Exponential Decay Calculator

Online calculator and formulas to calculate exponential decay for a given decay rate per period

Exponential Decay Calculator

Exponential Decay

This function calculates either the time periods needed to reach a specific reduction factor at a given decay rate, or the decay rate required to reach that factor in a specified time.

Select Calculation Mode
Input Values
0.5 = half, 0.25 = quarter, 0.1 = 10% of original
Result precision
Example: 5 for 5% decay per period
Decay Calculation Result
Periods:
About Exponential Decay

Exponential decay is used for radioactive materials (half-life), disease reduction, pollution degradation, and many natural processes. The decay factor represents the fraction remaining (0.5 = half remaining = 50% decay rate).

Decay Visualization

Exponential decay shows rapid initial decline that gradually slows.
Higher decay rates result in faster reduction to the target factor.

Exponential Decay Chart 0 0.5 1.0 Target Factor 100% 50% Decay Curve Initial (100%) Target Factor


What is Exponential Decay?

Exponential decay describes how quantities decrease at a rate proportional to their current value:

  • Definition: The value decreases by a constant percentage each period
  • Proportional Decay: The amount lost is proportional to what remains
  • Factor-Based: Reduction factor specifies the remaining fraction (0.5 = half remaining)
  • Never Zero: Theoretically never reaches zero, only approaches it
  • Real-World: Radioactivity, drug metabolism, pollution degradation
  • Half-Life: Time for decay factor of 0.5 is called the half-life

Mathematical Foundations

The exponential decay calculation is based on decay mathematics:

Calculate Decay Periods
t = log(f) / log(1 - d/100)

Calculates periods needed to reach factor f at decay rate d%. Decay rate is given as positive percentage.

Calculate Decay Rate
d = -(e^(ln(f)/t) - 1) × 100

Calculates the required decay rate % to reach factor f in t periods.

Common Decay Factors

The decay factor determines what fraction remains after one period:

Factor Examples
  • 0.5 = Half remains (50% decay rate)
  • 0.25 = Quarter remains (75% decay rate)
  • 0.1 = 10% remains (90% decay rate)
  • 0.9 = 90% remains (10% decay rate)
Half-Life (Factor = 0.5)
  • Carbon-14: ~5,730 years
  • Uranium-238: ~4.5 billion years
  • Iodine-131: ~8 days
  • Aspirin (body): ~20 minutes

Applications of Exponential Decay

Exponential decay applies to many natural and practical processes:

Nuclear Physics & Medicine
  • Radioactive isotope decay
  • Half-life calculations
  • Medical imaging (PET scans)
  • Radiocarbon dating
Chemistry & Environmental
  • Chemical reaction rates
  • Pollutant degradation
  • Drug metabolism
  • Atmospheric cooling
Epidemiology & Biology
  • Disease prevalence decline
  • Bacterial population reduction
  • Virus inactivation
  • Immune response fading
Engineering & Physics
  • Capacitor discharge
  • Heat loss over time
  • Signal attenuation
  • Material degradation

Calculation Examples

Example 1: Carbon-14 Dating
Given

Carbon-14 decays with a half-life of 5,730 years. What is the annual decay rate?

Question: What is the decay rate per year?

Solution
\[d = -(e^{\ln(0.5)/5730} - 1) \times 100\] \[d = -(e^{-0.000121} - 1) \times 100 = 0.0121\%\]
Result

Carbon-14 decays at approximately 0.0121% per year. After 5,730 years, half remains.

Example 2: Drug Metabolism
Given

A drug is eliminated from the body at 20% per hour. How long until only 25% remains?

Question: How many hours to reach 25% of initial dose?

Solution
\[t = \frac{\log(0.25)}{\log(1 - 0.20)}\] \[t = \frac{\log(0.25)}{\log(0.8)} = \frac{-0.602}{-0.097} = 6.21 \text{ hours}\]
Result

At 20% hourly elimination, it takes approximately 6.21 hours for the drug concentration to drop to 25% of the original level.

Example 3: Pollution Degradation
Given

A pollutant must be reduced to 10% of current levels in 5 years through natural degradation.

Question: What annual degradation rate is needed?

Solution
\[d = -(e^{\ln(0.1)/5} - 1) \times 100\] \[d = -(e^{-0.461} - 1) \times 100 = 36.84\%\]
Result

A degradation rate of 36.84% per year is needed to reduce pollution to 10% of current levels within 5 years.

Key Points About Exponential Decay

Core Concepts
  • Proportional: Decay rate is constant percentage per period
  • Factor vs. Rate: Factor 0.5 = 50% decay rate
  • Half-Life: Time to reach factor of 0.5
  • Never Zero: Theoretically approaches zero asymptotally
Practical Applications
  • Radioactive material dating and safety
  • Drug dosing and pharmacokinetics
  • Environmental remediation timelines
  • Disease and infection control
Summary

Exponential decay is a fundamental phenomenon in nature, describing how quantities decrease at rates proportional to their current values. From radioactive materials with million-year half-lives to drugs eliminated from the body in hours, exponential decay provides the mathematical framework for understanding these processes. The ability to calculate decay periods or rates is essential for fields ranging from nuclear physics and archaeology to medicine and environmental science.