Mass and activity
Grams to becquerels and back for 147 nuclides, from half-life and evaluated molar mass.
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Where the number comes from
Specific activity follows from two definitions and two measured quantities. A gram of a pure nuclide contains NA/M atoms, where M is the molar mass in grams per mole. Each atom decays with probability λ = ln 2 / T½ per second. Multiplying gives a = ln 2 · NA / (T½ · M) becquerel per gram, with nothing left to choose: no geometry, no efficiency, no convention. The two measured inputs are the half-life and the molar mass, and both are evaluated values with published uncertainties. The molar mass here is the evaluated atomic mass from AME2020, not the mass number — the two differ by the nuclear binding energy, which reaches 0.53% for tritium and changes sign across the chart of nuclides.
The inverse relationship with half-life is the whole story. Tritium, with a half-life of 12.3 years, has a specific activity of 3.6 × 10¹⁴ Bq/g. Uranium-238, at 4.5 billion years, has 1.2 × 10⁴ Bq/g — ten orders of magnitude lower. A microgram of tritium is a serious source; a kilogram of depleted uranium is a paperweight that happens to be radioactive.
Why plutonium and uranium need care
The calculator returns the mass of the pure isotope. Real nuclear material never is one. Weapons-grade plutonium is mostly Pu-239 with several percent Pu-240 and smaller quantities of Pu-238, Pu-241 and Am-241 grown in from Pu-241 decay. Reactor-grade plutonium has far more of the heavier isotopes. Because Pu-238 and Pu-241 have specific activities thousands of times higher than Pu-239, a few percent of them dominates the total activity of the mixture even though they contribute almost nothing to its mass.
The consequence for accountancy is that grams and becquerels are not interchangeable through a single number unless the isotopic vector is known. To convert a real sample, apply this calculation to each isotope with its own mass fraction and add the activities. The reverse direction — inferring a total mass from a measured total activity — requires the vector too, and is much more sensitive to it.
Americium ingrowth
Pu-241 beta-decays with a 14-year half-life to Am-241, which is a strong 59.5 keV photon emitter. Plutonium that was almost purely an alpha and low-energy problem when it was separated becomes a measurable gamma source over a couple of decades. Any mass-to-activity figure for aged plutonium that omits the americium is wrong in a way that grows with time.
Practical uses
Three come up constantly. Converting a waste inventory between the mass basis used by a chemical process and the activity basis used by a disposal limit. Checking whether a quoted source strength is physically possible for the stated mass — a claim that fails this check is usually a units error. And estimating the carrier mass in a tracer preparation, where the radioactive atoms may be a vanishing fraction of the material actually present.
The last case is where intuition fails most often. A 37 MBq Tc-99m generator elution contains about two nanograms of Tc-99m. Everything else in the vial is carrier, saline and the long-lived Tc-99 ground state that the metastable state decays into.
Worked examples
Checking a caesium-137 source certificate
A certificate describes a Cs-137 source of 37 GBq (1 Ci). How much caesium-137 is actually in it? Choose “Activity → mass” and Cs-137, enter 37 GBq, and read the mass.
| Result | Value |
|---|---|
| Mass of Cs-137, mg | 11.52 |
| Specific activity, Ci/g | 86.81 |
About 11.52 mg of Cs-137 makes one curie, so no more than about 87 Ci can be packed into a gram of the pure nuclide. That gives a quick plausibility check in both directions. A document claiming 1 Ci in a speck of 1 mg has a units error somewhere. A real source capsule weighs far more than 11.52 mg because the caesium is a chloride or glass compound that also contains stable caesium isotopes from the fission product, so the source's own specific activity is lower than the nuclide's. The figure from this calculator is always the upper limit for the pure nuclide.
Why a gram of plutonium is not like a gram of uranium
Choose “Mass → activity” and enter 1 g, first for U-238 and then for Pu-239. Then switch to “Activity → mass”, choose Am-241 and enter 37 kBq, the typical activity in an ionisation smoke detector.
| Input | Result |
|---|---|
| 1 g of U-238, kBq | 12.44 |
| 1 g of Pu-239, GBq | 2.295 |
| 37 kBq of Am-241, µg | 0.292 |
Gram for gram, Pu-239 is about 185000 times more active than U-238, because its half-life is about 185 000 times shorter and the molar masses are almost equal. Only 5.42 µg of Pu-239 has the activity of a whole gram of U-238. The smoke-detector source is a third of a microgram of americium. Masses this small are why alpha emitters are controlled by activity and not by weight, and why a weighing balance tells you nothing useful about them. It works the other way too: a kilogram of depleted uranium shielding holds only about 12 MBq of U-238, which is why its main hazard is chemical toxicity and the beta dose from its decay products at the surface, not the alpha activity of the uranium itself.
FAQ
How is specific activity calculated?
Specific activity is the activity per gram of the pure nuclide: SA = λ·N_A / M = ln 2 · N_A / (T½ · M), with T½ in seconds, N_A Avogadro's number and M the molar mass in g/mol. This calculator uses the AME2020 atomic mass rather than the mass number, which changes the result by up to a few tenths of a percent.
How many grams are in one curie of a radionuclide?
One curie is 3.7 × 10¹⁰ Bq, so the mass is 3.7 × 10¹⁰ divided by the specific activity in Bq/g. For Cs-137 that is 11.52 mg; for radium-226 it is very close to 1 g, because the curie was originally defined from radium.
Why does the source I bought weigh far more than the calculated mass?
The calculation is for the pure nuclide. Real sources contain the nuclide in a chemical compound, mixed with stable isotopes of the same element, a carrier and the capsule itself. The calculated mass is a lower limit on the material, and the nuclide's specific activity is an upper limit on what the source can reach.
Why is plutonium so much more active than uranium per gram?
Specific activity is inversely proportional to half-life. Pu-239 has a half-life of about 24 100 years against 4.47 billion years for U-238, so a gram of it is about 185000 times more active: 2.295 GBq against 12.44 kBq.
Does the specific activity of natural uranium match U-238?
No. Natural uranium also contains U-234 in radioactive equilibrium with U-238 and about 0.72% U-235, so its specific activity is roughly twice that of U-238 alone, before counting any decay products. For a mixture, add the activity of each nuclide in proportion to its mass fraction.