Exponential Decay Calculator
Calculate decay amounts, half-life, and decay constants for radioactive decay, drug concentration, and other exponential decay processes
N(t) = N₀ × e-λt
Exponential decay formula where λ is the decay constant and t is time
Starting quantity (must be positive)
Rate of decay (per unit time)
Time elapsed (same unit as λ)
Time for quantity to halve
Quick Examples
Invalid Input
Remaining Amount After time units
remaining ( decayed)
Half-Life for λ =
time units for quantity to halve
Decay Constant for t½ =
per time unit
Decay Visualization
Initial Amount
Final Amount
Half-Life
# of Half-Lives
Step-by-Step Solution
Identify values
N₀ = , λ = , t =
Apply the formula
N(t) = N₀ × e-λt = × e-( × )
Calculate exponent
-λt = -( × ) =
Final result
N() =
Common Half-Lives Reference
| Substance | Half-Life | Application |
|---|---|---|
| Carbon-14 | 5,730 years | Archaeological dating |
| Uranium-238 | 4.5 billion years | Geological dating |
| Iodine-131 | 8.02 days | Medical imaging |
| Caffeine | 5 hours | Pharmacology |
| Polonium-210 | 138 days | Nuclear physics |
| Radon-222 | 3.82 days | Environmental monitoring |
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About Exponential Decay Calculator
What is Exponential Decay?
Exponential decay describes a process where a quantity decreases over time at a rate proportional to its current value. This mathematical model is fundamental in physics, chemistry, biology, and many other fields.
Mathematical Formula
The exponential decay formula is:
N(t) = N₀ × e^(-λt)
Where:
- N(t) = Quantity remaining after time t
- N₀ = Initial quantity (at time t = 0)
- e = Euler's number (approximately 2.71828)
- λ (lambda) = Decay constant (rate of decay)
- t = Time elapsed
Half-Life Relationship
The half-life (t½) is the time required for a quantity to reduce to half of its initial value:
t½ = ln(2) / λ ≈ 0.693 / λ
Converting between half-life and decay constant:
- λ = ln(2) / t½
- t½ = ln(2) / λ
Alternative Forms
Using Half-Life
N(t) = N₀ × (1/2)^(t/t½)
Using Decay Rate (r)
N(t) = N₀ × (1 - r)^t
Where r is the decay rate per time period.
Applications
1. Radioactive Decay
Radioactive isotopes decay at a constant exponential rate. Examples:
- Carbon-14 (t½ = 5,730 years) - Used in carbon dating
- Uranium-238 (t½ = 4.5 billion years) - Geological dating
- Iodine-131 (t½ = 8 days) - Medical treatments
2. Drug Concentration
The elimination of drugs from the bloodstream follows exponential decay, characterized by the drug's biological half-life.
3. Newton's Law of Cooling
The rate of cooling of an object is proportional to the temperature difference with its surroundings: T(t) = T_ambient + (T₀ - T_ambient) × e^(-kt)
4. Capacitor Discharge
In electrical circuits, capacitors discharge exponentially: V(t) = V₀ × e^(-t/RC)
5. Population Decay
Populations with declining birth rates or resource limitations may follow exponential decay patterns.
Common Half-Lives
| Substance | Half-Life | Application |
|---|---|---|
| Carbon-14 | 5,730 years | Archaeological dating |
| Uranium-238 | 4.5 billion years | Geological dating |
| Iodine-131 | 8.02 days | Medical imaging |
| Caffeine | 5 hours | Pharmacology |
| Polonium-210 | 138 days | Nuclear physics |
Key Properties
- Continuous Process: Decay happens continuously, not in discrete steps
- Never Reaches Zero: Mathematically, the quantity never fully disappears
- Constant Ratio: Same fraction decays in each equal time period
- Independent of Amount: Half-life is independent of the initial quantity
Important Notes
- The decay constant λ must be positive
- Time and half-life must use the same units
- For very small decay constants, the process is slow; for large constants, it's fast
- The formula assumes no external additions to the quantity