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Specific activity (symbol ''a'') is the activity per unit mass of a
radionuclide A radionuclide (radioactive nuclide, radioisotope or radioactive isotope) is a nuclide that has excess numbers of either neutrons or protons, giving it excess nuclear energy, and making it unstable. This excess energy can be used in one of three ...
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 Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is conside ...
, activity or total activity (symbol ''A'') is a
physical quantity A physical quantity (or simply quantity) is a property of a material or system that can be Quantification (science), quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ''nu ...
defined as the number of radioactive transformations per second that occur in a particular
radionuclide A radionuclide (radioactive nuclide, radioisotope or radioactive isotope) is a nuclide that has excess numbers of either neutrons or protons, giving it excess nuclear energy, and making it unstable. This excess energy can be used in one of three ...
. The unit of activity is the ''
becquerel The becquerel (; symbol: Bq) is the unit of radioactivity in the International System of Units (SI). One becquerel is defined as an activity of one per second, on average, for aperiodic activity events referred to a radionuclide. For applicatio ...
'' (symbol Bq), which is defined equivalent to
reciprocal second The inverse second or reciprocal second (s−1), also called ''per second'', is a unit defined as the multiplicative inverse of the second (a unit of time). It is applicable for physical quantities of dimension reciprocal time, such as frequency ...
s (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 Ionizing (ionising) radiation, including Radioactive decay, nuclear radiation, consists of subatomic particles or electromagnetic waves that have enough energy per individual photon or particle to ionization, ionize atoms or molecules by detaching ...
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 Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is conside ...
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 T1/2

Radioactivity is expressed as the decay rate of a particular radionuclide with decay constant ''λ'' and the number of atoms ''N'': : -\frac = \lambda N. The integral solution is described by
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda Lambda (; uppe ...
: : N = N_0 e^, where ''N''0 is the initial quantity of atoms at time ''t'' = 0.
Half-life Half-life is a mathematical and scientific description of exponential or gradual decay. Half-life, half life or halflife may also refer to: Film * Half-Life (film), ''Half-Life'' (film), a 2008 independent film by Jennifer Phang * ''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 e^. Taking the natural logarithm of both sides, the half-life is given by : T_ = \frac. Conversely, the decay constant ''λ'' can be derived from the half-life ''T''1/2 as : \lambda = \frac.


Calculation of specific activity

The mass of the radionuclide is given by : = \frac
text Text may refer to: Written word * Text (literary theory) In literary theory, a text is any object that can be "read", whether this object is a work of literature, a street sign, an arrangement of buildings on a city block, or styles of clothi ...
\times
text Text may refer to: Written word * Text (literary theory) In literary theory, a text is any object that can be "read", whether this object is a work of literature, a street sign, an arrangement of buildings on a city block, or styles of clothi ...
where ''M'' is
molar mass In chemistry, the molar mass () (sometimes called molecular weight or formula weight, but see related quantities for usage) of a chemical substance ( element or compound) is defined as the ratio between the mass () and the amount of substance ...
of the radionuclide, and ''N''A is the
Avogadro constant The Avogadro constant, commonly denoted or , is an SI defining constant with an exact value of when expressed in reciprocal moles. It defines the ratio of the number of constituent particles to the amount of substance in a sample, where th ...
. Practically, the
mass number The mass number (symbol ''A'', from the German word: ''Atomgewicht'', "atomic weight"), also called atomic mass number or nucleon number, is the total number of protons and neutrons (together known as nucleons) in an atomic nucleus. It is appro ...
''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
text Text may refer to: Written word * Text (literary theory) In literary theory, a text is any object that can be "read", whether this object is a work of literature, a street sign, an arrangement of buildings on a city block, or styles of clothi ...
= \frac = \frac. Thus, specific radioactivity can also be described by : a = \frac. This equation is simplified to : a
text Text may refer to: Written word * Text (literary theory) In literary theory, a text is any object that can be "read", whether this object is a work of literature, a street sign, an arrangement of buildings on a city block, or styles of clothi ...
\approx \frac. When the unit of half-life is in years instead of seconds: : a
text Text may refer to: Written word * Text (literary theory) In literary theory, a text is any object that can be "read", whether this object is a work of literature, a street sign, an arrangement of buildings on a city block, or styles of clothi ...
= \frac \approx \frac.


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
text Text may refer to: Written word * Text (literary theory) In literary theory, a text is any object that can be "read", whether this object is a work of literature, a street sign, an arrangement of buildings on a city block, or styles of clothi ...
= \frac \approx 3.7 \times 10^
text Text may refer to: Written word * Text (literary theory) In literary theory, a text is any object that can be "read", whether this object is a work of literature, a street sign, an arrangement of buildings on a city block, or styles of clothi ...
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 Half-life is a mathematical and scientific description of exponential or gradual decay. Half-life, half life or halflife may also refer to: Film * Half-Life (film), ''Half-Life'' (film), a 2008 independent film by Jennifer Phang * ''Half Life: ...
of a radionuclide. Where decay constant ''λ'' is related to specific radioactivity ''a'' by the following equation: : \lambda = \frac. Therefore, the half-life can also be described by : T_ = \frac.


Example: half-life of Rb-87

One gram of
rubidium-87 Rubidium (37Rb) has 36 isotopes, with naturally occurring rubidium being composed of just two isotopes; 85Rb (72.2%) and the radioactive 87Rb (27.8%). 87Rb has a half-life of . It readily substitutes for potassium in minerals, and is therefore ...
and a radioactivity count rate that, after taking
solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poin ...
effects into account, is consistent with a decay rate of 3200 decays per second corresponds to a specific activity of . Rubidium
atomic mass Atomic mass ( or ) is the mass of a single atom. The atomic mass mostly comes from the combined mass of the protons and neutrons in the nucleus, with minor contributions from the electrons and nuclear binding energy. The atomic mass of atoms, ...
is 87 g/mol, so one gram is 1/87 of a mole. Plugging in the numbers: : T_ = \frac \approx \frac \approx 1.5 \times 10^\text \approx 47\text.


Other calculations

For a given mass m (in grams) of an isotope with
atomic mass Atomic mass ( or ) is the mass of a single atom. The atomic mass mostly comes from the combined mass of the protons and neutrons in the nucleus, with minor contributions from the electrons and nuclear binding energy. The atomic mass of atoms, ...
m_\text (in g/mol) and a
half-life Half-life is a mathematical and scientific description of exponential or gradual decay. Half-life, half life or halflife may also refer to: Film * Half-Life (film), ''Half-Life'' (film), a 2008 independent film by Jennifer Phang * ''Half Life: ...
of t_ (in s), the radioactivity can be calculated using: :A_\text = \frac N_\text \frac With N_\text = , the
Avogadro constant The Avogadro constant, commonly denoted or , is an SI defining constant with an exact value of when expressed in reciprocal moles. It defines the ratio of the number of constituent particles to the amount of substance in a sample, where th ...
. Since m/m_\text is the number of moles (n), the amount of radioactivity A can be calculated by: :A_\text = nN_\text \frac For instance, on average each gram of
potassium Potassium is a chemical element; it has Symbol (chemistry), symbol K (from Neo-Latin ) and atomic number19. It is a silvery white metal that is soft enough to easily cut with a knife. Potassium metal reacts rapidly with atmospheric oxygen to ...
contains 117 micrograms of 40K (all other naturally occurring isotopes are stable) that has a t_ 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


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.


References


Further reading

* * * {{Authority control Radioactivity quantities