section_path: Mathematics/Decay timing: definitions and relations/Half-life
section_name: Half-life
paragraph_text: Half-life
  
  A more commonly used parameter is the half-life T. Given a sample of a particular radionuclide, the half-life is the time taken for half the radionuclide's atoms to decay. For the case of one-decay nuclear reactions:
  
  the half-life is related to the decay constant as follows: set N = N /2 and t = T to obtain
  
  This relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent, while those that radiate weakly endure longer. Half-lives of known radionuclides vary by almost 54 orders of magnitude, from more than 2.25(9) × 10 years (6.9 × 10 sec) for the very nearly stable nuclide Te, to 8.6(6) × 10 seconds for the highly unstable nuclide H.
  
  The factor of ln(2) in the above relations results from the fact that the concept of "half-life" is merely a way of selecting a different base other than the natural base e for the lifetime expression. The time constant τ is the e −1 -life, the time until only 1/ e remains, about 36.8%, rather than the 50% in the half-life of a radionuclide. Thus, τ is longer than t. The following equation can be shown to be valid:
  
  Since radioactive decay is exponential with a constant probability, each process could as easily be described with a different constant time period that (for example) gave its "(1/3)-life" (how long until only 1/3 is left) or "(1/10)-life" (a time period until only 10% is left), and so on. Thus, the choice of τ and t for marker-times, are only for convenience, and from convention. They reflect a fundamental principle only in so much as they show that the same proportion of a given radioactive substance will decay, during any time-period that one chooses.
  
  Mathematically, the n life for the above situation would be found in the same way as above — by setting N = N /n, t = T and substituting into the decay solution to obtain