# Radioactive Decay Calculator

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD and Rijk de Wet
Last updated: Jan 05, 2023

Radiation is quite dangerous, and in order to understand it and its risks, scientists introduced many measurement units: our radioactive decay calculator will teach you how the activity of a substance is measured. Here you will learn:

• A short history of the discovery of radioactivity;
• What the radioactive decay is;
• What types of radioactive decay are known;
• How to calculate radioactive decay, plus how to measure it; and
• What is and how to calculate specific activity.

We will also see a few examples of radioactive decay!

## The discovery of radioactivity

Sometimes, important discoveries happen by chance: this was particularly true at the end of the XIX century when decades of experiments with electricity and photography finally led to the initial understanding of the phenomenon of radioactivity.

While experimenting with cathodic tubes and "cathode rays" at the end of 1895, Wilhelm Röntgen discovered radiation (that is, a means of transmission of energy either by means of a wave or of a particle) that we now know is made of electrons. He discovered that shielding that radiation was not stopping another kind of ray: X-rays.

The origin of X-rays was not known at the time, and the French physicist Henri Becquerel was investigating the possibility of using solar-stimulated uranium salts to emit X-rays. He did this by trying to impress a photographic film after the salts were exposed to the sun and wrapping them in black paper.

🔎 The dangers of radiation were not known at the time, and many experimenters ended up suffering after years of exposition to ionizing radiation. Maria Skłodowska-Curie's notebooks are still heavily contaminated, and it has to be stored in lead-lined boxes. Those were truly the Age of Exploration in physics.

Here is where two coincidences happened. Speculating on what would have happened if one or both of them didn't happen is pretty much useless: humanity was on the path of discovering the power of radiation anyway.

The first coincidence is an overcast sky. The clouds made it impossible for Becquerel to conduct his experiment, so he put his uranium salt sample in a drawer in proximity to the photographic film.

A few days later, the sun still hiding, Becquerel decided to develop the film anyway: no one knows why. This is our second coincidence. He found out that the film was neatly exposed, even without the presence of the sun: Becquerel quickly understood that the uranium salt themselves emitted rays able to pass through the opaque paper and expose the photographic paper.

So, what Becquerel discovered was a spontaneous emission of radiation that the physicist Rutherford correctly identified as a kind of decay coming from the atomic nuclei.

## What is radioactive decay? Types of radioactive decay

The time has come to discuss the definition of radioactive decay. Massive atomic nuclei or nuclei with a high imbalance of neutrons and protons can undergo processes of decay to reach a less energetic state. In these processes, the atoms emit radiation: waves or particles (or both? check our De Broglie wavelength calculator). Many types of radioactive decay are known to scientists:

• Alpha decay, where two protons and two neutrons (bound together) are released from the nucleus. It's indistinguishable from a helium nucleus.

• Beta decay, where two processes are identified:

• Beta minus, where a neutron becomes a proton, with an emission of an electron and an antineutrino; and

• Beta plus where a proton turns into a neutron, emitting a positron (the positively charged electron antiparticle) and a neutrino.

• Gamma decay, where a high energy photon with the shortest wavelength possible is emitted by the nucleus as a result of one of the previously mentioned decays. You can discover the relationship between energy and wavelength at Omni's photon energy calculator.

• Neutron emission, in which a neutron-rich nucleus emits one or more neutrons to reach a more stable isotope of the same element.

• Decays where more particles are released at the same time (cluster decay and nuclear fission).

💡 Other radioactive decay mechanisms are known, but they are of secondary importance: the ones listed here are the most likely to happen — even if we hope you will not stumble upon them!

A few notes are required here. Notice how in the beta processes, the total charge is conserved. This is one of the strongest universal principles, the conservation of charge! And it's worth remembering that two general types of phenomena can be identified: one in which the daughter atom (the one resulting from the decay) is of a different type than the original one (transmutation) and one where the two atoms belong to the same element. The difference lies in whether the process involves a change in the number of protons and neutrons (alpha, beta: transmutation) or not (gamma, neutron emission: no transmutation).

## Examples of radioactive decay

As you're now familiar with the definition of radioactive decay, here we are going to lay down some examples of decay in the notation used by nuclear physicists!

• Alpha:

$_{84}^{210}\text{Po}\rightarrow _{82}^{206}\text{Pb} + _{2}^{4}\text{He}$

An unstable polonium atom decays in a stable lead isotope emitting an alpha particle (a helium nucleus).

• Beta minus:

$_{6}^{14}\text{C}\rightarrow _{7}^{14}\text{N}+\text{e}^{-}+\nu^{*}$

A radioactive isotope of carbon transmutes in a nitrogen atom. $\text{e}^{-}$ is an electron and $\nu^{*}$ is an antineutrino.

