Specific activity

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

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

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:

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

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).

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

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

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:

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

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.

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]}

• 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.