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Adsorption sites and the Langmuir isotherm

Inspect the periodic potential wells on a solid surface — the adsorption sites — and connect their finite number to the saturation shape of the Langmuir isotherm.

Goal

Derive the Langmuir isotherm $\theta=\frac{Kp}{1+Kp}$ assumptions from the picture: fixed sites, monolayer, no lateral interaction.

Apparatus and reagents

Rotatable 3D model of a corrugated adsorption surface; reference data for a Langmuir fit (coverage vs pressure).

Procedure

Rotate the surface and locate the periodic wells; count the marked sites.
Imagine molecules landing one per site: list what the model forbids (double occupancy, hopping between sites).
Reason that coverage θ is proportional to pressure at low p but approaches 1 when every site is full — the Langmuir shape.
Contrast with multilayer condensation (BET) when the adsorbate wets the surface above the first layer.

What to observe

The surface potential is periodic: molecules see a regular array of equivalent binding wells rather than a smooth plane.
Because the number of wells is finite, coverage must saturate: θ → 1 as p → ∞, never growing unboundedly.

Explanation

Langmuir’s 1918 model assumes identical independent sites, at most one molecule per site, and no interaction between adsorbates. Balancing adsorption rate $k_a p (1-\theta)$ against desorption $k_d\theta$ gives $\theta=\frac{Kp}{1+Kp}$ , the equation behind heterogeneous catalysis (Sabatier principle: bind neither too weakly nor too strongly).

History and context

Irving Langmuir received the 1932 Nobel Prize for surface chemistry after deriving his isotherm at General Electric; Gerhard Ertl (2007) turned the same picture of surface sites into the molecular toolkit that explains catalytic converters.

Related chemists

Irving Langmuir Gerhard Ertl Paul Sabatier