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)

Hard-disk gas: elastic hits vs an activation speed threshold; T and Eₐ vs exp(−Eₐ/RT).

About this model

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.