Half-Life Kinetics Calculator

Half-Life Kinetics Calculator: t½ for Any Reaction Order

This half-life kinetics calculator finds the half-life, the rate constant, or the initial concentration for a chemical reaction — whether it follows zero-order, first-order, or second-order kinetics. Just pick the reaction order, enter the two values you already know, and the calculator solves for the third one instantly, with every step of the working shown in plain language.

It also does something the order-specific rate calculators don't: it lines up the successive half-lives of a reaction in one place, so you can see exactly how the time between halvings changes — or doesn't — as the reaction order changes. And with the Advanced Tools tab, you're not limited to 50% remaining at all: you can find the time to reach any target percentage, from 90% left down to a trace amount.

What Is Half-Life in Chemical Kinetics?

The half-life of a reaction (t½) is the time it takes for the concentration of a reactant to drop to exactly half of whatever it was at the start of that period. It's one of the simplest, most intuitive ways to describe how fast a reaction runs — instead of quoting an abstract rate constant with awkward units, you can just say 'half of it is gone in 20 minutes' and everyone immediately understands the pace.

Half-life shows up everywhere kinetics matters: drug metabolism in the body, the decay of unstable atoms, the breakdown of pollutants in the environment, and the shelf life of chemicals in storage. But the exact formula for t½ — and whether it stays the same value every time or keeps changing — depends entirely on the reaction's order.

Half-Life Formulas by Reaction Order

Each reaction order has its own half-life equation, derived directly from its integrated rate law:

• Zero-order: t½ = [A]0 / (2k) — half-life is directly proportional to the starting concentration.
• First-order: t½ = ln(2) / k ≈ 0.693 / k — half-life depends only on k, never on concentration.
• Second-order: t½ = 1 / (k [A]0) — half-life is inversely proportional to the starting concentration.

Why Half-Life Depends on Reaction Order

First-order half-life is special: it's a true constant. No matter how much reactant you start with, the time to lose half of it is always the same, which is why it's the only kind of half-life that behaves the way most people intuitively picture (like radioactive decay, which is a first-order process). Cut the starting amount in half and it still takes the same t½ to lose half of what's left.

Zero-order and second-order reactions don't work that way. For a zero-order reaction, the concentration drops in a straight line, so each successive half-life is shorter than the last — the reaction actually speeds up (in the sense of taking less time per halving) as it runs out of material. For a second-order reaction, the opposite happens: every successive half-life is exactly double the one before it, since the rate depends on the square of what's left, and there's less of it to react as time goes on.

How to Use This Calculator

On the Standard Solver tab, choose the reaction order, choose what you want to solve for (t½, k, or [A]0), pick a time unit, and fill in the two remaining fields. The result updates instantly, along with a chart showing how long each of the next six half-lives will take for your reaction.

On the Advanced Tools tab, enter the reaction order, k, and (for zero- and second-order reactions) the initial concentration, then set a target percentage remaining — it doesn't have to be 50%. The calculator finds the time to reach that exact point, plus the equivalent number of half-lives, and plots the full decay curve with your target marked on it.

Worked Example: First-Order Half-Life

A drug is eliminated from the bloodstream by a first-order process with a rate constant k = 0.0693 per hour. Using t½ = ln(2) / k = 0.693 / 0.0693 = 10.0 hours. It doesn't matter whether the starting dose was 100 mg or 10 mg — either way, half of whatever is present clears every 10 hours, which is exactly why pharmacokinetics relies so heavily on first-order half-lives to schedule repeat doses.

Worked Example: Second-Order Half-Life Doubling

A second-order reaction starts at [A]0 = 1.00 M with k = 0.50 M⁻¹min⁻¹. The first half-life is t½ = 1 / (k[A]0) = 1 / (0.50 × 1.00) = 2.0 minutes. Once the concentration drops to 0.50 M, the next half-life becomes 1 / (0.50 × 0.50) = 4.0 minutes — exactly double. The one after that, starting from 0.25 M, takes 8.0 minutes. Each halving takes twice as long as the one before it, which is the signature behavior of second-order kinetics.

Worked Example: Zero-Order Half-Life Shrinking

A zero-order reaction starts at [A]0 = 1.00 M with k = 0.050 M/min. The first half-life is t½ = [A]0 / (2k) = 1.00 / (2 × 0.050) = 10.0 minutes. After that first half-life, the remaining concentration is 0.50 M, and the next half-life is 0.50 / (2 × 0.050) = 5.0 minutes — half as long. This shrinking pattern continues until the reactant runs out completely, which is why zero-order reactions have a finite, calculable time to total completion.

Fractional Life: Going Beyond 50% Remaining

Half-life is just one specific case of a more general idea: the time it takes to reach any target fraction remaining, sometimes called fractional life. The Advanced Tools tab generalizes each order's half-life formula to any percentage you choose, using the fraction remaining f (10% remaining means f = 0.10):

• Zero-order: t = [A]0 (1 − f) / k
• First-order: t = −ln(f) / k
• Second-order: t = (1 − f) / (f k [A]0)

Half-Lives Elapsed and Percent Remaining

One relationship holds true no matter the reaction order: after n half-lives, exactly (1/2)ⁿ of the original amount is left. This table is worth memorizing — it's the same for radioactive decay, drug clearance, or any other exponential-style falloff:

• 1 half-life: 50% remaining
• 2 half-lives: 25% remaining
• 3 half-lives: 12.5% remaining
• 4 half-lives: 6.25% remaining
• 5 half-lives: 3.125% remaining
• 10 half-lives: about 0.098% remaining (essentially gone)

Common Mistakes to Avoid

A few slip-ups come up again and again with half-life problems:

• Using the first-order half-life formula (0.693/k) for a zero- or second-order reaction — each order has its own formula, and they're not interchangeable.
• Assuming half-life is constant for every order — it's only truly constant for first-order reactions.
• Mixing up units for k — zero-order k has concentration/time units, first-order k has 1/time units, and second-order k has 1/(concentration·time) units.
• Forgetting that '5 half-lives' is only ever exactly 3.125% remaining, not a rough guess — it's an exact power of one-half every time.

Where Half-Life Kinetics Shows Up in Real Life

Half-life calculations sit behind drug dosing schedules and elimination charts in pharmacology, shelf-life and expiration dating for chemicals and food, environmental persistence of pesticides and pollutants, catalyst deactivation studies in chemical engineering, and radioactive decay and dating (a special case of first-order kinetics covered by this site's dedicated Radioactive Decay Calculator).

Limitations to Keep in Mind

This calculator assumes a clean, single reactant following an ideal zero-, first-, or second-order rate law at constant temperature — real reactions can shift order under different conditions, involve multiple steps with different rate-determining stages, or be affected by temperature changes that alter k itself (see the Arrhenius Activation Energy Solver for that relationship). For reactions with more complex or mixed-order behavior, treat the result here as a useful approximation rather than an exact prediction.

Quick Reference: Every Formula on This Page

Zero-order: t½ = [A]0 / (2k), fractional life t = [A]0(1 − f)/k. First-order: t½ = ln(2)/k ≈ 0.693/k, fractional life t = −ln(f)/k. Second-order: t½ = 1/(k[A]0), fractional life t = (1 − f)/(fk[A]0). Universal: after n half-lives, fraction remaining = (1/2)ⁿ, so n = −log2(f).