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Langmuir Isotherm Calculator

Chemistry

Calculate surface coverage (θ), amount adsorbed (q), and saturation capacity from Langmuir adsorption isotherm: q = qₘ × K × C / (1 + K × C).

50
0.05
20

Amount Adsorbed (q_e, mg/g)

25
Surface Coverage (θ)
0.5
Separation Factor (R_L)
0.5
Isotherm Type
Favorable isotherm (0 < R_L < 1)

This calculator computes your Amount Adsorbed (q_e, mg/g), Surface Coverage (θ), Separation Factor (R_L), Isotherm Type from the values you enter.

Inputs
Maximum Adsorption Capacity (qₘ, mg/g)Langmuir Constant (K_L, L/mg)Equilibrium Concentration (C_e, mg/L)
Outputs
Amount Adsorbed (q_e, mg/g)Surface Coverage (θ)Separation Factor (R_L)Isotherm Type

What is a Langmuir?

The Langmuir Isotherm Calculator computes the equilibrium amount adsorbed (q_e), surface coverage (θ), separation factor (R_L), and isotherm favorability from the Langmuir equation: q_e = qₘ × K_L × C_e / (1 + K_L × C_e). Enter the maximum adsorption capacity (qₘ), Langmuir constant (K_L), and equilibrium concentration (C_e).

The Langmuir isotherm (Irving Langmuir, 1916, Nobel Prize Chemistry 1932) is the theoretical framework for monolayer adsorption on uniform surfaces. It is the most widely used model in adsorption research for water treatment, catalysis, and surface science. When the isotherm is favorable (0 < R_L < 1), the adsorbate preferentially partitions onto the surface — the basis of activated carbon filters, zeolite adsorption, and affinity chromatography.

For context on the chemical oxygen demand reduced by adsorption, the Chemical Oxygen Demand Calculator quantifies the pollutant loading. The Activity Coefficient Calculator and Ionic Strength Calculator influence adsorption in electrolyte solutions through activity effects.

How to use this Langmuir calculator

  1. Fit experimental (C_e, q_e) data to the linearised Langmuir equation to obtain qₘ and K_L. Or use published values from literature.
  2. Enter qₘ (mg/g) — the monolayer saturation capacity.
  3. Enter K_L (L/mg) — the Langmuir affinity constant.
  4. Enter C_e (mg/L) — the equilibrium concentration at the operating condition.
  5. Read q_e and θ for the adsorption loading, and R_L for favorability.

Formula & Methodology

Langmuir isotherm equation:

q_e = qₘ × K_L × C_e / (1 + K_L × C_e) θ = q_e / qₘ = K_L × C_e / (1 + K_L × C_e) R_L = 1 / (1 + K_L × C_0)   [C_0 = initial concentration]  Linearised: C_e/q_e = 1/(qₘ × K_L) + C_e/qₘ Slope = 1/qₘ; Intercept = 1/(qₘ × K_L)

Worked example — fluoride removal by activated alumina:

qₘ = 12 mg F⁻/g; K_L = 0.08 L/mg. Initial groundwater F⁻ = 5 mg/L (WHO limit: 1.5 mg/L; Indian drinking water standard BIS IS:10500: 1.0 mg/L maximum permissible).

q_e = 12 × 0.08 × 5 / (1 + 0.08 × 5) = 4.8/1.4 = 3.43 mg/g θ = 3.43/12 = 0.286 (28.6% surface covered) R_L = 1/(1 + 0.08 × 5) = 1/1.4 = 0.714 (favorable isotherm)

For a 1000 L/day treatment capacity with 5 mg/L initial fluoride target to 1 mg/L effluent: fluoride to remove = (5−1) × 1000 L/day = 4000 mg/day. At q_e = 3.43 mg/g at equilibrium C_e: adsorbent needed ≈ 4000/3.43 ≈ 1165 g/day. India has 27 million people in fluorosis-endemic districts (Rajasthan, UP, Bihar, AP, Telangana) — defluoridation using the Nalgonda technique and activated alumina is a major public health intervention under JJBY/Jal Jeevan Mission.

