Specific activity

Specific activity (symbol a) is the activity per unit mass of a radionuclide and is a physical property of that radionuclide. 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). 1 Ci/g = 37 TBq/kg.

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. The unit of activity is the becquerel (symbol Bq), which is defined equivalent to reciprocal seconds (symbol s−1). The older, non-SI unit of activity is the curie (Ci), which is 3.7×1010 radioactive decays per second (37 GBq). Another unit of activity is the rutherford (Rd), which is defined as 1×106 radioactive decays per second (1 MBq).

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 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

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:

− d N d t = λ N . {\displaystyle -{\frac {dN}{dt}}=\lambda N.}

The integral solution is described by exponential decay:

N = N 0 e − λ t , {\displaystyle 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:

N 0 2 = N 0 e − λ t 1 / 2 . {\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

t 1 / 2 = ln ⁡ 2 λ . {\displaystyle t_{1/2}={\frac {\ln 2}{\lambda }}.}

Conversely, the decay constant λ can be derived from the half-life t1/2 as

λ = ln ⁡ 2 t 1 / 2 . {\displaystyle \lambda ={\frac {\ln 2}{t_{1/2}}}.}

Calculation of specific activity

The mass of the radionuclide is given by

m = N N A [ mol ] × M [ g/mol ] , {\displaystyle {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.

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

a [ Bq/g ] = λ N M N / N A = λ N A M . {\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

a = N A ln ⁡ 2 t 1 / 2 × M . {\displaystyle a={\frac {N_{\text{A}}\ln 2}{t_{1/2}\times M}}.}

This equation is simplified to

a [ Bq/g ] ≈ 4.17 × 10 23 [ mol − 1 ] t 1 / 2 [ s ] × M [ g/mol ] . {\displaystyle a[{\text{Bq/g}}]\approx {\frac {4.17\times 10^{23}[{\text{mol}}^{-1}]}{t_{1/2}[{\text{s}}]\times M[{\text{g/mol}}]}}.}

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

a [ Bq/g ] = 4.17 × 10 23 [ mol − 1 ] t 1 / 2 [ year ] × 365 × 24 × 60 × 60 [ s/year ] × M ≈ 1.32 × 10 16 [ mol − 1 ⋅ s − 1 ⋅ year ] t 1 / 2 [ year ] × M [ g/mol ] . {\displaystyle {\begin{aligned}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}}\\[1ex]&\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}}]}}.\end{aligned}}}

Example: specific activity of Ra-226

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

a Ra-226 [ Bq/g ] = 1.32 × 10 16 1600 × 226 ≈ 3.7 × 10 10 [ Bq/g ] . {\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:

λ = a × M N A . {\displaystyle \lambda ={\frac {a\times M}{N_{\text{A}}}}.}

Therefore, the half-life can also be described by

t 1 / 2 = N A ln ⁡ 2 a × M . {\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 3.2×106Bq/kg. Rubidium atomic mass is 87gmol−1, so one gram is 1/87 of a mole. Plugging in the numbers:

t 1 / 2 = N A × ln ⁡ 2 a × M ≈ 6.022 × 10 23 mol − 1 × 0.693 3200 s − 1 ⋅ g − 1 × 87 g/mol ≈ 1.5 × 10 18 s ≈ 47 billion years . {\displaystyle {\begin{aligned}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}}}}\\[1ex]&\approx 1.5\times 10^{18}{\text{ s}}\approx 47{\text{ billion years}}.\end{aligned}}}

Other calculations

For a given mass m (in grams) of an isotope with atomic mass ma (in g mol−1) and a half-life of t1/2 (in s), the radioactivity can be calculated using:

A Bq = m m a N A ln ⁡ 2 t 1 / 2 {\displaystyle A_{\text{Bq}}={\frac {m}{m_{\text{a}}}}N_{\text{A}}{\frac {\ln 2}{t_{1/2}}}}

With N A {\displaystyle N_{\text{A}}} = 6.02214076×1023mol−1, the Avogadro constant.

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

A Bq = n N A ln ⁡ 2 t 1 / 2 {\displaystyle A_{\text{Bq}}=nN_{\text{A}}{\frac {\ln 2}{t_{1/2}}}}

For instance, on average each gram of potassium contains 117micrograms of 40K (all other naturally occurring isotopes are stable) that has a t1/2 of 1.277×109years = 4.030×1016s, and has an atomic mass of 39.964g/mol, so the amount of radioactivity associated with a gram of potassium is 30Bq.

Examples

Isotope	Half-life	Mass of 1 curie	Specific Activity (a) (activity per 1kg)
232Th	1.405×1010 years	9.1 tonnes	4.07 MBq (110 μCi or 4.07 Rd)
238U	4.471×109 years	2.977 tonnes	12.58 MBq (340 μCi, or 12.58 Rd)
235U	7.038×108 years	463kg	79.92 MBq (2.160 mCi, or 79.92 Rd)
40K	1.25×109 years	140kg	262.7 MBq (7.1 mCi, or 262.7 Rd)
129I	15.7×106 years	5.66kg	6.66 GBq (180 mCi, or 6.66 kRd)
99Tc	211×103 years	58 g	629 GBq (17 Ci, or 629 kRd)
239Pu	24.11×103 years	16 g	2.331 TBq (63 Ci, or 2.331 MRd)
240Pu	6563 years	4.4 g	8.51 TBq (230 Ci, or 8.51MRd)
14C	5730 years	0.22 g	166.5 TBq (4.5 kCi, or 166.5 MRd)
226Ra	1601 years	1.01 g	36.63 TBq (990 Ci, or 36.63 MRd)
241Am	432.6 years	0.29 g	126.91 TBq (3.43 kCi, or 126.91 MRd)
238Pu	88 years	59mg	629 TBq (17 kCi, or 629 MRd)
137Cs	30.17 years	12mg	3.071 PBq (83 kCi, or 3.071 GRd)
90Sr	28.8 years	7.2mg	5.143 PBq (139 kCi, or 5.143 GRd)
241Pu	14 years	9.4mg	3.922 PBq (106 kCi, or 3.922 GRd)
3H	12.32 years	104 μg	355.977 PBq (9.621 MCi, or 355.977 GRd)
228Ra	5.75 years	3.67mg	10.101 PBq (273 kCi, or 10.101 GRd)
60Co	1925 days	883 μg	41.884 PBq (1.132 MCi, or 41.884 GRd)
210Po	138 days	223 μg	165.908 PBq (4.484 MCi, or 165.908 GRd)
131I	8.02 days	8 μg	4.625 EBq (125 MCi, or 4.625 TRd)
123I	13 hours	518ng	71.41 EBq (1.93 GCi, or 71.41 TRd)
212Pb	10.64 hours	719ng	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.