Integrated Rate Law Calculator
Determine reaction order (0th, 1st, or 2nd) from time-concentration data by linearizing [A], ln[A], and 1/[A] vs. time t, comparing R² and RMSE, and calculating rate constant k and half-life.
Open toolReaction Rate Calculator
Calculate the rate of reactant disappearance (`-Δ[A]/Δt`), stoichiometrically normalized reaction rate (`-Δ[A]/(a·Δt)`), integrated rate constant `k`, and half-life `t₁/₂` from initial and final molarity.
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Enter the inputs, then select Calculate Reaction Rate.
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How the Chemical Reaction Rate Calculator Works
The calculator evaluates the concentration change `Δ[A] = [A]ₜ - [A]₀` over elapsed time `Δt`, normalizes by the balanced equation's stoichiometric coefficient `a`, and solves the integrated rate law for your chosen reaction order `n`.
Rate_norm = ([A]₀ - [A]ₜ) / (a * Δt); k_1st = ln([A]₀ / [A]ₜ) / (a * Δt); t_half_1st = ln(2) / (a * k_1st).
Reports rates in both decimal and scientific notation (`M/s` and `M/min`) for fast lab-report verification.
Chemical Kinetics Rate Laws, Integrated Equations & Half-Life Table
Compare differential rate laws, integrated concentration equations, linear graph plots, and half-life formulas for 0th, 1st, and 2nd order reactions (`A -> Products`).
| Order (n) | Differential Rate Law | Integrated Rate Law | Units of k | Half-Life (t₁/₂) |
|---|---|---|---|---|
| Zero-Order (`n = 0`) | `Rate = k` | `[A]ₜ = [A]₀ - k·t` | `M·s⁻¹` (`mol·L⁻¹·s⁻¹`) | `[A]₀ / (2k)` |
| First-Order (`n = 1`) | `Rate = k[A]` | `ln([A]ₜ) = ln([A]₀) - k·t` | `s⁻¹` | `ln(2) / k ≈ 0.693 / k` |
| Second-Order (`n = 2`) | `Rate = k[A]²` | `1/[A]ₜ = 1/[A]₀ + k·t` | `M⁻¹·s⁻¹` (`L·mol⁻¹·s⁻¹`) | `1 / (k·[A]₀)` |
How to Calculate Reaction Rate from Concentration Data
Solve kinetics homework and spectrophotometry lab data in three steps.
Enter initial [A]₀ and final [A]ₜ molarity
Supply the starting and ending concentrations in `mol/L` (`M`).
![Reaction Rate Calculator step 1: Enter initial [A]₀ and final [A]ₜ molarity](/tools/assets/standalone-diagrams/reaction-rate-calculator-step-1.svg)
Input elapsed time Δt and stoichiometric coefficient a
Enter the time interval `Δt` in seconds and the reactant's balancing coefficient `a` from `aA -> products`.
Select reaction order (0th, 1st, or 2nd)
Click Calculate Reaction Rate to view the rate of disappearance, normalized reaction rate, integrated rate constant `k` with exact units, and half-life `t₁/₂`.
Worked examples for Reaction Rate Calculator
Compare baseline and alternative runs for Reaction Rate Calculator before exporting the result.
Baseline example
Calculate average chemical reaction rate, reactant consumption rate, apparent rate constant k, and half-life from initial and final molarity and Δt.
Comparison scenario
Rate_norm = ([A]₀ - [A]ₜ) / (a * Δt); k_1st = ln([A]₀ / [A]ₜ) / (a * Δt); t_half_1st = ln(2) / (a * k_1st).
Boundary check before exporting
Reports rates in both decimal and scientific notation (`M/s` and `M/min`) for fast lab-report verification.
How to interpret the Reaction Rate Calculator output
Review the calculated output alongside your entered parameters and unit conventions.
Reports rates in both decimal and scientific notation (`M/s` and `M/min`) for fast lab-report verification.
Input limits, assumptions, and related tools
Check the model scope and boundary conditions before applying the numbers to a real project.
Rate_norm = ([A]₀ - [A]ₜ) / (a * Δt); k_1st = ln([A]₀ / [A]ₜ) / (a * Δt); t_half_1st = ln(2) / (a * k_1st). Reports rates in both decimal and scientific notation (`M/s` and `M/min`) for fast lab-report verification.
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Open toolFrequently Asked Questions
Answers about using Reaction Rate Calculator and understanding its results.
What is the formula for the average rate of a chemical reaction?
For a balanced reaction aA + bB -> cC + dD, the unique normalized reaction rate is Rate = -(1/a)*(Δ[A]/Δt) = -(1/b)*(Δ[B]/Δt) = +(1/c)*(Δ[C]/Δt) = +(1/d)*(Δ[D]/Δt), expressed in M/s (mol·L⁻¹·s⁻¹). The negative sign for reactants ensures the overall reaction rate is always positive when reactants are consumed.
Why do you divide by the stoichiometric coefficient when calculating reaction rate?
In a reaction like 2 N₂O₅ -> 4 NO₂ + O₂, NO₂ is produced four times as fast as O₂ and twice as fast as N₂O₅ disappears. Dividing each species' rate of change by its stoichiometric coefficient (1/2 for N₂O₅, 1/4 for NO₂, 1/1 for O₂) yields a single, unambiguous reaction rate regardless of which species is measured.
How do the units of the rate constant k change with reaction order?
Because Reaction Rate always has units of M·s⁻¹ and Rate = k·[A]ⁿ, the units of k are M^(1-n)·s⁻¹: M·s⁻¹ (mol·L⁻¹·s⁻¹) for zero-order (n=0), s⁻¹ for first-order (n=1), and M⁻¹·s⁻¹ (L·mol⁻¹·s⁻¹) for second-order (n=2).
What is the difference between average reaction rate and instantaneous reaction rate?
The average reaction rate is the secant slope -Δ[A]/Δt between two time points t₁ and t₂. Because reactant concentration drops over time, the instantaneous rate -d[A]/dt (the tangent slope at time t) starts higher than the average rate and decreases as the reaction proceeds (for n >= 1).
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