Frequently Asked Questions

The Langmuir isotherm is a theoretical model for adsorption of molecules onto a surface under the following assumptions: (1) Monolayer adsorption only — once a surface site is occupied, no further adsorption occurs at that site. (2) All adsorption sites are equivalent and equally energetic. (3) No interaction between adsorbed molecules (molecules don't attract or repel each other on the surface). The Langmuir equation is: q_e = qₘ × K_L × C_e / (1 + K_L × C_e), where q_e = amount adsorbed at equilibrium (mg/g), qₘ = maximum monolayer capacity (mg/g), K_L = Langmuir constant (L/mg), C_e = equilibrium concentration (mg/L).
The Langmuir constant K_L (also written K_a or b in different notations, with units L/mg or L/mol) is related to the affinity of the adsorbate for the adsorbent surface. Higher K_L → stronger affinity → the isotherm plateaus at lower concentrations (nearly all surface sites occupied at low C_e). The equilibrium constant K_L = k_ads/k_des, the ratio of adsorption to desorption rate constants (Langmuir kinetics). At low concentrations: q_e ≈ qₘ × K_L × C_e (linear, Henry's law regime). At high concentrations: q_e → qₘ (saturation, surface fully covered).
Enter the Maximum Adsorption Capacity (qₘ, mg/g — the plateau value from the isotherm), Langmuir Constant (K_L, L/mg — from the linearised Langmuir plot), and Equilibrium Concentration (C_e, mg/L). The calculator returns q_e (amount adsorbed), surface coverage θ, separation factor R_L, and isotherm favorability classification. Default: qₘ=50 mg/g, K_L=0.05 L/mg, C_e=20 mg/L (typical for dye removal by activated carbon in Indian textile wastewater treatment).
Linearised Langmuir (most common): C_e/q_e = 1/(qₘ × K_L) + C_e/qₘ. Plot C_e/q_e (y-axis) vs C_e (x-axis): slope = 1/qₘ; intercept = 1/(qₘ × K_L) → K_L = slope/intercept. The linearised form is used for graphical determination of qₘ and K_L from batch adsorption experiment data: prepare solutions of different concentrations, equilibrate with adsorbent, measure residual concentration, calculate q_e = (C₀ − C_e) × V/m (V = solution volume, m = adsorbent mass), then plot C_e/q_e vs C_e.
The dimensionless separation factor (or equilibrium parameter) R_L = 1 / (1 + K_L × C₀), where C₀ is the initial concentration. R_L indicates isotherm favourability: R_L > 1: unfavourable (convex isotherm — poor adsorption). R_L = 1: linear (Henry's law, very dilute). 0 < R_L < 1: favourable (concave isotherm — good adsorption). R_L = 0: irreversible (very high K_L → essentially complete adsorption). Favorable isotherms mean the adsorbate preferentially partitions onto the surface rather than staying in solution — desirable for water treatment applications.
Langmuir isotherm: based on monolayer adsorption on homogeneous surface with finite sites; has a definite maximum capacity qₘ; q_e = qₘ × K × C_e / (1 + K × C_e). Freundlich isotherm: empirical model for multilayer adsorption on heterogeneous surfaces; no saturation limit; q_e = K_F × C_e^(1/n), where K_F = Freundlich capacity factor and n = heterogeneity factor (1/n < 1 favourable; 1/n > 1 unfavourable). Langmuir is more physically meaningful; Freundlich fits a wider range of systems. Many real systems (activated carbon, zeolites) follow Langmuir at lower concentrations and deviate at higher concentrations.
Common Indian adsorption research: (1) Fluoride removal: activated alumina, bone char, hydroxyapatite — Langmuir qₘ 5–15 mg/g F⁻ for groundwater defluoridation (critical in Bihar, Rajasthan, AP — endemic fluorosis areas). (2) Textile dye removal: activated carbon, zeolites, agricultural waste — qₘ 50–500 mg/g for reactive dyes (major issue in Tiruppur Tamil Nadu, Surat Gujarat textile clusters). (3) Heavy metal removal: Pb²⁺, Cr⁶⁺, Cd²⁺ from industrial effluents using modified biochar, chitosan (CPCB red-category industry requirement). (4) Arsenic: iron-based adsorbents for groundwater treatment in West Bengal (highest global arsenic-affected population).
Experimental procedure: (1) Prepare a series of standard solutions at different initial concentrations C₀ (e.g., 10, 25, 50, 100, 200, 500 mg/L). (2) Add a fixed adsorbent dose (e.g., 1 g/L) to each. (3) Agitate at constant temperature (usually 25°C) for sufficient time to reach equilibrium (typically 24–48 hours). (4) Filter and measure residual concentration C_e by UV-Vis or HPLC. (5) Calculate q_e = (C₀ − C_e) × V/m. (6) Plot q_e vs C_e (non-linear) or C_e/q_e vs C_e (linear) and fit to Langmuir equation. (7) Read qₘ from the plateau or from the slope of the linearised plot.
The Brunauer-Emmett-Teller (BET) theory extends Langmuir's monolayer adsorption to multilayer adsorption of gas molecules. In BET surface area measurement, N₂ gas is adsorbed on a solid at 77 K (liquid nitrogen temperature); the volume adsorbed at various pressures is measured. The linear BET equation gives the monolayer volume, from which total surface area is computed: A_BET = V_mono × N_A / (22414) × A_N₂, where A_N₂ = 0.162 nm² per N₂ molecule. BET surface area is reported in m²/g — activated carbon: 500–2000 m²/g; zeolites: 100–800 m²/g; metal-organic frameworks (MOFs): up to 7000 m²/g. NABL-accredited laboratories in India (CIPET, IICT) perform BET surface area analysis under ISO 9277.