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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).

Reviewed by the thecalcu.com team · Last updated January 11, 2025

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.

Why Use a Langmuir Isotherm Calculator?

Computing q_e = qₘKC/(1+KC) with units (mg/g, L/mg, mg/L) is straightforward but the surface coverage θ = q_e/qₘ and the separation factor R_L = 1/(1+KC) require additional steps. This calculator computes all four outputs simultaneously and provides the isotherm type classification.

For fitting experimental data, the linearised Langmuir plot (C_e/q_e vs C_e) gives qₘ and K_L; enter those fitted values here to predict q_e at any equilibrium concentration on the isotherm.

Who Should Use This Calculator?

Environmental engineers and researchers studying water treatment by adsorption, dye, fluoride, arsenic, or heavy metal removal in batch and column systems.

Chemical engineering students learning mass transfer, adsorption equilibria, and unit operations, Langmuir isotherm is a standard topic in GATE chemical engineering (Paper CH).

Pharmaceutical scientists studying drug adsorption onto excipients, active packaging, and drug delivery carriers.

Soil scientists and agricultural researchers studying nutrient adsorption-desorption on Indian soils (phosphate adsorption on red laterite soils, pesticide adsorption on black cotton soils).

What Insights Does the Langmuir Isotherm Calculator Give You?

Amount Adsorbed (q_e, mg/g) is the primary output, the equilibrium loading at the given C_e. Compare to qₘ to assess how close the surface is to saturation.

Surface Coverage (θ) = q_e/qₘ, fraction of surface sites occupied (0 = empty surface; 1 = fully saturated monolayer).

Separation Factor (R_L) = 1/(1+K_L × C_e), dimensionless favorability indicator.

Isotherm Type classifies favorability: favorable (0 < R_L < 1), linear (R_L = 1), unfavorable (R_L > 1), or irreversible (R_L → 0).

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.
Show formula & methodology ↓Show less ↑

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

What is the Langmuir adsorption isotherm?
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).
What is the Langmuir constant K_L and what does it represent?
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).
How do I use the Langmuir Isotherm Calculator?
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).
How is the Langmuir isotherm linearised for curve fitting?
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.
What is the separation factor R_L and what does it indicate?
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.
What is the difference between Langmuir and Freundlich adsorption isotherms?
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.
What adsorbents and pollutants are studied with Langmuir isotherm in India?
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).
How is qₘ determined experimentally for Langmuir isotherm?
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.
What is the Langmuir BET surface area measurement?
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.