Second-Order Reaction Rate Calculator

Free Second-Order Reaction Rate Calculator

This calculator solves the second-order integrated rate law, the equation chemists use whenever a reaction's speed depends on the square of a single reactant's concentration — or on two reactants present in equal amounts. 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 reciprocal of each concentration, fits a straight line through the (time, 1/[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 second-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 Second-Order Reaction?

In chemical kinetics, a second-order reaction is one whose rate depends on the concentration of a reactant raised to the second power, or on the product of two reactant concentrations, each raised to the first power. In the most common textbook case — a single reactant A — the rate law is rate = k[A]². Double the concentration of A and the reaction goes four times as fast; triple it and the rate goes up ninefold.

Second-order kinetics shows up whenever a reaction depends on two molecules meeting and reacting with each other, rather than on a single molecule acting alone. That makes it common in gas-phase collision reactions, radical recombination steps, many organic substitution and elimination reactions, and any reaction where two identical reactant molecules must collide for the reaction to proceed.

The Second-Order Integrated Rate Law

Because rate = k[A]² describes how fast the concentration is falling at any instant, integrating that relationship over time gives the equation this calculator solves:

1/[A]t = 1/[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. Unlike a first-order reaction, plotting [A] or ln[A] against time gives a curve for a second-order reaction — only a plot of the reciprocal, 1/[A], against time gives a straight line, with slope k and y-intercept 1/[A]0. That straight-line test is exactly what the Advanced Tools regression on this page performs on your own data.

The units of k for a second-order reaction are 1/(concentration·time) — such as M⁻¹s⁻¹, M⁻¹min⁻¹, or L·mol⁻¹·s⁻¹ (the same thing written differently) — unlike a first-order rate constant, which carries no concentration unit at all.

Half-Life of a Second-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:

t₁/₂ = 1/(k[A]0)

Unlike a first-order half-life, this result depends directly on the starting concentration [A]0 — and unlike a zero-order half-life, which shrinks over time, a second-order half-life grows over time. Each successive half-life is exactly double the one before it: if the first half-life takes 20 minutes, the second half-life (taking the concentration from a quarter down to an eighth of the original) takes 40 minutes, the third takes 80 minutes, and so on.

This happens because the reaction rate depends on the square of the concentration — as the reactant is used up, the collisions that drive the reaction become steadily rarer, so it takes longer and longer to remove each successive half of what's left. This calculator reports both the first half-life (from the starting concentration you entered) and the second half-life (double the first), so you can see this stretching-out effect directly.

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 current half-life, the next (doubled) 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 reciprocal of every concentration and performs a linear regression against time. The slope of that line is your rate constant k, the reciprocal of the y-intercept is your fitted [A]0, and the R² value tells you how well a straight line actually describes your 1/[A]-vs-time data, which is the standard way chemists confirm a reaction really is second order before trusting the rate constant that comes out of it.

Worked Example: Finding Concentration After a Given Time

A second-order reaction starts with [A]0 = 1.00 M and has a rate constant of k = 0.0500 M⁻¹min⁻¹. What is the concentration remaining after 30 minutes?

Apply 1/[A]t = 1/[A]0 + kt = 1/1.00 + (0.0500)(30) = 1.00 + 1.50 = 2.50, so [A]t = 1/2.50 = 0.400 M.

So after 30 minutes, 60.0% of the original reactant has been used up, leaving 0.400 M behind. The first half-life here is t₁/₂ = 1/(kA0) = 1/(0.0500 × 1.00) = 20.0 minutes, and the second half-life (getting from 0.25 M down to 0.125 M) would take twice that, or 40.0 minutes.

Worked Example: Finding the Rate Constant from Two Readings

A reaction starts at [A]0 = 0.500 M and after 40 minutes the concentration has dropped to [A]t = 0.200 M. What is the rate constant k?

