Langmuir Isotherm Calculator
Half coverage sits at exactly one over K, which is how the constant is read off an experiment.
The formula
theta = K P / (1 + K P)
One layer, identical sites, no interaction
The Langmuir isotherm rests on three assumptions: adsorption stops at a single molecular layer, every site is equally attractive, and adsorbed molecules ignore one another. Real surfaces satisfy none of these exactly, and the isotherm still describes a great many of them well, which is the usual fate of a good idealisation.
Half coverage locates K
Set theta to a half and the equation gives P equal to 1/K directly. That makes the constant easy to read off an experiment: find the pressure at which the surface is half covered and take its reciprocal. A large K means strong adsorption, saturating at low pressure.
Saturation is the point
At low pressure coverage rises almost linearly; at high pressure it flattens towards one and no further pressure helps, because there is nowhere left to adsorb. That plateau is what distinguishes monolayer adsorption from the multilayer kind, where the amount taken up keeps climbing.
Where these models stop
Michaelis-Menten assumes a single substrate and a steady state; Langmuir assumes one layer on identical sites with no interaction between them; Stokes-Einstein assumes a hard sphere much larger than the solvent molecules around it. Each is an idealisation that happens to describe real systems well over a useful range, and each fails predictably outside it. Knowing which assumption a number rests on is usually more valuable than the number.