The Quick Answer
Half-life is the time required for a quantity to reduce to half of its initial value, most commonly applied to radioactive decay and drug metabolism. It is a fixed property of the substance -- not the amount present -- and produces a characteristic exponential decay curve.
The formula: N(t) = N0 x (1/2)^(t / t_half)
Where N(t) is the amount remaining at time t, N0 is the initial amount, and t_half is the half-life.
Quick example: If you start with 100 mg of a substance with a 4-hour half-life, after 4 hours you have 50 mg. After 8 hours: 25 mg. After 12 hours: 12.5 mg.
The Half-Life Formula
Half-life decay is a form of exponential decay. There are two equivalent mathematical representations.
Form 1: Half-life form
N(t) = N0 x (1/2)^(t / t_half)
This is the most intuitive version. It directly says: every time t increases by one half-life, the quantity is multiplied by 1/2.
Form 2: Exponential decay with decay constant
N(t) = N0 x e^(-lambda x t)
Where lambda (the decay constant) relates to half-life by:
lambda = ln(2) / t_half = 0.6931 / t_half
Both forms produce identical results. Form 1 is easier to work with for whole or simple fractions of half-lives. Form 2 is preferred in physics and engineering for calculus-based analysis.
Useful Rearrangements
Finding time to reach a target amount:
t = t_half x log2(N0 / N)
Finding half-life from two measurements:
t_half = t x ln(2) / ln(N0 / N)
Finding initial amount:
N0 = N(t) x 2^(t / t_half)
Worked Example 1: Radioactive Decay (Carbon-14)
Carbon-14 has a half-life of 5,730 years. A sample starts with 100 grams of C-14.
| Time Elapsed | Half-Lives Passed | C-14 Remaining | Calculation |
|---|---|---|---|
| 0 years | 0 | 100.0 g | Starting amount |
| 5,730 years | 1 | 50.0 g | 100 x (1/2)^1 |
| 11,460 years | 2 | 25.0 g | 100 x (1/2)^2 |
| 17,190 years | 3 | 12.5 g | 100 x (1/2)^3 |
| 22,920 years | 4 | 6.25 g | 100 x (1/2)^4 |
| 28,650 years | 5 | 3.125 g | 100 x (1/2)^5 |
After 5 half-lives (28,650 years), only 3.125% of the original C-14 remains. This predictable decay is the basis of radiocarbon dating, used to determine the age of organic materials up to approximately 50,000 years old.
Worked Example 2: Drug Metabolism (Pharmacology)
A medication produces a peak blood concentration of 200 mg. Its half-life is 4 hours.
| Time | Half-Lives | Drug Remaining | % of Peak |
|---|---|---|---|
| 0 h | 0 | 200.0 mg | 100% |
| 4 h | 1 | 100.0 mg | 50% |
| 8 h | 2 | 50.0 mg | 25% |
| 12 h | 3 | 25.0 mg | 12.5% |
| 16 h | 4 | 12.5 mg | 6.25% |
| 20 h | 5 | 6.25 mg | 3.125% |
After 5 half-lives (20 hours), approximately 97% of the drug has been eliminated. This is why pharmacologists and the FDA use the "5 half-lives rule" as the standard for considering a drug effectively cleared from the body.
This principle determines:
- Dosing schedules: How often to take a medication to maintain therapeutic levels
- Washout periods: How long to wait between stopping one drug and starting another
- Steady state: After approximately 5 half-lives of regular dosing, drug levels stabilize
Worked Example 3: Calculating Time to Reach a Target Amount
Problem: How long until only 10% of a substance remains?
Setup: We need N(t) = 0.10 x N0
0.10 = (1/2)^(t / t_half)
Solve by taking log base 2 of both sides:
log2(0.10) = t / t_half
t / t_half = log2(10) (since log2(1/0.10) = log2(10))
t / t_half = 3.322
t = 3.322 x t_half
Application: For the 4-hour drug above, reaching 10% remaining takes 3.322 x 4 = 13.3 hours.
Verification: N(13.3) = 200 x (1/2)^(13.3/4) = 200 x (1/2)^3.322 = 200 x 0.100 = 20.0 mg. That is 10% of 200 mg. Confirmed.
Quick Reference: Half-Lives to Key Percentages
| Remaining % | Half-Lives Required | Formula |
|---|---|---|
| 50% | 1.000 | By definition |
| 25% | 2.000 | log2(4) |
| 10% | 3.322 | log2(10) |
| 5% | 4.322 | log2(20) |
| 1% | 6.644 | log2(100) |
| 0.1% | 9.966 | log2(1000) |
Notable Half-Lives
Radioactive Isotopes
| Isotope | Half-Life | Common Application |
|---|---|---|
| Fluorine-18 | 110 minutes | PET scans in medical imaging |
| Iodine-131 | 8.02 days | Thyroid cancer treatment |
| Cobalt-60 | 5.27 years | Radiation therapy, food irradiation |
| Tritium (H-3) | 12.3 years | Self-luminous signs, fusion research |
| Carbon-14 | 5,730 years | Radiocarbon dating |
| Plutonium-239 | 24,100 years | Nuclear weapons, nuclear fuel |
| Uranium-235 | 704 million years | Nuclear reactors, geological dating |
| Uranium-238 | 4.47 billion years | Geological dating (age of Earth) |
Source: IAEA Nuclear Data Services
Substances in the Human Body
| Substance | Biological Half-Life | Notes |
|---|---|---|
| Alcohol (ethanol) | ~4-5 hours | Varies with weight, metabolism |
| Aspirin | 3.5-4.5 hours | Active metabolite (salicylic acid) has longer half-life |
| Caffeine | 5-6 hours | Varies by individual; longer in pregnancy |
| Ibuprofen | 2-4 hours | Explains typical 4-6 hour dosing |
| Nicotine | ~2 hours | Cotinine (metabolite) has ~16 hour half-life |
Applications of Half-Life
Carbon Dating
Living organisms continuously exchange carbon with the environment, maintaining a stable ratio of radioactive C-14 to stable C-12. When an organism dies, it stops absorbing carbon, and C-14 begins to decay with its 5,730-year half-life. By measuring the remaining C-14-to-C-12 ratio and comparing it to the atmospheric ratio, scientists calculate time since death.
