Radioactive Decay Calculator — Activity, Half-life & Specific Activity

Radioactive Decay Calculator — Activity, Half-life & Specific Activity

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 and how to calculate radioactive decay. Here you will learn:

• A short history of the discovery of radioactivity;
• What radioactive decay is and its types;
• How to calculate radioactive decay and measure activity;
• What specific activity is and its formula.

The Discovery of Radioactivity

Sometimes, important discoveries happen by chance. While experimenting with cathode tubes at the end of 1895, Wilhelm Röntgen discovered X-rays — a form of radiation able to pass through opaque materials. This sparked further research into the phenomenon of radiation.

French physicist Henri Becquerel was investigating whether uranium salts could emit X-rays after solar exposure. By a double coincidence — an overcast sky forced him to store his samples near photographic film, and he developed the film unexpectedly early — he discovered that uranium salts spontaneously exposed the film without any solar activation. This was the first evidence of spontaneous radioactive decay.

The physicist Rutherford later identified this phenomenon as emission originating from atomic nuclei.

⚠️ The dangers of radiation were unknown at the time. Maria Skłodowska-Curie's notebooks are still so radioactive they must be stored in lead-lined boxes.

What is Radioactive Decay? Types of Radioactive Decay

Massive or unstable atomic nuclei undergo radioactive decay — a spontaneous transformation to reach a lower energy state. During this process, atoms emit radiation (waves or particles).

The main types of radioactive decay are:

• Alpha (α) decay — two protons and two neutrons (a helium-4 nucleus) are emitted.
• Beta minus (β⁻) decay — a neutron becomes a proton, emitting an electron and an antineutrino.
• Beta plus (β⁺) decay — a proton turns into a neutron, emitting a positron and a neutrino.
• Gamma (γ) decay — a high-energy photon is emitted following alpha or beta decay.
• Neutron emission — a neutron-rich nucleus emits one or more neutrons.
• Cluster decay & nuclear fission — multiple particles or large fragments are emitted simultaneously.

How to Calculate Radioactive Decay

Radioactive decay follows an exponential law. If A₀ is the initial activity and T½ is the half-life, then the activity at time t is:

A(t) = A₀ × 2−t/T½   or equivalently   A(t) = A₀ × e−λt

Where the decay constant λ (lambda) relates to the half-life by:

λ = ln(2) / T½  ≈  0.6931 / T½

The number of half-lives elapsed is simply:

n = t / T½

And the percentage of the original substance remaining is:

% remaining = (1/2)n × 100%

Example: Iodine-131 Decay

Iodine-131 has a half-life of 8.02 days. Starting with 100 MBq of activity, after 30 days:

• Number of half-lives: n = 30 / 8.02 ≈ 3.74
• Remaining fraction: (1/2)³·⁷⁴ ≈ 0.0742 (7.42%)
• Current activity: 100 × 0.0742 ≈ 7.42 MBq

Activity in Radioactive Materials

Activity (A) measures how many radioactive decays occur per unit time. It is defined as:

A = λ × N

Where N is the total number of radioactive nuclei present.

How to Measure Activity

Activity is measured in two unit systems:

Metric (SI) Units

US / Imperial Units

The Curie was originally defined as the activity of 1 gram of radium-226: 1 Ci = 3.7 × 10¹⁰ Bq.

What is Specific Activity? Specific Activity Formula

Specific activity is the activity per unit mass of a radioactive substance:

a = A / m

For a pure radionuclide, it can be calculated directly from the half-life and molar mass:

a = NA × ln(2) / (M × T½)

Where:

• N_A = 6.022 × 10²³ mol⁻¹ (Avogadro constant)
• M = molar mass in g/mol
• T½ = half-life in seconds
• Result in Bq/g

Example: Carbon-14 Specific Activity

C-14: T½ = 5,730 years = 1.807 × 10¹¹ s, M = 14.003 g/mol

a = (6.022×10²³ × 0.6931) / (14.003 × 1.807×10¹¹) ≈ 1.65 × 10¹¹ Bq/g ≈ 4.46 Ci/g

Radioactive Decay in Real Life

• ☢ Medical imaging (PET/SPECT) — isotopes like F-18 (T½ = 110 min) or Tc-99m (T½ = 6 h) decay quickly after use.
• ☢ Cancer radiotherapy — I-131 (T½ = 8.02 d) treats thyroid cancer.
• ☢ Nuclear power — U-235 (T½ = 7.04 × 10⁸ y) fissions in reactors.
• ☢ Radiocarbon dating — C-14 (T½ = 5,730 y) dates organic material up to ~50,000 years old.
• ☢ Smoke detectors — Am-241 (T½ = 432 y) ionizes air to detect smoke.
• ☢ Geological dating — U-238 (T½ = 4.47 × 10⁹ y) dates rocks and meteorites.

How to Use Our Radioactive Decay Calculator

The calculator offers two modes:

1. Activity Decay — Enter the initial activity, half-life, and elapsed time. The calculator returns the current activity, decay constant λ, number of half-lives elapsed, and the fraction remaining. Choose Metric (SI) for Bq units or US/Imperial for Curie units.
2. Specific Activity — Enter the half-life and molar mass of the isotope. The calculator returns the specific activity in both Bq/g and Ci/g.

FAQs

Q: What is the half-life?
A: The half-life T½ is the time it takes for half of the radioactive nuclei in a sample to decay. After n half-lives, (1/2)ⁿ of the original amount remains.

Q: What is the difference between Bq and Ci?
A: The Becquerel (Bq) is the SI unit — 1 Bq = 1 decay per second. The Curie (Ci) is the older US unit: 1 Ci = 3.7 × 10¹⁰ Bq (activity of 1 gram of Ra-226).

Q: Can activity ever become zero?
A: Mathematically, exponential decay never reaches exactly zero. In practice, when the activity drops below measurable levels (typically below 1 Bq), the sample is considered spent.

Q: What is the decay constant λ?
A: λ is the probability per unit time that a single nucleus decays. It relates to half-life by λ = ln(2) / T½ ≈ 0.6931 / T½.