Adsorption isotherms

 Adsorption isotherms describe how molecules adhere to a solid surface as a function of their concentration in a surrounding fluid. These isotherms are crucial in understanding surface interactions in various fields, such as catalysis, environmental science, and material science.

1. Langmuir Isotherm:

   • Assumes a fixed number of identical sites on the surface.
   • Adsorption is limited to a monolayer (one molecule thick).
   • Describes the equilibrium between adsorbed molecules and free molecules in the fluid.
   • Mathematical expression: $\frac{q_e}{q_m}=\frac{K_L{C}_e}{1+K_L{C}_e}$
     • $q_e$ = amount adsorbed per unit mass of adsorbent
     • $q_m$ = maximum adsorption capacity
     • $K_L$ = Langmuir constant
     • $C_e$ = equilibrium concentration of the adsorbate

2. Freundlich Isotherm:

   • Describes adsorption on heterogeneous surfaces with varied affinities for the adsorbate.
   • Allows for multilayer adsorption.
   • Mathematical expression: $q_e=K_F{C}_e^{1\mathrm{/}n}$
     • $K_F$ = Freundlich constant (related to the adsorption capacity)
     • $1\mathrm{/}n$ = adsorption intensity

3. BET Isotherm (Brunauer-Emmett-Teller):

   • Extends the Langmuir model to multilayer adsorption.
   • Applicable for porous materials and surfaces.
   • Mathematical expression: $\frac{C_e}{q_e}=\frac{1}{q_m}(\frac{C_e}{K_B}+\frac{C_e-C_0}{K_B})$
     • $K_B$ = BET constant
     • $C_0$ = constant related to the surface energy

4. Temkin Isotherm:

   • Assumes that the heat of adsorption decreases linearly with coverage.
   • Accounts for the interactions between adsorbate molecules.
   • Mathematical expression: $q_e=B\ln (A{C}_e)$
     • $A$ = Temkin constant related to the adsorption capacity
     • $B$ = Temkin constant related to the adsorption energy

Each isotherm provides different insights depending on the nature of the adsorbent and adsorbate, and they can be used to design and optimize processes involving adsorption.