Editing Specific activity

{{Short description|Activity per unit mass of a radionuclide}}
{{About|specific activity radioactivity|the use in biochemistry|Enzyme assay#Specific activity}}
{{technical|date=January 2014}}
{{Infobox physical quantity
| name = Activity
| image = Radium 226 radiation source 1.jpg
| caption = Ra 226 radiation source. Activity 3300 Bq (3.3 kBq)
| unit = [[becquerel]]
| otherunits = [[Rutherford (unit)|rutherford]], [[Curie (unit)|curie]]	
| symbols = ''A''
| baseunits = s<sup>−1</sup>
| dimension =
| extensive =
| intensive =
| derivations =
}}
{{Infobox physical quantity
| name = Specific activity
| image = 
| caption = 
| unit = [[becquerel]] per [[kilogram]]
| otherunits = [[Rutherford (unit)|rutherford]] per [[gram]], [[Curie (unit)|curie]] per gram
| symbols = ''a''
| baseunits = s<sup>−1</sup>⋅kg<sup>−1</sup>
| dimension =
| extensive =
| intensive =
| derivations =
}}
'''Specific activity''' (symbol ''a'') is the activity [[per unit mass]] of a [[radionuclide]] and is a physical property of that radionuclide.<ref name="BreemanJong2003">{{cite journal |last1=Breeman |first1=Wouter A. P. |last2=Jong |first2=Marion |last3=Visser |first3=Theo J. |last4=Erion |first4=Jack L. |last5=Krenning |first5=Eric P. |title=Optimising conditions for radiolabelling of DOTA-peptides with <sup>90</sup>Y, <sup>111</sup>In and <sup>177</sup>Lu at high specific activities |journal=European Journal of Nuclear Medicine and Molecular Imaging |volume=30 |issue=6 |year=2003 |pages=917–920 |issn=1619-7070 |doi=10.1007/s00259-003-1142-0 |pmid=12677301|s2cid=9652140 }}</ref><ref name="de GoeijBonardi2005">{{cite journal |last1=de Goeij |first1=J. J. M. |last2=Bonardi |first2=M. L. |title=How do we define the concepts specific activity, radioactive concentration, carrier, carrier-free and no-carrier-added? |journal=Journal of Radioanalytical and Nuclear Chemistry |volume=263 |issue=1 |year=2005 |pages=13–18 |issn=0236-5731 |doi=10.1007/s10967-005-0004-6|s2cid=97433328 }}</ref>
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]].<ref>{{cite journal |title=SI units for ionizing radiation: becquerel |journal=Resolutions of the 15th CGPM |date=1975 |issue=Resolution 8 |access-date=3 July 2015 |url=http://www.bipm.org/en/CGPM/db/15/8/}}</ref> The unit of activity is the ''[[becquerel]]'' (symbol Bq), which is defined equivalent to [[reciprocal second]]s (symbol s<sup>−1</sup>). The older, non-SI unit of activity is the [[Curie (unit)|''curie'']] (Ci), which is {{val|3.7|e=10}} radioactive decays per second. Another unit of activity is the [[Rutherford (unit)|''rutherford'']], which is defined as {{val|1|e=6}} 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 [[Radioactive decay#Changing rates|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 ==
{{see also|Radioactive decay#Rates}}

===Relationship between ''λ'' and T<sub>1/2</sub>===

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

: <math>-\frac{dN}{dt} = \lambda N.</math>

The integral solution is described by [[exponential decay]]:

: <math>N = N_0 e^{-\lambda t},</math>

where ''N''<sub>0</sub> is the initial quantity of atoms at time ''t'' = 0.

[[Half-life]] '''T<sub>1/2</sub>''' is defined as the length of time for half of a given quantity of radioactive atoms to undergo radioactive decay:

: <math>\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}.</math>

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

: <math>T_{1/2} = \frac{\ln 2}{\lambda}.</math>

Conversely, the decay constant ''λ'' can be derived from the half-life ''T''<sub>1/2</sub> as

: <math>\lambda = \frac{\ln 2}{T_{1/2}}.</math>

===Calculation of specific activity===

The mass of the radionuclide is given by

: <math>{m} = \frac{N}{N_\text{A}} [\text{mol}] \times {M} [\text{g/mol}],</math>

where ''M'' is [[molar mass]] of the radionuclide, and ''N''<sub>A</sub> 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:

: <math>a [\text{Bq/g}] = \frac{\lambda N}{M N/N_\text{A}} = \frac{\lambda N_\text{A}}{M}.</math>

Thus, specific radioactivity can also be described by

: <math>a = \frac{N_\text{A} \ln 2}{T_{1/2} \times M}.</math>

This equation is simplified to

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

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

: <math>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}]}.</math>

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

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

: <math chem>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}].</math>

This value derived from radium-226 was defined as unit of radioactivity known as the [[Curie (unit)|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:

