Langmuir Adsorption Isotherm Calculator

Calculate surface coverage, adsorption constant, and monolayer capacity using Langmuir isotherm.

Equilibrium concentration C (mol/L or Pa) (M or Pa)

Langmuir constant K_L (L/mol or 1/Pa) (L/mol)

Monolayer capacity q_max (mg/g or mmol/g) (mg/g)

Measured adsorption q for linearization (mg/g) (mg/g)

Optional: for linear plot

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The Four Assumptions Behind Langmuir

The Langmuir isotherm is a model, and it rests on four idealisations. Knowing them tells you exactly when it will fail.

Assumption	Meaning	Fails when
Monolayer only	Adsorption stops at one molecular layer	Multilayers form — use BET instead
Uniform surface	Every site has identical binding energy	Real surfaces are heterogeneous
No lateral interaction	Adsorbed molecules ignore each other	Coverage is high and crowding matters
Dynamic equilibrium	Adsorption and desorption rates balance	System has not equilibrated

θ = K_LC / (1 + K_LC) q = q_maxθ

The equation follows directly from setting adsorption rate equal to desorption rate. Adsorption is proportional to concentration and to vacant sites (1 − θ); desorption is proportional to occupied sites (θ). Solving that balance gives the expression above — which is why the same functional form appears in Michaelis–Menten enzyme kinetics, built from the identical logic.

The Three Regimes

Condition	Behaviour	θ
K_LC « 1	Linear, Henry's law region	θ ≈ K_LC
K_LC ≈ 1	Transition	θ = 0.5 — half the sites filled
K_LC » 1	Saturation plateau	θ → 1, q → q_max

Note that θ = 0.5 when C = 1/K_L. That concentration is a direct measure of binding affinity, exactly as K_M is in enzyme kinetics. A large K_L means saturation is reached at low concentration.

Linearised Forms

Fitting is usually done by linearising. The most common form plots C/q against C, giving slope 1/q_max and intercept 1/(K_Lq_max). Be aware that different linearisations weight the errors differently and can yield noticeably different parameters from the same data — non-linear regression on the original equation is more reliable where software allows.

Worked Examples

Example 1: Activated carbon: qm=200mg/g, K=10L/mg, C=5mg/L

θ=10×5/(1+50)=0.98

Result: q=196mg/g — near saturation

High K = strong adsorption affinity

Example 2: Freundlich comparison: low concentration

Linear region K×C<<1

Result: θ≈KC (Henry's law region)

Langmuir reduces to Henry's law at low C

Example 3: Finding K_L from half-saturation

Adsorption reaches half of q_max at C = 0.02 mol/L

Result: K_L = 1/0.02 = 50 L/mol

Half-saturation concentration is the reciprocal of the binding constant — the same relationship as K_M in Michaelis–Menten kinetics.

Example 4: The linear region

q_max = 200 mg/g, K_L = 10 L/mg, C = 0.001 mg/L

Result: θ = 0.00990, essentially linear

With K_LC = 0.01, the denominator is nearly 1 and adsorption is proportional to concentration. This is the Henry's law region.

Example 5: Linearised fitting

Plot of C/q versus C gives slope 0.005 and intercept 0.0005

Result: q_max = 200 mg/g, K_L = 10 L/mg

q_max is the reciprocal of the slope, and K_L is the slope divided by the intercept.

Common Mistakes

⚠️

Using Langmuir where multilayers form

The model assumes a single layer. Physical adsorption of gases below their critical temperature usually builds multilayers, requiring the BET isotherm instead.

⚠️

Assuming a good linear fit proves the mechanism

Linearised plots often look convincing even when the model is wrong, because the transformation compresses deviations. Check the fit against the original non-linear data.

⚠️

Ignoring surface heterogeneity

Real adsorbents have sites of varying energy. The Freundlich isotherm, which assumes a distribution of binding energies, often fits heterogeneous surfaces better.

⚠️

Confusing q_max with total surface area

q_max is monolayer capacity for that particular adsorbate. A larger molecule gives a smaller q_max on the same surface.

Frequently Asked Questions

Langmuir assumptions?

1. Monolayer adsorption only. 2. Identical sites (homogeneous surface). 3. No lateral interactions between adsorbates. 4. Langmuir equilibrium constant K = k_ads/k_des. Real surfaces: heterogeneous → use Freundlich or Temkin isotherms.

Applications?

Wastewater treatment (heavy metal removal, dye adsorption). Catalysis (BET surface area measurement). Drug loading on nanoparticles. Protein adsorption to biosensors. Chromatography stationary phase. Separation factor R_L predicts isotherm shape: R_L<1 favorable.

What are the Langmuir assumptions?

Monolayer coverage, energetically uniform sites, no interaction between adsorbed molecules, and dynamic equilibrium between adsorption and desorption.

What does the Langmuir constant K_L mean?

It measures binding affinity. Its reciprocal is the concentration at which half the sites are occupied, so a large K_L means saturation at low concentration.

When should I use BET instead?

When multilayer adsorption occurs — typically gas adsorption below the critical temperature. BET extends the Langmuir approach to successive layers and is the standard for surface area measurement.

How does Langmuir relate to Michaelis–Menten?

They share the same mathematical form because both derive from a site-binding equilibrium. q_max corresponds to V_max, and 1/K_L to K_M.

Why does the Freundlich isotherm sometimes fit better?

Because it allows for a distribution of binding site energies rather than assuming a uniform surface. Real adsorbents such as activated carbon are usually heterogeneous.

Formula Explorer connections

Interpretation: This formula describes how reactants, products, ions or phases distribute when opposing processes reach equilibrium. Assumption: Use equilibrium rather than initial concentrations, correct stoichiometric exponents, and the specified temperature; activities may replace concentrations in nonideal systems.

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