Last updated: July 3, 2026
Langmuir Isotherm Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The Langmuir isotherm describes monolayer adsorption on identical independent sites. Fractional coverage is θ = KC/(1 + KC), and amount adsorbed is q = q_maxθ. K controls affinity, C is the equilibrium concentration, and q_max is the saturation capacity.
The Langmuir isotherm calculates fractional coverage as K times C divided by one plus K times C. The amount adsorbed equals q max times that fractional coverage.
Key Takeaways
- The Langmuir isotherm predicts θ = KC/(1 + KC) and q = q_maxθ.
- K × C must be dimensionless, so K and C units must be paired consistently.
- At K × C = 1, coverage is 0.5 and the loading is half of q_max.
- q approaches q_max asymptotically and does not exceed it for valid nonnegative inputs.
- The model assumes identical independent sites and monolayer adsorption.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
θ = (K × C) / (1 + K × C); q = (q_max × K × C) / (1 + K × C) = q_max × θ
Where:
- θ=Fractional surface coverage(dimensionless)
- q=Amount adsorbed at equilibrium(mg/g or chosen loading unit)
- q_max=Maximum monolayer adsorption capacity(mg/g or chosen loading unit)
- K=Langmuir equilibrium constant(L/mg or reciprocal concentration)
- C=Equilibrium adsorbate concentration(mg/L or chosen concentration unit)
Worked Examples
Half coverage and half capacity
K × C equals 1, so the surface reaches exactly half of the monolayer capacity.
- 1Compute the dimensionless product: K × C = 0.5 × 2 = 1.
- 2Calculate fractional coverage: θ = 1 / (1 + 1) = 0.5.
- 3Calculate amount adsorbed: q = q_max × θ = 100 × 0.5 = 50.
Moderate-to-high affinity adsorption
A larger K value gives two-thirds surface coverage at C = 1.
- 1Compute K × C = 2 × 1 = 2.
- 2Calculate fractional coverage: θ = 2 / (1 + 2) = 2/3 ≈ 0.6667.
- 3Calculate q = 200 × 0.6667 ≈ 133.33.
Lower affinity or dilute equilibrium concentration
A smaller K × C value gives only one-third coverage.
- 1Compute K × C = 0.1 × 5 = 0.5.
- 2Calculate fractional coverage: θ = 0.5 / (1 + 0.5) = 0.5/1.5 ≈ 0.3333.
- 3Calculate q = 80 × 0.3333 ≈ 26.67.
Introduction
The Langmuir isotherm models adsorption onto a surface with a fixed number of identical sites. It predicts fractional coverage θ = KC/(1 + KC) and loading q = q_maxKC/(1 + KC) from the equilibrium concentration C, affinity constant K, and monolayer capacity q_max. Use it when adsorption rises quickly at low concentration and then levels off toward a saturation plateau. It pairs naturally with the Michaelis–Menten equation calculator, because both share the same saturation-curve mathematics, and with the concentration calculator for preparing or interpreting equilibrium solutions. For terminology, compare IUPAC adsorption definitions in the Gold Book and the original Langmuir adsorption paper at DOI.org.
Langmuir isotherm equation
The Langmuir model assumes dynamic equilibrium between molecules adsorbing from solution or gas and molecules desorbing from a single layer of surface sites. The key dimensionless term is K × C. When K × C is small, θ is approximately K × C and adsorption is nearly linear. When K × C is large, θ approaches 1 and q approaches q_max.
θ is fractional coverage, so it ranges from 0 to just under 1 for finite positive C.
q is the adsorbed amount and has the same units as q_max.
K and C must use reciprocal units so K × C has no units.
At K × C = 1, θ = 0.5 and q = q_max/2.
Variables and units
Use any consistent concentration units. In water-treatment adsorption, K is often reported as L/mg when C is in mg/L, and q_max is commonly mg of adsorbate per g of adsorbent. For gas adsorption, pressure may replace concentration if K is expressed as reciprocal pressure. The calculator does not convert units; it preserves q in the same loading unit as q_max.
If K is fitted with C in mmol/L, do not enter C in mg/L unless you refit or convert K accordingly.
