Half-Life Calculator
Solve any unknown variable in the exponential decay equation. Model radioactive isotopes, drug elimination curves, and organic radiocarbon dating structures.
Known Variables
Solved decay value
25units
Step-by-Step Exponential Decay Calculations
Remaining Amount Formula:
N(t) = N₀ × (0.5)^(t / t_½)
100 × (0.5)^(16 days / 8 days) = 25.0000
Five-Cycle Decay Schedule Ledger
| Cycles | Time Elapsed | Remaining | % Left | Decayed |
|---|---|---|---|---|
| 0 t½ | 0.00 days | 100 | 100% | 0 |
| 1 t½ | 8.00 days | 50 | 50% | 50 |
| 2 t½ | 16.00 days | 25 | 25% | 75 |
| 3 t½ | 24.00 days | 12.5 | 12.5% | 87.5 |
| 4 t½ | 32.00 days | 6.25 | 6.25% | 93.75 |
| 5 t½ | 40.00 days | 3.125 | 3.13% | 96.875 |
Understanding Half-Life and Exponential Decay
The concept of **half-life** ($t_0.5$) represents the time required for a physical quantity to decay or decrease to exactly half of its initial value. This is the hallmark of first-order exponential decay, where the rate of breakdown is proportional to the current amount remaining.
The Algebraic Exponential Decay Formula
The core algebraic formula relating initial quantity ($N_0$) to remaining quantity ($N(t)$) over elapsed time ($t$) is:
N(t) = N₀ × (0.5)^(t / t_half)
Alternatively, this can be written using Euler's number ($e$) and the **decay constant** ($\lambda$), where $\lambda = \ln(2) / t_0.5$:
N(t) = N₀ × e^(-λt)
Real-Life Presets & Isotopes
Exponential decay plays a pivotal role across scientific disciplines:
- Radioactive decay: Heavy isotopes like Carbon-14, Cesium-137, or Uranium-235 decay at constant rates over days, years, or millennia. Carbon-14 is widely used in archeology to date organic remains.
- Pharmacology elimination: Medications (like Caffeine, Ibuprofen, or Acetaminophen) are metabolised by the kidneys and liver. The half-life describes how quickly the drug's concentration is halved in blood plasma.
Frequently Asked Questions
What is half-life?
Half-life is the duration required for a quantity to decrease to half of its initial value. This is used to describe radioactive decay or pharmacology clearance rates.
How is the decay constant calculated from half-life?
The decay constant (λ) is calculated by dividing the natural logarithm of 2 (approximately 0.693147) by the half-life: λ = ln(2) / t1/2.
How is radiocarbon dating used to determine fossil ages?
Radiocarbon dating measures the remaining ratio of unstable Carbon-14 inside organic remains. Using the exponential decay formula, we solve for the elapsed time to find the age.
Can this solve exponential decay problems?
Yes. Calculate remaining amount, initial quantity, half-life, or elapsed time.