Apply k = (1/[A]t − 1/[A]0) / t = (1/0.200 − 1/0.500) / 40 = (5.00 − 2.00) / 40 = 3.00 / 40 = 0.0750 M⁻¹min⁻¹.

With k known, the first half-life for this reaction (starting from the original 0.500 M) is t₁/₂ = 1/(0.0750 × 0.500) ≈ 26.7 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, 0.667 M at t = 10 min, 0.500 M at t = 20 min, 0.400 M at t = 30 min, and 0.333 M at t = 40 min.

Taking the reciprocal of each concentration and fitting a straight line through the five (t, 1/[A]) points by least squares gives a slope of about 0.0500 and a y-intercept very close to 1.00, with R² essentially equal to 1 — confirming the data really does follow a straight line in reciprocal form. Since slope = k, the rate constant is k ≈ 0.0500 M⁻¹min⁻¹, and since the intercept is 1/[A]0, the fitted starting concentration works out to 1/1.00 ≈ 1.000 M — both matching the reaction's true values, exactly what a good second-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 reciprocal form, and fitted with a line, giving both a more reliable rate constant and, through the R² value, direct evidence for whether second-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 over time (each one is double the last) and is inversely proportional to [A]0.
• The quickest way to identify which order a reaction follows is exactly what the Advanced Tools tab on this page does for second order: plot your data the way each order predicts and see which plot actually comes out straight (highest R²).

Real-World Examples of Second-Order Reactions

Gas-phase decomposition of nitrogen dioxide. The reaction 2NO₂ → 2NO + O₂ is a classic textbook second-order reaction, with rate = k[NO₂]², since two NO₂ molecules must collide with enough energy for the reaction to proceed.

The reaction of hydrogen iodide, 2HI → H₂ + I₂. Like NO₂ decomposition, this gas-phase reaction depends on two HI molecules colliding, giving it a rate law of rate = k[HI]².

Many radical recombination and dimerization reactions. When two identical reactive radicals or monomer units combine to form a single product, the rate of disappearance of the starting species naturally depends on the square of its own concentration.

SN2 nucleophilic substitution reactions in organic chemistry. Unlike SN1 reactions, an SN2 reaction's rate-determining step involves both the substrate and the nucleophile colliding simultaneously, so the reaction is first order in each reactant — second order overall — with rate = k[substrate][nucleophile].

Common Mistakes to Avoid

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

• Using [A] or ln[A] directly in a straight-line plot instead of 1/[A] — a plot of raw concentration or its natural log against time for a second-order reaction is a curve, not a line; only the reciprocal 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.
• Treating the second-order half-life as a constant — unlike first-order kinetics, a second-order half-life depends on the starting concentration and keeps doubling with every successive half-life, so a half-life measured early in a reaction cannot be reused later on.
• Forgetting the units of k — a second-order rate constant carries units of 1/(concentration·time), not just 1/time (first-order) or concentration/time (zero-order); dropping or misreading these units is one of the most common sources of error when comparing rate constants across reaction orders.
• 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 second order at all.

Limitations to Keep in Mind

Second-order kinetics describes a genuine, common class of reactions — but it isn't universal, and this calculator's simple form applies specifically to the case of a single reactant A (rate = k[A]²) or two reactants present in exactly equal, matched concentrations. A true two-reactant second-order reaction (rate = k[A][B]) with unequal starting concentrations follows a different integrated rate law that this calculator does not cover. This calculator also assumes true second-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 first-order plots of the same data before concluding the reaction is genuinely second order.

Quick Reference: Every Formula on This Page

1/[A]t = 1/[A]0 + kt — the second-order integrated rate law used in the Standard Solver. t₁/₂ = 1/(k[A]0) — the half-life, which depends on the starting concentration and doubles with each successive half-life. Percent reacted = 100 × (1 − [A]t/[A]0). Rate law: rate = k[A]². Units of k: 1/(concentration·time) (for example M⁻¹s⁻¹, M⁻¹min⁻¹, or L·mol⁻¹·s⁻¹).