reactions have the same order, the faster reaction will have a shorter half-life, and the slower reaction will have a longer half-life.
The half-life of a first-order reaction under a given set of reaction conditions is a constant.
Using this method implicitly assumes that the ratio in the atmosphere is constant, which is not strictly correct.