Specific Activity Calculator

Release notes

Version 1.0  The Ionactive Specific Activity Calculator has two operational modes:

• Custom / manual entry mode: Here you enter the half-life and molar mass (approximates to mass number in most cases).
• Radionuclide mode: Here you choose the radionuclide from the menu where half-life and molar mass are populated for you. 

Choosing either of the following you just press calculate and some standard data will be provided such as:

• Specific activity (Bq/g).
• The activity of 1g of material.
• The mass  (g) required for 1 MBq of material.

Note that this calculator does not store the specific activity data, it is calculated and this includes the preset radionuclides. The calculation basics are shown later below. 

Other selections can be made as noted next:

• Output unit system - choose from SI or Non SI units.
• Output scale - choose the desired output scale (e.g. micro, milli, base, kilo, mega, giga etc).
• Output format - choose normal or use the scientific notation option.
• Mass for conversion - Optional. Choose a numerical mass and mass multiplier and convert this to activity for your chosen radionuclide (or half-life / molar mass combination).
• Activity for conversation - Optional. Choose a numerical activity value and unit multiplier and convert this to mass of material. 

Specific activity calculator use notes

Here are a couple of examples on how to use the calculator (at least one of them borrows some ideas from our companion calculator 'Radioactive decay heat calculator').

Example 1

You wish to use Pu-238 as a power source (RTG) to develop about 150W of electrical power (assuming a RTG efficiency of about 6%). Your calcs, and / or the Ionactive decay heat calculator predict you will need 2.85 PBq of Pu-238 (that is 2850000 GBq). Note for this example that this is the activity of the Pu-238, and we are going to calculate the mass of Pu-238 required, which will be less than the total mass of (for example) Pu-239 oxide. What mass of Pu-238 do you need? 

The specific activity calculator can be used by selecting:

• 87.7 years (half-life)
• 238 as the molar mass (238.04956 g/mol might be better - but 238 is good enough for this example)

The results are: 4.497 kg which is bang on the reported mass for the Pu-238 component used in one of the three Voyager 2 probe 150W RTGs (which is what this example is based on). If you are curious to know why 2.85 PBq of Pu-238 then visit the 'Radioactive decay heat calculator' resource! 

The calculation process

The specific activity of a radioactive material is the activity per unit mass (e.g. Bq/g or Ci /g). It describes how radioactive a material is for its size, rather than how much activity it contains in total.

Two radioactive materials with the same activity can have very different specific activities depending on how much physical material is present. A small quantity of a short-lived radionuclide may be extremely radioactive per gram (e.g. F-18), whereas a large mass of a very long-lived radionuclide may have a low specific activity (e.g. U-238).

Radioactive decay is governed by the decay constant 𝜆, which is related to the half-life T_1/2 by:

\[
\lambda = \frac{\ln 2}{T_{1/2}}
\]The number of atoms N per gram of a pure radionuclide (i.e. not compound) is determined by its molar mass 𝑀:\[
N = \frac{N_A}{M}
\](where N_A is the Avogadro constant which is 6.022 × 10²³ atoms per mole).

The specific activity A_s (activity per gram) can then be written as:

\[
A_s = \lambda \, \frac{N_A}{M}
\]
Or more completely as:\[
A_s = \frac{\ln 2}{T_{1/2}} \, \frac{N_A}{M}
\]
This full expression shows that specific activity depends only on the half-life and molar mass of the radionuclide.

If you know the specific activity, it can be used to convert between mass and activity.

The activity A associated with a mass m of radioactive material can be written as follows:

\[
A = A_s \, m
\]
 Or, if you need m we can write the following: 

\[
m = \frac{A}{A_s}
\]

This explains why very large masses of long-lived radionuclides may still correspond to relatively modest activities. Consider the activity of 1000 kg of natural uranium compared to 1000 kg of Co-60. Use the Ionactive calculator to work this out, the results are:

• 1000 kg of natural uranium  - 25.4 GBq  [T_1/2 = of the order of 4,468,000,000  years]
• 1000 kg of Co-60  - 41,873,099 TBq  (41,873,099,000 GBq)   [T_1/2 = 5.73 years]

The  two examples are rather wild (at least Co-60 is),  but it illustrates the point rather well!  

The use of natural uranium in our example above is interesting - it is a mixture, not a single radionuclide.  We have hard wired this as depleted uranium into the calculator. 

In such cases, the specific activity is calculated as a mass-fraction-weighted sum of the component isotopes:

\[
A_{s,\text{mix}} = \sum_i w_i \, A_{s,i}
\] In the above expression w_i  is the mass fraction of isotope i, and A_s,i is its specific activity.

This expression explains why trace amounts of short-lived isotopes (such as U-234 in natural uranium) can make a significant contribution to the overall activity. Try the calculator and see for yourself (i.e look at U-238, U-235 and U-234 in isolation first, then look at the natural uranium present results). 

Overall, specific activity is a fundamental concept in radiation protection, nuclear medicine, radioactive materials handling and radioactive waste management. It explains why activity alone does not describe radiological significance, and why both quantity of material and nuclear properties must be considered together.