Half-Life Calculator

Complete Half-Life Calculator Guide & Information

1. What is Half-Life?

Half-life (symbol t_1/2) is the time required for a quantity to reduce to half of its initial value. The term is most commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay, but it also applies to any exponential decay process in chemistry, biology, pharmacology and finance.

Half-life is a characteristic constant for each decaying substance. It does not depend on the initial amount — after one half-life, 50% remains; after two half-lives, 25% remains; after three half-lives, 12.5% remains, and so on. Theoretically, the quantity never reaches exactly zero; it approaches zero asymptotically.

2. The Half-Life Formula

Where:

• N(t) — remaining quantity after time t
• N₀ — initial quantity at t = 0
• t_1/2 — half-life of the substance
• t — elapsed time

Alternative form using the decay constant λ:

3. Relationships Between Parameters

• Decay constant (λ): Probability of decay per unit time.
• Mean lifetime (τ): Average time before a nucleus decays. τ = 1/λ ≈ 1.4427 × t_1/2
• ln(2) ≈ 0.69314718056

4. Solving for Each Variable

• Find N(t): N(t) = N₀ × 0.5^{(t / t_1/2)}
• Find N₀: N₀ = N(t) × 2^{(t / t_1/2)}
• Find t_1/2: t_1/2 = t / log₂(N₀ / N(t))
• Find t: t = t_1/2 × log₂(N₀ / N(t))

5. Common Radioisotopes and Half-Lives

Isotope	Half-Life	Common Use
Carbon-14	5,730 years	Radiocarbon dating
Uranium-238	4.47 billion years	Nuclear fuel, geologic dating
Cobalt-60	5.27 years	Radiotherapy, industrial radiography
Iodine-131	8.02 days	Thyroid treatment, medical imaging
Technetium-99m	6.01 hours	Nuclear medicine diagnostic scans
Caesium-137	30.17 years	Industrial gauges, fallout marker
Radon-222	3.82 days	Indoor air health hazard
Strontium-90	28.8 years	Nuclear fission product

6. Types of Radioactive Decay

• Alpha decay: Nucleus emits an alpha particle (helium nucleus). Reduces atomic number by 2 and mass by 4.
• Beta-minus decay: Neutron converts to proton, emits electron and antineutrino. Increases atomic number by 1.
• Gamma decay: Nucleus drops from excited state to ground state, emitting a gamma photon.
• Beta-plus decay: Proton converts to neutron, emits positron and neutrino.
• Electron capture: Nucleus captures an inner electron, converting proton to neutron.
• Spontaneous fission: Heavy nucleus splits into two lighter nuclei.

7. Input & Control Definitions

• Main Calculator section:
  • N_t (quantity remains): Amount of substance remaining after elapsed time.
  • N₀ (initial quantity): Starting amount at time zero.
  • t (time): Elapsed time period.
  • t_1/2 (half-life): Characteristic half-life of the substance.
  • Enter any three values; the fourth will be calculated.
• Conversion section:
  • Enter any one of half-life, mean lifetime or decay constant.
  • The other two equivalent values are computed automatically.
• Calculate Button: Runs the calculation and updates all results.
• Clear Button: Clears all input fields in the section.

8. Real-World Applications

• Radiometric dating: Carbon-14 and uranium-lead dating determine ages of samples.
• Radiation therapy: Calculating delivered dose and timing for cancer treatment.
• Nuclear power: Reactor core design, fuel burn-up and waste management.
• Pharmacokinetics: Drug half-life determines dosing frequency in medicine.
• Environmental monitoring: Assessing radioactive contamination and cleanup timelines.
• Archaeology and geology: Establishing chronological timelines.

9. Important Notes & Limitations

• Half-life describes statistical behavior of large numbers of atoms. For small samples, actual decay may deviate due to statistical variation.
• This calculator assumes first-order exponential decay, which applies to most radioactive decay processes.
• Results are estimates. Actual decay rates may be affected by extreme conditions (generally negligible under normal circumstances).
• This tool is for educational and informational purposes only. Not for medical or safety-critical calculations.

10. Related Concepts

• Exponential growth: The inverse process — quantity doubles over each doubling time.
• Doubling time: Time for a quantity to double (growth equivalent of half-life).
• Decay chain: Series of sequential decays from parent isotope through daughters to stable end product.
• Effective half-life: Combined physical and biological half-life in living organisms.
• Biological half-life: Time for substance to reduce by half through biological excretion.
• Radioactive equilibrium: When production rate equals decay rate of a daughter isotope.

11. References

1. Rutherford, Ernest. "Radioactivity." Cambridge University Press. 1904.
2. Krane, Kenneth S. "Introductory Nuclear Physics." Wiley. 1987.
3. National Institute of Standards and Technology (NIST). "Table of Isotopes."
4. International Commission on Radiological Protection (ICRP). "Publication 107: Nuclear Decay Data for Dosimetric Calculations." 2008.
5. Libby, Willard F. "Radiocarbon Dating." University of Chicago Press. 1952.
6. Turner, J. E. "Atoms, Radiation, and Radiation Protection." Wiley. 2007.