# Reaction Mechanism Simulator - Elementary Steps, Steady-State & Pre-Equilibrium

> Simulate reaction mechanisms from elementary steps: concentration–time curves, steady-state and pre-equilibrium rate laws checked against the full kinetics.

- URL: https://qilingtools.com/en/chemistry/mechanism
- Category: Chemistry Tools (https://qilingtools.com/en/chemistry)
- Tags: [Chemistry](https://qilingtools.com/en/tags/chemistry)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

A complex reaction is a sequence of elementary steps, and each step obeys the law of mass action: its order equals the stoichiometric coefficients of its reactants. This simulator lets you write the steps line by line — "A -> B ; k1 = 1", "B -> C ; k2 = 0.5" or a reversible "A <=> B ; kf = 1, kr = 0.25" — enter initial concentrations, and it builds the rate equations d[X]/dt and integrates them with an adaptive Dormand–Prince RK45 solver. Stiff systems, where rate constants differ by a factor of a million, are detected automatically and finished with an implicit Rosenbrock method, so the curves stay stable and accurate. Take consecutive first-order reactions A → B → C with k₁ = 1 s⁻¹, k₂ = 0.5 s⁻¹ and [A]₀ = 1 M: the intermediate B peaks at t_max = ln(k₁/k₂)/(k₁ − k₂) = 1.386 s with [B]max = 0.5 M, and the simulation matches the analytic solution to better than 10⁻⁶. Ten classic textbook mechanisms are built in — parallel reactions (product ratio k₁/k₂), a reversible reaction approaching equilibrium (K = k_f/k_r), Lindemann–Hinshelwood unimolecular fall-off, the Michaelis–Menten enzyme mechanism, the H₂ + Br₂ chain reaction, 2NO + O₂ pre-equilibrium and ozone decomposition — each with the rate law derived by the steady-state or pre-equilibrium approximation and a point-by-point comparison with the full simulation that shows when the approximation holds. You also get a rate table that flags equilibrated and slowest steps, maxima and half-lives, conserved quantities from the stoichiometric matrix with an atom-balance check, optional Arrhenius temperature dependence, log axes, hover readout and CSV export.

## How to use

1. Pick a classic mechanism or type your own elementary steps, one per line, as "A + B -> C ; k = 0.5"
2. Enter initial concentrations and the time span; optionally turn on Arrhenius temperature dependence with Ea values
3. Read the concentration–time plot, rate table, derived rate law and approximation test, and export CSV or SVG

## Features

- Line-by-line elementary steps with reversible steps, termolecular steps and third bodies (M)
- Rate equations generated automatically from the law of mass action
- Adaptive RK45 with stiffness detection and automatic switch to an implicit Rosenbrock solver — stable for k ratios of 10⁶
- 10 presets: consecutive, parallel, reversible, second order, Lindemann, Michaelis–Menten, H₂–Br₂ chain, 2NO + O₂, ozone and Arrhenius
- Steady-state and pre-equilibrium rate laws compared with the full simulation (maximum deviation)
- Forward and reverse rates of every step at a chosen time, with equilibrium and slowest steps highlighted
- Intermediate maxima, half-lives, conserved quantities and per-step atom balance
- Log time / concentration axes, hover readout, CSV and SVG export

## FAQ

### What is the steady-state approximation and when is it valid?

The steady-state approximation assumes that a reactive intermediate (a radical, an enzyme–substrate complex) stays at a low, nearly constant concentration, so d[I]/dt ≈ 0. That lets you express the intermediate in terms of reactants and derive an overall rate law. It requires the intermediate to be consumed much faster than it is formed — k₂ ≫ k₁ in A → B → C, or [E]₀ ≪ [S]₀ + K_M for Michaelis–Menten. The simulator compares the SSA prediction with the full numerical solution and rates deviations below 1% as excellent.

### How does the pre-equilibrium approximation differ from the steady-state approximation?

The pre-equilibrium approximation assumes a fast reversible step reaches equilibrium (forward and reverse rates nearly equal) before a slow step controls the overall rate — for example 2NO ⇌ N₂O₂ (fast) followed by N₂O₂ + O₂ → 2NO₂ (slow), giving d[NO₂]/dt = 2k₂K[NO]²[O₂]. It needs k₋₁ ≫ k₂[O₂]. The steady-state approximation only requires a constant intermediate concentration; when k₋₁ ≫ k₂ both give the same result. The rate table marks steps whose forward and reverse rates differ by less than 5% as "≈ equilibrium".

### How do you find the maximum concentration of B in A → B → C?

[B] = k₁[A]₀/(k₂ − k₁)·(e^(−k₁t) − e^(−k₂t)). Setting the derivative to zero gives t_max = ln(k₁/k₂)/(k₁ − k₂), and substituting back gives [B]max = [A]₀(k₂/k₁)^(k₂/(k₁−k₂)). With k₁ = 1, k₂ = 0.5 and [A]₀ = 1, t_max = 1.386 and [B]max = 0.5. If k₁ = k₂, t_max = 1/k₁ and [B]max = [A]₀/e.

### Why is the H₂ + Br₂ reaction three-halves order?

It is a chain reaction: Br₂ → 2Br initiates, Br + H₂ → HBr + H and H + Br₂ → HBr + Br propagate, H + HBr → H₂ + Br inhibits, and 2Br → Br₂ terminates. Applying the steady-state approximation to Br and H gives [Br] = (k₁[Br₂]/k₅)^½ and d[HBr]/dt = 2k₂(k₁/k₅)^½[H₂][Br₂]^½/(1 + (k₄/k₃)[HBr]/[Br₂]). Early on [HBr] ≈ 0, so the rate is proportional to [H₂][Br₂]^½ — order 3/2 — and accumulating HBr slows the reaction. In the simulator you can see the radicals sit at only 10⁻⁴–10⁻⁷ M and reach steady state almost immediately.

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