The theoretical basis for the Freundlich adsorption isotherm

The Freundlich isotherm has for more than a century been known to describe many adsorption situations, ranging from nutrient uptake on soil particles during the agricultural use of fertilizers over adsorption on heterogeneous catalysts and fuel cell electrodes to adsorption of CO 2 for carbon capture processes. Understanding this isotherm is thus essential for the rational development of materials and processes for adsorption applications, many of which are important for the sustainable transition. It is known that the Freundlich isotherm emerges, if the adsorption energy declines exponentially with the amount of adsorbates taken up, but because there has been no theoretical explanation of the reason for this exponential relationship, the isotherm has remained purely empirical. Here the first complete theoretical derivation of the Freundlich isotherm that can be validated against experiments is provided. When combining the use of equilibrium thermodynamics to describe an equilibrated, multi-faceted surface with theoretical relations that correlate the adsorption energy and surface tension via the coordination number of the surface atoms, the result is shown to be the Freundlich isotherm. Thereby it is determined that the isotherm describes surface heterogeneity and that it is only formally valid at low adsorbate coverages. The theoretical derivation is validated against experiments by showing examples of how multifaceted surfaces give Freundlich behavior, whereas the adsorption on uniform single crystal surfaces of the same material does not. With this new theoretical understanding the Freundlich isotherm can be used to assist rational development of new processes and materials for adsorption applications.