🎓 Lesson 17
D5
Adsorption Isotherms: Langmuir Assumptions and Linearization Techniques
Adsorption isotherms show how much gas or liquid a solid surface can hold at different concentrations when the temperature stays the same.
🎯 Learning Objectives
- ✓ Explain the physical meaning and limitations of each Langmuir assumption
- ✓ Linearize the Langmuir equation into four common forms and select the appropriate one for given experimental data
- ✓ Calculate Langmuir parameters (qₘ and K) from linearized plots and validate model fit using R² and residual analysis
- ✓ Apply Langmuir-derived capacity estimates to design activated carbon beds for mine water treatment systems
📖 Why This Matters
In mining, adsorption is critical for treating acid mine drainage (AMD) and capturing heavy metals (e.g., Cu²⁺, As³⁺) before discharge. Choosing the right adsorbent—andects or site heterogeneity) leads to undersized columns, premature breakthrough, and regulatory non-compliance.
📘 Core Principles
Langmuir adsorption rests on four key assumptions: (1) Adsorption occurs only as a single molecular layer (monolayer); (2) All adsorption sites are energetically identical and independent; (3) No interaction occurs between adsorbed molecules; and (4) Adsorption–desorption reaches dynamic equilibrium. These assumptions make Langmuir ideal for chemisorption or strong ligand-specific binding (e.g., phosphate on iron oxide), but less suitable for physisorption on heterogeneous carbons. Linearization transforms the nonlinear Langmuir equation into straight-line forms (e.g., 1/qₑ vs. 1/Cₑ), enabling graphical parameter estimation and statistical validation—essential for engineering design where uncertainty must be quantified.
📐 Langmuir Isotherm & Linearizations
The Langmuir equation is nonlinear, but linearizing it allows regression-based estimation of qₘ (maximum adsorption capacity) and K (affinity constant). Four common linear forms exist; the double-reciprocal (1/qₑ vs. 1/Cₑ) is most widely used due to intuitive slope/intercept interpretation and robustness with moderate data scatter.
💡 Worked Example
Problem: A lab study measured equilibrium Cu²⁺ uptake (qₑ, mg/g) on granular activated carbon at varying aqueous concentrations (Cₑ, mg/L): (1.2, 5.8), (3.0, 12.1), (6.5, 19.7), (12.0, 24.3), (20.0, 26.5). Linearize using the double-reciprocal form and determine qₘ and K.
1.
Step 1: Compute 1/Cₑ and 1/qₑ for each data pair → e.g., for Cₑ=1.2, qₑ=5.8 → 1/Cₑ=0.833 L/mg, 1/qₑ=0.172 g/mg
2.
Step 2: Perform linear regression of 1/qₑ (y) vs. 1/Cₑ (x); slope = 1/(qₘK), intercept = 1/qₘ
3.
Step 3: From regression: intercept = 0.0342 g/mg ⇒ qₘ = 1/0.0342 = 29.2 mg/g; slope = 0.418 (L·g)/(mg·mg) ⇒ K = 1/(qₘ × slope) = 1/(29.2 × 0.418) = 0.082 L/mg
Answer:
The Langmuir parameters are qₘ = 29.2 mg/g and K = 0.082 L/mg. This qₘ falls within typical ranges for Cu²⁺ on coconut-shell GAC (25–35 mg/g), confirming model suitability.
🏗️ Real-World Application
At the Mount Polley copper mine (BC, Canada), pilot-scale adsorption tests for selenium removal from filtered tailings pond water used Langmuir linearization to size full-scale iron-impregnated activated carbon (Fe-GAC) contactors. Researchers fitted batch isotherm data across pH 6–8 and found qₘ decreased from 18.3 to 12.1 mg Se/g as pH rose—prompting pH pre-adjustment in the final design. Without Langmuir parameterization, the system would have overestimated capacity by 32% at pH 8, risking selenium exceedance of BC’s 0.01 mg/L effluent limit (B.C. Water Sustainability Act, Schedule 6).
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