Editing Specific activity (section)

== Formulation ==
{{see also|Radioactive decay#Rates}}

===Relationship between ''λ'' and T<sub>1/2</sub>===

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

: <math>-\frac{dN}{dt} = \lambda N.</math>

The integral solution is described by [[exponential decay]]:

: <math>N = N_0 e^{-\lambda t},</math>

where ''N''<sub>0</sub> is the initial quantity of atoms at time ''t'' = 0.

[[Half-life]] '''T<sub>1/2</sub>''' is defined as the length of time for half of a given quantity of radioactive atoms to undergo radioactive decay:

: <math>\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}.</math>

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

: <math>T_{1/2} = \frac{\ln 2}{\lambda}.</math>

Conversely, the decay constant ''λ'' can be derived from the half-life ''T''<sub>1/2</sub> as

: <math>\lambda = \frac{\ln 2}{T_{1/2}}.</math>

===Calculation of specific activity===

The mass of the radionuclide is given by

: <math>{m} = \frac{N}{N_\text{A}} [\text{mol}] \times {M} [\text{g/mol}],</math>

where ''M'' is [[molar mass]] of the radionuclide, and ''N''<sub>A</sub> 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:

: <math>a [\text{Bq/g}] = \frac{\lambda N}{M N/N_\text{A}} = \frac{\lambda N_\text{A}}{M}.</math>

Thus, specific radioactivity can also be described by

: <math>a = \frac{N_\text{A} \ln 2}{T_{1/2} \times M}.</math>

This equation is simplified to

: <math>a [\text{Bq/g}] \approx \frac{4.17 \times 10^{23} [\text{mol}^{-1}]}{T_{1/2} [s] \times M [\text{g/mol}]}.</math>

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

: <math>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}]}.</math>

==== Example: specific activity of Ra-226 ====

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

: <math chem>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}].</math>

This value derived from radium-226 was defined as unit of radioactivity known as the [[Curie (unit)|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:

: <math>\lambda = \frac{a \times M}{N_\text{A}}.</math>

Therefore, the half-life can also be described by

: <math>T_{1/2} = \frac{N_\text{A} \ln 2}{a \times M}.</math>

==== Example: half-life of Rb-87 ====
One gram of [[Isotopes of rubidium|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 {{val|3.2|e=6|u=Bq/kg}}. Rubidium [[atomic mass]] is 87&nbsp;g/mol, so one gram is 1/87 of a mole. Plugging in the numbers:

: <math>
 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}.
</math>

===Other calculations===
{{cleanup merge|Becquerel|21=section|date=July 2023}}

For a given mass <math>m</math> (in grams) of an isotope with [[atomic mass]] <math>m_\text{a}</math> (in g/mol) and a [[half-life]] of <math>t_{1/2}</math> (in s), the radioactivity can be calculated using:

:<math>A_\text{Bq} = \frac{m} {m_\text{a}} N_\text{A} \frac{\ln 2} {t_{1/2}}</math>

With <math>N_\text{A}</math> = {{val|6.02214076|e=23|u=mol-1}}, the [[Avogadro constant]].

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

:<math>A_\text{Bq} = nN_\text{A} \frac{\ln 2} {t_{1/2}}</math>

For instance, on average each gram of [[potassium]] contains 117&nbsp;micrograms of [[Potassium-40|<sup>40</sup>K]] (all other naturally occurring isotopes are stable) that has a <math>t_{1/2}</math> of {{val|1.277|e=9|u=years}} = {{val|4.030|e=16|u=s}},<ref>{{cite web|url=http://nucleardata.nuclear.lu.se/toi/nuclide.asp?iZA=190040 |title=Table of Isotopes decay data |publisher=[[Lund University]] |date=1990-06-01 |access-date=2014-01-12}}</ref> and has an atomic mass of 39.964&nbsp;g/mol,<ref>{{cite web|url=http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl?ele=&ascii=html&isotype=some |title=Atomic Weights and Isotopic Compositions for All Elements |publisher=[[NIST]] |access-date=2014-01-12}}</ref> so the amount of radioactivity associated with a gram of potassium is 30&nbsp;Bq.