Exponential Decay Calculator

P(t) = P₀ × e^−rt — Solve for any variable • Half-life • Decay curve • Decay table

Decay Model

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Quick Examples

What Is Exponential Decay?

Exponential decay describes any process where a quantity decreases at a rate proportional to its current value. The larger the quantity, the faster it shrinks — yet the fractional loss per unit time remains constant. This produces the characteristic smooth, ever-flattening curve that never quite reaches zero.

Mathematically, the continuous exponential decay formula is:

P(t) = P₀ × e^−rt
P₀ = initial amount • r = decay rate • t = time • e = Euler's number ≈ 2.71828

Understanding the Formula Components

P₀ (Initial Amount) — The quantity at time t = 0. Could be grams of a radioactive isotope, milligrams of a drug, number of organisms, or any measurable quantity.
r (Decay Rate / Decay Constant λ) — A positive constant expressing how fast the quantity shrinks per unit of time. Often called λ (lambda) in physics. Higher r means faster decay.
t (Time) — Elapsed time measured in consistent units matching the decay rate.
e^−rt — The exponential factor; always between 0 and 1, it scales P₀ downward.

Continuous vs. Discrete Exponential Decay

Two models describe exponential decay:

Model	Formula	When Used	Example (r=0.1, t=5)
Continuous	P₀ × e^−rt	Physics, pharmacology, cooling	P(5) = P₀ × e^−0.5 ≈ 0.6065 P₀
Discrete	P₀ × (1−r)^t	Annual statistics, period-by-period	P(5) = P₀ × 0.9⁵ ≈ 0.5905 P₀

Continuous decay is slightly slower than discrete for the same rate r, because the continuous model effectively re-compounds the reduction at infinitely small intervals. The difference is analogous to continuous vs. annual compounding in finance.

Half-Life: The Key Concept

The half-life (t½) is the time required for the quantity to reach half its current value. Setting P(t) = P₀/2 in the formula and solving:

t½ = ln(2) / r ≈ 0.6931 / r

The most important property of half-life: it is constant, regardless of the starting amount P₀. Whether you start with 1 kg or 1 tonne of a substance, it always takes exactly t½ to reduce to half. Each successive half-life halves whatever remains:

Half-lives elapsed	Fraction remaining	% remaining
0	1	100%
1	1/2	50%
2	1/4	25%
3	1/8	12.5%
5	1/32	3.125%
10	1/1024	~0.098%

Real-World Applications

1. Radioactive Decay & Carbon Dating

Every radioactive isotope decays with a characteristic half-life. Carbon-14 has a half-life of ~5,730 years (r ≈ 0.0001209 yr⁻¹), which archaeologists use to date organic remains up to ~50,000 years old. Below is a reference table for common isotopes:

Isotope	Half-life	Decay rate r (per year)	Application
Carbon-14 (¹⁴C)	5,730 years	0.0001209	Radiocarbon dating
Uranium-238 (²³⁸U)	4.47 billion years	1.55 × 10⁻¹⁰	Geological dating
Iodine-131 (¹³¹I)	8.02 days	0.0864	Medical therapy
Technetium-99m	6.01 hours	2.77	Medical imaging
Cesium-137 (¹³⁷Cs)	30.17 years	0.02297	Nuclear fallout
Radon-222 (²²²Rn)	3.82 days	0.1814	Indoor air quality

2. Drug Pharmacokinetics

The concentration of a drug in the bloodstream typically follows first-order exponential decay after the absorption phase. A drug with elimination half-life of 4 hours (r = ln(2)/4 ≈ 0.1733 hr⁻¹) will fall to ~50% after 4 h, ~25% after 8 h, and <3% after 20 h. This informs dosing intervals.

3. Newton's Law of Cooling

The temperature difference between an object and its surroundings decays exponentially: T(t) − T_env = (T₀ − T_env) × e^−kt, where k depends on surface area, thermal conductivity, and the medium. Forensic scientists use this to estimate time of death.

4. Population Decline

Species loss, rural population emigration, or business customer churn can all follow exponential decay. If a fish population falls 3% per year continuously, after 20 years only e^{−0.03 × 20} ≈ 54.9% of the original population remains.

5. Capacitor Discharge

When a capacitor discharges through a resistor, the voltage follows V(t) = V₀ e^−t/RC where RC is the time constant (τ). The voltage reaches ≈ 36.8% (1/e) of its initial value after one time constant, and <1% after 5 time constants.

Solving for Each Variable

Solve for	Formula	Notes
P(t)	P₀ × e^−rt	Direct calculation
P₀	P(t) × e^rt	Reverse: given final amount, find initial
r	−ln(P(t)/P₀) / t	Requires two measurements and elapsed time
t	−ln(P(t)/P₀) / r	Find when a specific level is reached

Frequently Asked Questions

What is the exponential decay formula?

The standard formula is P(t) = P₀ × e^−rt, where P₀ is the initial amount, r is the positive decay rate constant, t is time, and e is Euler's number (≈2.71828). The discrete version is P(t) = P₀ × (1−r)^t, which applies the percentage loss once per period.

What is half-life and how is it calculated?

Half-life (t½) is the time for a quantity to fall to 50% of its current value. Setting P(t) = P₀/2 in the formula: t½ = ln(2)/r ≈ 0.6931/r. The half-life is constant — after each additional t½, exactly half of whatever remains decays. After 2 half-lives: 25%; after 3: 12.5%; after 10: <0.1%.

What is the difference between continuous and discrete exponential decay?

Continuous: P(t) = P₀e^−rt — decay happens at every instant, producing a perfectly smooth curve. Used in physics, pharmacology, and thermodynamics.

Discrete: P(t) = P₀(1−r)^t — decay is applied once per period (annually, daily, etc.). For the same r, discrete decay is slightly more aggressive than continuous because the full rate is applied at each step rather than being split across infinitely many sub-intervals.

How do I find the decay rate r from two measurements?

Rearrange the formula to isolate r: r = −ln(P(t)/P₀) / t. You need the initial amount P₀, the final amount P(t), and the elapsed time t. Example: sample decays from 100 to 50 g in 5.73 years → r = −ln(0.5)/5.73 ≈ 0.121/yr. Select the Find r mode in the calculator above to automate this.

How is exponential decay used in radiocarbon dating?

Carbon-14 (¹⁴C) in a living organism is in equilibrium with the atmosphere. After death, it decays with half-life ≈ 5,730 years (r ≈ 0.0001209 yr⁻¹). By measuring the ratio of ¹⁴C remaining vs. the expected initial ratio and applying t = −ln(P(t)/P₀)/r, scientists can date samples up to ∼50,000 years old. Uncertainties in the initial ¹⁴C/¹²C ratio are calibrated using dendrochronology (tree rings).

What happens to exponential decay after many half-lives?

The quantity is halved with each half-life: 50% → 25% → 12.5% → ... After 10 half-lives only (1/2)¹⁰ ≈ 0.098% remains. Mathematically it approaches zero asymptotically but never reaches it. In nuclear safety, materials are generally considered safe after 10 half-lives. In pharmacology, a drug is considered fully eliminated after 5 half-lives (<3.1% remaining).

How do I find the time for a quantity to reach a specific value?

Use the rearranged formula: t = −ln(P(t)/P₀) / r. For example, if P₀ = 100, r = 0.05 per year, and you want P(t) = 10: t = −ln(10/100)/0.05 = −ln(0.1)/0.05 = 2.3026/0.05 ≈ 46.05 years. Select the Find t mode above to compute this instantly.