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Since the probability of radioactive decay for a given radionuclide is a fixed physical quantity (with some slight exceptions, see Changing decay rates), the number of decays that occur in a given time of a specific number of atoms of that radionuclide is also a fixed physical quantity (if there are large enough numbers of atoms to ignore statistical fluctuations).
Thus, specific activity is defined as the activity per quantity of atoms of a particular radionuclide. It is usually given in units of Bq/g, but another commonly used unit of activity is the curie (Ci) allowing the definition of specific activity in Ci/g.
Radioactivity is expressed as the decay rate of a particular radionuclide with decay constant λ and the number of atoms N:
Mass of the radionuclide is given by
Specific radioactivity a is defined as radioactivity per unit mass of the radionuclide:
In addition, decay constant λ is related to the half-life T1/2 by the following equation:
Thus, specific radioactivity can also be described by
This equation is simplified by
When the unit of half-life converts a year
For example, specific radioactivity of radium 226 with a half-life of 1600 years is obtained by
This value derived from radium 226 was defined as unit of radioactivity known as Curie (Ci).
Experimentally-measured specific activity can be used to calculate the half-life of a radionuclide.
First, radioactivity is expressed as the decay rate of a particular radionuclide with decay constant λ and the number of atoms N:
The integral solution is described by exponential decay
where N0 is the initial quantity of atoms at time t = 0.
Half-life (T1/2) is defined as the length of time for half of a given quantity of radioactive atoms to undergo radioactive decay:
Taking the natural log of both sides, the half-life is given by
Where decay constant λ is related to specific radioactivity a by the following equation:
Therefore, the half-life can also be described by
Example: half-life of Rb-87
One gram of rubidium-87 and a radioactivity count rate that, after taking solid angle effects into account, is consistent with a decay rate of 3200 decays per second corresponds to a specific activity of ×106 Bq/kg. Rubidium's atomic weight is 87, so one gram is one 87th of a mole. Plugging in the numbers: 3.2
The specific activity of radionuclides is particularly relevant when it comes to select them for production for therapeutic pharmaceuticals, as well as for immunoassays or other diagnostic procedures, or assessing radioactivity in certain environments, among several other biomedical applications.
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