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 K | [47] |

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 | [47] |

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. R | [48] |

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 | [49] |