Decay Calculator
Solve exponential decay for amount, time, or half-life
How to Use
- Select what you want to calculate (final amount, time, half-life, or decay constant)
- Enter the known values with appropriate units
- Choose between using half-life or decay constant (λ)
- Click calculate to see the detailed results
- Review the step-by-step calculation breakdown
What is Exponential Decay?
Exponential decay is a process where a quantity decreases at a rate proportional to its current value. This pattern appears in many natural phenomena, from radioactive decay to population decline and chemical reactions.
The key characteristic of exponential decay is that the quantity decreases by the same proportion over equal time intervals, creating a characteristic curve that never quite reaches zero but approaches it asymptotically.
Decay Formulas
The fundamental exponential decay equation can be expressed in two equivalent forms:
Using half-life: A = A₀ · (1/2)^(t/T½)
Using decay constant: A = A₀ · e^(−λt)
Where:
- A₀ = initial amount
- A = final amount
- t = time elapsed
- T½ = half-life
- λ = decay constant (λ = ln(2)/T½)
- e = Euler's number (approximately 2.71828)
These formulas can be rearranged to solve for any variable:
- Time: t = T½ · log₂(A₀/A)
- Half-life: T½ = t · log₂(A₀/A)
- Decay constant: λ = ln(A₀/A) / t
Understanding Half-life
Half-life is the time required for a quantity to reduce to exactly half of its initial value. After one half-life, 50% remains. After two half-lives, 25% remains. After three half-lives, 12.5% remains, and so on.
This concept is crucial in radioactive dating, medical applications, and nuclear physics. Different isotopes have vastly different half-lives, from fractions of a second to billions of years.
Real-World Applications
Exponential decay calculations are essential in many fields:
- Nuclear physics: Radioactive decay and dating techniques
- Medicine: Radioisotope therapy and diagnostic imaging
- Chemistry: Reaction kinetics and drug metabolism
- Finance: Depreciation of assets and compound interest
- Biology: Population dynamics and drug elimination
- Environmental science: Pollutant decay and carbon dating
Decay Constant vs. Half-life
Both parameters describe the same decay process but in different ways:
| Parameter | Half-life | Decay Constant |
|---|---|---|
| Definition | Time for 50% decay | Probability of decay per unit time |
| Units | Time units | 1/time units |
| Relationship | T½ = ln(2)/λ | λ = ln(2)/T½ |
| Usage | Intuitive, widely used | Mathematical convenience |
Frequently Asked Questions
- What's the difference between exponential and linear decay?
- Linear decay decreases by a constant amount over time, while exponential decay decreases by a constant percentage. Exponential decay creates a curve that levels off, while linear decay creates a straight line.
- Can decay ever reach zero?
- Mathematically, exponential decay approaches zero asymptotically but never quite reaches it. In practice, we consider the process complete when the remaining amount is negligible for the application.
- Why is natural logarithm (ln) used in decay calculations?
- Natural logarithm appears because exponential decay is based on e^x. The ln function is the inverse of e^x, making it essential for solving time-related decay equations.
- How accurate are these calculations for radioactive dating?
- The mathematical model is highly accurate, but real-world radioactive dating must account for additional factors like contamination, initial conditions, and measurement uncertainties.
- Can this calculator be used for population decline?
- Yes, exponential decay can model population decline under constant conditions. However, real populations often have variable rates due to environmental factors, birth rates, and migration.
- What does a negative decay constant mean?
- A negative decay constant would indicate growth rather than decay. In our calculations, we assume positive decay constants for decay processes.
- How does temperature affect decay rates?
- For radioactive decay, temperature has virtually no effect. For chemical decay processes, higher temperatures generally increase the decay rate (decrease half-life).