How to calculate Langmuir adsorption
Start by multiplying the Langmuir constant K by the equilibrium concentration C. Divide that product by 1 plus the same product to obtain fractional coverage. Finally multiply by q_max to obtain the amount adsorbed. This direct workflow is useful for checking fitted isotherm parameters, estimating removal capacity, and comparing sorbents before more detailed experiments.
Enter K in reciprocal concentration units.
Enter the measured equilibrium concentration C after adsorption equilibrium is reached.
Enter q_max from a fitted isotherm or reported monolayer capacity.
Read q as the predicted equilibrium loading at that concentration.
Saturation-curve interpretation
The curve is hyperbolic. A high K shifts the curve left, meaning less concentration is needed to reach the same coverage. A high q_max raises the plateau without changing fractional coverage. This is mathematically similar to enzyme saturation in the Michaelis–Menten equation calculator, where [S]/(Km + [S]) plays the same role as KC/(1 + KC).
| K × C | θ | Interpretation |
|---|---|---|
| 0.1 | 0.091 | Low coverage, mostly vacant sites |
| 1 | 0.5 | Half of sites occupied |
| 4 | 0.8 | High coverage |
| 9 | 0.9 | Near monolayer saturation |
Assumptions and limitations
The Langmuir isotherm works best for a uniform surface, one molecule per site, no lateral interaction between adsorbed molecules, and a single monolayer. Real activated carbons, soils, catalysts, and nanoparticles may have heterogeneous sites, pore filling, multilayer adsorption, or competitive adsorption. If those effects dominate, Freundlich, Temkin, BET, or competitive isotherm models may fit better.
Plot residuals after fitting K and q_max; systematic curvature is a warning that the Langmuir assumptions are incomplete.
Using the calculator with adsorption experiments
Batch adsorption studies usually measure the initial and equilibrium concentrations, then calculate q from a mass balance. The Langmuir equation is then fitted to q versus C_e. This calculator is the forward prediction step: given fitted K and q_max, it predicts q at any chosen equilibrium concentration. Use the molarity calculator for stock solutions, the Beer–Lambert law calculator when absorbance is used to determine C, and adsorption guidance from LibreTexts surface chemistry or NIST chemistry resources.
Use equilibrium, not initial, concentration for C.
Keep pH, ionic strength, temperature, and contact time consistent when comparing K values.
Report q_max with adsorbent mass basis, such as mg/g.
Do not extrapolate far beyond the concentration range used to fit the isotherm.
Quick Reference Card
Langmuir Isotherm — Quick Reference
Quick reference • Langmuir Isotherm Calculator
θ = KC/(1+KC); q = q_maxKC/(1+KC)Valid range: Use nonnegative K, C, and q_max with consistent units; best for monolayer adsorption on uniform independent sites.
Common Values
⚠ Watch Out
- •Use equilibrium concentration C, not the initial concentration before adsorption.
- •Do not mix units; K × C must be dimensionless.
- •Do not extrapolate far outside the fitted concentration range.
- •Avoid the simple model for multilayer, competitive, or strongly heterogeneous adsorption.
Pro Tips
- →Check the half-saturation point: C = 1/K when θ = 0.5.
- →Compare q_max to judge capacity and K to judge affinity.
- →Fit q versus equilibrium concentration using nonlinear regression when possible.
- →Report temperature, pH, ionic strength, adsorbent dose, and contact time with fitted parameters.
FAQs
What does the Langmuir isotherm calculate?
It calculates fractional surface coverage θ and amount adsorbed q at equilibrium from the Langmuir constant K, equilibrium concentration C, and maximum monolayer capacity q_max.
Why must K and C use consistent units?
The product K × C must be dimensionless. If C is in mg/L, K should be in L/mg; if C is in mol/L, K should be in L/mol or M⁻¹.
What is q_max?
q_max is the maximum monolayer adsorption capacity predicted by the Langmuir model. It is the plateau loading approached as concentration becomes very large.
When is θ equal to 0.5?
θ equals 0.5 when K × C = 1. At that point, q is half of q_max and the concentration is the reciprocal of K.
Can q exceed q_max in this model?
No. For nonnegative finite K and C, θ is less than 1, so q approaches q_max asymptotically but does not exceed it.
When should I avoid the Langmuir model?
Avoid it when adsorption is clearly multilayer, strongly heterogeneous, cooperative, competitive, or controlled by pore filling rather than independent identical surface sites.