2D Adsorption Isotherm Comparison

Langmuir show

Saturation q_max 120

Affinity K 4.0

Freundlich show

Coefficient K_f 40

Heterogeneity 1/n 0.45

BET show

Monolayer q_m 70

BET constant C 60

Coverage animation focus

Langmuir

Freundlich

BET

Readouts

Rel. pressure P

0.00

Uptake q

0.0

The science

An adsorption isotherm gives the uptake q of a gas or solute on a surface as the relative pressure P (0 to 1) rises at constant temperature. Three classic models are overlaid:

Langmuir:   q = qmax * K * P / (1 + K * P)
Freundlich: q = Kf * P^(1/n)
BET:        q = qm * C * P / [(1-P) * (1 - P + C*P)]

Langmuir: identical, independent sites, one molecule each; K = k_ads/k_des. Plateau at q_max.
Freundlich: empirical power law for heterogeneous surfaces; no plateau.
BET: molecules also stack on top of the first layer; the curve diverges as P approaches 1 (condensation).

The surface picture is a real kinetic Monte Carlo, not a lookup. Each cell is an adsorption site that randomly captures and releases molecules. Langmuir: identical sites, so coverage settles at KP/(1+KP). Freundlich: every site gets its own affinity K_i drawn from a heavy-tailed (Pareto) distribution, so the sum of many Langmuir sites reproduces a power law whose exponent is 1/n. BET: each cell is a column; the first molecule is held strongly (leaves at k/C), later ones like liquid (leave at k), which is exactly the chain that gives the BET formula. "Occupied" and "layers" in the readout are counted from the molecules on screen; "theory" is the exact equilibrium of the same model, so the small difference is statistical noise plus a short lag behind the moving pressure.

Simplifications: no lateral interactions between molecules, one temperature, sweep slow enough for near-equilibrium; BET is stopped at P = 0.8 because near P = 1 the multilayer needs very long to equilibrate (the curve itself runs to 1, and the 0.985 cap in the plot avoids the singularity). The Freundlich prefactor K_f is empirical, so the lattice reproduces the exponent 1/n, not K_f. Molecule and surface sprites were rendered in Blender.

Independent numerical check

Headless run of the same kinetic Monte Carlo on 20,000 sites, compared with closed-form results:

Waiting…

Langmuir Freundlich BET — sweep marker follows relative pressure

Surface coverage (focused model)0%