Specific activity

 ( Radioactivity Quantities ) 

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).

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 radioactive decays per second. Another unit of activity is the rutherford, which is defined as 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 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:

$-\backslash frac\{dN\}\{dt\}\; =\; \backslash lambda\; N.$

The integral solution is described by exponential decay:

$N\; =\; N\_0\; e^\{-\backslash lambda\; t\},$

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:

$\backslash frac\{N\_0\}\{2\}\; =\; N\_0\; e^\{-\backslash lambda\; T\_\{1/2\}\}.$

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

$T\_\{1/2\}\; =\; \backslash frac\{\backslash ln\; 2\}\{\backslash lambda\}.$

Conversely, the decay constant λ can be derived from the half-life T_1/2 as

$\backslash lambda\; =\; \backslash frac\{\backslash ln\; 2\}\{T\_\{1/2\}\}.$

---

Calculation of specific activity The mass of the radionuclide is given by

$\{m\}\; =\; \backslash frac\{N\}\{N\_\backslash text\{A\}\}\; \backslash text\{mol\}\; \backslash times\; \{M\}\; \backslash text\{g/mol\},$

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:

$a\; \backslash text\{Bq/g\}\; =\; \backslash frac\{\backslash lambda\; N\}\{M\; N/N\_\backslash text\{A\}\}\; =\; \backslash frac\{\backslash lambda\; N\_\backslash text\{A\}\}\{M\}.$

Thus, specific radioactivity can also be described by

$a\; =\; \backslash frac\{N\_\backslash text\{A\}\; \backslash ln\; 2\}\{T\_\{1/2\}\; \backslash times\; M\}.$

This equation is simplified to

$a\; \backslash text\{Bq/g\}\; \backslash approx\; \backslash frac\{4.17\; \backslash times\; 10^\{23\}\; \backslash text\{mol\}^\{-1\}\}\{T\_\{1/2\}\; s\; \backslash times\; M\; \backslash text\{g/mol\}\}.$

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

$a\; \backslash text\{Bq/g\}\; =\; \backslash frac\{4.17\; \backslash times\; 10^\{23\}\; \backslash text\{mol\}^\{-1\}\}\{T\_\{1/2\}\backslash text\{year\}\; \backslash times\; 365\; \backslash times\; 24\; \backslash times\; 60\; \backslash times\; 60\; \backslash text\{s/year\}\; \backslash times\; M\}\; \backslash approx\; \backslash frac\{1.32\; \backslash times\; 10^\{16\}\; \backslash text\{mol\}^\{-1\}\{\backslash cdot\}\backslash text\{s\}^\{-1\}\{\backslash cdot\}\backslash text\{year\}\}\{T\_\{1/2\}\; \backslash text\{year\}\; \backslash times\; M\; \backslash text\{g/mol\}\}.$

---

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

$a\_\backslash text\{Ra-226\}\backslash text\{Bq/g\}\; =\; \backslash frac\{1.32\; \backslash times\; 10^\{16\}\}\{1600\; \backslash times\; 226\}\; \backslash approx\; 3.7\; \backslash times\; 10^\{10\}\; \backslash text\{Bq/g\}.$

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:

$\backslash lambda\; =\; \backslash frac\{a\; \backslash times\; M\}\{N\_\backslash text\{A\}\}.$

Therefore, the half-life can also be described by

$T\_\{1/2\}\; =\; \backslash frac\{N\_\backslash text\{A\}\; \backslash ln\; 2\}\{a\; \backslash times\; M\}.$

---

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 . 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}.
     

---

Other calculations For a given mass $m$ (in grams) of an isotope with atomic mass $m\_\backslash text\{a\}$ (in g/mol) and a half-life of $t\_\{1/2\}$ (in s), the radioactivity can be calculated using:

$A\_\backslash text\{Bq\}\; =\; \backslash frac\{m\}\; \{m\_\backslash text\{a\}\}\; N\_\backslash text\{A\}\; \backslash frac\{\backslash ln\; 2\}\; \{t\_\{1/2\}\}$

With $N\_\backslash text\{A\}$ = , the Avogadro constant.

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

$A\_\backslash text\{Bq\}\; =\; nN\_\backslash text\{A\}\; \backslash frac\{\backslash ln\; 2\}\; \{t\_\{1/2\}\}$

For instance, on average each gram of potassium contains 117 micrograms of ⁴⁰K (all other naturally occurring isotopes are stable) that has a $t\_\{1/2\}$ of = , and has an atomic mass of 39.964 g/mol, so the amount of radioactivity associated with a gram of potassium is 30 Bq.

---

Examples

4.07 MBq (110 μCi or 4.07 Rd)

12.58 MBq (340 μCi, or 12.58 Rd)

79.92 MBq (2.160 mCi, or 79.92 Rd)

262.7 MBq (7.1 mCi, or 262.7 Rd)

6.66 GBq (180 mCi, or 6.66 kRd)

629 GBq (17 Ci, or 629 kRd)

2.331 TBq (63 Ci, or 2.331 MRd)

8.51 TBq (230 Ci, or 8.51MRd)

166.5 TBq (4.5 kCi, or 166.5 MRd)

36.63 TBq (990 Ci, or 36.63 MRd)

126.91 TBq (3.43 kCi, or 126.91 MRd)

629 TBq (17 kCi, or 629 MRd)

3.071 PBq (83 kCi, or 3.071 GRd)

5.143 PBq (139 kCi, or 5.143 GRd)

3.922 PBq (106 kCi, or 3.922 GRd)

355.977 PBq (9.621 MCi, or 355.977 GRd)

10.101 PBq (273 kCi, or 10.101 GRd)

41.884 PBq (1.132 MCi, or 41.884 GRd)

165.908 PBq (4.484 MCi, or 165.908 GRd)

4.625 EBq (125 MCi, or 4.625 TRd)

71.41 EBq (1.93 GCi, or 71.41 TRd)

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.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.

---

Further reading

Post Comment