• Beta plus:

$_{6}^{11}\text{C}\rightarrow _{5}^{11}\text{B}+\text{e}^{+}+\nu$

A different isotope of carbon loses a proton, transmuting in an atom of boron, with the emission of a positron ($\text{e}^{+}$) and a neutrino.

• Gamma:

$^{60}_{28}\text{Ni}\rightarrow^{60}_{28}\text{Ni}+\gamma$

The unstable isotope on the left decays by emitting a high-energy photon to the stable state on the right.

• Neutron emission:

$^{13}_{4}\text{Be}\rightarrow ^{12}_{4}\text{Be}+^1_0\text{n}$

Beryllium isotope with a high number of neutrons can emit one of them. The number of protons doesn't change: there is no transmutation.

The decay processes are usually linked in longer chains that start from a radioactive nucleus (radionuclides) and, through various steps, reach a final stable state.

🔎 Radiations are a natural occurrence, and they come from every direction! The granite in the New York Grand Central Station emits enough radiation to exceed the safety regulations for nuclear power plants (but it's safe to pass through it, don't worry). And the sky around us is filled with gamma rays coming from space, from the ground, and... from lightning! The amount of radiation increases with altitude: flying a lot can be dangerous! Check out our flight radiation calculator if you want to know more!

## Activity in radioactive materials

Physicists define the activity of a sample of radioactive material as the number of disintegrations per unit of time: it is a measure of the decay of a certain radionuclide.

🔎 The activity of a sample depends on its size. The bigger the sample, the higher the activity because more nuclei will decay in the same time period.

Activity is inversely proportional to the quantity called half-life, t½. The half-life is the time required to halve the quantity of a certain radioactive species in a sample. Notice that it doesn't depend on the size of a sample! If you are interested in this topic, check out our half-life calculator!

One can define activity utilizing the following equation:

$A = \lambda N,$

where:

• $A$ – Activity (decays per unit time);
• $N$ – Number of radionuclides in a given sample; and
• $\lambda$ (the decay constant) – Probability of one of them decaying.

The quantity $\lambda$ is associated with the half-life through the following formula:

$λ = \frac{\ln 2 }{ t_½}$

## How to measure activity?

The SI measurement unit of activity is the Becquerel, symbol $\text{Bq}$. A Becquerel equals a decay per second, regardless of the process involved: notice how it corresponds to a unit of frequency.

Another unit widely used for the activity was the Curie, $\text{Ci}$, defined as the activity of a gram of radium. This quantity is extremely large, and the conversion between the two units reads:

$1\ \textrm{Ci} = 3.7 \times 10^{10}\ \textrm{Bq} = 37\ \textrm{GBq}$

🔎 The Curie is named in honor of Maria Skłodowska-Curie, the famous Polish physicist. When the unit was proposed, someone asked to reduce the mass of radium used to 1 nanogram: that would have reduced its value by a factor $10^{-9}$. Maria Skłodowska opposed the decision: for her, that would diminish the efforts put into the research. We agree with her!

🔎 Did you know that Maria Skłodowska is also known for other scientific discoveries? Check out our Curie constant calculator and Curie's Law calculator. She is one of the few people to win two Nobel Prizes and the only one to win them in different fields — chemistry and physics!

## How to calculate radioactive decay?

Here, we introduce you to the radioactive decay formula. Namely, the radioactivity of a sample with mass $m$, composed by a chemical species with molar mass $m_\text a$ and half-life $t_½$ is:

$A = N_\text A \cdot \frac{m}{m_\text a} \cdot \frac{\ln2}{t_{½}}$

where $N_\text A$ is Avogadro's number.

Let's take a closer look at the above formula for radioactive decay. The first part of the equation, $N_\text A \cdot \frac{m}{m_\text a}$ is equal to the number of atoms in a sample. The mass over the molar mass equals the number of moles, which, when multiplied by the Avogadro number $N_\text A$, gives the number of atoms of the given species. The second part is the decay rate $\lambda$. In fact, this equation is the same as the more abstract $A = \lambda N$.

💡 The Avogadro number is defined as the number of atoms in a mole of a certain element. It equals $6.022 \times 10^{23}$, and it's a huge number!

That's it! Knowing how to calculate radioactive decay requires you to know only the weight of the specimen you are studying. All the other quantities are written down in tables.

## What is specific activity? Specific activity formula

Specific activity is a useful quantity defined as the activity per quantity of radionuclide, meaning that the units of specific activity are $\text{Bq}/\text{g}$.

