Isotherms Isotherm Equation Uses and notes Ref. One-Parameter Isotherm Henry’s ${q}_{e}={K}_{HE}{C}_{e}$ Simplest adsorption isotherm Two-Parameter Isotherm Hill-Deboer $\mathrm{ln}\left[\frac{{C}_{e}\left(1-\theta \right)}{\theta }\right]-\frac{\theta }{1-\theta }=-\mathrm{ln}{k}_{1}-\frac{{k}_{2}\theta }{RT}$ Math_9# o Defines the case where there is mobile adsorption and lateral interaction and K2 is the energetic constant of the interaction between adsorbed molecules (kJ∙mol−1)  Fowler- Guggenheim $\mathrm{ln}\left[\frac{{C}_{e}\left(1-\theta \right)}{\theta }\right]=-\mathrm{ln}{k}_{FG}+\frac{2w\theta }{RT}$ Math_11# o Side contact between adsorbed molecules and the heat of adsorption (w) varies linearly with loading. If attractive, w = positive, repulsive w = negative and w = 0 no interaction between absorbate. But applicable only if θ < 0.6  Langmuir ${q}_{e}=\frac{{q}_{m}{k}_{L}{C}_{e}}{1+{k}_{L}{C}_{e}}$ $\frac{{C}_{e}}{{q}_{e}}=\frac{1}{{q}_{m}{k}_{L}}+\frac{{C}_{e}}{{q}_{m}}$ ${R}_{L}=\frac{1}{1+{k}_{L}{C}_{o}}$ o Homogeneous binding sites (same affinities), alike sorption energies, and no interactions between adsorbed species. RL is separation aspect which decides whether the adsorption is un-favorable when its value is >1, linear (=1), favourable (0 < RL < 1), and irreversible (RL = 0)  Freundlich ${q}_{e}={k}_{f}{c}_{e}^{1/n}$ $\mathrm{log}{q}_{e}=\mathrm{log}{k}_{F}+\frac{1}{n}\mathrm{log}{c}_{e}$ o Heterogonous surfaces(varied affinities), 1/n (adsorption intensity) that specifies the energy and the heterogeneity of the adsorbent sites (1/n < 1= Langmuir, 1/n > 1cooperative adsorption and generally, n in b/n 2 - 10 indicates good favorability of sorption