Freundlich equation

The Freundlich adsorption isotherm is mathematically expressed as

${\displaystyle {\frac {x}{m}}=K\cdot c_{\,\mathrm {eq} }^{\,1/n}}$

(1)

In Freundlich's notation (used for his experiments dealing with the adsorption of organic acids on coal in aqueous solutions), ${\displaystyle x/m}$ signifies the ratio between the adsorbed mass or adsorbate ${\displaystyle x}$ and the mass of the adsorbent ${\displaystyle m}$, which in Freundlich's studies was coal. In the figure above, the x-axis represents ${\displaystyle c_{\mathrm {\,eq} }}$, which denotes the equilibrium concentration of the adsorbate within the solvent.

Freundlich's numerical analysis of the three organic acids for the parameters ${\displaystyle K}$ and ${\displaystyle n}$ according to equation 1 were:

Freundlich's experimental data can also be used in a contemporary computer based fit. These values are added to appreciate the numerical work done in 1907.

△ K and △ n values are the error bars of the computer based fit. The K and n values itself are used to calculate the dotted lines in the figure.

Equation 1 can also be written as

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

Sometimes also this notation for experiments in the gas phase can be found:

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

x = mass of adsorbate

m = mass of adsorbent

p = equilibrium pressure of the gaseous adsorbate in case of experiments made in the gas phase (gas/solid interaction with gaseous 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).

K = distribution coefficient

n = correction factor

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]

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 P_s is reached. Beyond that point, the rate of adsorption saturates even after applying higher pressure. Thus, the Freundlich adsorption isotherm fails at higher pressure.

• Langmuir adsorption model

• Jaroniec, M. (1975). "Adsorption on heterogeneous surfaces: The exponential equation for the overall adsorption isotherm". Surface Science. 50 (2): 553–564. Bibcode:1975SurSc..50..553J. doi:10.1016/0039-6028(75)90044-8.
• Levan, M. Douglas; Vermeulen, Theodore (1981). "LeVan, M. Douglas, and Theodore Vermeulen. "Binary Langmuir and Freundlich isotherms for ideal adsorbed solutions." The Journal of Physical Chemistry 85.22 (1981): 3247–3250". The Journal of Physical Chemistry. 85 (22): 3247–3250. doi:10.1021/j150622a009.
• "Freundlich Equation". Archived from the original on 3 March 2016.

• "Freundlich equation solver".
• "Freundlich Adsorption Isotherm". Archived from the original on 2 March 2012.