# Half-Life by Reaction Order

t½ = a₀/(2k), ln 2/k or 1/(k·a₀) for zero-, first- and second-order reactions, with timelines and depletion time.

> Canonical page: https://elysiatools.com/en/tools/half-life-reaction-order

- **Category:** Science & Education

- **Keywords:** half life, reaction order, zero order, first order, second order, kinetics, carbon 14

## Overview

The Half-Life by Reaction Order calculator computes the half-life and concentration depletion timelines for zero-, first-, and second-order chemical reactions. By selecting the reaction order and providing the rate constant alongside initial reactant concentrations when required, you can instantly determine half-life values, evaluate decay profiles, and analyze how reaction rates evolve over time.

## Inputs

- **Reaction order** (select)
- **Rate constant k** (number): e.g. 1.21e-4 yr⁻¹ for C-14
- **Initial concentration \[A\]₀ (required for 0th and 2nd order)** (number): e.g. 0.4

## When to use

- Determining half-life durations and total depletion times for zero-order, surface-catalyzed, or enzyme-saturated processes.
- Evaluating first-order exponential decay constants and half-lives for radioactive isotopes or unimolecular decomposition reactions.
- Calculating concentration-dependent half-lives for second-order dimerization or bimolecular kinetics.

## How it works

- Select the reaction order (zero order, first order, or second order) corresponding to the kinetic model.
- Enter the reaction rate constant k in units matching your timeframe and concentration dimensions.
- Input the initial concentration \[A\]₀ for zero-order (t½ = a₀/(2k)) or second-order (t½ = 1/(k·a₀)) reactions; this field is omitted for first-order kinetics (t½ = ln 2/k).
- View the computed half-life value alongside a step-by-step timeline of remaining reactant concentrations or fractions over multiple half-life intervals.

## Use cases

- Chemistry students verifying chemical kinetics homework problems and plotting reactant concentration decay curves.
- Researchers calculating radioactive dating benchmarks such as Carbon-14 decay intervals.
- Process engineers estimating batch reaction completion times and reactant consumption rates in industrial reactors.

## Frequently asked questions

### Why is the initial concentration not required for first-order reactions?

First-order half-life depends exclusively on the rate constant (t½ = ln 2 / k), meaning the time required to halve the reactant remains constant regardless of starting concentration.

### How does half-life change over time in a second-order reaction?

In second-order kinetics, half-life is inversely proportional to concentration (t½ = 1 / (k·[A]₀)), causing each successive half-life period to double in duration as the reactant is consumed.

### When does a zero-order reaction reach complete depletion?

A zero-order reaction reaches full depletion at time t = [A]₀ / k, which is exactly twice its half-life value.

### What units should be used for rate constant k and initial concentration?

Units must be consistent: zero-order k uses concentration/time, first-order k uses 1/time, and second-order k uses 1/(concentration·time). The resulting half-life matches the time unit of k.

### Can this tool calculate radioactive decay half-lives?

Yes. Radioactive decay follows first-order kinetics, allowing you to compute isotope half-lives directly from the decay constant k.

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