# Freundlich equation

Freundlich's original data for adsorption of acetic acid (page 392 in [3]) and a fit according to Freundlich's exponential law.

The Freundlich adsorption isotherm is mathematically expressed as

${\displaystyle {\frac {x}{m}}=Kp^{1/n}}$

It is also written as

${\displaystyle \log {\frac {x}{m}}=\log K+{\frac {1}{n}}\log p}$

or

${\displaystyle {\frac {x}{m}}=Kc^{1/n}}$

It is also written as

${\displaystyle \log {\frac {x}{m}}=\log K+{\frac {1}{n}}\log c}$

where

p = equilibrium pressure of the gaseous adsorbate in case of experiments made in the gas phase (gas/solid interaction with gaseous species/adsorbed species)
c = equilibrium concentration of adsorbate in case of experiments made with an aqueous solution in contact with a dispersed solid phase (dissolved species/adsorbed species).

K and n are constants for a given adsorbate and adsorbent at a given temperature (from there, the term isotherm needed to avoid significant gas pressure fluctuations due to uncontrolled temperature variations in the case of adsorption experiments of a gas onto a solid phase).

At high pressure 1/n = 0, hence extent of adsorption becomes independent of pressure.

The Freundlich equation is unique; consequently, if the data fit the equation, it is only likely, but not proved, that the surface is heterogeneous. The heterogeneity of the surface can be confirmed with calorimetry. Homogeneous surfaces (or heterogeneous surfaces that exhibit homogeneous adsorption (single site)) have a constant ΔH of adsorption [4]. On the other hand, heterogeneous adsorption (multi-site) have a variable ΔH of adsorption depending on the percent of sites occupied. When the adsorbate pressure in the gas phase (or the concentration in solution) is low, high-energy sites will be occupied first. As the pressure in the gas phase (or the concentration in solution) increases, the low-energy sites will then be occupied resulting in a weaker ΔH of adsorption.[5]

## Limitation of Freundlich adsorption isotherm

Experimentally it was determined that extent of gas adsorption varies directly with pressure, and then it directly varies with pressure raised to the power 1/n until saturation pressure Ps is reached. Beyond that point, the rate of adsorption saturates even after applying higher pressure. Thus, the Freundlich adsorption isotherm fails at higher pressure.

## References

1. ^ Freundlich, Herbert. Kapillarchemie, eine Darstellung der Chemie der Kolloide und verwandter Gebiete. Akademische Verlagsgesellschaft, 1909.
2. ^ Adamson, A.W (1997). Physical chemistry of surfaces. p. 393.
3. ^ Freundlich, Herbert (1907). " Über die Adsorption in Lösungen." Zeitschrift für Physikalische Chemie – Stöchiometrie und Verwandschaftslehre. 57 (4), 385–470.
4. ^ Burke GM, Wurster DE, Buraphacheep V, Berg MJ, Veng-Pedersen P, Schottelius DD. Model selection for the adsorption of phenobarbital by activated charcoal. Pharm Res. 1991 Feb;8(2):228-31. doi: 10.1023/a:1015800322286. PMID: 2023872.
5. ^ Adamson, A.W (1997). Physical chemistry of surfaces. p. 699.