An aerogel's insulating power comes from trapping gas inside pores that are only tens of nanometres wide — comparable to, or smaller than, the distance a gas molecule normally travels between collisions with other gas molecules (its mean free path, λ). Kinetic theory gives, for a gas at pressure P and temperature T with molecular collision diameter σ:
λ = k_B T / (√2 π σ² P)
Kn = λ / d (Knudsen number, d = pore diameter)
In bulk gas (Kn ≪ 1) molecules mostly hit each other, and remarkably the thermal conductivity κ_gas of an ideal gas does not depend on pressure — as λ shrinks with rising P, the number density of carriers rises to compensate. But once the pore is smaller than λ (Kn ≳ 1), molecules hit the solid pore walls far more often than each other. Each wall collision thermalises the molecule to the (poorly conducting) silica skeleton instead of relaying energy ballistically, so conduction collapses. The standard confined-gas model (Kaganer 1969; used throughout the aerogel literature, e.g. Lu et al. 1992) is:
κ_gas(d) = κ_gas,0 / (1 + 2β·Kn), β ≈ 2 (accommodation)
This is why silica aerogels — 90–99.8% air by volume — can beat still air itself as an insulator: their characteristic pore size (~20–40 nm) is deliberately built smaller than air's mean free path (~68 nm at 1 atm, 300 K), pushing Kn above 1 and suppressing the gas contribution well below its free value. The solid silica backbone is itself a poor conductor because it is a tortuous, nanometre-thin necklace of particles (large Kapitza-type boundary resistance at every neck), and the open structure keeps radiative transfer small too — the three effects together give aerogels thermal conductivities as low as 12–20 mW/m·K, below still air's 26 mW/m·K.
- Pore diameter — sets d directly; shrinking it (denser silica cage, more nanoparticle "necks" visible) raises Kn and suppresses gas κ.
- Pressure — lowering P stretches λ (fewer molecules to collide with) even in a wide pore, which is exactly how an evacuated panel pushes Kn ≫ 1 without needing nanoscale pores.
- Temperature — raises λ slightly and raises the intrinsic gas κ; the two partly offset.
- Gas species — a larger molecule (bigger collision cross-section σ) has a shorter λ, so it needs an even smaller pore to reach the same suppression.
The wall-hits/gas-hits counters in the "Live Simulation" panel come from an actual bouncing-ball kinetic simulation in the 3D view (elastic collisions, momentum-conserving) — watch that ratio swing toward wall hits as you shrink the pore, the same qualitative shift the Knudsen formula predicts.