Biodegradation Half‑Life Calculator
Solve first-order decay in any direction — half-life from a rate constant, rate constant from half-life, or the time required to reach a target remaining fraction. Includes a decay curve, persistence gauge and companion DT50/DT90 metrics.
Exponential decay
Concentration falls as C(t) = C₀·e^(−kt). Each half-life removes exactly half of whatever remains — so the curve never quite reaches zero.
k and t½ are reciprocal
Doubling the rate constant halves the half-life. The product k · t½ is a universal constant, ln(2) ≈ 0.6931, regardless of the substance.
Targets scale with half-life
Time to 10% remaining (DT90) is ≈ 3.32 × t½. Time to 1% remaining is ≈ 6.64 × t½. Every target fraction maps to a fixed multiple of the half-life.
First-order assumption
Real biodegradation only follows first-order kinetics at low substrate concentrations. Above ~1–10 mg/L, microbial saturation slows the decay and the simple model over-predicts loss.
How to Use This Calculator
The first-order decay relation links three quantities: the rate constant k, the half-life t½, and the time t required to reach a target remaining fraction. Pick which one you want the tool to solve, then fill in the others.
1. Choose what to solve for
Pick t½ (half-life from k), k (rate constant from t½), or t (time to a target remaining percentage) — the tool rearranges the formula automatically.
2. Enter the rate constant or half-life
Type k and pick its unit (1/day, 1/hour, 1/year), or type t½ directly in days. Quick presets cover common laboratory and field values.
3. Set the target remaining percentage
Use 50% for the half-life itself, 10% for DT90 (a common regulatory endpoint), or 1% for near-complete removal.
4. Read the result and steps
The result card shows the solved value with engineering notation, and the step-by-step panel shows exactly how the number was reached.
5. Check the curve and persistence gauge
The decay curve plots C(t)/C₀ with half-life markers, and the gauge places your substance on the very-fast → very-persistent scale.
6. Switch to Advanced for more depth
Advanced mode adds a 1% sensitivity check plus DT50, DT90, time-to-1% and cross-unit k values, alongside the parameter-relationship breakdown.
What This Tool Offers
Built around the standard first-order decay relation t½ = ln(2)/k, the workhorse equation behind every environmental persistence and biodegradation estimate.
Solve any variable
Go from rate constant to half-life, from half-life to rate constant, or work out how long it takes to reach any target remaining fraction.
Flexible k units
Enter the rate constant in 1/day, 1/hour or 1/year — the tool normalises internally and reports the result in whichever unit you chose.
Live parameter diagram
A visual t½ = ln(2) ÷ k diagram updates with your inputs, alongside a mini decay curve so the relationship is always visible.
Persistence gauge
A five-band very-fast → very-persistent scale (log-scaled) that instantly places your substance in a recognised environmental-fate category.
Built-in sensitivity check
Advanced mode shows exactly how much a 1% error in k, t½ or the target fraction propagates into the solved value.
Companion metrics
Every result is cross-reported as DT50, DT90, time to 1% remaining, and k in alternative units — no manual conversions needed.
Reference Tables & Conversions
A quick t½ ↔ k converter, plus lookup tables for decay-rate units, persistence classification, and time-to-target multiples.
Quick t½ ↔ k converter
Half-Life ↔ Rate Constant Reference
For a given half-life, the corresponding first-order rate constant in several common time units. The product k · t½ is always ln(2) ≈ 0.6931.
| t½ (days) | k (1/day) | k (1/hour) | k (1/year) | Class |
|---|---|---|---|---|
| 0.5 | 1.386 | 0.0578 | 506 | Very fast |
| 1 | 0.693 | 0.0289 | 253 | Very fast |
| 5 | 0.139 | 0.00577 | 50.6 | Readily biodegradable |
| 10 | 0.0693 | 0.00289 | 25.3 | Readily biodegradable |
| 30 | 0.0231 | 0.000963 | 8.43 | Inherently biodegradable |
| 60 | 0.0116 | 0.000481 | 4.22 | Persistent |
| 180 | 0.00385 | 0.000160 | 1.41 | Very persistent |
| 365 | 0.00190 | 0.0000791 | 0.693 | Very persistent |
Persistence Classification Bands
Half-life bands commonly used in environmental fate assessments. Exact thresholds vary by regulation; these follow the widely-cited OECD / REACH style guidance.
