Why does the cumulative activated/total ratio not equal exp(−Eₐ/RT)?

The threshold rule in model units is a cartoon, collisions are counted in 2D disks without orientation or tunneling, and statistics are noisy at small N. exp(−Eₐ/RT) is shown as a **reference trend**, not a fit to the simulation.

What does recoloring particles to yellow mean?

Particles whose **speed** is near or above a rough fraction of the activation threshold are tinted warmer—purely visual intuition for who is “hot enough,” not a separate chemistry species.

Are these real molecule sizes and masses?

No. Radii and the mapping from Eₐ (kJ/mol) to speed threshold are chosen for on-screen behavior, not for a specific compound.

Why 2D instead of 3D?

Fewer degrees of freedom and simpler overlap tests make a smooth classroom animation on the web; the qualitative T dependence is the teaching goal.

Collision Theory (2D Particles)

**Collision theory** connects reaction rates to **collision frequency** and to the **fraction of collisions** that carry enough **energy** (and the right **geometry**, omitted here) to surmount an **activation barrier Eₐ**. This page is a **two-dimensional hard-disk** toy gas: disks undergo **elastic** billiard collisions in a box, with **Gaussian random velocities** whose scale grows like **√T** so that **mean kinetic energy** tracks temperature in a Maxwell–Boltzmann spirit. When the **relative speed** of a colliding pair exceeds a **threshold** tied to **Eₐ**, the encounter is tallied as an **“activated” collision**. The panel also shows the **Boltzmann factor** exp(−Eₐ/RT) for the same numbers—**same qualitative message** as **Arrhenius**, but the particle ratio is **not calibrated** to reproduce that exponential exactly (no orientation factor, no real pair potential, finite count, discrete time steps). A small **speed histogram** gives a visual for how **heating** fattens the high-speed tail. **Reset** reshuffles positions and clears cumulative counters.

Who it's for: High school and introductory college chemistry linking qualitative collision ideas to Arrhenius plots; complements the existing reaction-rate and Arrhenius sliders without replacing molecular dynamics.

Key terms

Collision theory
Activation energy
Arrhenius equation
Boltzmann factor
Maxwell–Boltzmann distribution
Elastic collision
Reaction rate

How it works

**Collision theory** links reaction rate to **how often** molecules collide and **how often** those collisions carry enough **energy** along the reaction coordinate. This toy **2D hard-disk** tank gives a **visual** sense that **raising T** spreads speeds upward so more encounters exceed a **threshold** tied to **Eₐ**. The dimensionless **Boltzmann factor** exp(−Eₐ/RT) is shown for comparison — it is **not** fitted to the particle counts here.

Key equations

k ∝ exp(−Eₐ / RT) (Arrhenius-style temperature dependence)

Fraction with enough energy ~ exp(−Eₐ / RT) (qualitative)

Frequently asked questions

Why does the cumulative activated/total ratio not equal exp(−Eₐ/RT)?: The threshold rule in model units is a cartoon, collisions are counted in 2D disks without orientation or tunneling, and statistics are noisy at small N. exp(−Eₐ/RT) is shown as a **reference trend**, not a fit to the simulation.
What does recoloring particles to yellow mean?: Particles whose **speed** is near or above a rough fraction of the activation threshold are tinted warmer—purely visual intuition for who is “hot enough,” not a separate chemistry species.
Are these real molecule sizes and masses?: No. Radii and the mapping from Eₐ (kJ/mol) to speed threshold are chosen for on-screen behavior, not for a specific compound.
Why 2D instead of 3D?: Fewer degrees of freedom and simpler overlap tests make a smooth classroom animation on the web; the qualitative T dependence is the teaching goal.