Reference ID: MET-3FB5 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Langmuir Isotherm Model is a fundamental mathematical framework used in process engineering to describe the equilibrium distribution of adsorbate molecules between a liquid phase and a solid adsorbent surface. Understanding the underlying classification of adsorption mechanisms helps in designing and optimizing unit operations such as wastewater treatment, protein purification, and gas separation.
In practice, the model assumes that adsorption occurs at specific, localized sites on the adsorbent surface, that each site can hold only one molecule (monolayer adsorption), and that there are no lateral interactions between adsorbed molecules. By determining the maximum adsorption capacity and the affinity constant, engineers can predict the performance of fixed‑bed adsorbers and batch contactors under varying process conditions, and they often compare these results with those obtained from the BET model for specific surface area to gain a more comprehensive understanding of surface characteristics.
Methodology & Formulas
The calculation process relies on determining the equilibrium adsorption capacity based on mass balance and subsequently linearizing the Langmuir equation to extract characteristic parameters, while a similar approach is used for the Freundlich isotherm model calculation.
First, the equilibrium adsorption capacity x is calculated for each experimental data point using the mass balance of the adsorbate:
\[ x = \frac{(C_{0} - C) \cdot V}{m} \]
To determine the model parameters xm and K, the Langmuir equation is rearranged into a linear form, allowing for the application of linear regression (y = slope·x + intercept), as explained in the single‑stage batch adsorption material balance methodology.
The parameters are extracted from the regression results as follows:
\[ x_{m} = \frac{1}{\text{intercept}} \]
\[ K = \frac{\text{intercept}}{\text{slope}} \]
Parameter
Condition/Regime
Criteria
Mass Balance
Physical Validity
C < C_{0}
Equilibrium Concentration
Physical Validity
C > 0
Langmuir Constant
Empirical Range
0.001 ≤ K ≤ 10.0
Regression Quality
Model Fit
R^{2} > 0.95
To extract the Langmuir constants from batch adsorption data, follow these steps:
Collect equilibrium concentration (C) and amount adsorbed (x) at several points.
Linearize the Langmuir equation using the form \( \frac{1}{x} = \frac{1}{x_{m} \cdot K} \cdot \frac{1}{C} + \frac{1}{x_{m}} \).
Plot \( 1/x \) versus \( 1/C \). The slope equals \( 1/(x_{m} \cdot K) \) and the intercept equals \( 1/x_{m} \).
Calculate x_{m} from the intercept (x_{m} = 1/\text{intercept}) and then determine K from the slope (K = \text{intercept} / \text{slope}).
Validate the fit by checking the correlation coefficient (R^{2}) of the linear plot.
The Langmuir isotherm is expressed as:
\( x = \frac{x_{m} \cdot K \cdot C}{1 + K \cdot C} \)
Where:
x – equilibrium adsorption capacity (mass of adsorbate per mass of adsorbent, e.g., mg/g).
C – equilibrium concentration of adsorbate in the fluid phase (e.g., mg/L).
x_{m} – maximum monolayer adsorption capacity (same units as x).
K – Langmuir affinity constant (related to adsorption energy, units are the inverse of C, e.g., L/mg).
To use the equation, insert the known values of C, x_{m}, and K to calculate the predicted x. This value can then be compared with experimental data or incorporated into mass-balance calculations for design of adsorption columns.
Evaluate model suitability with the following criteria:
Fit quality: Perform linear regression on the Langmuir linear form and examine the correlation coefficient (R^{2}). Values close to 1 indicate a good fit.
Residual analysis: Plot residuals (experimental – predicted x) versus C. Randomly distributed residuals suggest the model is appropriate.
Physical plausibility: Verify that the calculated x_{m} is realistic for the adsorbent’s surface area and that K is positive.
Compare with alternative isotherms (e.g., Freundlich, Temkin). If the Langmuir fit is superior in terms of R^{2} and residuals, it is likely the best representation.
Dimensionless parameters simplify integration into simulation software. Convert as follows:
Define a reference concentration, C_{\text{ref}} (often the inlet concentration or a characteristic concentration of the system).
Calculate the dimensionless affinity constant: K^{*} = K \cdot C_{\text{ref}}.
Define a reference adsorption capacity, x_{\text{ref}} (e.g., x_{m} or a characteristic loading). Calculate the dimensionless capacity: X^{*} = x_{m} / x_{\text{ref}}.
Insert K^{*} and X^{*} into the dimensionless Langmuir expression: x^{*} = \frac{X^{*} \cdot K^{*} \cdot C^{*}}{1 + K^{*} \cdot C^{*}}, where C^{*} = C / C_{\text{ref}} and x^{*} = x / x_{\text{ref}}.
Use these dimensionless forms to maintain consistency across unit systems and to reduce numerical stiffness in dynamic simulators.
Worked Example: Determining Langmuir Isotherm Parameters for Protein Adsorption
A batch adsorption experiment is conducted to study the equilibrium adsorption of bovine serum albumin (BSA) onto a synthetic ion-exchange resin at 25°C. The goal is to determine the maximum monolayer adsorption capacity x_{m} and the Langmuir equilibrium constant K for the Langmuir isotherm model.
Measured equilibrium BSA concentrations, C: 2.0, 5.0, 15.0, 40.0 mg/L (corresponding to each C_{0})
Step-by-Step Calculation:
Calculate the equilibrium adsorption capacity x for each data point using x = (C_{0} - C) \cdot V / m. For the data point with C_{0} = 100.0 \, \text{mg/L} and C = 40.0 \, \text{mg/L}:
\[ x = \frac{(100.0 - 40.0) \cdot 0.1}{0.1} = 60.0 \, \text{mg/g} \]
Similar calculations yield: for (10.0, 2.0): x = 8.0 mg/g; for (20.0, 5.0): x = 15.0 mg/g; for (50.0, 15.0): x = 35.0 mg/g.
Linearize the data by computing 1/x and 1/C for each point. For the point with x = 60.0 \, \text{mg/g} and C = 40.0 \, \text{mg/L}:
\[ 1/x = \frac{1}{60.0} = 0.01667 \, \text{g/mg} \]
\[ 1/C = \frac{1}{40.0} = 0.02500 \, \text{L/mg} \]
The full dataset is:
C (mg/L)
x (mg/g)
1/C (L/mg)
1/x (g/mg)
2.0
8.0
0.5000
0.12500
5.0
15.0
0.2000
0.06667
15.0
35.0
0.06667
0.02857
40.0
60.0
0.02500
0.01667
Perform linear regression on the plot of 1/x vs. 1/C using all four data points. The regression calculations using the sums of the data yield:
Sum of 1/C, \( \Sigma (1/C) = 0.79167 \)
Sum of 1/x, \( \Sigma (1/x) = 0.23691 \)
Sum of (1/C) \cdot (1/x), \( \Sigma \left( \frac{1}{C} \cdot \frac{1}{x} \right) = 0.07815 \)
Sum of (1/C)^{2}, \( \Sigma \left( (1/C)^{2} \right) = 0.29507 \)
The Langmuir isotherm parameters for BSA adsorption on the synthetic resin at 25°C, derived from linear regression of the provided data, are:
x_{m} = 69.0 \, \text{mg/g} (rounded to three significant figures)
K = 0.0642 \, \text{L/mg}
Note: The regression fit for this example data yields an R^{2} value of approximately 0.998, indicating an excellent linear fit and validating the use of the Langmuir model for this system.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
