# Specific activity

Specific activity is the activity per quantity of a radionuclide and is a physical property of that radionuclide.[1][2]

Activity is a quantity related to radioactivity. The SI unit of activity is the becquerel (Bq), equal to one reciprocal second.[3]

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:

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

Mass of the radionuclide is given by

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

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

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

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

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

Thus, specific radioactivity can also be described by

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

This equation is simplified by

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

When the unit of half-life converts a year

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

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

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

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:

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

The integral solution is described by exponential decay

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

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:

${\displaystyle {\frac {N_{0}}{2}}=N_{0}e^{-\lambda T_{1/2}}\,}$

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

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

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

${\displaystyle {\lambda }={\frac {a\times {m}}{N_{A}}}}$

Therefore, the half-life can also be described by

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

### 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:

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

## 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.[4][5][6][7][8][9]

## References

1. ^ Breeman, Wouter A. P.; Jong, Marion; Visser, Theo J.; Erion, Jack L.; Krenning, Eric P. (2003). "Optimising conditions for radiolabelling of DOTA-peptides with 90Y, 111In and 177Lu at high specific activities". European Journal of Nuclear Medicine and Molecular Imaging. 30 (6): 917–920. doi:10.1007/s00259-003-1142-0. ISSN 1619-7070.
2. ^ de Goeij, J. J. M.; Bonardi, M. L. (2005). "How do we define the concepts specific activity, radioactive concentration, carrier, carrier-free and no-carrier-added?". Journal of Radioanalytical and Nuclear Chemistry. 263 (1): 13–18. doi:10.1007/s10967-005-0004-6. ISSN 0236-5731.
3. ^ "SI units for ionizing radiation: becquerel". Resolutions of the 15th CGPM (Resolution 8). 1975. Retrieved 3 July 2015.
4. ^ Duursma, E. K. "Specific activity of radionuclides sorbed by marine sediments in relation to the stable element composition." Radioactive contamination of the marine environment (1973): 57-71.
5. ^ Wessels, Barry W. (1984). "Radionuclide selection and model absorbed dose calculations for radiolabeled tumor associated antibodies". Medical Physics. 11 (5): 638. Bibcode:1984MedPh..11..638W. doi:10.1118/1.595559. ISSN 0094-2405.
6. ^ I. Weeks, I. Beheshti, F. McCapra, A. K. Campbell & J. S. Woodhead (August 1983). "Acridinium esters as high-specific-activity labels in immunoassay". Clinical chemistry. 29 (8): 1474–1479. PMID 6191885.
7. ^ Neves, M; Kling, A; Lambrecht, R.M (2002). "Radionuclide production for therapeutic radiopharmaceuticals". Applied Radiation and Isotopes. 57 (5): 657–664. doi:10.1016/S0969-8043(02)00180-X. ISSN 0969-8043.
8. ^ Mausner, Leonard F. (1993). "Selection of radionuclides for radioimmunotherapy". Medical Physics. 20 (2): 503. Bibcode:1993MedPh..20..503M. doi:10.1118/1.597045. ISSN 0094-2405.
9. ^ Murray, A. S.; Marten, R.; Johnston, A.; Martin, P. (1987). "Analysis for naturally occuring radionuclides at environmental concentrations by gamma spectrometry". Journal of Radioanalytical and Nuclear Chemistry Articles. 115 (2): 263–288. doi:10.1007/BF02037443. ISSN 0236-5731.