Mathematics · Applied Mathematics

Exponential Decay

Reference entry · last updated August 24, 2026

Exponential decay is a pattern of change in which a quantity's rate of decrease is proportional to its current size, so the quantity loses a constant percentage of itself each period rather than a constant amount. A quantity N decaying exponentially follows \( N(t) = N_0 e^{-\lambda t} \), where \( \lambda \) is the decay constant. It is the mirror image of exponential growth: the same functional form, with a negative rate, so the quantity shrinks toward zero instead of compounding upward. Like growth, it is characterized by a fixed multiplicative rate rather than a fixed absolute change.

Half-life

Exponential decay is usually characterized by its half-life: the time it takes for the quantity to fall to half its current value, which is the same regardless of what that current value happens to be. Solving \( N_0 e^{-\lambda t} = \tfrac{1}{2}N_0 \) for t gives \( t_{1/2} = \ln(2)/\lambda \approx 0.693/\lambda \).[1] This is the exact mirror of the doubling-time formula for exponential growth: same constant, opposite sign on the exponent.

\( t_{1/2} = \dfrac{\ln(2)}{\lambda} \approx \dfrac{0.693}{\lambda} \)

Half-lives elapsed	Fraction remaining
1	50%
2	25%
3	12.5%
4–5	≈3–6% (commonly treated as "negligible")
10	≈0.1%

Because half-life is a fixed property of the process, not of the current quantity, a sample never technically reaches zero under this model. It only ever gets closer, by half each period. Cobalt-60, with a half-life of about 5.27 years, leaves half its original radioactivity after 5.27 years regardless of whether the starting sample was one gram or ten.[1]

Carbon-14 dating and its limit

Radiocarbon dating is the standard applied example of half-life. Living organisms continuously exchange carbon with the atmosphere (respiration in animals, photosynthesis in plants), which keeps the ratio of radioactive carbon-14 to stable carbon-12 in their tissue matched to the atmospheric ratio. At death, that exchange stops, and the carbon-14 already present decays on its own schedule, with a half-life of 5,730 years.[2] Because the technique depends on measuring how much carbon-14 activity remains, it has a hard practical ceiling: after roughly ten half-lives (about 58,000 to 62,000 years), so little carbon-14 is left that it can no longer be distinguished from instrument noise. Older material has to be dated with an isotope of longer half-life, matched to the timescale being measured; carbon-14's short half-life is what makes it useful for archaeological and recent-geological dating specifically, not a general-purpose clock.[2]

Drug elimination

Pharmacokinetics applies the same model to how the body clears a medication. A drug cleared by first-order kinetics has a half-life (the time for its plasma concentration to fall by half) that stays constant regardless of the current dose, because the elimination rate is proportional to the concentration present.[3] Clinicians use this to justify a standard rule of thumb: a drug is considered substantially eliminated after four to five half-lives, by which point roughly 94–97% of it is gone. Morphine, with a half-life of about 120 minutes, is treated as negligible in the body after roughly eight to ten hours: four to five halvings of its two-hour clock.[3] Not every drug follows first-order kinetics: some are cleared at a fixed rate regardless of concentration (zero-order kinetics), in which case half-life is not constant and this shortcut does not apply.

Newton's Law of Cooling

Newton's Law of Cooling models how a hot object's temperature falls toward the temperature of its surroundings, and the detail easiest to miss is what is decaying: not the object's temperature itself, but the gap between the object's temperature and the ambient temperature. The solution is \( T(t) = T_a + (T_0 - T_a)e^{-kt} \), where \( T_a \) is the constant ambient temperature, \( T_0 \) is the starting temperature, and k is a positive cooling-rate constant.[4] Because the exponential term is attached to the difference \( (T_0 - T_a) \) rather than to \( T_a \) itself, it is that difference which shrinks toward zero; the object's temperature therefore approaches \( T_a \) asymptotically rather than continuing past it, matching what is observed. A cup of coffee 10° above room temperature and one 50° above room temperature both lose the same percentage of their respective gaps in the same time, even though the hotter cup loses far more heat in absolute terms early on.

Decay vs. growth

Exponential decay and exponential growth are the same equation with the sign of the rate flipped: \( N(t) = N_0 e^{kt} \) grows when k is positive and decays when k is negative. Doubling time and half-life are computed with the identical formula, \( \ln(2)/|k| \), just answering opposite questions: how long to double versus how long to halve. Both share the property that the rate applies to the current quantity rather than the starting quantity, which is why both a growing population and a decaying radioactive sample can be fully described by a single constant rate that never has to change as the quantity itself changes.

References

^ "21.3 Radioactive Decay," Chemistry 2e, OpenStax — https://openstax.org/books/chemistry-2e/pages/21-3-radioactive-decay
^ "Radiocarbon Dating," Chemistry LibreTexts — https://chem.libretexts.org/.../Radiocarbon_Dating
^ "Pharmacokinetics," StatPearls, NIH National Library of Medicine — https://www.ncbi.nlm.nih.gov/books/NBK557744/
^ "4.7: Exponential and Logarithmic Models," Precalculus, Mathematics LibreTexts — https://math.libretexts.org/.../4.07:_Exponential_and_Logarithmic_Models