Levella

Meet Levella, a crowd of particles on two steps, where the heat decides how many can climb up.

Particles in a warm system keep swapping energy as they collide. Most sit in low-energy states, but random kicks lift a few into higher ones. The higher a state's energy, the rarer it is, and the hotter the system, the more particles make the climb.

The ratio of particles in two states is N₂/N₁ = e^(−ΔE/kT). At room temperature, 300 K, kT ≈ 0.026 eV. For a step of ΔE = 0.1 eV the ratio is about 0.021, so roughly 2 particles in every 100 are up.

Double the temperature to 600 K and the ratio grows to about 0.14. That is seven times as many, even though the temperature only doubled.

This simple factor explains why chemical reactions speed up so much when warmed. It also tells you how many atoms in a star are excited enough to make each spectral line.