The coverage equation, and what each term means
Fractional coverage θ is the share of surface sites filled. P is the partial pressure of a gas (or Ce, the equilibrium concentration, for a solution) and K is the equilibrium constant that measures how strongly the adsorbate binds. At low pressure θ rises almost linearly; as KP grows it levels off toward 1, a full monolayer.
The full form, in amount adsorbed
In terms of measurable quantities, the amount adsorbed qe approaches the monolayer capacity qmax as concentration rises. Coverage is just qe divided by qmax.
- qmax — the monolayer capacity: the most the surface can hold, set by the number of active sites (mg/g).
- KL — the affinity constant. Higher KL means stronger binding and faster approach to saturation (L/mg).
- Ce — equilibrium concentration left in solution once adsorption has settled (mg/L).
How KP drives the curve
The dimensionless product KP (or KLCe) is what sets coverage. It is what makes the isotherm bend from near-linear to nearly flat.
| KP (or KLCe) | θ (coverage) | State |
|---|---|---|
| 0.1 | 9.1% | Sparse, near-linear regime |
| 0.5 | 33.3% | Filling up |
| 1 | 50.0% | Half the sites occupied |
| 10 | 90.9% | Approaching a full monolayer |
| 100 | 99.0% | Effectively saturated |
Coverage hits 50% exactly when KP = 1, i.e. when the concentration equals 1/K. That midpoint is a quick sanity check on any fitted K.
Reading a fit, and where it fails
- Linearize to check the fit. Plotting Ce/qe against Ce should give a straight line; slope is 1/qmax and intercept is 1/(KLqmax). An R² of 0.95 or better supports the model.
- Keep units consistent. qmax in mg/g, KL in L/mg, Ce in mg/L. Mixing them is the most common error.
- Watch the assumptions. Real surfaces are often heterogeneous and can adsorb in multiple layers; then Freundlich or BET describes the data better. K also shifts with temperature.
Common questions
What does the Langmuir isotherm assume?
A single layer of molecules on a uniform surface, one molecule per fixed site, no interaction between adsorbed molecules, and adsorption in equilibrium with desorption. When those hold, coverage follows a simple curve that flattens out at full saturation.
What is the fractional coverage theta?
Theta is the fraction of surface sites that are occupied, running from 0 (bare) to 1 (a full monolayer). It equals the amount adsorbed divided by the maximum capacity, or KP divided by (1 plus KP).
When should I use Freundlich instead of Langmuir?
Use Freundlich when the surface is heterogeneous or the data suggest multilayer adsorption with no clear maximum capacity. Langmuir fits best when the data plot straight in linearized form and a real saturation limit exists.
