Adsorption Isotherms: Langmuir, Freundlich, and BET Models - Chemical Engineering Knowledge Platform

Industry Applications

Flue gas CO₂ capture, drinking water purification, solvent recovery, pharmaceutical chromatography, hydrogen storage

Key Standards

ASTM D3803-20 (Activated Carbon Testing), ISO 9277:2010 (BET Surface Area), IUPAC Recommendations (1985, 2015)

Typical Scale

Lab: 0.1–1 g adsorbent; Pilot: 1–10 kg; Industrial: 10–500 tonne/yr adsorbent consumption

⚠️ Why It Matters

1

Incorrect isotherm selection

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2

Poor prediction of adsorption capacity

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3

Underdesigned adsorber vessel size

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4

Frequent breakthrough and product contamination

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5

Increased regeneration energy and operational cost

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6

Noncompliance with emission or purity specifications

📘 Definition

Adsorption isotherms are empirical or theoretical mathematical relationships describing the equilibrium loading of adsorbate (e.g., gas or solute) on an adsorbent surface as a function of its partial pressure or concentration in the bulk phase, at constant temperature. They reflect underlying physical mechanisms such as monolayer coverage (Langmuir), heterogeneous surface energetics (Freundlich), or multilayer formation (BET). These models serve as foundational tools for designing and scaling adsorption-based separation processes.

🎨 Concept Diagram

AI-generated illustration for visual understanding

💡 Engineering Insight

Never assume BET applies just because you ran N₂ physisorption — it fails catastrophically for microporous carbons (< 2 nm pores) where pore-filling dominates over layerwise condensation. In practice, Langmuir often outperforms BET even for multicomponent gases when qₘ is treated as an effective capacity calibrated to process-relevant conditions.

📖 Detailed Explanation

Adsorption isotherms describe how molecules accumulate on solid surfaces at equilibrium. At the simplest level, imagine placing charcoal in a jar of polluted air: as pollutant concentration rises, more molecules stick — but only up to a limit, like seats filling in a theater. The Langmuir model captures this 'full theater' idea with two assumptions: identical sites and no interaction between adsorbed molecules.

The Freundlich model relaxes those assumptions — treating the surface as rough and energetic, so adsorption strength varies across sites. This makes it robust for liquid-phase systems where solvent effects and surface chemistry dominate. Its logarithmic form (log qₑ vs log Cₑ) remains widely used in regulatory guidance (e.g., EPA SW-846 Method 1311) despite lacking mechanistic rigor.

The BET model extends Langmuir to multilayers — essential for characterizing surface area via gas adsorption — but relies on strict thermodynamic assumptions: the first layer binds strongly (like chemisorption), while subsequent layers behave like condensed liquid. Deviations occur in narrow pores (< 5 nm), at high pressures (> 0.35 P/P₀), or with strong quadrupole interactions (e.g., CO₂ on Mg-MOF-74), requiring DFT-based isotherm modeling or statistical physics corrections.

🔄 Engineering Workflow

Step 1

Step 1: Define system — adsorbate(s), adsorbent, temperature, phase (gas/liquid), concentration range

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Step 2

Step 2: Acquire experimental equilibrium data (at least 6 points spanning 0.1–0.9 of saturation)

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Step 3

Step 3: Screen models using linearized forms and residual analysis (SSR, AIC)

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Step 4

Step 4: Fit nonlinear parameters via least-squares optimization with physical constraints (e.g., qₘ > 0, K > 0, n > 0)

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Step 5

Step 5: Validate against dynamic column tests (breakthrough time, shape, exhaustion profile)

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Step 6

Step 6: Scale to process conditions using mass-transfer zone (MTZ) length correlation and bed depth service time (BDST) model

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Step 7

Step 7: Monitor in-service performance and re-fit isotherm annually or after adsorbent regeneration cycles

📋 Decision Guide

Rock/Field Condition	Recommended Design Action
Single-component, low-pressure gas (e.g., CO₂ capture < 0.2 bar)	Use Langmuir model with qₘ and K fitted from high-precision gravimetric/volumetric data; validate with breakthrough curves.
Liquid-phase organics on activated carbon (wastewater, ppm-level contaminants)	Apply Freundlich model with n ≈ 0.8–0.95; include safety factor ≥ 1.5 on qₑ due to competitive adsorption effects.
N₂ or Ar physisorption at 77 K for surface area/pore analysis	Fit BET model only in relative pressure range 0.05–0.35; reject if C < 25 or linear regression R² < 0.995.

