First-order elimination follows the exponential decay equation A(t) = A0 x e^(-kt), where A0 is the starting amount, t is elapsed time, k is the elimination rate constant, and k relates to half-life by k = ln(2) / half-life. This post is a mathematics explainer, not medical or dosing guidance: the equations below describe a one-compartment, standard textbook mathematical idealization, not a claim about how any real substance behaves in a real body, and nothing here is a recommended dose or schedule. Every calculator this post links to, including the half-life calculator that runs this model end to end, states the same thing on its own page.
What is the first-order elimination half-life formula?
First-order elimination assumes that the amount remaining decreases at a rate proportional to how much is currently present. That assumption, applied to a single well-mixed compartment, produces exponential decay:
A(t) = A0 x e^(-k x t)A0 is the starting amount, t is elapsed time, k is the elimination rate constant, and A(t) is the amount remaining at time t. This is the same functional form used for radioactive decay, capacitor discharge, and any number of other first-order processes; it is a general mathematical pattern, not something specific to pharmacology.
Where the rate constant comes from
The rate constant k and the half-life describe the same decay from two angles, and the relationship between them is fixed:
k = ln(2) / half-life
half-life = ln(2) / kGiven either one, the other follows directly. The elimination rate constant calculator computes k from a half-life, and the half-life from rate constant calculator runs the same relationship in reverse. Both are pure algebra on the same constant, ln(2), which is the mathematical fact that a quantity decaying at a constant proportional rate takes a fixed amount of time to halve, regardless of the starting amount.
Plugging both pieces together
The first-order elimination calculator evaluates A(t) = A0 x e^(-kt) directly: given a starting amount, a rate constant (or a half-life, converted to k first), and an elapsed time, it returns the modeled amount remaining at that time. It is a direct evaluation of the formula above, nothing added.
Working in the other direction
Two related questions invert this equation to solve for time instead of amount.
How long until a specific fraction remains?
time = half-life x log2(1 / fraction)The time to percent remaining calculator solves for t given a target fraction, using log base 2 because each full half-life is exactly one doubling of that log.
What fraction remains after a specific time?
The percent remaining calculator answers the reverse question, wrapping the same decay fraction function used throughout this model family: given an elapsed time and a half-life, it returns what fraction of the starting amount the model predicts remains.
Why the framing matters
A one-compartment, first-order model is a simplification even within pharmacokinetics, where real elimination can involve multiple compartments, saturable processes, or rates that are not actually constant over the full range being modeled. Whether this simple model is even a reasonable approximation for any specific real substance is a question this site does not answer, and it never assigns a half-life value to any substance; every calculator here takes a half-life as an input you supply, not something it looks up or asserts. These are equations, evaluated exactly, applied to whatever inputs are typed in. Nothing about entering a number here is guidance on what that number should be.
The same model, with repeated doses
Everything above describes a single starting amount decaying once. When doses repeat at a fixed interval, the same exponential model extends into questions about accumulation and steady state, covered separately in steady-state accumulation, the math, which builds directly on the equations in this post.
In short: first-order elimination is A(t) = A0 x e^(-kt), and the rate constant k relates to half-life by k = ln(2) / half-life.
Primary scientific literature and citations
- Rowland, M., & Tozer, T. N. (2019). Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications (4th ed.). Wolters Kluwer / Lippincott Williams & Wilkins. Chapter 3: Elimination Kinetics, Rate Constants, and Clearance; Chapter 5: One-Compartment Intravenous and Extravascular Disposition.
- Gibaldi, M., & Perrier, D. (1982). Pharmacokinetics (2nd ed.). Marcel Dekker, New York. Definitive classical text detailing one-compartment exponential disposition, the relationship k = ln(2)/t(1/2), and linear kinetic foundations.
- Grogan, S., & Preuss, C. V. (2023). Pharmacokinetics. In StatPearls. StatPearls Publishing. PubMed PMID: 32491677. NCBI Bookshelf ID: NBK557744. Peer-reviewed clinical overview of first-order clearance, volume of distribution, and elimination rate constants.