| Half-life (t½) | Class | Interpretation |
|---|---|---|
| < 2 days | Very fast | Rapid mineralisation; unlikely to persist or bioaccumulate |
| 2 – 10 days | Readily biodegradable | Meets the OECD 301 “readily biodegradable” 10-day window |
| 10 – 60 days | Inherently biodegradable | Degrades but not at the rate required for “readily” classification |
| 60 – 180 days | Persistent | Meets the P criterion in most regulatory frameworks |
| > 180 days | Very persistent | Meets the vP criterion; long-range transport concern |
Time to Reach Target vs Half-Life Multiples
The time required to reach any remaining fraction is a fixed multiple of the half-life. These are the values behind the DT50, DT90 and DT99 endpoints.
| Remaining % | Fraction f | Multiplier of t½ | Endpoint name |
|---|---|---|---|
| 50% | 0.50 | 1.000 × t½ | DT50 (= t½) |
| 25% | 0.25 | 2.000 × t½ | — |
| 10% | 0.10 | 3.322 × t½ | DT90 |
| 5% | 0.05 | 4.322 × t½ | — |
| 1% | 0.01 | 6.644 × t½ | DT99 |
| 0.1% | 0.001 | 9.966 × t½ | — |
What Does t½ = ln(2) / k Measure?
First-order decay describes any process where the rate of loss is proportional to the amount remaining. Radioactive decay, many chemical hydrolyses, and the biodegradation of dilute organic contaminants all follow this pattern. The half-life t½ is the time required for the concentration to fall to exactly half its starting value, and it is set entirely by the rate constant k.
The First-Order Decay Formula
The defining relationship is:
The same relation rearranged to solve for the other two variables:
where f is the target remaining fraction (0 < f ≤ 1). Because the underlying concentration follows C(t) = C₀·e^(−kt), the half-life is independent of the starting concentration — the same t½ applies whether you begin at 1,000 mg/L or 1 mg/L.
Decay Calculation Variables
| Symbol | Meaning | Typical values |
|---|---|---|
| k | First-order decay rate constant | 0.001 – 1.5 per day |
| t½ | Half-life | 0.5 – 365+ days |
| t | Time to reach a target remaining fraction | 0 – 2,000+ days |
| f | Target remaining fraction (C/C₀) | 0.001 – 0.999 |
Why the Half-Life Is Independent of Concentration
Because the rate of loss is proportional to the amount present, the fraction lost per unit time is constant. That is what makes the half-life such a useful number: 50% of whatever is there disappears in t½, then 50% of the remainder disappears in the next t½, and so on. The concentration never reaches exactly zero, which is why regulatory endpoints are defined at 90% or 99% removal rather than at “complete” disappearance.
How the Rate Constant Affects Half-Life
The relationship is exactly reciprocal. A substance with k = 0.0693 per day has t½ = 10 days; double k to 0.1386 per day and the half-life drops to 5 days. This reciprocity makes the product k · t½ a universal constant, ln(2) ≈ 0.6931, no matter what is decaying — which is precisely why the equation is so widely used across radiochemistry, pharmacology and environmental science.
Decay Calculation Examples
t½ = ln(2) / 0.0693 = 0.6931 / 0.0693
t½ = 10.0 days
k = ln(2) / 30 = 0.6931 / 30
k = 0.0231 /day
t = ln(1/0.10) / k = ln(10) / 0.0693 = 2.3026 / 0.0693
t = 33.2 days (DT90)
Frequently Asked Questions
It is the time required for half of the initial amount of a substance to disappear, under the assumption that the loss rate is proportional to the amount present. The number ln(2) ≈ 0.6931 appears because solving C₀/2 = C₀·e^(−k·t½) for t½ gives t½ = ln(2)/k.
