0.714286
71.43
%
71.4286
mg/g
28.5714
mg/g
1,428.5714
L/g
0.285714
0.714286
71.43
%
71.4286
mg/g
28.5714
mg/g
1,428.5714
L/g
0.285714
The Langmuir Isotherm Calculator models the adsorption of molecules onto a surface using the Langmuir adsorption equation, one of the most fundamental models in surface chemistry and catalysis. Developed by Irving Langmuir in 1918, this model describes monolayer adsorption onto a surface with a finite number of identical sites. The calculator determines the fractional surface coverage (θ), the amount adsorbed (q), and the separation factor (Rₗ) given the equilibrium constant (K), adsorbate concentration or pressure, and maximum adsorption capacity. This model is widely applied in heterogeneous catalysis, chromatography, water treatment, gas storage, drug delivery, and environmental remediation, where understanding the relationship between bulk concentration and surface coverage is essential for process design and optimization.
The Langmuir isotherm equation for adsorption from solution is:
$$\theta = \frac{KC}{1 + KC} \quad \text{or equivalently} \quad q = \frac{q_m KC}{1 + KC}$$
where θ is the fractional surface coverage (0 to 1), K is the Langmuir equilibrium constant (L/mol), C is the adsorbate concentration (mol/L), q is the amount adsorbed (mg/g), and q_m is the maximum monolayer capacity. For gas-phase adsorption, replace C with pressure P.
The model assumes: (1) all adsorption sites are equivalent, (2) each site can hold at most one molecule, (3) there are no lateral interactions between adsorbed molecules, and (4) the surface is energetically homogeneous.
The separation factor (also called the dimensionless equilibrium parameter):
$$R_L = \frac{1}{1 + KC_0}$$
classifies the isotherm: R_L > 1 (unfavorable), R_L = 1 (linear), 0 < R_L < 1 (favorable), R_L = 0 (irreversible).
The fractional coverage θ ranges from 0 (empty surface) to approaching 1 (nearly saturated monolayer). At low concentrations where KC << 1, adsorption is approximately linear (θ ≈ KC), resembling Henry's law. At high concentrations where KC >> 1, the surface approaches saturation (θ → 1). The half-coverage point occurs when C = 1/K, meaning the equilibrium constant determines how readily the surface fills. A larger K indicates stronger adsorption affinity. The separation factor Rₗ between 0 and 1 indicates favorable adsorption — the closer to 0, the more strongly the adsorbate is bound. The amount adsorbed q gives the practical loading in mass units.
Inputs
Results
Activated carbon with K = 50 L/mol and qₘ = 100 mg/g achieves 71.4% surface coverage at 0.05 mol/L dye concentration. The Rₗ = 0.29 indicates favorable adsorption, suitable for water treatment applications.
Inputs
Results
A catalyst with high affinity (K = 500) at 0.01 mol/L reactant concentration achieves 83.3% surface coverage. The low Rₗ = 0.17 confirms strong, favorable adsorption — beneficial for catalytic performance.
The Langmuir model assumes: (1) monolayer adsorption only, (2) all surface sites are energetically equivalent, (3) no lateral interactions between adsorbed molecules, (4) each site accommodates one molecule, (5) dynamic equilibrium between adsorption and desorption. Real surfaces often violate these assumptions, which is why other models (Freundlich, BET) exist.
K is the ratio of adsorption to desorption rate constants (K = k_ads/k_des). A large K means strong binding — molecules prefer to be adsorbed rather than in solution. K is related to the free energy of adsorption: ΔG° = -RT ln(K). Temperature affects K through the van't Hoff equation.
The most common linearization is: C/q = C/qₘ + 1/(qₘK). Plot C/q vs. C: the slope gives 1/qₘ and the y-intercept gives 1/(qₘK). Alternative forms include 1/q vs. 1/C (double reciprocal) and q vs. q/C (Scatchard plot). Non-linear regression is preferred for accuracy.
Langmuir assumes a homogeneous surface with a saturation limit (monolayer). Freundlich (q = KfC^(1/n)) assumes a heterogeneous surface with no saturation limit. Freundlich is empirical and often fits data better at intermediate concentrations, while Langmuir has a stronger theoretical foundation and a meaningful maximum capacity.
For physisorption (exothermic), increasing temperature decreases K and thus decreases adsorption. For chemisorption, the effect depends on the activation energy. qₘ generally remains constant (it depends on surface area), while K changes with temperature following the van't Hoff equation: d(ln K)/dT = ΔH/(RT²).
The separation factor Rₗ = 1/(1+KC₀) classifies isotherm favorability: Rₗ = 0 (irreversible), 0 < Rₗ < 1 (favorable), Rₗ = 1 (linear), Rₗ > 1 (unfavorable). For adsorption processes like water treatment, Rₗ between 0 and 1 confirms that the adsorbent is effective. Lower Rₗ values indicate stronger adsorption preference.
Yes. The competitive Langmuir isotherm for two species is: θᵢ = KᵢCᵢ / (1 + K₁C₁ + K₂C₂). Each species competes for the same sites. The species with higher K (stronger binding) will preferentially adsorb, reducing the other's coverage. This is important in catalysis and chromatography.
In heterogeneous catalysis, the Langmuir-Hinshelwood mechanism describes reactions between two species both adsorbed on the surface: rate = k × θ_A × θ_B. Using Langmuir isotherms for each species gives rate expressions that depend on partial pressures and temperature. This is the most common mechanism for surface catalytic reactions.
Measure the equilibrium amount adsorbed (q) at several concentrations (C). Use non-linear least squares fitting of q = qₘKC/(1+KC) to find K and qₘ. Alternatively, linearize as C/q = C/qₘ + 1/(qₘK) and use linear regression. Non-linear fitting is statistically more rigorous.
Applications include: activated carbon water purification, gas mask adsorbent design, catalytic reactor modeling, chromatographic separation design, protein binding assays (Scatchard analysis), drug adsorption in pharmacy, heavy metal removal from wastewater, hydrogen storage on metal surfaces, and sensor surface design.
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