Exponential Decay Calculator
Input Parameters
Using Decay Constant (λ)
About This Tool
Understanding Exponential Decay
Exponential decay is a mathematical model describing how quantities decrease over time at a rate proportional to their current value. This universal pattern appears throughout nature—from radioactive decay in physics to drug metabolism in medicine, population decline in biology, and even depreciation in finance. Our exponential decay calculator helps you compute key parameters including remaining quantity, decay time, decay constants, and half-lives with precision and clarity. Whether you're a scientist analyzing isotope decay, a pharmacist calculating drug clearance, or a student learning about exponential functions, this tool provides instant, accurate results.
The Exponential Decay Formula Explained
The fundamental exponential decay equation is N = N₀ × e^(-λt), where N represents the remaining quantity, N₀ is the initial quantity, λ (lambda) is the decay constant, t is time elapsed, and e is Euler's number (approximately 2.718). This formula reveals that decay follows a continuous, smooth curve rather than linear steps. The decay constant λ determines how quickly the substance decays—higher values mean faster decay.
An alternative formulation uses half-life: N = N₀ × (1/2)^(t/T½), where T½ is the half-life period when exactly 50% remains.
Decay Constant vs. Half-Life: What's the Difference?
The decay constant (λ) and half-life (T½) are two ways to express the same decay rate. The decay constant represents the probability of decay per unit time—a rate that stays constant regardless of how much material remains. Half-life, conversely, is the time required for half of any amount to decay, making it more intuitive for many applications.
These parameters are mathematically related through λ = ln(2) / T½ ≈ 0.693 / T½. If you know one, you can calculate the other. For instance, a substance with a 10-year half-life has a decay constant of approximately 0.0693 per year.
Real-World Applications of Exponential Decay
Radioactive Decay:
Nuclear physicists use exponential decay to predict how radioactive isotopes like Carbon-14 (half-life: 5,730 years) or Uranium-238 (half-life: 4.5 billion years) diminish over time, essential for carbon dating and nuclear waste management.
Pharmacokinetics:
Medical professionals calculate how drugs are eliminated from the bloodstream. Most medications follow first-order kinetics (exponential decay), helping determine proper dosing intervals.
Population Biology:
Ecologists model population decline when death rates exceed birth rates, such as endangered species recovery planning or bacterial die-off after antibiotic treatment.
Atmospheric Science:
Air pressure decreases exponentially with altitude. Temperature cooling also approximates exponential decay in Newton's Law of Cooling.
Finance and Economics:
Asset depreciation often follows exponential decay patterns. The purchasing power of money also decays exponentially during inflation.
Physics Experiments:
Capacitor discharge in RC circuits follows exponential decay, as does the amplitude of damped oscillations in mechanical systems.
Step-by-Step: Calculating Remaining Quantity
To find how much substance remains after time t, use these steps:
1. Identify your initial quantity (N₀) and decay parameter—either decay constant λ or half-life T½
2. If you have half-life, convert it to decay constant using λ = 0.693/T½
3. Multiply λ by elapsed time t to get the exponent
4. Calculate e raised to this negative exponent (most calculators have an e^x button)
5. Finally, multiply your initial quantity by this result
How to Use This Calculator Effectively
Our exponential decay calculator offers three calculation modes to solve different problem types:
Calculate Remaining Quantity (N):
Provide initial quantity, time elapsed, and either decay constant or half-life. The calculator returns how much remains after the specified time period.
Calculate Initial Quantity (N₀):
If you know the current amount and how long it has been decaying, this mode works backward to find the original quantity.
Calculate Time Elapsed (t):
When you know both the starting and ending quantities plus the decay rate, this mode determines how much time has passed.
You can toggle between using decay constant (λ) or half-life (T½) as your input parameter—the calculator automatically converts between them. Select appropriate time units (seconds, minutes, hours, days, or years) to match your specific application. Results include percentage remaining, percentage decayed, step-by-step calculations, and a visual decay curve showing how the quantity changes over time.
Key Properties of Exponential Decay
Exponential decay has several distinctive characteristics:
Memoryless: The future decay depends only on the current amount, not on how long it has been decaying.