Yes — DT50 (“disappearance time to 50%”) is the environmental-fate term for the half-life. DT90 and DT99 are the times to 90% and 99% disappearance, and they are simply fixed multiples of DT50: 3.322 × DT50 and 6.644 × DT50.
At low substrate concentrations, microbial enzymes are not saturated, so the degradation rate is roughly proportional to the contaminant concentration — exactly the condition for first-order kinetics. At high concentrations the enzymes saturate and the reaction becomes zero-order (constant rate), which the simple model does not capture.
Under true first-order kinetics, no — the half-life is a constant for a given substance and environment. In practice, however, higher concentrations can inhibit or saturate the microbial community, which slows the apparent decay and makes the measured half-life concentration-dependent. Always check that your data actually follow a straight line on a log-concentration plot before trusting a single t½.
Most regulatory frameworks (REACH, Stockholm Convention, UNEP) treat a half-life above 60 days in soil or water as “persistent” (P) and above 180 days as “very persistent” (vP). Below 10 days is generally considered readily biodegradable. The exact thresholds vary by compartment (soil, water, sediment) and by regulation.
Mathematically, no — an exponential decay curve approaches zero asymptotically. In practice, once the remaining concentration falls below the analytical detection limit or below the threshold at which microbes can scavenge it, the substance is treated as effectively gone. That is why regulatory endpoints use 90% or 99% removal rather than 100%.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| t1/2 | Biodegradation half-life | days |
| k | First-order biodegradation rate constant | 1/day |
What it means
The half‑life is a measure of the persistence of a substance in the environment. For biodegradation, it is the time needed for the concentration of a chemical to decrease by half through microbial action. It is calculated as the natural logarithm of 2 divided by the first‑order degradation rate constant k. A short half‑life indicates rapid degradation, while a long half‑life suggests persistence, which is important for assessing the risk of pollutants. This concept is used in modelling contaminant transport, designing bioremediation systems, and setting regulatory limits. Example: If a pollutant has a first‑order degradation rate of 0.01 day⁻¹, its half‑life is ln(2)/0.01 ≈ 69.3 days. This helps in estimating how long it will take for the pollutant to decrease to safe levels in the environment.
Worked example
Biodegradation Half‑Life – Two Examples
Real‑World| Parameter | Value |
|---|---|
| k | 0.05 day⁻¹ |
| Parameter | Value |
|---|---|
| k | 0.02 day⁻¹ |
Common mistakes
- Rate constant k: The first‑order degradation rate constant – in units of 1/time (e.g., day⁻¹, year⁻¹).
- ln(2): Natural logarithm of 2 (≈0.693).
- Half‑life: The time required for the concentration of a substance to decrease by half – in the same time units as k.
- Assumptions: First‑order kinetics (exponential decay) – applicable to many biodegradation and photolysis processes.
- Environmental conditions: k depends on temperature, pH, moisture, and microbial activity – use site‑specific values.
Applications
Biodegradation half‑life (t₁/₂) is the time required for the concentration of a compound to be reduced by half through biological degradation, calculated as t₁/₂ = ln(2) / k, where k is the first‑order degradation rate constant. This parameter is essential for assessing the persistence of organic pollutants in the environment, for designing bioremediation strategies, and for regulatory risk assessment. Environmental chemists and engineers use half‑life to predict the fate of pesticides, petroleum hydrocarbons, and other contaminants in soil, water, and sediments. Shorter half‑lives indicate faster natural attenuation, which influences remediation feasibility and cleanup timeframes. By understanding half‑life, professionals can evaluate the effectiveness of bioremediation and set realistic cleanup goals.
- Risk assessment for organic contaminants in soil and water
- Design of bioremediation systems (e.g., bioventing, bioaugmentation)
- Evaluation of natural attenuation for site closure
- Predictive modelling of pollutant persistence in the environment
- Regulatory decision‑making on chemical safety and persistence
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