Specific activity

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

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 for which the SI unit is the becquerel (Bq), equal to one reciprocal second.[3] 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

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

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

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

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

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

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

Thus, specific radioactivity can also be described by

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

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

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

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:

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

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:

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

Therefore, the half-life can also be described by

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

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:

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

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

Quantity Unit Symbol Derivation Year SI equivalence
Activity (A) becquerel Bq s−1 1974 SI unit
curie Ci 3.7 × 1010 s−1 1953 3.7×1010 Bq
rutherford Rd 106 s−1 1946 1,000,000 Bq
Exposure (X) coulomb per kilogram C/kg C⋅kg−1 of air 1974 SI unit
röntgen R esu / 0.001293 g of air 1928 2.58 × 10−4 C/kg
Absorbed dose (D) gray Gy J⋅kg−1 1974 SI unit
erg per gram erg/g erg⋅g−1 1950 1.0 × 10−4 Gy
Dose equivalent (H) sievert Sv J⋅kg−1 × WR 1977 SI unit
röntgen equivalent man rem 100 erg⋅g−1 1971 0.010 Sv

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. PMID 12677301.
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): 638–645. Bibcode:1984MedPh..11..638W. doi:10.1118/1.595559. ISSN 0094-2405. PMID 6503879.
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.CS1 maint: multiple names: authors list (link)
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): 503–509. Bibcode:1993MedPh..20..503M. doi:10.1118/1.597045. ISSN 0094-2405. PMID 8492758.
9. ^ Murray, A. S.; Marten, R.; Johnston, A.; Martin, P. (1987). "Analysis for naturally occuring [sic] 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.