The Langmuir Adsorption Isotherm: Derivation and Its Limits

The assumptions, stated first

These are worth learning before the equation, because questions frequently ask which assumption fails in a given situation:

• Adsorption forms a monolayer only — no stacking.
• All surface sites are energetically identical.
• There is no interaction between molecules adsorbed on neighbouring sites.
• Each site holds at most one molecule, and adsorption is reversible.

The derivation

Let θ be the fraction of surface sites occupied. Adsorption requires an empty site and a gas molecule, so its rate is proportional to the pressure and to the free fraction:

Desorption requires an occupied site:

At equilibrium the two are equal:

Writing K = k_a/k_d and rearranging:

The two limits

That saturation plateau is the model's most characteristic prediction and the clearest way to recognise Langmuir behaviour in experimental data.

The linear form used for plotting

Taking reciprocals gives a straight-line form suitable for testing the model:

A plot of p/x against p is linear if the model applies; the slope gives the monolayer capacity x_m and the intercept gives K. Being asked what the slope and intercept represent is a routine question, and answering it requires knowing which form you plotted.

Where the model breaks, and what replaces it

The Freundlich isotherm, θ ∝ p^1/n, is empirical rather than derived and predicts no saturation at all — which is precisely why it fails at high pressure while working well over a middle range. The BET isotherm extends Langmuir to multilayers and is the basis of the standard method for measuring surface area, since the first-layer capacity it returns converts directly to area if the cross-section of the adsorbate is known.

Physisorption and chemisorption

The temperature row is the discriminating one, and the reason is worth stating: chemisorption requires activation energy, so raising temperature initially helps, until desorption begins to dominate.

Why this matters for catalysis

Heterogeneous catalysis proceeds by adsorption of reactants onto the surface, reaction there, then desorption of products. The Langmuir picture supplies the coverage term, and combining it with a surface reaction step gives Langmuir–Hinshelwood kinetics, in which both reactants must be adsorbed.

The result explains an initially puzzling observation: catalytic rate can fall as reactant pressure rises. If one reactant adsorbs so strongly that it crowds the other off the surface, increasing its pressure reduces the rate. Producing that explanation is a standard higher-order question, and it follows directly from the coverage expression.

Frequently asked questions

What does K represent physically?

The ratio of adsorption to desorption rate constants, so it measures how strongly the adsorbate binds. A large K means saturation is reached at low pressure.

Can the isotherm apply to adsorption from solution?

Yes, with concentration in place of pressure. The same assumptions apply and fail in the same ways, and solution-phase questions are common in analytical contexts.

Why does Freundlich work when Langmuir fails?

Because it is empirical and its exponent absorbs the effect of a heterogeneous surface. It has no saturation limit, so it should not be used at high pressure — a limitation questions test directly.

How is BET used to measure surface area?

It returns the monolayer capacity, which is the number of molecules needed to cover the surface once. Multiplying by the cross-sectional area of the adsorbate gives the total area, and nitrogen at low temperature is the standard choice.