Calculate Langmuir adsorption isotherm parameters. Find maximum adsorption capacity, equilibrium constant, and separation factor from experimental data.
The Langmuir adsorption isotherm is the most widely used model for describing monolayer adsorption of molecules onto a surface. Proposed by Irving Langmuir in 1918, it assumes that adsorption occurs at specific homogeneous sites on the surface, each site can hold only one molecule, there are no interactions between adsorbed molecules, and adsorption is reversible. The model gives the relationship: q = qmax × KL × Ce / (1 + KL × Ce), where q is the amount adsorbed, qmax is the maximum monolayer capacity, KL is the Langmuir constant, and Ce is the equilibrium concentration.
The Langmuir model is fundamental in environmental engineering (water treatment with activated carbon), catalysis (heterogeneous catalysis on solid surfaces), biochemistry (receptor-ligand binding follows identical mathematics), and materials science (gas storage on porous materials). Its simplicity and physical interpretability make it the starting point for all adsorption studies.
The separation factor RL = 1/(1 + KL × C0), where C0 is the initial concentration, indicates the favorability of adsorption: RL = 0 is irreversible, 0 < RL < 1 is favorable, RL = 1 is linear, and RL > 1 is unfavorable. Understanding these parameters helps engineers design adsorption systems for pollutant removal, gas purification, and chromatographic separations.
Essential for environmental engineers designing water treatment systems, researchers studying surface chemistry, and students learning adsorption theory. Analyze isotherm data, estimate design parameters, and compare adsorption models efficiently. This langmuir isotherm calculator helps you compare outcomes quickly and reduce avoidable mistakes when making day-to-day care decisions. Use the estimate as a planning baseline and confirm final decisions with a qualified professional when risk is high.
Langmuir Isotherm: qe = (qmax × KL × Ce) / (1 + KL × Ce). Linearized form: Ce/qe = (1/qmax)Ce + 1/(qmax × KL). Separation factor: RL = 1/(1 + KL × C₀). Surface coverage: θ = qe/qmax = KL × Ce/(1 + KL × Ce). Freundlich comparison: qe = KF × Ce^(1/n).
Result: qe = 75.0 mg/g, θ = 50.0%
With qmax = 150 mg/g and KL = 0.05 L/mg: qe = (150 × 0.05 × 20)/(1 + 0.05 × 20) = 150/2 = 75 mg/g. Surface coverage θ = 75/150 = 50%. At Ce = 20 mg/L, half the available sites are occupied.
Four linearized forms of the Langmuir equation exist: (1) Ce/qe vs Ce (most common, best for large Ce values), (2) 1/qe vs 1/Ce (Lineweaver-Burk, familiar to biochemists), (3) qe vs qe/Ce, and (4) qe/Ce vs qe. Each linearization gives different error distributions, so the "best" form depends on the data structure. Modern practice favors nonlinear least squares fitting using the original equation, which avoids distortion of error structure inherent in linearization. Software like Origin, MATLAB, or Python scipy.optimize provides straightforward nonlinear fitting capabilities.
The Langmuir constant KL is related to the standard free energy of adsorption: ΔG° = −RT ln(KL × Mw × 1000), where the unit conversion depends on the concentration units used. By measuring isotherms at multiple temperatures, the van't Hoff plot (ln KL vs 1/T) yields ΔH° and ΔS° of adsorption. Negative ΔH° indicates exothermic adsorption (physisorption < −40 kJ/mol; chemisorption typically −40 to −800 kJ/mol). Physical adsorption is driven by van der Waals forces, while chemisorption involves chemical bond formation.
In water treatment, activated carbon adsorbers are designed using isotherm data combined with mass transfer zone analysis. The design procedure involves: (1) measuring batch isotherms to determine qmax and KL, (2) running column studies to determine mass transfer coefficients, (3) modeling breakthrough curves, and (4) scaling up to full design. The Langmuir isotherm determines the equilibrium capacity — the maximum amount of pollutant that can be removed per gram of carbon. This sets the minimum carbon usage rate and the required column regeneration frequency.
qmax is the maximum monolayer adsorption capacity — the amount adsorbed when every available surface site is occupied. Typical values for activated carbon adsorbing organics range from 50-500 mg/g, depending on the adsorbent and adsorbate.
KL (L/mg or L/mol) reflects the affinity between adsorbent and adsorbate. Higher KL means stronger binding and steeper initial rise in the isotherm. It's related to the thermodynamic equilibrium constant and the free energy of adsorption.
The Langmuir model fails for: heterogeneous surfaces (use Freundlich), multilayer adsorption (use BET), surfaces with lateral interactions (use Temkin), and when adsorption is irreversible. Real systems often deviate at very high or very low coverage.
Plot Ce/qe vs Ce. The slope = 1/qmax and the intercept = 1/(qmax × KL). From the slope: qmax = 1/slope. From the intercept: KL = slope/intercept. Alternatively, use nonlinear curve fitting directly.
RL between 0 and 1 indicates favorable adsorption. RL close to 0 means nearly irreversible (very favorable). RL = 1 means linear (Henry's law region). RL > 1 means unfavorable adsorption. Most practical systems have RL = 0.01-0.5.
Langmuir assumes monolayer adsorption on homogeneous surface with a finite maximum capacity. Freundlich is empirical, works for heterogeneous surfaces, and has no maximum capacity (qe = KF × Ce^(1/n)). Neither model captures the full range of behavior in most real systems.