: <math>\lambda = \frac{a \times M}{N_\text{A}}.</math>

Therefore, the half-life can also be described by

: <math>T_{1/2} = \frac{N_\text{A} \ln 2}{a \times M}.</math>

==== Example: half-life of Rb-87 ====
One gram of [[Isotopes of rubidium|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 {{val|3.2|e=6|u=Bq/kg}}. Rubidium [[atomic mass]] is 87&nbsp;g/mol, so one gram is 1/87 of a mole. Plugging in the numbers:

: <math>
 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}.
</math>

===Other calculations===
{{cleanup merge|Becquerel|21=section|date=July 2023}}

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

:<math>A_\text{Bq} = \frac{m} {m_\text{a}} N_\text{A} \frac{\ln 2} {t_{1/2}}</math>

With <math>N_\text{A}</math> = {{val|6.02214076|e=23|u=mol-1}}, the [[Avogadro constant]].

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

:<math>A_\text{Bq} = nN_\text{A} \frac{\ln 2} {t_{1/2}}</math>

For instance, on average each gram of [[potassium]] contains 117&nbsp;micrograms of [[Potassium-40|<sup>40</sup>K]] (all other naturally occurring isotopes are stable) that has a <math>t_{1/2}</math> of {{val|1.277|e=9|u=years}} = {{val|4.030|e=16|u=s}},<ref>{{cite web|url=http://nucleardata.nuclear.lu.se/toi/nuclide.asp?iZA=190040 |title=Table of Isotopes decay data |publisher=[[Lund University]] |date=1990-06-01 |access-date=2014-01-12}}</ref> and has an atomic mass of 39.964&nbsp;g/mol,<ref>{{cite web|url=http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl?ele=&ascii=html&isotype=some |title=Atomic Weights and Isotopic Compositions for All Elements |publisher=[[NIST]] |access-date=2014-01-12}}</ref> so the amount of radioactivity associated with a gram of potassium is 30&nbsp;Bq.

