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2014-08-12T06:20:16Z

TitusSecombe:

{{About|specific activity radioactivity|the use in biochemistry|Enzyme assay#Specific activity}}
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{{technical|date=January 2014}}
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In [[nuclear sciences]] and technologies, "activity" is the SI quantity related to the phenomenon of natural and artificial [[radioactivity]]. The SI unit of "activity" is [[becquerel]] (Bq) while that of "'''specific activity'''" is Bq/g. The old unit of "activity" was [[curie]] (Ci) making "specific activity", Ci/g.

== Half-life ==

Experimentally-measured specific activity can be used to calculate the [[half-life]] of a radioactive element.

The definition of half-life ('''T1/2''') is that
the half life T1/2 of an isotope is the length of time at which half
of a given quantity has decayed into another isotope, usually of a different element:
Or more generally:
Starting with '''''N0''''', atoms of an element, the number of atoms, '''''N''''',
remaining after time, '''''t''''', is given by:

: <math>N=N_0\left(\frac{1}{2}\right)^{t \over T_{1/2} }</math>

The natural log of both sides

: <math>\ln(N)=\ln(N_0)+\left(\frac{t}{T_{1/2} }\right)\ln\left(\frac{1}{2}\right)</math>

The derivative with respect to time, ''t''

: <math>\frac{1}{N}\frac{dN}{dt}=\frac{\ln\left(\frac{1}{2}\right)}{T_{1/2} }</math>

Multiplying both sides by ''N''

: <math>\frac{dN}{dt}=\frac{N\ln\left(\frac{1}{2}\right)}{T_{1/2} }</math>

Yields

: <math>\frac{dN}{dt}=\frac{-0.693\,N}{T_{1/2} }</math>

''dN''/''dt'' represents the decay rate of atoms. The negative sign shows that the rate is negative, so the number of atoms is decreasing with time. Rearranging terms:

: <math>T_{1/2}=\frac{-0.693\,N}{\frac{dN}{dt} }</math>

=== 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 {{val|3.2|e=6|u=Bq/kg}}. Rubidium's atomic weight is 87, so one gram is one 87th of a mole, or ''N''={{val|6.9|e=21}} atoms. Plugging in the numbers:

: <math>T_{1/2}=\frac{-0.693(6.9\times 10^{21})}{-3200\text{ s}^{-1} }=1.5\times 10^{18}\text{ s or 47 billion years}</math>

== Formulation ==

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>

Mass of the radionuclide is given by

: <math>\frac{N}{N_A} [\text{mol}] \times {m} [\text {g } \text{mol}^{-1}]</math>

where ''m'' is [[mass number]] of the radionuclide and ''NA'' is [[Avogadro's constant]].

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

: <math>S [\text {Bq/g}] = \frac{\lambda N}{m N/N_A} = \frac{\lambda N_A}{m}</math>

In addition, decay constant ''λ'' is related to the half-life ''T1/2'' by the following equation:

: <math>{\lambda} = \frac{ln2}{T_{1/2}}</math>

Thus, specific radioactivity can also be described by

: <math>S = \frac{ln2 \times {N_A}}{T_{1/2} \times {m}}</math>

This equation is simplified by

: <math>S [\text {Bq/g}] \simeq \frac{4.17\times 10^{23}}{T_{1/2} [s]\times m}</math>

When the unit of half-life converts a year

: <math> S [\text {Bq/g}] = \frac{ln2 \times {N_A}}{T_{1/2} [s] \times {m}} = \frac{ln2 \times {N_A}}{T_{1/2}[year] \times365\times24\times60\times60 \times m} \simeq \frac{1.32\times 10^{16} }{T_{1/2}[year] \times m}</math>

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

: <math> \frac{1.32\times 10^{16} }{1600[year] \times 226} \simeq {3.7} \times 10^{10} [\text {Bq/g}] </math>

This value derived from radium 226 was defined as unit of radioactivity known as [[Curie]] (Ci).

[[Category:Units of radioactivity]]

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