Continuous and Smooth: Decay occurs without sudden jumps.
Asymptotic: The quantity never reaches exactly zero in finite time, only approaching it asymptotically.
Proportional Decrease: Equal time intervals result in equal proportional (not absolute) decreases: each half-life period always reduces quantity by 50%, regardless of the starting amount.
Constant Rate: The decay rate itself remains constant even as the absolute amount decreases.
Common Mistakes and How to Avoid Them
• Mixing up decay constant and half-life (remember they're inversely related)
• Using inconsistent time units (always ensure time and rate use matching units)
• Confusing percentage remaining versus percentage decayed (they sum to 100%)
• Attempting to calculate decay with zero or negative decay constants (physically meaningless)
• Expecting linear rather than exponential behavior
• Forgetting that exponential decay assumes a constant decay rate—some real-world processes deviate from this ideal model
Always verify your input values make physical sense before trusting the output.
Advanced Topics: Decay Chains and Effective Half-Life
In complex systems, you may encounter decay chains where one radioactive isotope decays into another radioactive isotope, creating sequential exponential decays. The mathematics becomes more complex, requiring differential equations to solve.
Another advanced concept is effective half-life in pharmacology, which combines biological elimination (drug metabolism/excretion) with physical decay (for radioactive drugs), calculated as 1/T_eff = 1/T_bio + 1/T_phys. These scenarios often require specialized software, but understanding basic exponential decay provides the foundation for grasping these more sophisticated models.
Why Exponential Decay Matters in Science
Exponential decay is one of nature's most fundamental patterns, appearing whenever a rate of change is proportional to the current quantity. This mathematical elegance makes it predictable and powerful for modeling countless phenomena.
From helping archaeologists date ancient artifacts through carbon-14 decay to enabling doctors to prescribe safe medication dosages, exponential decay calculations are essential across scientific disciplines.
Understanding this concept not only aids in specific calculations but also provides insight into the underlying principles governing change over time in natural systems. By mastering exponential decay, you gain a versatile analytical tool applicable to an extraordinary range of real-world problems.
Frequently Asked Questions
Yes, Exponential Decay Calculator is totally free :)
Yes, you can install the webapp as PWA.
Yes, any data related to Exponential Decay Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.
Exponential decay describes how a quantity decreases over time at a rate proportional to its current value. The formula N = N₀ × e^(-λt) shows that the remaining quantity (N) equals the initial quantity (N₀) multiplied by e raised to the power of negative decay constant (λ) times time (t). This pattern appears in radioactive decay, drug metabolism, temperature cooling, and population decline.
The decay constant (λ) and half-life (T½) are inversely related through the formula λ = ln(2) / T½, where ln(2) ≈ 0.693. A larger decay constant means faster decay and shorter half-life. For example, if T½ = 10 years, then λ = 0.693 / 10 = 0.0693 per year. You can use either λ or T½ to describe the same decay process.
To find the elapsed time (t), rearrange the decay formula: t = -ln(N/N₀) / λ, where N is the remaining quantity, N₀ is the initial quantity, and λ is the decay constant. Alternatively, using half-life: t = T½ × log₂(N₀/N). For example, if a substance decays from 100g to 25g with λ = 0.1/hour, then t = -ln(25/100) / 0.1 ≈ 13.86 hours.
Half-life is a specific measure of exponential decay—the time for a quantity to reduce to 50% of its original value. Exponential decay is the broader mathematical model describing continuous proportional decrease over time. Half-life is easier to visualize (after one half-life: 50%, two half-lives: 25%, three: 12.5%), while the decay constant provides the instantaneous rate of change.
Yes! Exponential decay applies to many real-world scenarios beyond radioactivity. In pharmacology, it models drug concentration decreasing in the bloodstream. In biology, it describes population decline when death rate exceeds birth rate. In finance, it can represent depreciation. The same mathematical formula works across all these contexts—just ensure your units are consistent.
When λ = 0, there is no decay—the quantity remains constant over time since e^(0) = 1, making N = N₀. This represents a stable, non-decaying system. Conversely, as λ approaches infinity, decay becomes instantaneous. Negative values of λ would represent exponential growth instead of decay.