🎓 Lesson 15
D5
Langmuir and Freundlich Isotherms: Assumptions and Fitting Methods
Langmuir and Freundlich isotherms are simple mathematical rules that tell us how much gas or liquid sticks to a solid surface—like how much explosive fume gets trapped on activated carbon in ventilation filters.
🎯 Learning Objectives
- ✓ Calculate Langmuir parameters (qₘ and Kₗ) from experimental adsorption data using linearized regression
- ✓ Apply the Freundlich equation to fit non-linear adsorption data and interpret K_f and 1/n values for surface heterogeneity
- ✓ Analyze and compare the physical assumptions of both isotherms to select the appropriate model for a given adsorbent-adsorbate system
- ✓ Design a lab-scale adsorption column by estimating equilibrium loading using fitted isotherm parameters
📖 Why This Matters
📘 Core Principles
Langmuir assumes: (1) adsorption forms a single molecular layer, (2) all sites are identical and independent, (3) no interaction between adsorbed molecules, and (4) adsorption/desorption reach dynamic equilibrium. Its derivation yields a hyperbolic relationship between adsorbed amount (qₑ) and equilibrium concentration (Cₑ). Freundlich abandons site uniformity—it assumes power-law dependence reflecting energetic heterogeneity and possible multilayer formation, making it more flexible but less mechanistic. While Langmuir gives thermodynamic insight (e.g., affinity constant Kₗ ≈ K_eq), Freundlich’s 1/n < 1 indicates favorable adsorption; values near 0.1–0.5 are typical for carbon-DPM systems.
📐 Key Calculations & Linearization
Both isotherms are commonly linearized for parameter estimation via least-squares regression. Langmuir linearization uses the double-reciprocal form; Freundlich uses log-log transformation. Misapplying linearization (e.g., forcing Langmuir fit on strongly heterogeneous data) introduces systematic error—non-linear regression is preferred when high accuracy is required.
💡 Worked Example
Problem: A lab test on activated carbon exposed to NO₂ yielded: at Cₑ = 25 mg/m³, qₑ = 82 mg/g; at Cₑ = 100 mg/m³, qₑ = 118 mg/g; at Cₑ = 200 mg/m³, qₑ = 132 mg/g. Fit the Langmuir isotherm using linear regression.
1.
Step 1: Compute 1/Cₑ and 1/qₑ: e.g., for Cₑ=25 → 1/Cₑ = 0.04 m³/mg; qₑ=82 → 1/qₑ = 0.0122 g/mg
2.
Step 2: Perform linear regression of 1/qₑ vs. 1/Cₑ → slope = 1/qₘKₗ = 0.0021, intercept = 1/qₘ = 0.0068
3.
Step 3: Solve: qₘ = 1/0.0068 = 147 mg/g; Kₗ = (1/0.0068)/0.0021 = 70.2 L/mg
Answer:
The result is qₘ = 147 mg/g and Kₗ = 70.2 L/mg, consistent with typical activated carbon–NO₂ systems (qₘ: 120–180 mg/g; Kₗ: 40–100 L/mg).
🏗️ Real-World Application
At Newmont’s Boddington Gold Mine (WA), carbon-in-leach (CIL) circuit optimization used Freundlich isotherms to model Au(CN)₂⁻ adsorption onto coconut-shell carbon. Batch tests across [Au] = 0.02–0.2 ppm revealed K_f = 2.9 (mg/g)(L/mg)^1/n and 1/n = 0.38—indicating highly favorable, heterogeneous adsorption. This validated extending carbon residence time by 12%, increasing gold recovery from 94.1% to 96.7% and avoiding $4.2M annual losses.
📋 Case Connection
📋 Rare Earth Element Recovery from Acid Mine Drainage
Ultra-low REE concentrations (<10 mg/L), high Fe³⁺/Al³⁺ interference, pH-sensitive extraction
📋 Food-Grade Citric Acid Purification via Liquid-Liquid Extraction
High viscosity broth, emulsion formation with tertiary amines, difficult phase separation