Freundlich Isotherm Model Calculation

Reference ID: MET-8841 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The Freundlich isotherm model describes equilibrium adsorption of a solute from a liquid phase onto a solid adsorbent. It is widely used in batch adsorption studies, fixed-bed design, and contaminant removal assessments because it captures the empirically observed non-linear relationship between the equilibrium aqueous concentration x (mass of solute per unit volume) and the adsorbed amount y^* (mass of solute per unit mass of adsorbent). The model is applicable when the surface is heterogeneous and adsorption sites have a distribution of energies, which is typical for activated carbon, silica gels, and many polymeric adsorbents.

Methodology & Formulas

The calculation follows the logical sequence implemented in the reference Python script, expressed here with standard engineering symbols and LaTeX notation.

1. Data Preconditions

All experimental concentrations must be strictly positive to permit logarithmic transformation:

\[ x_i > 0,\qquad y_i^{*} > 0\quad \text{for } i = 1,\dots ,N \]

2. Log-log Transformation

Convert the raw data to base-ten logarithms:

\[ X_i = \log_{10}\!\left(x_i\right),\qquad Y_i = \log_{10}\!\left(y_i^{*}\right) \]

3. Linear Regression on Transformed Data

Define the following summations over all N experimental points:

\[ S_X = \sum_{i=1}^{N} X_i,\qquad S_Y = \sum_{i=1}^{N} Y_i,\qquad S_{XX}= \sum_{i=1}^{N} X_i^{2},\qquad S_{XY}= \sum_{i=1}^{N} X_i\,Y_i \]

Compute the denominator of the regression coefficients:

\[ D = N\,S_{XX} - S_X^{2} \]

If \(|D|\) approaches zero, the data are collinear in log-log space and regression is undefined.

The slope of the best-fit line corresponds to the Freundlich exponent n:

\[ n = \frac{N\,S_{XY} - S_X\,S_Y}{D} \]

The intercept provides \(\log_{10} m\):

\[ \log_{10} m = \frac{S_Y - n\,S_X}{N} \]

4. Back-Transformation to Original Parameters

Recover the Freundlich coefficient m by exponentiation:

\[ m = 10^{\log_{10} m} \]

5. Prediction of Adsorbed Amount

For any admissible liquid concentration \(x_{\text{pred}}\), the equilibrium adsorbed amount is obtained from the non-linear Freundlich expression:

\[ y_{\text{pred}}^{*} = m \; x_{\text{pred}}^{\,n} \]

6. Validity and Empirical Checks

The model is considered reliable only when the following criteria are satisfied:

Criterion	Mathematical Condition	Interpretation
Exponent range	\(0.1 \le n \le 1.0\)	Ensures physically realistic adsorption intensity; values greater than 1 suggest data or model issues.
Coefficient positivity	\(m > 0\)	Negative m would imply negative adsorption capacity.
Prediction domain	\( \dfrac{x_{\min}}{10} \le x_{\text{pred}} \le 10\,x_{\max}\)	Limits extrapolation to one order of magnitude beyond the experimental concentration range \([x_{\min},\,x_{\max}]\).

7. Summary of Computational Flow

Verify positivity of all x and y* data.
Transform each datum to \(\log_{10}\) space, yielding \(\{X_i, Y_i\}\).
Calculate the summations \(S_X, S_Y, S_{XX}, S_{XY}\) and the denominator \(D\).
Obtain the slope n and intercept \(\log_{10} m\) using the regression formulas.
Back-transform to acquire the Freundlich coefficient m.
Check that n lies within the typical range, that m is positive, and that the prediction concentration respects the extrapolation limits.
Compute the predicted adsorbed amount \(y_{\text{pred}}^{*}\) with the Freundlich equation.

To determine the constants for the Freundlich isotherm model, you must linearize the empirical equation and perform regression:

Transform each datum to base-ten logarithms: compute \(X_i = \log_{10}(x_i)\) and \(Y_i = \log_{10}(y_i^{*})\).
Plot \(\log_{10}(y^{*})\) against \(\log_{10}(x)\).
Perform a linear regression analysis on the resulting plot.
The slope of the line is the Freundlich exponent n (the adsorption intensity).
The y-intercept of the line is \(\log_{10}(m)\), where m is the Freundlich coefficient (adsorption capacity).
Obtain m by raising ten to the power of the intercept: \(m = 10^{\,\text{intercept}}\).

