**11. Biosorption equilibrium isotherms**

Sorption isotherms explain the equilibrium relationships between biosorbent and biosorbate and the mass of the biosorbed component per unit mass of biosorbent and the concentration of biosorbate in the medium under a given set of conditions (temperature and concentration). It also determines the equilibrium distribution of metal ions and how selective retention takes place when two or more biosorbent components are present [185]. The term "isotherm" can be defined as a curve explaining the retention of a substance on a solid at various concentrations [82]. The determination of equilibrium parameters is the basic requirement for designing a good biosorption system. For determination of the best-fitting sorption isotherm, linear regression is frequently used. In order to predict the isotherm parameters, the method of least squares is applied.

The biosorption capacities of different biosorbents for different pollutants can be best explained by biosorption equilibrium isotherms. Several isotherm models are available to describe the mechanism of the biosorption process and the equilibrium biosorption distribution. Some of the isotherms used in biosorption studies are Langmuir, Freundlich, and Temkin isotherms. However, the biosortion process may show better fit with a specific isotherm.

Biosorption isotherm data of Pb (II) and Cu (II) ions onto green algal species, *Spirogyra* and *Cladophora,* were in good agreement with the Langmuir isotherm demonstrating the formation of the monolayer coverage of metal ions on the outer surface of the biosorbent [95]. The Langmuir model fitted well with the biosorption of Pb (II), Zn (II), and Ni (II) ions onto *Bacillus subtilis* [186]. Freundlich isotherm showed the best fit for the biosorption of Cu (II) ions onto lactic acid bacterium, *Enterococcus faecium* [106]. Biosorption of Cr (VI) ions onto *Bacillus thuringiensis* also shows the better fit with Freundlich isotherm [187].
