Exponential Decay Calculator
Calculate exponential decay and growth with step-by-step solutions
Exponential Decay Calculator
Calculate exponential decay or growth using the formula N(t) = N₀ × e-λt. Provide any three of the four values to compute the missing one. This calculator supports both exponential decay (decreasing quantities) and exponential growth (increasing quantities).
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📊 Exponential Decay Calculator
Please provide any three of the following to calculate the fourth value:
📊 Result
Formula Used:
N(t) = N₀ × e-λt
Where N(t) is the final value, N₀ is the initial value, λ is the decay constant, and t is time elapsed.
📋 Common Exponential Decay Examples
| Phenomenon | Half-Life / Time Constant | Description |
|---|---|---|
| ☢️ Carbon-14 Decay | t₁/₂ = 5,730 years | Used in radiocarbon dating of organic materials |
| 💊 Drug Metabolism (Caffeine) | t₁/₂ ≈ 5 hours | Time for body to eliminate half of caffeine |
| 🌡️ Newton's Cooling | Varies by material | Rate of temperature change of an object |
| 🔋 Capacitor Discharge | τ = R × C | Voltage decay in RC circuits |
| ☢️ Uranium-238 Decay | t₁/₂ = 4.47 billion years | One of the longest known radioactive half-lives |
| 💊 Drug Metabolism (Aspirin) | t₁/₂ ≈ 15–20 minutes | Rapid elimination from bloodstream |
| 📉 Atmospheric Pressure | Scale height ≈ 8.5 km | Pressure decreases exponentially with altitude |
| 💡 Light Absorption (Beer-Lambert) | Varies by medium | Light intensity decreases through absorbing media |
📚 Understanding Exponential Decay
What is Exponential Decay?
Exponential decay describes the process where a quantity decreases at a rate proportional to its current value. This means the larger the quantity, the faster it decreases. The quantity approaches zero asymptotically but never quite reaches it.
Exponential decay is found throughout nature and science — from the decay of radioactive isotopes to the discharge of electrical capacitors, the cooling of hot objects, and the metabolism of drugs in the human body. The mathematical model is remarkably consistent across these diverse phenomena.
The Exponential Decay Formula
The general exponential decay formula can be expressed in several equivalent forms:
N(t) = N₀ × e-λt
N(t) = N₀ × e-t/τ
N(t) = N₀ × (1/2)t/t₁/₂
Where:
- N(t) — the quantity at time t
- N₀ — the initial quantity (at t = 0)
- λ (lambda) — the decay constant (positive value)
- τ (tau) — the mean lifetime, where τ = 1/λ
- t₁/₂ — the half-life, where t₁/₂ = ln(2)/λ ≈ 0.693/λ
- e — Euler's number ≈ 2.71828
Exponential Growth vs. Decay
While exponential decay describes decreasing quantities, exponential growth describes quantities that increase at a rate proportional to their current value. The formula for growth removes the negative sign from the exponent:
Growth: N(t) = N₀ × e+λt
Decay: N(t) = N₀ × e-λt
Examples of exponential growth include population growth (under ideal conditions), compound interest, bacterial reproduction, and the spread of viral infections in their early stages.
Relationship Between Constants
The decay constant, mean lifetime, and half-life are all related:
t₁/₂ = ln(2) / λ ≈ 0.6931 / λ
τ = 1 / λ
t₁/₂ = τ × ln(2) ≈ 0.6931 × τ
Example Calculations
Example 1: Radioactive Decay
A sample contains 500 grams of a radioactive isotope with a decay constant λ = 0.0001 per year. How much remains after 1,000 years?
N(t) = N₀ × e-λt
N(1000) = 500 × e-0.0001 × 1000
N(1000) = 500 × e-0.1
N(1000) = 500 × 0.9048 = 452.42 grams
Example 2: Population Growth
A bacteria colony starts with 100 cells and grows at a rate of 0.3 per hour. How many cells will there be after 5 hours?
N(t) = N₀ × eλt
N(5) = 100 × e0.3 × 5
N(5) = 100 × e1.5
N(5) = 100 × 4.4817 = 448.17 cells