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11 Radiation for Radionuclide Users Decay Law The rate at which a quantity of radioactive material decays is directly proportional to the number of radioactive atoms present. This can be expressed mathematically by the equation: Eq. 1 Where dN/dt is the disintegration rate of the radioactive atoms, λ is the decay constant, and N is the number of radioactive atoms present at time t. Further mathematical treatment of this equation (i.e., by integration) yields: Eq. 2 Where N o is the initial number of radioactive atoms present and e is the base of the natural logarithms. Since activity (A) is proportional to N, the equation is often expressed as: Eq. 3 It can be shown mathematically that the half-life (T ½ ) of a particular radionuclide is related to the decay constant ( λ ) as follows: Eq. 4 Substituting this value of λ into Equation 3, one gets: Eq. 5 This is a very useful equation for determining the activity of a particular radionuclide after a particular period of time. Example 1: A researcher obtains 5 mCi of Phosphorus-32 (T1/2 = 14.3 days). How much activity will remain after 10 days? t = 10 d A = 5e-(0.0048)(10) = 3.1 mCi