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

⚠️ Why It Matters

1

Incorrect isotherm selection

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2

Underestimated adsorbent inventory

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3

Breakthrough before design life

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4

Product contamination or emissions violation

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5

Regulatory noncompliance and operational shutdown

📘 Definition

Adsorption isotherms are empirical or theoretical mathematical relationships describing the equilibrium distribution of adsorbate between the fluid phase and the solid adsorbent surface at constant temperature. They quantify adsorbed amount (qₑ, in mg/g or mmol/g) as a function of equilibrium concentration (Cₑ, in mg/L or atm) or partial pressure (P, in kPa or bar). These models underpin design of fixed-bed adsorbers, PSA systems, and catalytic reactors where surface occupancy governs performance.

🎨 Concept Diagram

AI-generated illustration for visual understanding

💡 Engineering Insight

Never rely on literature isotherm parameters for your adsorbent — even identical nominal grades (e.g., 'Calgon Filtrasorb 400') vary 20–35% in qₘₐₓ due to activation batch, ash content, and moisture history. Always run at least three equilibrium points spanning 10–90% of expected saturation, and validate with dynamic column data before finalizing vessel diameter.

📖 Detailed Explanation

Adsorption isotherms originate from the balance between molecular kinetic energy and surface binding forces. At low concentrations, adsorption increases nearly linearly with concentration — governed by site availability and affinity. As coverage rises, competition for sites intensifies, leading to curvature that distinguishes models: Langmuir assumes uniform sites and no interaction; Freundlich accounts for energetic disorder via power-law scaling.

The Langmuir model treats adsorption as reversible chemical equilibrium on identical sites, yielding hyperbolic saturation behavior. Its linear form (Cₑ/qₑ vs Cₑ) is convenient but obscures error structure — nonlinear regression is preferred for parameter accuracy. The Freundlich model lacks theoretical saturation, making it empirically robust for dilute aqueous systems but unsafe for design beyond its fitted range.

BET extends Langmuir to multilayer formation, requiring strict adherence to its foundational assumptions: (1) monolayer adsorption obeys Langmuir kinetics, (2) subsequent layers behave like condensed phase, and (3) no lateral interactions between adsorbed molecules. Deviations — especially at P/P₀ < 0.05 (micropore filling) or > 0.35 (capillary condensation) — invalidate BET-derived surface areas. Modern practice pairs BET with DFT pore-size distribution to resolve this limitation.

🔄 Engineering Workflow

Step 1

Step 1: Define system — adsorbate identity, phase (gas/aqueous), temperature, and concentration range

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Step 2

Step 2: Select candidate isotherms based on mechanism (monolayer vs multilayer, homogeneous vs heterogeneous)

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Step 3

Step 3: Conduct batch or dynamic equilibrium experiments (e.g., ASTM D3860 for carbon, ISO 15903 for gases)

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Step 4

Step 4: Fit linearized and nonlinear forms; compare AIC/BIC and residual plots — reject models with systematic bias

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Step 5

Step 5: Validate with breakthrough curve simulation (e.g., Thomas or Yoon–Nelson) using fitted parameters

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Step 6

Step 6: Scale up to fixed-bed design: calculate bed depth service time (BDST), minimum usable bed height, and regeneration frequency

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Step 7

Step 7: Monitor on-stream performance (inlet/outlet analyzers) and re-fit isotherm quarterly or after thermal/chemical aging

📋 Decision Guide

Rock/Field Condition	Recommended Design Action
Single-component, low-pressure vapor (e.g., VOC recovery < 5 kPa)	Use Langmuir model; verify linearized plot R² > 0.995 and qₑ/Cₑ vs qₑ linearity
Aqueous trace contaminants (e.g., heavy metals, pharmaceuticals) on heterogeneous biosorbents	Fit Freundlich first; if n_F < 0.4, consider Redlich–Peterson or Sips to avoid extrapolation error
Multilayer gas adsorption on porous solids (e.g., N₂, CO₂, H₂ for surface area/pore analysis)	Apply BET theory only within 0.05–0.35 P/P₀ range; exclude points outside with statistical validation (ISO 9277)

