# Specific activity

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:

$N=N_0\left(\frac{1}{2}\right)^{t \over T_{1/2} }$

The natural log of both sides

$\ln(N)=\ln(N_0)+\left(\frac{t}{T_{1/2} }\right)\ln\left(\frac{1}{2}\right)$

The derivative with respect to time, t

$\frac{1}{N}\frac{dN}{dt}=\frac{\ln\left(\frac{1}{2}\right)}{T_{1/2} }$

Multiplying both sides by N

$\frac{dN}{dt}=\frac{N\ln\left(\frac{1}{2}\right)}{T_{1/2} }$

Yields

$\frac{dN}{dt}=\frac{-0.693\,N}{T_{1/2} }$

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:

$T_{1/2}=\frac{-0.693\,N}{\frac{dN}{dt} }$

### 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 3.2×106 Bq/kg. Rubidium's atomic weight is 87, so one gram is one 87th of a mole, or N=6.9×1021 atoms. Plugging in the numbers:

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

## Formulation

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

$-\frac{dN}{dt}= \lambda N$

Mass of the radionuclide is given by

$\frac{N}{N_A} [\text{mol}] \times {m} [\text {g } \text{mol}^{-1}]$

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

$a [\text {Bq/g}] = \frac{\lambda N}{M N/N_A} = \frac{\lambda N_A}{M}$

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

${\lambda} = \frac{ln2}{T_{1/2}}$

Thus, specific radioactivity can also be described by

$a = \frac{ln2 \times {N_A}}{T_{1/2} \times {M}}$

This equation is simplified by

$a [\text {Bq/g}] \simeq \frac{4.17\times 10^{23} [\text{mol}^{-1}] }{T_{1/2} [s]\times M [\text {g } \text{mol}^{-1}]}$

When the unit of half-life converts a year

$a [\text {Bq/g}] = \frac{ln2 \times {N_A}}{T_{1/2} [s] \times {M [\text {g } \text{mol}^{-1}]}} = \frac{ln2 \times {N_A}}{T_{1/2}[year] \times365\times24\times60\times60 \times M [\text {g } \text{mol}^{-1}]} \simeq \frac{1.32\times 10^{16} [\text{mol}^{-1}] }{T_{1/2}[year] \times M [\text {g } \text{mol}^{-1}]}$

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

$a_{Ra}[\text {Bq/g}] = \frac{1.32\times 10^{16} [\text{mol}^{-1}] }{1600[year] \times 226 [\text {g } \text{mol}^{-1}] } \simeq {3.7} \times 10^{10} [\text {Bq/g}]$

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

Another two examples are specific radioactivity of thorium 232 and specific radioactivity of potassium 40:

$a_{Th}[\text {Bq/g}] = \frac{1.32\times 10^{16} [\text{mol}^{-1}] }{1.405\times 10^{10}[year] \times 232 [\text {g } \text{mol}^{-1}] } \simeq {4.059} \times 10^{3} [\text {Bq/g}]$
$a_{K}[\text {Bq/g}] = \frac{1.32\times 10^{16} [\text{mol}^{-1}] }{1.251\times 10^{9}[year] \times 40 [\text {g } \text{mol}^{-1}] } \simeq {2.63789} \times 10^{5} [\text {Bq/g}]$