Zero-Order Reaction Rate Calculator

Free Zero-Order Reaction Rate Calculator

This calculator solves the zero-order integrated rate law, the equation chemists use whenever a reaction proceeds at a fixed, constant speed no matter how much reactant is left. Give it any three of the four quantities — the initial concentration [A]0, the concentration remaining [A]t, the rate constant k, and the time elapsed t — and it instantly solves for the fourth, along with the reaction's half-life, the time it takes for the reactant to run out completely, and the percent reacted so far.

It also goes a step further with an Advanced Tools mode built for real lab data. If you've measured concentration at several time points during an experiment, paste those readings in as a table and the calculator fits a straight line through them, reads off the rate constant k and the initial concentration [A]0 directly from the slope and intercept, and reports an R² value so you can check whether the reaction actually behaves as zero order across the range you measured. Every calculation on this page comes with full, plain-language step-by-step working, completely free and with no sign-up required.

What Is a Zero-Order Reaction?

In chemical kinetics, the 'order' of a reaction describes how its rate depends on the concentration of the reactants involved. In most reactions, doubling the concentration of a reactant doubles (or otherwise changes) how fast the reaction goes. A zero-order reaction is the exception: its rate stays exactly the same no matter what the concentration of the reactant is.

That might sound strange at first, but it happens in a specific and common situation — when something other than the reactant's concentration is the real bottleneck. This is usually a fixed amount of a catalyst, an enzyme, or a light source that's already working at full capacity. Once that bottleneck is saturated, adding more of the actual reactant can't make the reaction go any faster, because the limiting factor isn't the reactant at all.

The Zero-Order Integrated Rate Law

For a zero-order reaction, the rate law is simply rate = k, a constant. Integrating that rate law with respect to time gives the equation this calculator solves:

[A]t = [A]0 − kt

Here [A]0 is the starting concentration of the reactant, [A]t is the concentration remaining after time t has passed, and k is the rate constant. Because the right-hand side is a straight line in t, a plot of concentration [A] against time gives a straight line with a negative slope of −k — this is the single clearest fingerprint of zero-order kinetics, and it's exactly what the Advanced Tools regression tool on this page tests for.

Note the units of k. For a first-order reaction k has units of 1/time, and for second-order it's 1/(concentration·time) — but for zero-order, k carries units of concentration/time, such as M/s or mM/min, since it directly represents how fast the concentration itself is falling.

Half-Life of a Zero-Order Reaction

The half-life t₁/₂ is the time it takes for the concentration to fall to exactly half of its starting value. Setting [A]t = [A]0/2 in the integrated rate law and solving for t gives:

t₁/₂ = [A]0 / (2k)

This is one of the most important differences between zero-order kinetics and every other reaction order: the zero-order half-life depends directly on the starting concentration. A more concentrated batch takes longer to reach its halfway point than a dilute one, even though k stays the same. This is the opposite of first-order kinetics, where the half-life is a fixed constant (ln2/k) no matter how much reactant you start with — which is why zero-order half-lives are always reported alongside the starting concentration they were measured from, never as a single constant number.

Because each successive half-life starts from half the previous concentration, the half-lives themselves keep shrinking: the second half-life is half as long as the first, the third is half the second, and so on, until the reactant runs out completely at t = [A]0/k — the same 'time to full consumption' value this calculator reports next to the half-life.

How to Use This Calculator

On the Standard Solver tab, pick what you want to solve for from the dropdown — most people either want the concentration remaining after a known reaction time, or the rate constant k from a before-and-after concentration reading. Fill in the other three values in whatever units you like (mol/L, mmol/L, or µmol/L for concentration; seconds, minutes, hours, or days for time), and the calculator converts everything internally, solves the equation, and converts the answer back into your chosen units. Alongside the main result, it always reports the half-life, the time until the reactant is fully consumed, and the percent of reactant used up so far.

The Advanced Tools tab is built for real experimental data. Enter a table of time and concentration readings from your own lab run — the calculator needs at least two points, but more points give a far more reliable fit — and it performs a linear regression of concentration against time. The negative of the slope is your rate constant k, the y-intercept is your fitted [A]0, and the R² value tells you how well a straight line actually describes your data, which is the standard way chemists confirm a reaction really is zero order before trusting the rate constant that comes out of it.

Worked Example: Finding Concentration After a Given Time

A zero-order reaction starts with [A]0 = 1.00 M and has a rate constant of k = 0.0500 M/min. What is the concentration remaining after 12 minutes?

Apply [A]t = [A]0 − kt = 1.00 − (0.0500)(12) = 1.00 − 0.600 = 0.400 M.

So after 12 minutes, 60% of the original reactant has been used up, leaving 0.400 M behind. The half-life here is t₁/₂ = [A]0/(2k) = 1.00/(2×0.0500) = 10 minutes, and the reactant runs out completely — 0.00 M — at t = [A]0/k = 1.00/0.0500 = 20 minutes.

Worked Example: Finding the Rate Constant from Two Readings

A reaction starts at [A]0 = 0.800 M and after 25 minutes the concentration has dropped to [A]t = 0.300 M. What is the rate constant k?

Apply k = ([A]0 − [A]t) / t = (0.800 − 0.300) / 25 = 0.500 / 25 = 0.0200 M/min.

With k known, the half-life for this particular run is t₁/₂ = 0.800/(2×0.0200) = 20 minutes, and the reaction would finish completely — assuming it keeps behaving as zero order the whole way — at t = 0.800/0.0200 = 40 minutes.