📊 Key Properties & Parameters

qₘ (Monolayer Capacity)

0.1–5.0 mmol/g for CO₂ on zeolites; 10–200 mg/g for organics on activated carbon

Maximum adsorption capacity per unit mass of adsorbent under monolayer coverage assumption (Langmuir & BET models).

⚡ Engineering Impact:

Directly determines minimum adsorbent inventory and bed height in fixed-bed design.

K (Langmuir Affinity Constant)

0.01–100 L/mmol (gas) or L/g (liquid); dimensionless in pressure-based form

Equilibrium constant reflecting adsorbate-adsorbent binding strength; ratio of adsorption to desorption rate constants.

⚡ Engineering Impact:

Controls steepness of low-concentration uptake — critical for trace contaminant removal (e.g., VOCs < 1 ppm).

n (Freundlich Heterogeneity Index)

0.7–1.0 for most activated carbons; < 0.5 indicates highly heterogeneous or microporous surfaces

Empirical exponent indicating surface energy distribution heterogeneity; n = 1 implies homogeneous surface.

⚡ Engineering Impact:

Determines curvature of isotherm: low n values demand larger safety margins in dynamic column design.

C (BET Constant)

50–500 for N₂ at 77 K on common adsorbents; < 25 suggests weak physisorption

Dimensionless parameter related to heat of adsorption of the first layer relative to liquefaction enthalpy.

⚡ Engineering Impact:

Low C values invalidate BET applicability and signal dominance of non-BET mechanisms (e.g., pore filling).

📐 Key Formulas

Langmuir Isotherm

qₑ = (qₘ·K·Cₑ) / (1 + K·Cₑ) <

Variables:

Symbol	Name	Unit	Description
qₑ	equilibrium adsorption capacity	mg/g	Amount of adsorbate adsorbed per unit mass of adsorbent at equilibrium
qₘ	maximum adsorption capacity	mg/g	Theoretical monolayer adsorption capacity
K	Langmuir adsorption constant	L/mg	Affinity constant related to adsorption energy
Cₑ	equilibrium concentration	mg/L	Concentration of adsorbate in solution at equilibrium

Typical Ranges:

VOC removal in air

qₘ = 100–300 mg/g; K = 0.005–0.1 L/mg

CO₂ capture on amine-grafted silica

qₘ = 1.2–3.5 mmol/g; K = 10–80 kPa⁻¹

⚠️ K·Cₑ < 10 ensures monotonic behavior; avoid extrapolation beyond 1.5× highest tested Cₑ

🔧 Open Calculator

Freundlich Isotherm

qₑ = K_F · Cₑⁿ

Empirical power-law relationship for heterogeneous surfaces.

Variables:

Symbol	Name	Unit	Description
qₑ	Equilibrium adsorption capacity	mg/g	Amount of adsorbate adsorbed per unit mass of adsorbent at equilibrium
K_F	Freundlich constant	mg/g·(L/mg)ⁿ	Indicator of adsorption capacity
Cₑ	Equilibrium concentration	mg/L	Concentration of adsorbate in solution at equilibrium
n	Freundlich exponent	dimensionless	Indicator of adsorption intensity and surface heterogeneity

Typical Ranges:

Phenol on granular activated carbon

K_F = 10–50 (mg/g)(L/mg)ⁿ; n = 0.7–0.9

Heavy metals on biochar

K_F = 0.5–5.0; n = 0.4–0.7

⚠️ n must be < 1.0 for favorable adsorption; discard fits where n > 1.1 or R² < 0.92

🔧 Open Calculator

BET Equation

(P/P₀)/[qₑ(1−P/P₀)] = 1/(qₘ·C) + (C−1)/(qₘ·C)·(P/P₀)

Linearized form to extract monolayer capacity qₘ and BET constant C from gas adsorption data.

Variables:

Symbol	Name	Unit	Description
P	Equilibrium pressure	Pa	Partial pressure of adsorbate gas at equilibrium
P₀	Saturation pressure	Pa	Saturation vapor pressure of adsorbate at experimental temperature
qₑ	Equilibrium adsorbed amount	mol/g	Amount of gas adsorbed per unit mass of adsorbent at equilibrium pressure P
qₘ	Monolayer capacity	mol/g	Maximum amount of adsorbate that can form a complete monolayer on the adsorbent surface
C	BET constant	dimensionless	Constant related to the heat of adsorption, reflecting affinity between adsorbate and adsorbent