==Examples==
{| class="wikitable"
!Isotope
!Half-life
!Mass of 1 curie
!Specific Activity (a) (activity per 1&nbsp;kg)
|-
| [[thorium-232|<sup>232</sup>Th]] || {{val|1.405|e=10}} years|| 9.1 tonnes || 4.07 MBq (110 μCi or 4.07 Rd)
|-
| [[uranium-238|<sup>238</sup>U]] ||{{val|4.471|e=9}} years|| 2.977 tonnes|| 12.58 MBq (340 μCi, or 12.58 Rd)
|-
| [[uranium-235|<sup>235</sup>U]] ||{{val|7.038|e=8}} years|| 463&nbsp;kg || 79.92 MBq (2.160 mCi, or 79.92 Rd)
|-
| [[potassium-40|<sup>40</sup>K]] ||{{val|1.25|e=9}} years||140&nbsp;kg || 262.7 MBq (7.1 mCi, or 262.7 Rd)
|-
| [[iodine-129|<sup>129</sup>I]] ||{{val|15.7|e=6}} years||5.66&nbsp;kg || 6.66 GBq (180 mCi, or 6.66 kRd)
|-
| [[technetium-99|<sup>99</sup>Tc]] ||{{val|211|e=3}} years||58 g || 629 GBq (17 Ci, or 629 kRd)
|-
| [[plutonium-239|<sup>239</sup>Pu]] ||{{val|24.11|e=3}} years||16 g|| 2.331 TBq (63 Ci, or 2.331 MRd)
|-
| [[plutonium-240|<sup>240</sup>Pu]] || 6563 years || 4.4 g || 8.51 TBq (230 Ci, or 8.51MRd)
|-
| [[carbon-14|<sup>14</sup>C]] ||5730 years||0.22 g || 166.5 TBq (4.5 kCi, or 166.5 MRd)
|-
| [[radium-226|<sup>226</sup>Ra]] || 1601 years || 1.01 g || 36.63 TBq (990 Ci, or 36.63 MRd)
|-
| [[americium-241|<sup>241</sup>Am]] || 432.6 years || 0.29 g || 126.91 TBq (3.43 kCi, or 126.91 MRd)
|-
| [[plutonium-238|<sup>238</sup>Pu]] || 88 years || 59&nbsp;mg || 629 TBq (17 kCi, or 629 MRd)
|-
| [[caesium-137|<sup>137</sup>Cs]] || 30.17 years || 12&nbsp;mg || 3.071 PBq (83 kCi, or 3.071 GRd)
|-
| [[strontium-90|<sup>90</sup>Sr]] || 28.8 years || 7.2&nbsp;mg || 5.143 PBq (139 kCi, or 5.143 GRd)
|-
| [[plutonium-241|<sup>241</sup>Pu]] || 14 years || 9.4&nbsp;mg || 3.922 PBq (106 kCi, or 3.922 GRd)
|-
| [[tritium|<sup>3</sup>H]] || 12.32 years || 104 μg || 355.977 PBq (9.621 MCi, or 355.977 GRd)
|-
| [[radium-228|<sup>228</sup>Ra]] ||5.75 years||3.67&nbsp;mg || 10.101 PBq (273 kCi, or 10.101 GRd)
|-
| [[cobalt-60|<sup>60</sup>Co]] ||1925 days||883 μg || 41.884 PBq (1.132 MCi, or 41.884 GRd)
|-
| [[polonium-210|<sup>210</sup>Po]] ||138 days||223 μg || 165.908 PBq (4.484 MCi, or 165.908 GRd)
|-
| [[iodine-131|<sup>131</sup>I]] ||8.02 days||8 μg || 4.625 EBq (125 MCi, or 4.625 TRd)
|-
| [[iodine-123|<sup>123</sup>I]] ||13 hours||518&nbsp;ng || 71.41 EBq (1.93 GCi, or 71.41 TRd)
|-
| [[lead-212|<sup>212</sup>Pb]] ||10.64 hours||719&nbsp;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.<ref>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.</ref><ref name="Wessels1984">{{cite journal |last1=Wessels |first1=Barry W. |title=Radionuclide selection and model absorbed dose calculations for radiolabeled tumor associated antibodies |journal=Medical Physics |volume=11 |issue=5 |year=1984 |pages=638–645 |issn=0094-2405 |doi=10.1118/1.595559 |pmid=6503879 |bibcode=1984MedPh..11..638W }}</ref><ref>{{Cite journal
|author=I. Weeks |author2=I. Beheshti |author3=F. McCapra |author4=A. K. Campbell |author5=J. S. Woodhead |title = Acridinium esters as high-specific-activity labels in immunoassay |journal = Clinical Chemistry |volume = 29 |issue = 8 |pages = 1474–1479 |date=August 1983 |doi = 10.1093/clinchem/29.8.1474 |pmid = 6191885}}</ref><ref name="NevesKling2002">{{cite journal |last1=Neves |first1=M. |last2=Kling |first2=A. |last3=Lambrecht |first3=R. M. |title=Radionuclide production for therapeutic radiopharmaceuticals |journal=Applied Radiation and Isotopes |volume=57 |issue=5 |year=2002 |pages=657–664 |issn=0969-8043 |doi=10.1016/S0969-8043(02)00180-X |pmid=12433039|citeseerx=10.1.1.503.4385 }}</ref><ref name="Mausner1993">{{cite journal |last1=Mausner |first1=Leonard F. |title=Selection of radionuclides for radioimmunotherapy |journal=Medical Physics |volume=20 |issue=2 |year=1993 |pages=503–509 |issn=0094-2405 |doi=10.1118/1.597045 |pmid=8492758 |bibcode = 1993MedPh..20..503M }}</ref><ref name="MurrayMarten1987">{{cite journal |last1=Murray |first1=A. S. |last2=Marten |first2=R. |last3=Johnston |first3=A. |last4=Martin |first4=P. |title=Analysis for naturally {{sic|occur|ing|nolink=y}} radionuclides at environmental concentrations by gamma spectrometry|journal=Journal of Radioanalytical and Nuclear Chemistry |volume=115 |issue=2 |year=1987 |pages=263–288 |issn=0236-5731 |doi=10.1007/BF02037443|s2cid=94361207 }}</ref>

==References==
{{Reflist}}

==Further reading==
* {{cite journal |last1=Fetter |first1=Steve |last2=Cheng |first2=E. T. |last3=Mann |first3=F. M. |title=Long-term radioactive waste from fusion reactors: Part II |journal=[[Fusion Engineering and Design]] |volume=13 |issue=2 |year=1990 |pages=239–246 |issn=0920-3796 |doi=10.1016/0920-3796(90)90104-E |citeseerx=10.1.1.465.5945}}
* {{cite journal |last1=Holland |first1=Jason P. |last2=Sheh |first2=Yiauchung |last3=Lewis |first3=Jason S. |title=Standardized methods for the production of high specific-activity zirconium-89 |journal=Nuclear Medicine and Biology |volume=36 |issue=7 |year=2009 |pages=729–739 |issn=0969-8051 |doi=10.1016/j.nucmedbio.2009.05.007 |pmid=19720285 |pmc=2827875}}
* {{cite journal |last1=McCarthy |first1=Deborah W. |last2=Shefer |first2=Ruth E. |last3=Klinkowstein |first3=Robert E. |last4=Bass |first4=Laura A. |last5=Margeneau |first5=William H. |last6=Cutler |first6=Cathy S. |last7=Anderson |first7=Carolyn J. |last8=Welch |first8=Michael J. |title=Efficient production of high specific activity <sup>64</sup>Cu using a biomedical cyclotron |journal=Nuclear Medicine and Biology |volume=24 |issue=1 |year=1997 |pages=35–43 |issn=0969-8051 |doi=10.1016/S0969-8051(96)00157-6 |pmid=9080473}}

{{Ionising radiation related quantities}}
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[[Category:Radioactivity quantities]]