Worked Example: Fitting k and [A]0 from Lab Data (Advanced Tools)

A student measures a reactant's concentration at five points during an experiment: [A] = 1.000 M at t = 0, 0.750 M at t = 5 min, 0.500 M at t = 10 min, 0.250 M at t = 15 min, and 0.010 M at t = 20 min.

Fitting a straight line through these five (t, [A]) points by least squares gives a slope of about −0.0498 M/min and a y-intercept of about 1.001 M, with R² very close to 1 — confirming the data really does follow a straight line. Since slope = −k, the rate constant is k ≈ 0.0498 M/min, and since the intercept is [A]0, the fitted starting concentration is about 1.001 M — both extremely close to the reaction's true values, which is exactly what a good zero-order fit should look like.

This is the real-world way chemists determine reaction order and rate constants: rather than trusting a single before-and-after measurement (which is sensitive to experimental error), a series of readings is taken over time and fitted with a line, giving both a more reliable rate constant and, through the R² value, direct evidence for whether zero-order kinetics is even the right model to use.

Zero-Order vs First-Order vs Second-Order Reactions

The three common reaction orders each have their own integrated rate law, their own straight-line plot, and their own half-life behavior — and telling them apart is one of the first things a kinetics experiment needs to establish.

• Zero-order: rate = k. Integrated law: [A]t = [A]0 − kt. A plot of [A] vs t is a straight line. Half-life shrinks over time and depends on [A]0.
• First-order: rate = k[A]. Integrated law: ln[A]t = ln[A]0 − kt. A plot of ln[A] vs t is a straight line. Half-life is constant (ln2/k) and does not depend on [A]0.
• Second-order: rate = k[A]². Integrated law: 1/[A]t = 1/[A]0 + kt. A plot of 1/[A] vs t is a straight line. Half-life increases as [A]0 decreases.
• The quickest way to identify which order a reaction follows is exactly what the Advanced Tools tab on this page does for zero order: plot your data the way each order predicts and see which plot actually comes out straight (highest R²).

Real-World Examples of Zero-Order Reactions

Enzyme-catalyzed reactions running at substrate saturation. When an enzyme is fully loaded with substrate, adding more substrate can't speed the reaction up any further, because every active site is already busy — the reaction becomes zero order in substrate, a case central to enzyme kinetics and the Michaelis-Menten model.

Reactions on a saturated catalyst surface. In heterogeneous catalysis, if every active site on a solid catalyst's surface is already occupied by reactant molecules, the reaction rate depends only on the catalyst's surface area, not on how much excess reactant is floating around in solution or in the gas phase.

Photochemical reactions limited by light intensity. When a reaction only happens after a photon is absorbed and the light source delivers photons at a fixed rate, the reaction proceeds at that same fixed rate regardless of how much reactant is present, as long as there's enough to keep absorbing every available photon.

Drug elimination at high, saturating doses. Some drugs — ethanol is the classic textbook example — are broken down by an enzyme system that becomes fully saturated at typical or high blood concentrations, so the body clears them at a roughly constant amount per hour rather than a constant percentage per hour, a genuinely zero-order elimination pattern used in pharmacokinetics.

Common Mistakes to Avoid

A handful of small errors account for most incorrect answers when working with zero-order kinetics by hand.

• Assuming every reaction that 'slows down over time' is automatically zero order — plenty of first- and second-order reactions also visibly slow down; the real test is whether [A] vs t is a straight line, not just whether the reaction is slowing.
• Mixing concentration or time units mid-calculation — keep k, [A]0, [A]t, and t in one consistent unit system throughout a hand calculation, or let this calculator's unit dropdowns handle the conversion for you.
• Forgetting that half-life for a zero-order reaction is not constant — unlike first-order kinetics, you can't reuse a zero-order half-life you calculated at one concentration for a different starting concentration.
• Extrapolating the straight-line rate law past the point where the reactant actually runs out — physically, concentration can never go negative, so the equation stops applying once t exceeds [A]0/k.
• Trusting a rate constant calculated from only two data points — real experimental noise means a single pair of readings can be misleading; whenever more than two time points are available, the Advanced Tools regression on this page gives a far more trustworthy value, along with an R² check on whether the reaction is really zero order at all.

Limitations to Keep in Mind

Zero-order kinetics is nearly always a special case that holds only under specific conditions — a saturated catalyst, a saturated enzyme, or a fixed light intensity — and typically only over part of a reaction's full lifetime. As a reactant is consumed and its concentration drops low enough, the bottleneck that caused zero-order behavior (an enzyme, a catalyst, a light source) often stops being saturated, and the reaction can shift toward first-order behavior near the very end. This calculator assumes true zero-order behavior holds across the entire time range you enter; for real experimental systems, always check the R² value from the Advanced Tools fit, and compare it against first-order and second-order plots of the same data before concluding the reaction is genuinely zero order.

Quick Reference: Every Formula on This Page

[A]t = [A]0 − kt — the zero-order integrated rate law used in the Standard Solver. t₁/₂ = [A]0/(2k) — the half-life, which depends on the starting concentration. t(complete) = [A]0/k — the time for the reactant to be fully consumed. Percent reacted = 100 × (1 − [A]t/[A]0). Rate law: rate = k, independent of concentration. Units of k: concentration/time (for example M/s, mM/min, or µM/h).