Radioactive nuclei decay by first-order kinetics.

where k is the rate constant and N is the number of atoms of the isotope in the sample

Activity - The rate at which a radioactive isotope decays.  Generally expressed in curies.

• Curie (Ci) - The amount of radioactive isotope required to achieve an activity of 3.700 x 10¹⁰ disintegrations per second.
  • Originally defined as the number of disintegrations per second in 1 g of ²²⁶Ra.

Relating rates of radioactive decay

1. Rate Constant (k)

   • The rate constant for ¹⁴C is much larger than the rate constant for ²³⁸U.

   ¹⁴C:    k = 1.210 x 10^-4 y^{-1
   238}U:  k = 1.54 x 10^-10 y^-1

   • Therefore, ¹⁴C decays much faster than ²³⁸U.

2. Half-life
   • The half life for the decay of ¹⁴C is much shorter than that of ²³⁸U.

     ¹⁴C:    t_1/2 = 5730 yr
     ²³⁸U:   t_1/2 = 4.51 x 10⁹ yr

   • Therefore, ¹⁴C decays much faster than ²³⁸U.

Half-life - The length of time it takes for exactly half of the nuclei of a radioactive sample to decay.

The integrated form of the first-order rate law in which N is substituted for the concentration of X:

Example:

Chart and plot the decay of a 120 g sample with a half life of 3 years.

Next:  "Carbon Dating"