11.7 Half-life and activity  (Page 6/15)

where $R_0$ is the activity at $t=0$ . This equation shows exponential decay of radioactive nuclei. For example, if a source originally has a 1.00-mCi activity, it declines to 0.500 mCi in one half-life, to 0.250 mCi in two half-lives, to 0.125 mCi in three half-lives, and so on. For times other than whole half-lives, the equation $R=R_0{e}^{-\mathrm{\lambda t}}$ must be used to find $R$ .

Section 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
  $\lambda =\frac{0\text{.}\text{693}}{t_{1/2}}.$
• 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 $R$ :
  $R=\frac{\text{Δ}N}{\text{Δ}t}.$
• 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
  $1\text{Ci}=3\text{.}\text{70}\times {\text{10}}^{\text{10}}\text{Bq.}$
• The activity $R$ of a source is related to $N$ and $t_{1/2}$ by
  $R=\frac{0\text{.}\text{693}N}{t_{1/2}}.$
• 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 $t=0$ .

Conceptual questions

Problems&Exercises

Data from the appendices and the periodic table may be needed for these problems.