# Specific activity

Activity
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 is the activity per unit mass of a radionuclide and is a physical property of that radionuclide.

Activity is a quantity (for which the SI unit is the becquerel) related to radioactivity, and is defined as the number of radioactive transformations per second that occur in a particular radionuclide. The unit of activity is the becquerel (Bq), which is defined as one radioactive decay per second. The older, non-SI unit of activity is the curie (Ci), which is 3.7×1010 radioactive decay per second. Another unit of activity is the Rutherford, which is defined as 1×106 radioactive decay per second.

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).

Thus, specific activity is defined as the activity per unit mass 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 specific activity to be given the unit 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 T1/2

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

$-{\frac {dN}{dt}}=\lambda N.$ The integral solution is described by exponential decay:

$N=N_{0}e^{-\lambda t},$ 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:

${\frac {N_{0}}{2}}=N_{0}e^{-\lambda T_{1/2}}.$ Taking the natural logarithm of both sides, the half-life is given by

$T_{1/2}={\frac {\ln 2}{\lambda }}.$ Conversely, the decay constant λ can be derived from the half-life T1/2 as

$\lambda ={\frac {\ln 2}{T_{1/2}}}.$ ### Calculation of specific activity

The mass of the radionuclide is given by

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

$a[{\text{Bq/g}}]={\frac {\lambda N}{MN/N_{\text{A}}}}={\frac {\lambda N_{\text{A}}}{M}}.$ Thus, specific radioactivity can also be described by

$a={\frac {N_{\text{A}}\ln 2}{T_{1/2}\times M}}.$ This equation is simplified to

$a[{\text{Bq/g}}]\approx {\frac {4.17\times 10^{23}[{\text{mol}}^{-1}]}{T_{1/2}[s]\times M[{\text{g/mol}}]}}.$ When the unit of half-life is in years instead of seconds:

$a[{\text{Bq/g}}]={\frac {4.17\times 10^{23}[{\text{mol}}^{-1}]}{T_{1/2}[{\text{year}}]\times 365\times 24\times 60\times 60[{\text{s/year}}]\times M}}\approx {\frac {1.32\times 10^{16}[{\text{mol}}^{-1}{\cdot }{\text{s}}^{-1}{\cdot }{\text{year}}]}{T_{1/2}[{\text{year}}]\times M[{\text{g/mol}}]}}.$ #### Example: specific activity of Ra-226

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

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

$\lambda ={\frac {a\times M}{N_{\text{A}}}}.$ Therefore, the half-life can also be described by

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

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

Isotope Half-life Mass of 1 curie Specific activity (Ci/g)
232Th 1.405×1010 years 9.1 tonnes 1.1×10−7 (110,000 pCi/g, 0.11 μCi/g)
238U 4.471×109 years 2.977 tonnes 3.4×10−7 (340,000 pCi/g, 0.34 μCi/g)
235U 7.038×108 years 463 kg 2.2×10−6 (2,160,000 pCi/g, 2.2 μCi/g)
40K 1.25×109 years 140 kg 7.1×10−6 (7,100,000 pCi/g, 7.1 μCi/g)
129I 15.7×106 years 5.66 kg 0.00018
99Tc 211×103 years 58 g 0.017
239Pu 24.11×103 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.