Half-Life Results
Remaining
250.0000
units
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Input Values
Result
Initial (N₀)
1,000
Remaining (N)
250
Half-Life (t½)
5,730 years
Time (t)
11,460 years
After 2.00 half-lives
25.0000% remaining
Formula & Calculation
N(t) = N₀ × (½)^(t/t½)
N(t) = 1,000 × (0.5)^(11460/5730)
N(t) = 1,000 × (0.5)^2.0000
N(t) = 1,000 × 0.25000000
N(t) = 250.0000
Decay Constant
λ = ln(2) / t½ = 1.2097e-4
Alternative Formula
N(t) = N₀ × e^(-λt)
Exponential Decay Chart
Decay Progress Table
| Half-Lives | Time | Remaining | % Remaining | Decayed |
|---|---|---|---|---|
| 0 | 0 | 1000.00 | 100.0000% | 0.0000% |
| 1 | 5,730 | 500.00 | 50.0000% | 50.0000% |
| 2 | 11,460 | 250.00 | 25.0000% | 75.0000% |
| 3 | 17,190 | 125.00 | 12.5000% | 87.5000% |
| 4 | 22,920 | 62.50 | 6.2500% | 93.7500% |
| 5 | 28,650 | 31.25 | 3.1250% | 96.8750% |
| 6 | 34,380 | 15.63 | 1.5625% | 98.4375% |
| 7 | 40,110 | 7.81 | 0.7813% | 99.2188% |
| 8 | 45,840 | 3.91 | 0.3906% | 99.6094% |
| 9 | 51,570 | 1.95 | 0.1953% | 99.8047% |
| 10 | 57,300 | 0.98 | 0.0977% | 99.9023% |
Common Radioactive Isotopes
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?How Do You Calculate Half-Life?
Half-life is the time for half of a substance to decay. Formula: N(t) = N0 * (1/2)^(t/t1/2), where N0 is initial amount, t is elapsed time, t1/2 is half-life. After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. The decay constant k = ln(2)/t1/2 approximately equals 0.693/t1/2.
What is Half-Life?
Half-life is the time required for exactly half of a substance to decay or transform. In radioactive decay, it describes how long until half the atoms have undergone nuclear decay. The concept applies to any exponential decay process including drug metabolism, chemical reactions, and population dynamics.
Key Facts About Half-Life
- Half-life: time for half the substance to decay
- Formula: N(t) = N0 * (1/2)^(t/t1/2)
- Decay constant: k = ln(2)/t1/2 approximately equals 0.693/t1/2
- After n half-lives: (1/2)^n of original remains
- Exponential decay: quantity never reaches zero
- Carbon-14 half-life: ~5,730 years (used in dating)
- Medical isotopes have short half-lives (hours to days)
- Also applies to drug elimination, population decline
Quick Answer
Half-life is the time for half of a substance to decay. Formula: N(t) = N0 * (1/2)^(t/t1/2), where N0 is initial amount, t is elapsed time, t1/2 is half-life. After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. The decay constant k = ln(2)/t1/2 approximately equals 0.693/t1/2.
Frequently Asked Questions
What is half-life?
Half-life (t½) is the time required for a quantity to reduce to half its initial value. It's constant regardless of the amount present. After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. Used in radioactive decay, pharmacology, and chemistry.
What is the half-life formula?
N(t) = N₀ × (1/2)^(t/t½), where N(t) = remaining amount, N₀ = initial amount, t = elapsed time, t½ = half-life. Alternative: N(t) = N₀ × e^(-λt), where λ = ln(2)/t½ is the decay constant.
How is half-life used in carbon dating?
Carbon-14 has a half-life of 5,730 years. Living organisms maintain constant C-14 levels. After death, C-14 decays. By measuring remaining C-14 compared to living organisms, we calculate time since death. Effective up to ~50,000 years.
What is biological half-life?
Biological half-life is the time for half of a substance to be eliminated from the body through metabolism and excretion. Different from radioactive half-life. Used in pharmacology to determine drug dosing schedules.
Why doesn't the amount reach zero?
Mathematically, exponential decay approaches but never reaches zero. In practice, after ~7 half-lives, less than 1% remains. After 10 half-lives, ~0.1% remains. Eventually, individual atom behavior becomes relevant (quantum effects).
Last updated: 2026-01-15
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How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
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