📊 Key Properties & Parameters

qₘₐₓ

10–500 mg/g for activated carbon; 0.5–5 mmol/g for zeolites

Maximum monolayer adsorption capacity — the theoretical saturation limit of adsorbent surface sites (mg/g or mol/kg)

⚡ Engineering Impact:

Directly determines minimum adsorbent mass required per unit feed flow, impacting vessel size and CAPEX

K_L

0.01–10 L/mg for organics on carbon; 10⁻³–10² bar⁻¹ for gases on zeolites

Langmuir affinity constant — reflects strength of adsorbate–adsorbent interaction (L/mg or bar⁻¹)

⚡ Engineering Impact:

Controls steepness of low-concentration uptake; low K_L necessitates longer contact time or higher bed depth

n_F

0.7–1.2 for homogeneous carbons; 0.3–0.8 for highly heterogeneous biochars or clays

Freundlich heterogeneity exponent — indicates surface energy distribution (dimensionless)

⚡ Engineering Impact:

Values < 0.5 indicate strong cooperative adsorption; values > 1 suggest unfavorable isotherm shape — both challenge fixed-bed modeling

C_BET

50–200 for N₂ at 77 K on microporous carbons; 10–60 for CO₂ on MOFs at 25 °C

BET constant — dimensionless parameter related to heat of adsorption relative to liquefaction enthalpy

⚡ Engineering Impact:

C < 2 invalidates BET applicability; C > 200 implies near-ideal monolayer behavior — affects surface area uncertainty by ±15%

📐 Key Formulas

Langmuir Isotherm

qₑ = (qₘₐₓ · K_L · Cₑ) / (1 + K_L · Cₑ)

Equilibrium adsorbed amount (qₑ) as function of aqueous concentration (Cₑ) or partial pressure (P)

Variables:

Symbol	Name	Unit	Description
qₑ	Equilibrium adsorbed amount	mass of adsorbate per mass of adsorbent (e.g., mg/g)	Amount of adsorbate adsorbed at equilibrium
qₘₐₓ	Maximum adsorption capacity	mass of adsorbate per mass of adsorbent (e.g., mg/g)	Theoretical maximum adsorption capacity of the adsorbent
K_L	Langmuir adsorption constant	L/mg or 1/(concentration unit)	Affinity constant related to adsorption energy
Cₑ	Equilibrium aqueous concentration	mg/L or other concentration unit	Concentration of adsorbate in solution at equilibrium
P	Partial pressure	atm or Pa	Gas-phase partial pressure of adsorbate (alternative to Cₑ for gas-phase systems)

Typical Ranges:

VOC removal on activated carbon (aqueous)

qₘₐₓ = 120–350 mg/g; K_L = 0.05–2.5 L/mg

CO₂ capture on amine-grafted silica (gas)

qₘₐₓ = 1.2–3.6 mmol/g; K_L = 10–85 bar⁻¹

⚠️ Use only when R² > 0.99 for nonlinear fit; reject if K_L uncertainty > ±25%

🔧 Open Calculator

Freundlich Isotherm

qₑ = K_F · Cₑⁿᶠ

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)ⁿᶠ	Indicator of adsorption capacity
Cₑ	Equilibrium concentration	mg/L	Concentration of adsorbate in solution at equilibrium
nᶠ	Freundlich exponent	dimensionless	Indicator of adsorption intensity and surface heterogeneity

Typical Ranges:

Pb²⁺ on bone char (aqueous)

K_F = 15–45 (mg/g)(L/mg)¹/ⁿ; n_F = 0.42–0.68

BTEX on coconut-shell carbon

K_F = 50–220; n_F = 0.85–1.05

⚠️ n_F must be 0.1–1.5; extrapolation beyond 2× highest experimental Cₑ invalidates prediction

🔧 Open Calculator

BET Equation

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

Linearized form to determine 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	Vapor pressure of adsorbate at the adsorption temperature
qₑ	Equilibrium adsorbed amount	mol/kg or cm³/g	Amount of adsorbate adsorbed per unit mass of adsorbent at equilibrium pressure P
qₘ	Monolayer capacity	mol/kg or cm³/g	Maximum adsorption capacity corresponding to complete monolayer coverage
C	BET constant	dimensionless	Temperature-dependent constant related to the heat of adsorption