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/kg). For its use in radiochemistry, see radioactivity.'' Moreover in thermodynamics, thermochemistry and electrochemistry the "activity" is the dimensionless ratio between the fugacity of a chemical species in actual to standard conditions.

One can use the experimentally measured specific activity to calculate the half-life of an element.

Starting from the definition of the half-life (T1/2) - with, N0, atoms of an element, the number of atoms, N, remaining after time, t, is giving by:


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

Take the natural log of both sides:


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

Take the derivative with respect to time, t:


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

Multiply both sides by N:


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

Which is:


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

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 as you would expect. Rearranging terms:


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

Example: determine the half-life of Rb-87

Suppose you have a gram of rubidium-87 and with your Geiger counter, you get a count rate which, after taking solid angle effects into account, is consistent with a decay rate of 3200 decays per second; this would correspond to a specific activity of 3.2&times;106 Bq/kg. The atomic weight is of the rubidium is 87, so one gram is one 87th of a mole, or N=6.9&times;1021 atoms. Plugging in the numbers:


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

Activitat d'una substància radioactiva Activité massique