Langmuir Isotherm Calculator

The Langmuir isotherm is a foundational model for adsorption at a solid or fluid interface. It relates the equilibrium amount adsorbed per unit mass of adsorbent, q, to the equilibrium concentration of adsorbate, C. The equation is q = qmax K C divided by (1 + K C), where qmax is the maximum monolayer adsorption capacity and K is the Langmuir affinity constant. The model assumes a uniform surface containing a fixed number of equivalent adsorption sites. Each site holds at most one adsorbate unit, adsorbed molecules do not interact with one another, and adsorption and desorption reach dynamic equilibrium. Under these assumptions, coverage approaches a single monolayer. At low concentration, q rises almost linearly with C. At high concentration, the surface saturates and q approaches qmax. The affinity constant K controls how quickly the isotherm approaches saturation. A larger K means stronger apparent affinity and substantial coverage at a lower equilibrium concentration. Its units are the reciprocal of the concentration units used for C. The units of q and qmax must match, commonly milligrams per gram or moles per kilogram. This calculator leaves units flexible so long as they are internally consistent. All four variables can be useful unknowns. Given qmax, K, and C, the standard equation returns q. Rearrangement can estimate qmax from a measured equilibrium point, or solve for K or C when q remains below qmax. A single point gives only an algebraic estimate; robust experimental parameter estimation normally fits many concentration and adsorption observations, preferably by nonlinear regression with uncertainty analysis. Langmuir analysis appears in environmental remediation, activated-carbon treatment, catalysis, gas storage, chromatography, biosorption, and surface chemistry. Researchers compare fitted capacity and affinity across materials, temperatures, pH conditions, or competing solutes. The model can also help estimate whether an adsorbent is operating in a concentration-sensitive region or close to saturation. Real surfaces may violate the ideal assumptions. They can contain sites with different energies, allow multilayer adsorption, exhibit pore filling, or show interactions between adsorbates. Freundlich, Sips, Temkin, BET, and other models may fit such systems better. A visually good fit alone does not prove the physical Langmuir mechanism. Use equilibrium rather than initial concentration for C, keep q and qmax in the same units, and ensure K is compatible with C. When solving for K or C, q must be less than qmax because the ideal model reaches capacity only asymptotically. This calculator performs the exact rearrangement for the selected unknown and is intended for transparent teaching, quick checks, and preliminary adsorption analysis.