Specific activity

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

Activity
Common symbols	A
SI unit	becquerel
Other units	rutherford, curie
In SI base units	s^−1

Specific activity
Common symbols	a
SI unit	becquerel per kilogram
Other units	rutherford per gram, curie per gram
In SI base units	s^−1 kg^−1

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/Kg, 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 T_1/2

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 T_1/2 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 T_1/2 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 N_A 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:

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

Example: specific activity of Ra-226

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

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

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:

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

Examples

Isotope	Half-life	Mass of 1 curie	Specific activity (Ci/g)
^232Th	1.405×10^10 years	9.1 tonnes	1.1×10^−7 (110,000 pCi/g, 0.11 μCi/g)
^238U	4.471×10^9 years	2.977 tonnes	3.4×10^−7 (340,000 pCi/g, 0.34 μCi/g)
^40K	1.25×10^9 years	140 kg	7.1×10^−6 (7,100,000 pCi/g, 7.1 μCi/g)
^235U	7.038×10^8 years	463 kg	2.2×10^−6 (2,160,000 pCi/g, 2.2 μCi/g)
^129I	15.7×10^6 years	5.66 kg	0.00018
^99Tc	211×10^3 years	58 g	0.017
^239Pu	24.11×10^3 years	16 g	0.063
^240Pu	6563 years	4.4 g	0.23
^14C	5730 years	0.22 g	4.5
^226Ra	1601 years	1.01 g	0.99
^241Am	432.6 years	0.29 g	3.43
^238Pu	88 years	59 mg	17
^137Cs	30.17 years	12 mg	83
^90Sr	28.8 years	7.2 mg	139
^241Pu	14 years	9.4 mg	106
^3H	12.32 years	104 μg	9,621
^228Ra	5.75 years	3.67 mg	273
^60Co	1925 days	883 μg	1,132
^210Po	138 days	223 μg	4,484
^131I	8.02 days	8 μg	125,000
^123I	13 hours	518 ng	1,930,000
^212Pb	10.64 hours	719 ng	1,390,000

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]

Ionizing radiation related quantities

Quantity	Unit	Symbol	Derivation	Year	SI equivalence
Activity (A)	becquerel	Bq	s^−1	1974	SI unit
	curie	Ci	3.7 × 10^10 s^−1	1953	3.7×10^10 Bq
	rutherford	Rd	10^6 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
	rad	rad	100 erg⋅g^−1	1953	0.010 Gy
Equivalent dose (H)	sievert	Sv	J⋅kg^−1 × WR	1977	SI unit
	röntgen equivalent man	rem	100 erg⋅g^−1 x WR	1971	0.010 Sv
Effective dose (E)	sievert	Sv	J⋅kg^−1 × WR x WT	1977	SI unit
	röntgen equivalent man	rem	100 erg⋅g^−1 x WR x WT	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. doi:10.1093/clinchem/29.8.1474. PMID 6191885.CS1 maint: uses authors parameter (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. PMID 12433039.
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.

Further reading

• Fetter, Steve; Cheng, E. T.; Mann, F. M. (1990). "Long-term radioactive waste from fusion reactors: Part II". Fusion Engineering and Design. 13 (2): 239–246. CiteSeerX 10.1.1.465.5945. doi:10.1016/0920-3796(90)90104-E. ISSN 0920-3796.
• Holland, Jason P.; Sheh, Yiauchung; Lewis, Jason S. (2009). "Standardized methods for the production of high specific-activity zirconium-89". Nuclear Medicine and Biology. 36 (7): 729–739. doi:10.1016/j.nucmedbio.2009.05.007. ISSN 0969-8051. PMC 2827875. PMID 19720285.
• McCarthy, Deborah W.; Shefer, Ruth E.; Klinkowstein, Robert E.; Bass, Laura A.; Margeneau, William H.; Cutler, Cathy S.; Anderson, Carolyn J.; Welch, Michael J. (1997). "Efficient production of high specific activity ⁶⁴Cu using a biomedical cyclotron". Nuclear Medicine and Biology. 24 (1): 35–43. doi:10.1016/S0969-8051(96)00157-6. ISSN 0969-8051. PMID 9080473.