The specific activity formula involves molar mass $m_\text a$, half-life $t_{½}$, and Avogadro's number $N_\text A$. It reads:

$a = \frac{N_\text A}{m_\text a} \cdot \frac{\ln2}{t_{½}}$

Note that the above formula for specific activity equals the activity expression divided by the mass of the sample, giving a more specific indication of the radioactivity of a chemical species.

Now that you know what the specific activity is and how to calculate it, it's time to learn when to use it! Specific activity is a fixed quantity for every different radioactive species. You can find .

## How to use our radioactive decay calculator?

Using our radioactive decay calculator is extremely easy! We ask you about the weight of your sample and the molar mass of its component (here is a handy table ), and then its half-life (you can find its values for many isotopes on ).

You can use our radioactive decay calculator in reverse too! Input the data you know, and find out the half-life of an element.

Let's see a couple of examples.

#### Example 1

The core of the nuclear bomb Fat Man, the second and last atomic weapon used in any conflict, was composed of two hollow hemispheres of plutonium (²³⁹Pu), weighing a total of 6.19 kg. A shockwave created by high-potential explosives was used to compress the core onto an initiator, increasing twofold the density of the metal itself. The reaction was then free to start.

Try to insert the values of the mass of the core, the molecular mass of the plutonium (239.05 g/mol), and its half-life (24,100 years) in the corresponding fields. The result is quite a high number: more than 14 TBq (that's Terabecquerel, a thousand billion Becquerels).

To give a scale of the immensity of that number, let's consider the classic example used in radiation courses, the banana.

#### Example 2

A banana contains around 0.5 grams of potassium. The isotope ⁴⁰K is a radioactive element that composes 0.012% of the total amount of potassium in a sample (this quantity is called abundance). Its half-life is 1.248 × 10⁹ years, and its molar mass is 39.96 g/mol.

Inserting the quantities in the calculator, we obtain the result 15.91 Bq. This is about 12 times smaller than the amount of radioactivity emitted by the core of a nuclear weapon.

## Radioactive decay in the real life

As we saw, a large banana has an activity of about 15 Bq. It means that 15 atoms (usually of an unstable potassium isotope) decay every second.

Only a radioactive gaseous atomic species is known to scientists: radon. It can reach the surface of the Earth and leak in basements, where it can be a major hazard. The highest value ever measured was a staggering 100,000 Bq/m³, measured in the basement of a nuclear power plant worker in the US. It was noticed because he was triggering every radiation alarm in the power plant, but the power plant was still being built. Imagine the confusion!

Half-life and activity play a fundamental role in historical dating: the amount of carbon-14 remaining in a sample allows us to determine its age since its specific activity and half-life are stable quantities. It works by measuring the amount of carbon-14 in a sample and then (knowing that going backward in time, its value doubles roughly every 5,730 years) establishing when the ratio between carbon-14 and carbon-12 (the stable isotope of carbon) was as expected. The radiocarbon dating calculator is spot-on for these kinds of problems. Try it out!

🔎 Nuclear testing after the 1950s affected the ratio of carbon-14 and carbon-12 by changing the relative abundances of the two isotopes. Fortunately, carbon-14 is not used for recent dating, and its spike in the '60s gave a tool for scientists to follow the growth of cells, trees, and other "real-time" phenomena!

## FAQ

### What is radioactive decay?

Radioactive decay is a process in which unstable nuclei reach more stable states by emitting particles or electromagnetic radiation.

### What is the activity of a radioactive substance?

The activity of a radioactive substance is the number of disintegrations per unit of time. This quantity tells you how dangerous a radioactive specimen is: the higher the activity, the more radiation the object is emitting.

### Which are the measurement units of radioactivity?

Radioactivity is measured in Becquerels. A Becquerel corresponds to the emission of any kind of radiation in a second. A million Becquerels (megabecquerel) is sometimes called a Rutherford, in honor of the physicist who first gave names to the types of radiations.

In the past, the Curie was used instead, but today it's not accepted in the International System.

### How do I calculate specific activity?

Specific activity refers to the activity of a given quantity of radioactive material. The unit of specific activity is Becquerel over grams.

The formula for a specific activity is a = Nₐ/mₐ × ln(2)/tₕ, where Nₐ is Avogadro's number, mₐ the molar mass of the substance, and tₕ the half-life.

### How do I calculate activity?

In order to calculate activity, use the specific activity of a substance, which is the amount of radiation of a radioactive sample. You can then multiply this quantity by the mass of the sample to get its activity.

Davide Borchia
Sample mass
g
Molar mass of the substance
g
/mol
Half-life
sec
Results
Activity
Bq
Specific activity
Bq
/kg
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