Summarizing Half-Life and Activity

Half-Life and Activity Summary

• Half-life \({t}_{1/2}\) is the time in which there is a 50% chance that a nucleus will decay. The number of nuclei \(N\) as a function of time is

  \(N={N}_{0}{e}^{-\mathrm{\lambda t}},\)

  where \({N}_{0}\) is the number present at \(t=0\), and \(\lambda \) is the decay constant, related to the half-life by

  \(\lambda =\cfrac{0\text{.}\text{693}}{{t}_{1/2}}.\)

• One of the applications of radioactive decay is radioactive dating, in which the age of a material is determined by the amount of radioactive decay that occurs. The rate of decay is called the activity \(R\):

  \(R=\cfrac{\text{Δ}N}{\text{Δ}t}.\)

• The SI unit for \(R\) is the becquerel (Bq), defined by

  \(\text{1 Bq}=\text{1 decay/s.}\)

• \(R\) is also expressed in terms of curies (Ci), where

  \(1\phantom{\rule{0.25em}{0ex}}\text{Ci}=3\text{.}\text{70}×{\text{10}}^{\text{10}}\phantom{\rule{0.25em}{0ex}}\text{Bq.}\)

• The activity \(R\) of a source is related to \(N\) and \({t}_{1/2}\) by

  \(R=\cfrac{0\text{.}\text{693}N}{{t}_{1/2}}.\)

• Since \(N\) has an exponential behavior as in the equation \(N={N}_{0}{e}^{-\mathrm{\lambda t}}\), the activity also has an exponential behavior, given by

  \(R={R}_{0}{e}^{-\mathrm{\lambda t}},\)

  where \({R}_{0}\) is the activity at \(t=0\).

Glossary

becquerel

SI unit for rate of decay of a radioactive material

half-life

the time in which there is a 50% chance that a nucleus will decay

radioactive dating

an application of radioactive decay in which the age of a material is determined by the amount of radioactivity of a particular type that occurs

decay constant

quantity that is inversely proportional to the half-life and that is used in equation for number of nuclei as a function of time

carbon-14 dating

a radioactive dating technique based on the radioactivity of carbon-14

activity

the rate of decay for radioactive nuclides

rate of decay

the number of radioactive events per unit time

curie

the activity of 1g of \({}^{\text{226}}\text{Ra}\), equal to \(\text{3.70}×{\text{10}}^{\text{10}}\phantom{\rule{0.25em}{0ex}}\text{Bq}\)