Specific activity

This article is about specific activity radioactivity. For the use in biochemistry, see Enzyme assay § Specific activity.

Specific activity is the activity per quantity of a radionuclide and is a physical property of that radionuclide.

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

Half-life

Experimentally-measured specific activity can be used to calculate the half-life of a radionuclide.

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: Or more generally: Starting with N₀, atoms of an element, the number of atoms, N, remaining after time, t, is given by:

$N=N_{0}\left({\frac {1}{2}}\right)^{t \over T_{1/2}}$

The natural log of both sides

$\ln(N)=\ln(N_{0})+\left({\frac {t}{T_{1/2}}}\right)\ln \left({\frac {1}{2}}\right)$

The derivative with respect to time, t

${\frac {1}{N}}{\frac {dN}{dt}}={\frac {\ln \left({\frac {1}{2}}\right)}{T_{1/2}}}$

Multiplying both sides by N

${\frac {dN}{dt}}={\frac {N\ln \left({\frac {1}{2}}\right)}{T_{1/2}}}$

Yields

${\frac {dN}{dt}}={\frac {-0.693\,N}{T_{1/2}}}$

dN/dt represents the decay rate of atoms. The negative sign shows that the rate is negative, so the number of atoms is decreasing with time. Rearranging terms:

$T_{1/2}={\frac {-0.693\,N}{\frac {dN}{dt}}}$

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, or N=6.9×10²¹ atoms. Plugging in the numbers:

$T_{1/2}={\frac {-0.693(6.9\times 10^{21})}{-3200{\text{ s}}^{-1}}}=1.5\times 10^{18}{\text{ s or 47 billion years}}$

Formulation

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

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

Mass of the radionuclide is given by

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

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

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

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

In addition, decay constant λ is related to the half-life T_1/2 by the following equation:

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

Thus, specific radioactivity can also be described by

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

This equation is simplified by

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

When the unit of half-life converts a year

$a[{\text{Bq/g}}]={\frac {ln2\times {N_{A}}}{T_{1/2}[s]\times {M[{\text{g }}{\text{mol}}^{-1}]}}}={\frac {ln2\times {N_{A}}}{T_{1/2}[year]\times 365\times 24\times 60\times 60\times M[{\text{g }}{\text{mol}}^{-1}]}}\simeq {\frac {1.32\times 10^{16}[{\text{mol}}^{-1}]}{T_{1/2}[year]\times M[{\text{g }}{\text{mol}}^{-1}]}}$

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

$a_{Ra}[{\text{Bq/g}}]={\frac {1.32\times 10^{16}[{\text{mol}}^{-1}]}{1600[year]\times 226[{\text{g }}{\text{mol}}^{-1}]}}\simeq {3.7}\times 10^{10}[{\text{Bq/g}}]$

This value derived from radium 226 was defined as unit of radioactivity known as Curie (Ci).

Another two examples are specific radioactivity of thorium 232 and specific radioactivity of potassium 40:

$a_{Th}[{\text{Bq/g}}]={\frac {1.32\times 10^{16}[{\text{mol}}^{-1}]}{1.405\times 10^{10}[year]\times 232[{\text{g }}{\text{mol}}^{-1}]}}\simeq {4.059}\times 10^{3}[{\text{Bq/g}}]$

$a_{K}[{\text{Bq/g}}]={\frac {1.32\times 10^{16}[{\text{mol}}^{-1}]}{1.251\times 10^{9}[year]\times 40[{\text{g }}{\text{mol}}^{-1}]}}\simeq {2.63789}\times 10^{5}[{\text{Bq/g}}]$

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.

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. 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.
• 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.