Nuclear Decay Calculator
Solve radioactive decay equations, half-life parameters, decay constants, and remaining parent/daughter isotope quantities over time. Select from common radioisotope presets or input custom element symbols, mass numbers, decay modes, and elapsed time to instantly render balanced nuclear equations and interactive exponential decay charts.
Compute atomic decay equations, half-life kinetic parameters, decay constants, and remaining mass or radio-activity. Plot dynamic decay curves, choose common radioisotopes, and view structured step-by-step kinetic derivations.
Load Common Radioisotopes
Decay Calculations Awaiting
Select a preset or input your isotope parameters and click "Calculate Decay" to generate equations, curves, and step-by-step kinetic breakdowns.
Nuclear Decay Physics Nomenclature
- Alpha Decay ($\alpha$): Release of a Helium-4 nucleus ($^4_2\text{He}$). Daughter isotope has $A - 4$ and $Z - 2$. Dominates in heavy unstable nuclei (e.g. Uranium, Radium).
- Beta-minus Decay ($\beta^-$): A neutron turns into a proton, emitting an electron ($^0_-1e$) and an antineutrino. Daughter isotope has $A$ and $Z + 1$. Common in neutron-rich nuclides (e.g. Carbon-14).
- Beta-plus Decay ($\beta^+$) / Positron Emission: A proton turns into a neutron, emitting a positron ($^0_1e$) and a neutrino. Daughter isotope has $A$ and $Z - 1$. Occurs in proton-rich nuclides.
- Electron Capture (EC): An inner orbital electron is captured by the nucleus, combining with a proton to form a neutron and neutrino. Has the same daughter output ($Z - 1$) as positron emission.
- Gamma Decay ($\gamma$): A metastable excited state nucleus ($^{A}_{Z}\text{X}^*$) releases excess energy as a high-frequency gamma photon without altering mass or atomic numbers.
Why Use Our Nuclear Decay Calculator?
Comprehensive Decay Modes
Evaluate all seven primary radioactive pathways: Alpha, Beta-minus, Beta-plus (positron), Gamma, Electron Capture (EC), Proton, and Neutron emission.
Dynamic Isotope Physics Solver
Enter any chemical symbol to auto-lookup atomic numbers, and automatically resolve daughter isotopes using balanced mass/charge equations.
Visual Decay Curve Plotting
Instantly renders interactive SVG graphs plotting the parent isotope’s exponential decay alongside the corresponding daughter isotope accumulation curve.
100% Client-Side Privacy
All equations, half-life kinetics, and activity factors are computed locally on your device. Your data is never uploaded to any remote server.
Common Use Cases for Nuclear Decay Calculator
Archaeology & Carbon Dating
Determine the age of organic artifacts by measuring remaining Carbon-14 activity and calculating the elapsed decay time.
Nuclear Medicine & Tracers
Calculate the remaining activity of radioisotopes like Technetium-99m or Iodine-131 to optimize diagnostic and therapeutic dosages.
Waste Decay Monitoring
Track the decline of long-term nuclear power plant radioactive waste (e.g. Cesium-137) to determine safe storage containment periods.
Physics & Chemistry Education
Instantly verify homework solutions for first-order kinetic rates, decay constants, and decay equation balancing problems.
Radiation Safety Audits
Evaluate the current activity of calibration sources in laboratories to ensure correct radiation exposure records and standards.
Geological Isochron Dating
Analyze rock and mineral ages by comparing parent-daughter isotope fractions (e.g., Uranium-238 to Thorium-234 pathways).
Understanding Nuclear Decay and Half-Life
What is Radioactive Decay?
In nuclear physics and chemistry, radioactive decay (also known as nuclear decay or radioactivity) is the process by which an unstable atomic nucleus loses energy by emitting radiation. A material containing unstable nuclei is considered radioactive. The original unstable atom is called the parent isotope, and the resulting atom after the transition is known as the daughter isotope. This process is completely random at the level of single atoms, but for a large population of atoms, it follows highly predictable first-order kinetics.
The Mathematics of the Radioactive Decay Law
Radioactive decay rate is directly proportional to the number of radioactive nuclei present. This leads to the standard exponential decay equation:
Where the terms represent:
- N(t): The remaining quantity (mass, activity, or number of atoms) of the parent radioisotope after elapsed time $t$.
- N₀: The initial quantity of the parent radioisotope at time $t = 0$.
