Freundlich equation

Lore & Background

Herbert Freundlich introduced this empirical adsorption isotherm in 1909 to describe the relationship between the quantity of gas adsorbed per unit mass of a solid adsorbent and the gas pressure, under constant temperature. The same mathematical form applies to the adsorption of a solute from a liquid phase onto a solid, where the solute concentration replaces pressure. The equation is expressed as x/m = K c^(1/n), where x is the mass of adsorbate, m the mass of adsorbent, c the equilibrium concentration (or pressure for gases), and K and n are constants specific to the adsorbate, adsorbent, and temperature. The constant n is a correction factor, while K acts as a distribution coefficient. Although the equation is empirical, it can be derived theoretically by attributing the variation in the equilibrium constant to surface heterogeneity and a changing heat of adsorption. Freundlich’s original experiments involved the adsorption of three organic acids onto coal from aqueous solutions; he determined K and n values through numerical analysis. A key characteristic is that at high pressures, the extent of adsorption becomes independent of pressure. The equation is unique: if data fit it, heterogeneity of the surface is likely but not proven; calorimetry can confirm this. Homogeneous surfaces have a constant heat of adsorption, while heterogeneous surfaces show variable heat depending on site occupancy—high-energy sites fill first at low pressure, then low-energy sites at higher pressure, weakening the heat of adsorption. A major limitation is that the isotherm fails at high pressures, where adsorption saturates and no longer increases with pressure.

Reader's Guide

The Freundlich equation is mathematically expressed as x/m = K * c_eq^(1/n) or, in logarithmic form, log(x/m) = log K + (1/n) log c_eq. For gas-phase experiments, pressure p replaces concentration. The constants K and n are specific to a given adsorbate and adsorbent at a given temperature. The equation is unique in that if data fit it, it suggests but does not prove surface heterogeneity; calorimetry can confirm this. A limitation is that at higher pressures, the equation fails because adsorption saturates. When adsorption behavior can be properly fit by theoretically based isotherms (e.g., Langmuir or BET), those are usually preferred.

Did You Know?

• The Freundlich equation is entirely empirical, though it can be derived from surface heterogeneity and variation in heat of adsorption.
• Freundlich's original experiments used the adsorption of organic acids on coal in aqueous solutions.
• At high pressure, the exponent 1/n approaches zero, making the extent of adsorption independent of pressure.
• The equation fails at higher pressures because adsorption saturates beyond a certain point.