First-Order Reaction Rate Calculator

Free First-Order Reaction Rate Calculator

This calculator solves the first-order integrated rate law, the equation chemists use whenever a reaction's speed depends directly on how much reactant is still 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, how many half-lives have passed, and the percent reacted so far.

It also includes an Advanced Tools mode built for real lab data. If you've measured concentration at several time points during an experiment, enter those readings as a table and the calculator takes the natural log of each concentration, fits a straight line through the (time, ln[A]) points, and reads off the rate constant k and the initial concentration [A]0 directly from the slope and intercept — with an R² value so you can see how well the data actually follows first-order behavior. 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 First-Order Reaction?

In chemical kinetics, a first-order reaction is one whose rate is directly proportional to the concentration of a single reactant: rate = k[A]. Double the concentration of A and the reaction goes exactly twice as fast; cut the concentration in half and the reaction slows to half speed.

This is one of the most common reaction orders in chemistry, and it shows up anywhere a single molecule has to do something on its own before a reaction can happen — break apart, rearrange, or lose a particle — without needing to collide with another reactant molecule first. Radioactive decay, many drug elimination processes in the body, and a huge range of decomposition reactions all follow first-order kinetics.

The First-Order Integrated Rate Law

Because rate = k[A] describes how fast the concentration is falling at any instant, integrating that relationship over time gives the equation this calculator solves:

[A]t = [A]0 · e^(−kt), equivalently written as ln[A]t = ln[A]0 − kt

Here [A]0 is the starting concentration, [A]t is the concentration remaining after time t, and k is the rate constant. Because the exponential form curves and flattens out, chemists almost always work with the logarithmic form instead — a plot of ln[A] against time gives a straight line with slope −k, and that straight-line test is exactly what the Advanced Tools regression on this page performs on your own data.

Unlike a zero-order rate constant, k for a first-order reaction has units of inverse time only — such as s⁻¹, min⁻¹, h⁻¹, or day⁻¹ — with no concentration unit attached, since concentration cancels out of both sides of the rate law.

Half-Life of a First-Order Reaction

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

t₁/₂ = ln2 / k ≈ 0.693 / k

Notice that [A]0 has completely cancelled out. This is the defining feature of first-order kinetics: the half-life is a fixed constant that never changes, no matter how much reactant you start with. A reaction that takes 10 minutes to lose half its concentration starting from 1.0 M will also take exactly 10 minutes to lose half its concentration starting from 0.1 M, or from 10.0 M. This is why radioactive half-lives — themselves a textbook example of first-order kinetics — are quoted as a single fixed number (like 'carbon-14 has a half-life of 5,730 years') rather than depending on how much material you started with.

Because the decay is exponential rather than linear, the concentration technically never reaches absolute zero — it just gets closer and closer with every half-life that passes. After 1 half-life, 50% remains; after 2 half-lives, 25%; after 5 half-lives, about 3.1%; after 10 half-lives, less than 0.1% — which is why '~7 half-lives' is often treated as 'essentially complete' in practice, even though the math never quite reaches zero.

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, how many half-lives have elapsed, 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 takes the natural log of every concentration and performs a linear regression against time. The negative of the slope is your rate constant k, the exponential of the y-intercept is your fitted [A]0, and the R² value tells you how well a straight line actually describes your ln[A]-vs-time data, which is the standard way chemists confirm a reaction really is first order before trusting the rate constant that comes out of it.

Worked Example: Finding Concentration After a Given Time

A first-order reaction starts with [A]0 = 1.00 M and has a rate constant of k = 0.0500 min⁻¹. What is the concentration remaining after 18 minutes?

Apply [A]t = [A]0 · e^(−kt) = 1.00 × e^(−0.0500 × 18) = 1.00 × e^(−0.900) ≈ 1.00 × 0.4066 ≈ 0.407 M.

