# Specific activity

Activity
Ra 226 radiation source. Activity 3300 Bq (3.3 kBq)
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 (symbol a) is the activity per unit mass of a radionuclide and is a physical property of that radionuclide.[1][2] It is usually given in units of becquerel per kilogram (Bq/kg), but another commonly used unit of specific activity is the curie per gram (Ci/g).

In the context of radioactivity, activity or total activity (symbol A) is a physical quantity defined as the number of radioactive transformations per second that occur in a particular radionuclide.[3] The unit of activity is the becquerel (symbol Bq), which is defined equivalent to reciprocal seconds (symbol s-1). The older, non-SI unit of activity is the curie (Ci), which is 3.7×1010 radioactive decays per second. Another unit of activity is the rutherford, which is defined as 1×106 radioactive decays per second.

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

Since the probability of radioactive decay for a given radionuclide within a set time interval is fixed (with some slight exceptions, see changing decay rates), the number of decays that occur in a given time of a given mass (and hence a specific number of atoms) of that radionuclide is also a fixed (ignoring statistical fluctuations).

## 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:

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

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

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

Conversely, the decay constant λ can be derived from the half-life T1/2 as

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

### Calculation of specific activity

The mass of the radionuclide is given by

${\displaystyle {m}={\frac {N}{N_{\text{A}}}}[{\text{mol}}]\times {M}[{\text{g/mol}}],}$

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.

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

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

Thus, specific radioactivity can also be described by

${\displaystyle a={\frac {N_{\text{A}}\ln 2}{T_{1/2}\times M}}.}$

This equation is simplified to

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

When the unit of half-life is in years instead of seconds:

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

#### Example: specific activity of Ra-226

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

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

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:

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

Therefore, the half-life can also be described by

${\displaystyle T_{1/2}={\frac {N_{\text{A}}\ln 2}{a\times M}}.}$

#### 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:

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

### Other calculations

For a given mass ${\displaystyle m}$ (in grams) of an isotope with atomic mass ${\displaystyle m_{\text{a}}}$ (in g/mol) and a half-life of ${\displaystyle t_{1/2}}$ (in s), the radioactivity can be calculated using:

${\displaystyle A_{\text{Bq}}={\frac {m}{m_{\text{a}}}}N_{\text{A}}{\frac {\ln 2}{t_{1/2}}}}$

With ${\displaystyle N_{\text{A}}}$ = 6.02214076×1023 mol−1, the Avogadro constant.

Since ${\displaystyle m/m_{\text{a}}}$ is the number of moles (${\displaystyle n}$), the amount of radioactivity ${\displaystyle A}$ can be calculated by:

${\displaystyle A_{\text{Bq}}=nN_{\text{A}}{\frac {\ln 2}{t_{1/2}}}}$

For instance, on average each gram of potassium contains 117 micrograms of 40K (all other naturally occurring isotopes are stable) that has a ${\displaystyle t_{1/2}}$ of 1.277×109 years = 4.030×1016 s,[4] and has an atomic mass of 39.964 g/mol,[5] so the amount of radioactivity associated with a gram of potassium is 30 Bq.

## Examples

Isotope Half-life Mass of 1 curie Specific Activity (a) (activity per 1 kg)
232Th 1.405×1010 years 9.1 tonnes 4.07 MBq (110 μCi or 4.07 Rd)
238U 4.471×109 years 2.977 tonnes 12.58 MBq (340 μCi, or 12.58 Rd)
235U 7.038×108 years 463 kg 79.92 MBq (2.160 mCi, or 79.92 Rd)
40K 1.25×109 years 140 kg 262.7 MBq (7.1 mCi, or 262.7 Rd)
129I 15.7×106 years 5.66 kg 6.66 GBq (180 mCi, or 6.66 kRd)
99Tc 211×103 years 58 g 629 GBq (17 Ci, or 629 kRd)
239Pu 24.11×103 years 16 g 2.331 TBq (63 Ci, or 2.331 MRd)
240Pu 6563 years 4.4 g 8.51 TBq (230 Ci, or 8.51MRd)
14C 5730 years 0.22 g 166.5 TBq (4.5 kCi, or 166.5 MRd)
226Ra 1601 years 1.01 g 36.63 TBq (990 Ci, or 36.63 MRd)
241Am 432.6 years 0.29 g 126.91 TBq (3.43 kCi, or 126.91 MRd)
238Pu 88 years 59 mg 629 TBq (17 kCi, or 629 MRd)
137Cs 30.17 years 12 mg 3.071 PBq (83 kCi, or 3.071 GRd)
90Sr 28.8 years 7.2 mg 5.143 PBq (139 kCi, or 5.143 GRd)
241Pu 14 years 9.4 mg 3.922 PBq (106 kCi, or 3.922 GRd)
3H 12.32 years 104 μg 355.977 PBq (9.621 MCi, or 355.977 GRd)
228Ra 5.75 years 3.67 mg 10.101 PBq (273 kCi, or 10.101 GRd)
60Co 1925 days 883 μg 41.884 PBq (1.132 MCi, or 41.884 GRd)
210Po 138 days 223 μg 165.908 PBq (4.484 MCi, or 165.908 GRd)
131I 8.02 days 8 μg 4.625 EBq (125 MCi, or 4.625 TRd)
123I 13 hours 518 ng 71.41 EBq (1.93 GCi, or 71.41 TRd)
212Pb 10.64 hours 719 ng 51.43 EBq (1.39 GCi, or 51.43 TRd)

## 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.[6][7][8][9][10][11]

## 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. S2CID 9652140.
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. S2CID 97433328.
3. ^ "SI units for ionizing radiation: becquerel". Resolutions of the 15th CGPM (Resolution 8). 1975. Retrieved 3 July 2015.
4. ^ "Table of Isotopes decay data". Lund University. 1990-06-01. Retrieved 2014-01-12.
5. ^ "Atomic Weights and Isotopic Compositions for All Elements". NIST. Retrieved 2014-01-12.
6. ^ 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.
7. ^ 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.
8. ^ 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. doi:10.1093/clinchem/29.8.1474. PMID 6191885.
9. ^ Neves, M.; Kling, A.; Lambrecht, R. M. (2002). "Radionuclide production for therapeutic radiopharmaceuticals". Applied Radiation and Isotopes. 57 (5): 657–664. CiteSeerX 10.1.1.503.4385. doi:10.1016/S0969-8043(02)00180-X. ISSN 0969-8043. PMID 12433039.
10. ^ 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.
11. ^ 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. 115 (2): 263–288. doi:10.1007/BF02037443. ISSN 0236-5731. S2CID 94361207.