Langmuir Isotherm Calculator
Find fractional surface coverage (theta) or equilibrium adsorption capacity (qe) from the Langmuir adsorption isotherm model.
🧲 What is the Langmuir Isotherm?
The Langmuir isotherm is a mathematical model that describes how molecules or ions from a gas or solution stick to, or adsorb onto, the surface of a solid at a fixed temperature. It predicts the fractional surface coverage, the proportion of available surface sites that are occupied, as a function of the equilibrium concentration of the adsorbing substance and a binding strength constant. Named after Irving Langmuir, who developed the model in 1918, it remains one of the most widely used equations in surface chemistry.
The Langmuir isotherm shows up across many fields. Environmental engineers use it to model how activated carbon or clay minerals adsorb pollutants and heavy metals from contaminated water. Catalysis researchers use it to describe how reactant gases adsorb onto a catalyst's surface before reacting. Pharmaceutical scientists apply the same equation to drug adsorption onto excipients and delivery vehicles. Biochemists recognize its identical mathematical form in the Michaelis-Menten equation, which describes enzyme kinetics.
A common point of confusion is treating the Langmuir isotherm as interchangeable with other adsorption models. It is not. The Langmuir model specifically assumes a single layer of adsorbate (a monolayer), a fixed and finite number of identical, independent sites, and no interaction between adsorbed molecules. Real surfaces with multiple layers, a range of site energies, or strong lateral interactions between adsorbed molecules are usually better described by the Freundlich isotherm or the BET (Brunauer-Emmett-Teller) model instead.
This calculator is useful because it lets you move quickly between the two most common questions asked about a Langmuir system: given a binding constant K and a concentration, how much of the surface is covered right now (fractional coverage mode), and given a known maximum capacity, how much material has actually adsorbed at equilibrium (adsorption capacity mode). Both modes plot the full saturation curve so you can see exactly where your current input sits relative to full surface saturation.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Moderate coverage on activated carbon
An adsorbent has a Langmuir constant K = 0.05 L/mg. The equilibrium concentration in solution is C = 20 mg/L.
Example 2 — Near-saturation coverage
A stronger-binding system has K = 0.1 L/mg at a higher equilibrium concentration, C = 50 mg/L.
Example 3 — Adsorption capacity from a known qmax
An activated carbon adsorbent has qmax = 100 mg/g and K = 0.05 L/mg, at an equilibrium concentration Ce = 20 mg/L.
Example 4 — Larger-capacity adsorbent
A higher-capacity adsorbent has qmax = 250 mg/g and a weaker binding constant K = 0.02 L/mg, at Ce = 100 mg/L.
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Langmuir isotherm equation?
The Langmuir isotherm gives fractional surface coverage as theta equals KC divided by (1 plus KC), where K is the Langmuir equilibrium constant and C is the equilibrium concentration of adsorbate in solution or gas phase. In capacity form it is qe equals qmax times K times Ce, all divided by (1 plus K times Ce), where qmax is the maximum monolayer adsorption capacity.
What does the Langmuir constant K represent?
K is the Langmuir equilibrium constant, a measure of the binding affinity between the adsorbate and the surface. A larger K value means adsorption saturates the surface at a lower equilibrium concentration, indicating stronger binding, while a smaller K means higher concentrations are needed to reach the same coverage.
What is fractional surface coverage theta?
Fractional surface coverage (theta) is the proportion of available adsorption sites that are occupied by adsorbate molecules, expressed as a number between 0 (empty surface) and 1 (fully saturated monolayer). A theta of 0.75 means 75% of the surface sites are occupied at that equilibrium concentration.
How do I find qmax from experimental adsorption data?
Qmax, the maximum monolayer adsorption capacity, is usually found by fitting a linearized form of the Langmuir equation (plotting Ce/qe against Ce gives a straight line whose slope is 1/qmax) to a series of measured qe values at different equilibrium concentrations Ce. This calculator assumes you already know qmax, K, and Ce, and computes qe directly rather than performing that regression.
What are the main assumptions of the Langmuir adsorption model?
The Langmuir model assumes adsorption forms a single molecular layer (monolayer) on a surface with a fixed number of identical, energetically equivalent sites, that each site holds exactly one adsorbate molecule, and that adsorbed molecules do not interact with each other. These assumptions make it accurate for chemisorption on uniform surfaces but less accurate for porous or heterogeneous materials.
Why does the Langmuir isotherm curve flatten out at high concentration?
As concentration C increases, more adsorption sites fill up, but the total number of sites is fixed. Once nearly all sites are occupied, adding more adsorbate in solution can no longer increase coverage significantly, so theta approaches 1 and the qe curve approaches qmax, producing the characteristic flattening or saturation plateau.
What is the difference between the Langmuir and Freundlich isotherms?
The Langmuir isotherm assumes a finite number of identical sites and predicts a maximum saturation plateau (qmax), making it appropriate for monolayer chemisorption on uniform surfaces. The Freundlich isotherm is an empirical power-law model with no defined maximum, often fitting adsorption on heterogeneous surfaces with a range of site energies better, but it does not predict a saturation limit.
Can theta or qe ever be negative?
No, for any physically valid Langmuir isotherm calculation with K greater than 0 and concentration C at or above 0, both theta and qe are always zero or positive, and theta can never exceed 1 while qe can never exceed qmax. A negative result indicates an invalid or negative input value.
What units does the Langmuir constant K use?
K is not dimensionless; its units are the inverse of whatever concentration unit is used for C. If C is measured in mg/L, K carries units of L/mg. If C is measured in mol/L (molarity), K carries units of L/mol. Always match the units of K and C when comparing values from different sources.
How is the Langmuir isotherm used in real applications?
The Langmuir isotherm is widely used to model gas adsorption onto catalyst surfaces, activated carbon adsorption of pollutants from water for environmental remediation, drug adsorption in pharmaceutical formulation, and enzyme-substrate binding kinetics, where its same mathematical form (a rectangular hyperbola) also appears as the Michaelis-Menten equation.
Is the Langmuir isotherm the same equation as Michaelis-Menten kinetics?
They share the identical mathematical form, a rectangular hyperbola that rises quickly then saturates, but describe different physical processes. Langmuir describes equilibrium surface coverage from concentration and a binding constant, while Michaelis-Menten describes enzyme reaction rate from substrate concentration and the Michaelis constant. The underlying algebra, and the shape of the resulting curve, are the same.
What happens to theta when concentration C is very small?
At low concentration, the term KC in the denominator is much smaller than 1, so the Langmuir equation approximately simplifies to theta equals KC, a straight line through the origin. This is why Langmuir adsorption looks nearly linear at low concentration and only curves noticeably as C increases and the surface starts filling up.