So after 18 minutes, about 59.3% of the original reactant has been used up, leaving roughly 0.407 M behind. The half-life here is t₁/₂ = ln2/k = 0.6931/0.0500 ≈ 13.9 minutes — a fixed number that would apply just the same if the reaction had started from any other concentration.

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 = ln([A]0/[A]t) / t = ln(0.800/0.300) / 25 = ln(2.667) / 25 ≈ 0.9808 / 25 ≈ 0.0392 min⁻¹.

With k known, the half-life for this reaction — and for any starting concentration — is t₁/₂ = ln2/0.0392 ≈ 17.7 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, 0.607 M at t = 10 min, 0.368 M at t = 20 min, 0.223 M at t = 30 min, and 0.135 M at t = 40 min.

Taking the natural log of each concentration and fitting a straight line through the five (t, ln[A]) points by least squares gives a slope of about −0.0500 and a y-intercept very close to 0, with R² essentially equal to 1 — confirming the data really does follow a straight line in log form. Since slope = −k, the rate constant is k ≈ 0.0500 min⁻¹, and since the intercept is ln[A]0, the fitted starting concentration works out to e^0 ≈ 1.000 M — both matching the reaction's true values, exactly what a good first-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, plotted in log form, and fitted with a line, giving both a more reliable rate constant and, through the R² value, direct evidence for whether first-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 first order: plot your data the way each order predicts and see which plot actually comes out straight (highest R²).

Real-World Examples of First-Order Reactions

Radioactive decay. Every radioactive isotope decays by first-order kinetics, which is exactly why isotopes are described by a single fixed half-life — carbon-14's 5,730-year half-life, for instance — that never changes no matter how much of the isotope is present.

Drug elimination in pharmacokinetics. Most drugs are cleared from the bloodstream by enzymes working well below saturation, so the body eliminates a constant fraction (not a constant amount) of the drug per hour — textbook first-order kinetics, and the reason drug half-lives are quoted as single fixed numbers on a package insert.

Unimolecular decomposition reactions. Many gas-phase reactions where a single, energized molecule falls apart into products — without needing to collide with a second reactant molecule first — follow first-order kinetics, since the decomposition rate depends only on how many of that one molecule are present.

SN1 nucleophilic substitution reactions in organic chemistry. The rate-determining step of an SN1 reaction is the spontaneous ionization of a single substrate molecule, so the overall reaction rate depends only on the substrate's concentration and is first order in that substrate, regardless of the nucleophile's concentration.

Common Mistakes to Avoid

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

• Using [A] directly in a straight-line plot instead of ln[A] — a plot of raw concentration against time for a first-order reaction is a curve, not a line; only the logarithmic form straightens out.
• 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 first-order half-life is a genuine constant — unlike zero-order kinetics, you can reuse a first-order half-life at any starting concentration, which makes it a very convenient number to quote once you know it.
• Assuming the concentration hits exactly zero at some finite time — mathematically, first-order decay only ever approaches zero and technically never reaches it, unlike zero-order kinetics where the reactant does run out completely.
• 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 first order at all.

Limitations to Keep in Mind

First-order kinetics describes a genuine, common class of reactions — but it isn't universal. Some reactions that look first order over a limited concentration or time range actually shift toward a different order once conditions change (for example, if a catalyst becomes saturated, or a second reactant that was in large excess starts to run low). This calculator assumes true first-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 zero-order and second-order plots of the same data before concluding the reaction is genuinely first order.

Quick Reference: Every Formula on This Page

[A]t = [A]0 · e^(−kt), equivalently ln[A]t = ln[A]0 − kt — the first-order integrated rate law used in the Standard Solver. t₁/₂ = ln2/k ≈ 0.693/k — the half-life, which is constant and does not depend on the starting concentration. Percent reacted = 100 × (1 − [A]t/[A]0). Rate law: rate = k[A]. Units of k: inverse time only (for example s⁻¹, min⁻¹, h⁻¹, or day⁻¹).