# Specific activity

Activity
Common symbols
A
SI unitbecquerel
Other units
rutherford, curie
In SI base unitss−1
Specific activity
Common symbols
a
SI unitbecquerel per kilogram
Other units
rutherford per gram, curie per gram
In SI base unitss−1 kg−1

Specific activity is the activity per quantity of a radionuclide and is a physical property of that radionuclide.

Activity is a quantity related to radioactivity, for which the SI unit is the becquerel (Bq), equal to one reciprocal second. The becquerel is defined as the number of radioactive transformations per second that occur in a particular radionuclide. The older, non-SI unit of activity is the curie (Ci), which is 3.7×1010 transformations per second.

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. The amount of specific activity should not be confused with level of exposure to ionizing radiation and thus the exposure or absorbed dose. The absorbed dose is the quantity important in assessing the effects of ionizing radiation on humans.

## Formulation

### Relationship between λ and T1/2

Radioactivity is expressed as the decay rate of a particular radionuclide with decay constant λ and the number of atoms N:

$-{\frac {dN}{dt}}=\lambda N.$ The integral solution is described by exponential decay:

$N=N_{0}e^{-\lambda t},$ 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:

${\frac {N_{0}}{2}}=N_{0}e^{-\lambda T_{1/2}}.$ Taking the natural logarithm of both sides, the half-life is given by

$T_{1/2}={\frac {\ln 2}{\lambda }}.$ Conversely, the decay constant λ can be derived from the half-life T1/2 as

$\lambda ={\frac {\ln 2}{T_{1/2}}}.$ ### Calculation of specific activity

The mass of the radionuclide is given by

${m}={\frac {N}{N_{\text{A}}}}[{\text{mol}}]\times {M}[{\text{g/mol}}],$ where M is molar mass of the radionuclide, and NA is the Avogadro constant. Practically, the mass number A of the radionuclide is within a fraction of 1% of the molar mass expressed in g/mol and can be used as an approximation.

$a[{\text{Bq/g}}]={\frac {\lambda N}{MN/N_{\text{A}}}}={\frac {\lambda N_{\text{A}}}{M}}.$ Thus, specific radioactivity can also be described by

$a={\frac {N_{\text{A}}\ln 2}{T_{1/2}\times M}}.$ This equation is simplified to

$a[{\text{Bq/g}}]\approx {\frac {4.17\times 10^{23}[{\text{mol}}^{-1}]}{T_{1/2}[s]\times M[{\text{g/mol}}]}}.$ When the unit of half-life is in years instead of seconds:

$a[{\text{Bq/g}}]={\frac {4.17\times 10^{23}[{\text{mol}}^{-1}]}{T_{1/2}[{\text{year}}]\times 365\times 24\times 60\times 60[{\text{s/year}}]\times M}}\approx {\frac {1.32\times 10^{16}[{\text{mol}}^{-1}{\text{s}}^{-1}{\text{year}}]}{T_{1/2}[{\text{year}}]\times M[{\text{g/mol}}]}}.$ #### Example: specific activity of Ra-226

For example, specific radioactivity of radium-226 with a half-life of 1600 years is obtained as

$a_{\text{Ra-226}}[{\text{Bq/g}}]={\frac {1.32\times 10^{16}}{1600\times 226}}\approx 3.7\times 10^{10}[{\text{Bq/g}}].$ This value derived from radium-226 was defined as unit of radioactivity known as the curie (Ci).

### Calculation of half-life from specific activity

Experimentally measured specific activity can be used to calculate the half-life of a radionuclide.

Where decay constant λ is related to specific radioactivity a by the following equation:

$\lambda ={\frac {a\times M}{N_{\text{A}}}}.$ Therefore, the half-life can also be described by

$T_{1/2}={\frac {N_{\text{A}}\ln 2}{a\times M}}.$ #### 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 3.2×106 Bq/kg. Rubidium atomic mass is 87 g/mol, so one gram is 1/87 of a mole. Plugging in the numbers:

$T_{1/2}={\frac {N_{\text{A}}\times \ln 2}{a\times M}}\approx {\frac {6.022\times 10^{23}{\text{ mol}}^{-1}\times 0.693}{3200{\text{ s}}^{-1}\,{\text{g}}^{-1}\times 87{\text{ g/mol}}}}\approx 1.5\times 10^{18}{\text{ s}}\approx 47{\text{ billion years}}.$ ## Applications

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.