qe is the amount of solute adsorbed per unit weight of adsorbent (mg g-1), Ce is the equilibrium concentration of solute in the bulk solution (mg L-1). Q0 is the monolayer adsorption capacity (mg g-1) and b is the constant related to the free energy of adsorption (b∝ e-∆G/RT) and represents the affinity of adsorbate and adsorbent. ∆G is the free energy, K is the Langmuir constant same as b. The sign of ∆G can be used to indicate the feasibility of adsorption process and the spontaneous nature of adsorption.
K LF Ce nLF qe 1 aLF Ce nLF
qe is the amount of solute adsorbed per unit weight of adsorbent (mg g-1), Ce is the equilibrium concentration of solute in the bulk solution (mg L-1). KLF, aLF, and nLF are the Slips constants.
RT qe (ln ACe ) b
qe is the amount of solute adsorbed per unit weight of adsorbent (mg g-1), Ce is the equilibrium concentration of solute in the bulk solution (mg L-1). R, A, T and b are Tempkin constants.
(establish one and change another one)
Sample after equilibrium (established parameter)
Amount absorbed (mmol/g)
Different temperature pH Ionic strength
qt qe (1 e k1t )
dqt k2 (qe qt )2 dt t 1 t 2 qt k2 qe qe
Time (h, d)
t is the adsorption time; qt is the amount of adsorbate adsorbed at time " t "; qe is the amont adsorbed at equilibrium; k0 , k1 , k 2 are the adsorption rate constant . 4
Q0
(Langmuir, 1918; Freundlich, 1906; Appelo & Postma, 2005)
6
Adsorption modelling
Adsorption isotherm or equilibrium experiment

Redlich-Peterson model
Байду номын сангаас
Tempkin considered the effects of some indirect adsorbate/adsorbent interactions and suggested that because of these interactions the heat of adsorption of all the molecules in the layer would decrease linearly with coverage
Redlich and Peterson, 1959; Allen, 2004
7
Adsorption modelling
Adsorption isotherm or equilibrium experiment

Tempkin model

Slips or Langmuir-Freundlich isotherm model
5
Adsorption modelling
Adsorption isotherm or equilibrium experiment

Langmuir isotherms

Freundlich isotherms
The adsorption capacity (Q0) of adsorbent can be calculated.
Ideal experiment design
Parameters may affect adsorption Solution: pH, Eh, temperature, ion strength,
competing or interaction with other ions, adsorbent dose, adsorbate concentration, contacting time.
2
Ideal experiment design
Adsorption isotherm or equilibrium experiment Experiments can be done under different temperature, pH,
ionic strength.
Vary adsorbate concentration to adsorbent dose ratio
Allen, 2004; Vijayaraghavan, 2006; Mohan, 2011
At low adsorbate concentrations it effectively reduces to the Freundlich isotherm. At high adsorbate concentrations, it predicts a monolayer sorption capacity characteristic of the Langmuir isotherm 8
Adsorbate: coordination number, ionic radius, electro
negativity, standard reduction potential, species in solution, and other physical and chemical property.
Adsorbent: surface functional groups, surface area,
pHzpc, elemental composition, morphology, pore structure, particle size distribution, and other physical and chemical property.
R-P model can be used to represent adsorption equilibria over a wide concentration range.
Toth model is derived from potential theory and is applicable to heterogeneous adsorption.
“Sorption extends infinitely as adsorbate concentrations increase.”
Amount absorbed (mmol/g)
Equilibrium concentration (mmol/L)
Amount absorbed (mmol/g) Equilibrium concentration (mmol/L)
1
Ideal experiment design
Adsorption kinetics experiment Experiments can be done under different
adsorbate concentrations, adsorbent dose, temperature, pH, etc.

Toth isotherm model
K RP Ce qe 1 aP Ce
qe is the amount of solute adsorbed per unit weight of adsorbent (mg g-1), Ce is the equilibrium concentration of solute in the bulk solution (mg L-1). KRP, aRP and β are Redlich-Peterson constants and the exponent, β, lies between 0 and 1.
Equilibrium concentration (mmol/L)
3
Adsorption modelling
Adsorption kinetics experiment
Mohan, 2011
Zero order kinetic model

Pseudo first order kinetic model

Pseudo second order kinetic model
Amount adsorbed (mmol/g)
Time (h, d)