- t: The elapsed decay time duration.
- T₁/₂: The half-life of the isotope—the duration required for half of the radioactive atoms in a sample to decay.
- λ (decay constant): The probability of decay per nucleus per unit time, solved as:λ = ln(2) / T1/2 ≈ 0.69315 / T1/2
Physics of Common Decay Modes
Depending on the ratio of neutrons to protons in the parent nucleus, different decay pathways are favored to reach stability:
1. Alpha Decay (α): Emits an alpha particle (Helium-4 nucleus, Consisting of 2 protons and 2 neutrons, $^4_2\text{He}$). This decreases the mass number $A$ by 4 and atomic number $Z$ by 2. It is common in heavy elements like Uranium and Radium.
2. Beta-minus Decay (β⁻): A neutron turns into a proton, emitting an electron ($e^-$) and an electron antineutrino. This increases the atomic number $Z$ by 1, leaving the mass number $A$ unchanged. It occurs in neutron-rich isotopes (e.g. Carbon-14).
3. Beta-plus / Positron Decay (β⁺): A proton converts into a neutron, emitting a positron ($e^+$) and a neutrino. This decreases the atomic number $Z$ by 1 while keeping mass number $A$ constant.
4. Electron Capture (EC): The nucleus captures an inner shell orbital electron, combining it with a proton to form a neutron. Like positron emission, it lowers the atomic number $Z$ by 1.
5. Gamma Decay (γ): Releases high-energy electromagnetic radiation (gamma photon) from an excited metastable nucleus. The atomic and mass numbers remain unchanged, but the nucleus drops to a lower, more stable energy configuration.
Parent Isotope Decay vs. Daughter Isotope Growth
As the parent isotope decays, the daughter isotope quantity grows over time. Assuming a direct 1-to-1 decay step, the quantity of the produced daughter isotope ($D(t)$) at elapsed time $t$ corresponds to the lost parent quantity:
D(t) = N₀ · (1 - e^(-λt))For example, after exactly one half-life ($t = T_0.5$), the remaining parent isotope quantity is 50%, and the generated daughter isotope quantity is 50%. After two half-lives ($t = 2T_0.5$), the remaining parent is 25%, and the daughter is 75%, matching the curves displayed on our interactive decay chart.
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Frequently Asked Questions About Nuclear Decay
The half-life (T₁/₂) is the time required for half of the radioactive nuclei in a given sample to undergo decay. It is a constant value characteristic of each specific radioisotope, ranging from fractions of a microsecond to billions of years, and is unaffected by external temperature, pressure, or chemical state.
Our calculator applies nuclear conservation rules based on the selected decay mode. It adjusts the mass number (A) and atomic number (Z) of the parent isotope, then looks up the new atomic number (Z) in our database containing elements 1 to 118 to retrieve the correct IUPAC chemical symbol and name.
The decay constant (λ) represents the probability of decay per nucleus per unit time. It is solved from the half-life using the formula λ = ln(2) / T₁/₂. A larger decay constant signifies that the isotope decays more rapidly (shorter half-life).
Alpha decay emits a Helium-4 nucleus (2 protons, 2 neutrons), reducing atomic number by 2 and mass number by 4. Beta-minus decay converts a neutron to a proton, emitting an electron and raising the atomic number by 1. Gamma decay is the release of high-energy photons from an excited nucleus, which releases energy without changing mass or atomic numbers.
Remaining parent quantity is solved using the radioactive decay law: N(t) = N₀ * e^(-λt). Alternatively, it can be computed using half-life counts: N(t) = N₀ * (0.5)^n, where n = elapsed time / half-life. Both methods yield identical, highly accurate kinetic results.
Mass represents the physical weight of the sample in grams (g) or milligrams (mg). Activity represents the rate of decay events occurring per second, measured in Becquerels (1 Bq = 1 decay/sec) or Curies (1 Ci = 3.7 × 10¹⁰ decays/sec). Since rate is directly proportional to quantity, the decay kinetics equation holds true for both mass and activity.
Yes. You can enter values in seconds, minutes, hours, days, or years. The calculator automatically standardizes all units into seconds during calculation to ensure mathematical accuracy, then formats the results back to the user's selected units.
No. The Nuclear Decay Calculator runs 100% locally in your web browser. No element symbols, half-lives, masses, or elapsed times are transmitted to external servers. Your work is kept private and secure on your local device.