# Specific activity

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

Specific activity
Common symbols
a
SI unitbecquerel
Other units
rutherford, curie
In SI base unitss−1

Activity is a quantity related to radioactivity. The SI unit of activity 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. Its related non-SI unit equivalent[citation needed] 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.

## Formulation

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

$-{\frac {dN}{dt}}=\lambda N$

Mass of the radionuclide is given by

${\frac {N}{N_{\text{A}}}}[{\text{mol}}]\times {m}[{\text{g }}{\text{mol}}^{-1}]$

where m is mass number of the radionuclide and NA is the Avogadro constant.

Specific radioactivity a is defined as radioactivity per unit mass of the radionuclide:

$a[{\text{Bq/g}}]={\frac {\lambda N}{{m}N/N_{\text{A}}}}={\frac {\lambda N_{\text{A}}}{m}}$

In addition, decay constant λ is related to the half-life T1/2 by the following equation:

${\lambda }={\frac {ln2}{T_{1/2}}}$

Thus, specific radioactivity can also be described by

$a={\frac {{N_{\text{A}}}ln2}{T_{1/2}\times {m}}}$

This equation is simplified by

$a[{\text{Bq/g}}]\simeq {\frac {4.17\times 10^{23}[{\text{mol}}^{-1}]}{T_{1/2}[s]\times {m}[{\text{g }}{\text{mol}}^{-1}]}}$

When the unit of half-life converts a year

$a[{\text{Bq/g}}]={\frac {4.17\times 10^{23}[{\text{mol}}^{-1}]}{T_{1/2}[year]\times 365\times 24\times 60\times 60[s/year]\times {m}}}\simeq {\frac {1.32\times 10^{16}[{\text{mol}}^{-1}{\text{s }}^{-1}{\text{year}}]}{T_{1/2}[year]\times {m}[{\text{g }}{\text{mol}}^{-1}]}}$

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

$a_{Ra-266}[{\text{Bq/g}}]={\frac {1.32\times 10^{16}}{1600[year]\times 226}}\simeq {3.7}\times 10^{10}[{\text{Bq/g}}]$

This value derived from radium 226 was defined as unit of radioactivity known as Curie (Ci).

## Half-life

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:

$-{\frac {dN}{dt}}=\lambda N$

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 log of both sides, the half-life is given by

${T_{1/2}}={\frac {ln2}{\lambda }}$

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

${T_{1/2}}={\frac {N_{\text{A}}ln2}{a\times {m}}}$

### 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's atomic weight is 87, so one gram is one 87th of a mole. Plugging in the numbers:

${T_{1/2}}={\frac {N_{\text{A}}\times ln2}{a\times {m}}}\simeq {\frac {6.022\times 10^{23}[{\text{mol}}^{-1}]\times 0.693}{3200[{\text{s}}^{-1}{\text{g}}^{-1}]\times 87[{\text{g }}{\text{mol}}^{-1}]}}\simeq 1.5\times 10^{18}{\text{ s or 47 billion years}}$

## 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.