Integrated Rate Law & Kinetics
Apply the integrated rate law for a zero, first or second order reaction — concentration after a time t and the reaction half-life — or paste concentration–time data and let the tool find the order and rate constant from the best linear fit.
How to use this tool
The Rate law tab projects a concentration forward in time for a known order and k. The Find order tab does the reverse: from your measured data it tests which order gives a straight line and reads k off the slope.
What to enter
- Order: 0, 1 or 2 (with respect to the reactant).
- k: the rate constant — units depend on order (0: M·s⁻¹, 1: s⁻¹, 2: M⁻¹·s⁻¹).
- [A]₀, t: initial concentration and the elapsed time.
- Find order: one time, concentration pair per line.
Reading the result
Rate-law mode gives [A] at time t, the fraction remaining, and the half-life. Find-order mode reports the R² of the zero-, first- and second-order plots; the highest R² is the order, and its slope gives k.
Worked example
First order, k = 0.10 s⁻¹, [A]₀ = 1.0 M, t = 10 s → [A] = 1.0·e⁻¹ = 0.368 M, half-life = ln2/k = 6.93 s.
Result
The integrated rate law links concentration to time; the order sets which plot of the data is a straight line.
Methodology
Integrated rate laws
Zero order: [A] = [A]₀ − kt, half-life [A]₀/2k. First order: [A] = [A]₀ e^(−kt) (equivalently ln[A] = ln[A]₀ − kt), half-life ln2/k — independent of concentration. Second order: 1/[A] = 1/[A]₀ + kt, half-life 1/(k[A]₀).
Finding the order
Each order is linear in a different transform of [A]: zero in [A], first in ln[A], second in 1/[A], all against t. The tool runs an ordinary least-squares fit of each against time and reports the coefficient of determination R²; the transform with R² closest to 1 indicates the order, and k comes from the magnitude of the slope.
Sources
- Atkins, P. & de Paula, J. Physical Chemistry — the rates of reactions.
Known limits
- Single-reactant integrated laws; for multi-reactant kinetics use initial rates or isolation.
- Order detection needs clean data over enough of the reaction; noisy or short runs make the R² values close.