The Freundlich exponent n indicates the intensity of the adsorption process and governs the shape of the isotherm:

If n is less than 1, the isotherm is concave downward and the adsorption process is considered favorable — high uptake is achieved at low equilibrium concentrations, which is desirable for most treatment applications.
If n is equal to 1, the adsorption is linear; the adsorbed amount is directly proportional to the liquid concentration.
If n is greater than 1, the isotherm is concave upward and the adsorption is considered unfavorable — the uptake at low concentrations is poor, requiring higher driving concentrations for effective removal.

In practice, values of n between 0.1 and 1.0 are typical for heterogeneous adsorbents such as activated carbon.

The Freundlich model is generally preferred in the following scenarios:

When the adsorbent surface is highly heterogeneous, meaning the adsorption sites have a continuous distribution of affinities for the adsorbate.
When the experimental data spans a wide concentration range and exhibits a non-linear trend that does not approach a clear plateau at high concentrations.
When multilayer adsorption or site-to-site interactions are suspected, conditions under which the Langmuir monolayer assumption may be violated.

Worked Example: Freundlich Isotherm for Pigment Adsorption

Scenario: Batch adsorption of a pigment from water onto activated carbon at 25°C under equilibrium conditions.

Knowns

Experimental data for liquid concentration x (mg/L) and adsorbed amount y* (mg/g):
- x = 0.1 mg/L, y* = 2.5 mg/g
- x = 0.3 mg/L, y* = 4.2 mg/g
- x = 1.0 mg/L, y* = 7.0 mg/g
- x = 3.0 mg/L, y* = 12.0 mg/g
- x = 10.0 mg/L, y* = 20.0 mg/g
Number of data points: N = 5
Prediction concentration: x = 5.0 mg/L

Step-by-Step Calculation

Tabulate experimental data. The five data pairs are listed above.
Transform to \(\log_{10}\) coordinates. The sums of the transformed values are computed: \[ \sum \log_{10}(x) = -0.046,\quad \sum \log_{10}(y^{*}) = 4.246, \] \[ \sum [\log_{10}(x)]^{2} = 2.501,\quad \sum [\log_{10}(x) \cdot \log_{10}(y^{*})] = 1.092. \]
Compute the regression denominator. \[ D = N \cdot \sum [\log_{10}(x)]^{2} - \left[\sum \log_{10}(x)\right]^{2} = 12.503. \]
Determine the slope n and intercept \(\log_{10}(m)\). \[ n = \frac{N \cdot \sum [\log_{10}(x) \cdot \log_{10}(y^{*})] - \sum \log_{10}(x) \cdot \sum \log_{10}(y^{*})}{D} = 0.452, \] \[ \log_{10}(m) = \frac{\sum \log_{10}(y^{*}) - n \cdot \sum \log_{10}(x)}{N} = 0.853. \]
Back-transform to obtain the Freundlich coefficient m. \[ m = 10^{\log_{10}(m)} = 7.136\;(\text{mg/g})(\text{L/mg})^{0.452}. \]
Validate the parameters. The exponent n = 0.452 falls within the typical range of 0.1 to 1.0. The coefficient m = 7.136 is positive. The prediction concentration x = 5.0 mg/L lies within the allowable extrapolation range (0.01 to 100.0 mg/L).
Predict the adsorbed amount at x = 5.0 mg/L. \[ y^{*} = m \cdot x^{n} = 7.136 \cdot (5.0)^{0.452} = 14.776\;\text{mg/g}. \]

Final Answer

The Freundlich isotherm parameters are m = 7.136 (mg/g)(L/mg)^0.452 and n = 0.452. At a liquid concentration of 5.0 mg/L, the predicted adsorbed amount is y* = 14.776 mg/g.

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