Example: A wooden artifact has 25% of the expected C-14. Since 25% = (1/2)^2, it has been through 2 half-lives: 2 x 5,730 = 11,460 years old.
Nuclear Medicine
Isotopes with short half-lives are used in medical imaging because they provide useful radiation for detection but decay quickly, limiting patient exposure. Technetium-99m (half-life: 6 hours) is the most commonly used isotope in diagnostic imaging, used in over 30 million procedures worldwide each year.
Drug Dosing
Drugs are typically dosed at intervals related to their half-life to maintain a therapeutic window -- above the minimum effective concentration but below toxic levels. For example, a drug with a 12-hour half-life might be prescribed as a twice-daily dose.
Environmental Science
Understanding the half-life of pollutants and radioactive waste informs cleanup timelines. Cesium-137 (half-life: 30 years) released from nuclear accidents like Chernobyl (1986) will take approximately 10 half-lives (300 years) to decay to 0.1% of initial levels.
The Decay Curve
Half-life decay produces a characteristic curve that is steep at first and flattens over time. Key properties:
- The curve never reaches zero -- it approaches zero asymptotically
- The curve is self-similar -- any section looks like a scaled version of any other section
- The rate of decay (amount lost per unit time) decreases as the quantity decreases
- The curve is entirely determined by two values: the initial amount and the half-life
This is why radioactive waste management requires such long time horizons. Even though most of the decay happens early, the remaining small fraction takes proportionally just as long to halve again.
Frequently Asked Questions
How many half-lives until a substance is gone?
Mathematically, a substance never fully reaches zero through half-life decay. Practically, after about 7 half-lives, less than 1% remains (0.78%). After 10 half-lives, less than 0.1% remains. In pharmacology, 5 half-lives (about 97% eliminated) is the standard threshold.
Why is the 5 half-lives rule used in pharmacology?
After 5 half-lives, approximately 96.875% of a drug has been eliminated from the body. The remaining 3.125% is generally considered clinically insignificant. This rule is used by the FDA and pharmacologists for drug clearance and steady-state calculations.
How is half-life used in carbon dating?
Living organisms maintain a constant ratio of carbon-14 to carbon-12. After death, C-14 decays with a 5,730-year half-life while C-12 remains stable. Measuring the remaining C-14 ratio reveals how long ago the organism died. The method is reliable for samples up to about 50,000 years old (roughly 9 half-lives).
What is the difference between half-life and decay constant?
They are mathematically related: lambda = ln(2) / t_half = 0.6931 / t_half. Half-life is more intuitive (time to halve). The decay constant is the probability per unit time that a given atom will decay, used in the exponential form N(t) = N0 x e^(-lambda x t).
Does half-life change with the amount of substance?
No. Half-life is a fixed property of the substance, independent of the amount present. Whether you start with 1 gram or 1 kilogram of carbon-14, it takes 5,730 years for half of it to decay.
Can you speed up or slow down radioactive decay?
Under normal conditions, no. Radioactive decay rates are determined by nuclear forces and are unaffected by temperature, pressure, chemical environment, or physical state. This constancy makes radioactive isotopes reliable for dating and timekeeping.
What is biological half-life vs physical half-life?
Physical (or radioactive) half-life is the time for half of a substance to decay. Biological half-life is the time for the body to eliminate half through metabolism and excretion. The effective half-life combines both: 1/t_effective = 1/t_physical + 1/t_biological.
Why do some isotopes have very long half-lives?
Half-life depends on nuclear stability. Uranium-238 has a half-life of 4.5 billion years because its nucleus is relatively stable -- the probability of any single atom decaying at any given moment is extremely small. Less stable nuclear configurations have shorter half-lives.
How do I calculate the time for a substance to reach a specific remaining percentage?
Use the formula t = t_half x log2(N0 / N). For 10% remaining: t = t_half x log2(10) = t_half x 3.322. For 1% remaining: t = t_half x log2(100) = t_half x 6.644.
What is the half-life of caffeine?
Caffeine has a half-life of approximately 5 to 6 hours in healthy adults. If you consume 200 mg of caffeine at noon, roughly 100 mg remains by 5-6 PM, and about 50 mg by 10-11 PM. This is why sleep experts recommend limiting caffeine after early afternoon.
Calculate Half-Life
Use the half-life calculator to compute remaining quantities, elapsed time, or half-life from any two known values. For the underlying math, try the exponent calculator or the logarithm calculator.