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Second-Order Concentration Calculator

Enter initial concentration, rate constant and time to find remaining concentration.

How it is calculated

For a reaction with rate = k[A]², the integrated law is 1/[A] = 1/[A]₀ + k·t. The tool computes [A] = 1 ÷ (1/[A]₀ + k·t) in mol/L, with [A]₀ the starting concentration in mol/L, k the rate constant in L/(mol·s) and t the time in seconds. It also shows the fraction remaining, [A] ÷ [A]₀ × 100, and the half-life t½ = 1 ÷ (k·[A]₀) in seconds. Unlike first order, this half-life depends on the starting concentration. The starting concentration must be above zero, and k and t cannot be negative, otherwise no result is shown. A rate constant of exactly 0 is allowed: the concentration stays unchanged and the half-life is shown as a dash because there is none. The concentration is printed to 4 decimals, and time must be in seconds to match k.

Worked example

Start at [A]₀ = 0.5 mol/L with k = 0.2 L/(mol·s) and wait t = 30 s. Step 1: 1/[A]₀ = 1 ÷ 0.5 = 2. Step 2: k·t = 0.2 × 30 = 6. Step 3: 1/[A] = 2 + 6 = 8. So [A] = 1 ÷ 8 = 0.1250 mol/L. Fraction remaining: 0.125 ÷ 0.5 = 25.00%. Half-life: 1 ÷ (0.2 × 0.5) = 10.00 s, and the numbers fit: with t = 10 s the tool gives 0.2500 mol/L, exactly half.

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Common mistakes

  • Using the first-order formula [A]₀·e^(−kt) with a second-order k: the units L/(mol·s) signal the second-order law, and the two give very different answers.
  • Mixing time units: if k is per minute, time must be in minutes, but the tool labels its half-life in seconds regardless.
  • Assuming a constant half-life: each half-life is twice as long as the one before, because the concentration has halved. The tool gives only the first one, 1 ÷ (k·[A]₀).
  • Entering a rate constant written for a different rate definition (for example the rate defined as d[A]/dt = −2k[A]²): the stoichiometric factor changes k.
  • Using very dilute solutions and reading the concentration literally. It is rounded to 4 decimals, so [A]₀ = 0.01 mol/L with k = 0.5 and t = 5000 s shows 0.0004 mol/L; rely on the fraction remaining (3.85%) for the real change.

About this calculator

For a second-order reaction — one whose rate depends on the square of a single reactant's concentration, rate = k[A]² — the integrated rate law is 1/[A] = 1/[A]₀ + kt, which is linear in 1/[A] versus time (unlike the exponential decay of a first-order reaction). Rearranging it lets you solve for the remaining concentration [A] at any elapsed time t, given the initial concentration [A]₀ and rate constant k.

This calculator takes [A]₀, k, and t and returns [A] directly from the integrated rate law, sparing you the algebra of inverting the equation by hand. Chemistry students and researchers studying reaction kinetics use it to model how a reactant depletes over time in processes that follow second-order kinetics — such as many gas-phase decomposition and dimerization reactions — and to compare predicted concentrations against experimental data when confirming a reaction's order.

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You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    Unlike first-order decay, a second-order half-life depends on the starting concentration. With k = 0.05 the tool gives 20.00 s at 1 mol/L and 10.00 s at 2 mol/L. The model treats rate as k times [A] squared for a single reactant, and some sources define k for 2A products with a factor of 2, so check yours. Does a plot of 1/[A] against time come out straight in your data?

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