Half-Life Calculator

Half-life (t₁/₂) is the time required for the concentration of a reactant to decrease to half of its initial value. It is one of the most important parameters in chemical kinetics and radioactive decay, providing a direct measure of how fast a reaction or decay process occurs. Unlike the rate constant, which has different units for different reaction orders, half-life is always expressed in units of time.

The half-life depends on the reaction order. For first-order reactions, the half-life is constant and independent of the initial concentration: t₁/₂ = ln(2)/k ≈ 0.693/k. This is the most common case in chemistry and includes radioactive decay, many decomposition reactions, and some enzyme-catalyzed reactions. A first-order reaction has a characteristic half-life regardless of how much material you start with.

For zero-order reactions, the half-life depends on the initial concentration: t₁/₂ = C₀/(2k). Higher initial concentrations take longer to halve because the reaction proceeds at a constant rate regardless of concentration. Surface-catalyzed reactions at saturation are often zero-order.

For second-order reactions, the half-life is inversely proportional to the initial concentration: t₁/₂ = 1/(kC₀). Higher initial concentrations actually result in shorter half-lives because the reaction rate depends on the square of the concentration. Many dimerization and radical recombination reactions follow second-order kinetics.

This calculator handles all three reaction orders and displays the appropriate formula for each case.

Each reaction order has its own half-life equation. The calculator selects the correct formula based on your input.

Follow these steps to calculate the half-life for any reaction order:

1. Enter the Rate Constant (k): Input the rate constant with appropriate units. For zero order: mol/(L·s). For first order: s⁻¹. For second order: L/(mol·s). Ensure the units match the selected reaction order.
2. Select Reaction Order: Choose 0 (zero order), 1 (first order), or 2 (second order) from the dropdown. The input fields adjust accordingly — for first order, only k is needed; for zero and second order, the initial concentration is also required.
3. Enter Initial Concentration (if needed): For zero-order and second-order reactions, enter C₀ in mol/L. This field is hidden for first-order reactions because the half-life does not depend on concentration.
4. View Results: The calculator displays the half-life in seconds with the appropriate formula shown.

The formula display adapts to the selected order, showing the exact equation used for the calculation.

The primary result is the half-life in seconds. The calculator displays the formula used so you can verify the calculation.

First-order reactions have the unique property that the half-life is constant. Whether you start with 1.0 M or 0.001 M, the time to halve is the same. This exponential decay behavior is characteristic of radioactive decay and many natural processes. After n half-lives, the remaining fraction is (1/2)^n.

Zero-order reactions show a linear decrease in concentration over time. The half-life increases as the reaction proceeds because each subsequent half-life starts from a lower concentration. This is common in surface-catalyzed reactions where the catalyst is saturated.

Second-order reactions show the opposite trend — the half-life gets shorter as concentration decreases. This means the reaction speeds up in relative terms as it proceeds, which is counterintuitive. This behavior is typical of bimolecular reactions where two molecules must collide.

For radioactive decay, the half-life determines the activity and is used to calculate dating ages (carbon-14 dating uses a half-life of 5,730 years). In pharmacology, drug half-lives determine dosing intervals.

Radioactive half-lives are fundamental to nuclear physics, medical imaging, and radiometric dating. Carbon-14 dating uses the 5,730-year half-life to determine the age of archaeological artifacts. Uranium-238 (half-life 4.5 billion years) is used to date geological formations. Technetium-99m (half-life 6 hours) is the most widely used medical radioisotope for diagnostic imaging.

Pharmaceutical kinetics relies heavily on drug half-lives for determining dosing schedules. A drug with a 4-hour half-life must be taken multiple times per day to maintain therapeutic levels, while a drug with a 24-hour half-life can be taken once daily. The elimination half-life determines how quickly the body clears a drug.

Chemical kinetics uses half-life measurements to determine reaction orders and rate constants. By measuring how the half-life changes with initial concentration, chemists can distinguish between zero, first, and second-order mechanisms. This is a standard technique in physical chemistry laboratories.

Environmental science uses half-life concepts to describe pollutant persistence. Pesticides, pharmaceuticals, and industrial chemicals all have environmental half-lives that determine how long they remain in soil, water, and air. The half-life concept helps predict the long-term impact of chemical releases.