1.1 Basic concepts and theory of adsorption
1.1.5 Adsorption isotherms for one-component systems
An adsorption isotherm describes the amount of component adsorbed on the adsorbent surface versus the adsorbate amount in the fluid phase at equilibrium. The fitting of adsorption isotherm equations is one of the main stages in data-analysis. The most commonly used adsorption isotherms are still the Langmuir and Freundlich models even though they were first introduced over 90 years ago [11-13].
Theoretical adsorption isotherms have originally been derived for the gas phase adsorption [10] and these equations were later applied with some modifications on the liquid-solid systems. For example, the simplest isotherm equation, Langmuir isotherm, is derived in this
section using the kinetic theory of gas as well as thermodynamic approach. Other isotherm models are presented directly for liquid-solid adsorption without derivation.
Langmuir isotherm
The assumptions of the Langmuir model are: (i) homogeneous surface, when adsorption energy of all adsorption sites is constant, (ii) adsorption of adsorbates is localized, and (iii) each of the adsorption sites can accommodate only one adsorbate. Kinetic theory of gas gives that the rate (Rs, mol/cm2s) of which a component collide on the surface is:
T MR R P
g
s = 2π (5)
where P is pressure (Pa), M molecular mass of the gas (kg/mol), Rg the universal gas constant, and T temperature (K). When gas molecules strike on the surface, a fraction of them will be adsorbed. In an ideal case this fraction would be unity, but Langmuir presented that a striking coefficient α, which accounts for non-perfect striking, should be added into the equation to obtain rate of the adsorption on bare surface:
T MR R P
g
ads 2π
= α (6)
When a gas molecule collides to the occupied adsorption site, it will reflect back to the gas phase very quickly. Therefore, the rate of the adsorption on the occupied surface can be written as:
(
1 θ)
π 2
α
g
a = −
T MR
R P (7)
where θ is a fractional coverage. The rate of desorption is the desorption rate from fully occupied surface (kdes) multiplied by a fractional coverage.
θ
where Edes is the activation energy for desorption and kdes∞ a rate constant of desorption at infinite temperature. The Langmuir isotherm can be obtained by equating the rates of the adsorption and desorption (Eq. 7 and 8):
bP
Parameter b is called affinity constant and Q is the heat of the adsorption, which is equal to the activation energy of desorption. Thus, when b is higher, surface is covered more with adsorbates and when Q increases, the adsorbed amount increases due to the higher energy barrier adsorbates have to overcome to be evaporated back into the gas phase [10].
In liquid-solid system, pressure is replaced by equilibrium concentration (Ce) and b by the Langmuir affinity constant KL. Furthermore, when both of the sides of Eq. (9) are multiplied by qm (mmol/g, maximum adsorption capacity), the Langmuir equation is transformed to:
e
(L/mmol) does not have a similar definition than in the gas-solid adsorption, but it still describes how strong the interactions between the adsorbate and surface are. At low concentrations, the Langmuir isotherm follows the Henry’s law i.e. adsorption capacity is linearly proportional to the adsorbate concentration in the solution. At high concentrations, the Langmuir isotherm approaches a constant capacity value [14].
Thermodynamic approach for the derivation of the Langmuir isotherm is based on the equilibrium equation of binding of adsorbate A to surface site S.
[ ][ ] [ ]
ASS A
AS S
A+ ↔ Keq =
K
(12)
where Keq is a thermodynamic equilibrium constant. The mass balance equation in this case is:
[ ] [ ] [ ]
Stot = S + AS (13)where [Stot] is a total amount of surface sites. By combining equations (12) and (13) and assuming that there are no changes of [AS] with respect of time one will obtain:
[ ] [ ] ( ) [ ] [ ]
eqeq tot
/ A 1
A / AS S
K K
= + (14)
which can be arranged to equation (11). The Langmuir model is a very commonly used isotherm equation mostly because it can be linearized and therefore easily fitted to the experimental data [13]. Most of the adsorption materials, however, have heterogeneous surface, which is taken into account in many other isotherm equations.
Freundlich isotherm
As the Langmuir isotherm, the Freundlich isotherm [11] contains only two parameters and is the simplest isotherm that takes the surface heterogeneity into account [13,14]. The Freundlich isotherm is given as:
/ F 1 e F e
C n
K
q = (15)
where KF ((mmol/g)/(L/mmol)nF) and nF are the Freundlich adsorption constants. The Freundlich isotherm is one of the earliest empirical isotherms applied for describing adsorption equilibrium.
It can be used for heterogeneous surfaces and for multilayer adsorption. The linearized form of
this equation is often used, but generally the Freundlich isotherm lacks the thermodynamic basis since not approaching the Henry’s law at low concentrations [14].
Sips isotherm (Langmuir-Freundlich)
The Sips isotherm is a combination of the Langmuir and Freundlich isotherms and can be derived using either equilibrium or thermodynamic approach [15]:
S S
) ( 1
) (
e S
e S m
e n
n
C K
C K q q
= + (16)
where KS (L/mmol) is the affinity constant and nS describes the surface heterogeneity. When nS
equals unity, the Sips isotherm returns to the Langmuir isotherm and predicts homogeneous adsorption. On the other hand, deviation of nS value from the unity indicates heterogeneous surface [16]. At high concentrations, the Sips isotherm approaches to a constant value and at low concentrations Freundlich type equation [14].
Redlich-Peterson isotherm
The Redlich-Peterson isotherm [17] also combines the features of the Langmuir and Freundlich models:
) RP
(
1 RP e
e RP m
e K C n
C K q q
= + (17)
where KRP and nRP are the Redlich-Peterson constants. There are some discrepancies in literature about the definitions of the Redlich-Peterson parameters. In some cases they are said to be constants without any physical meaning [14,18-21] and in some cases their meaning is assigned to be similar to the Sips model [13,22-24]. In this study the latter definition is used.
Toth isotherm
The Toth isotherm [25] is an empirical equation, which was derived to improve the Langmuir model fittings at both low and high concentration limits. The Toth model assumes an asymmetrical quasi-Gaussian energy distribution and is useful in the cases of heterogeneous adsorption [26-28].
T T
where aT is adsorptive potential constant (mmol/L) and mT Toth’s heterogeneity factor.
Temkin isotherm
The Temkin isotherm explicitly takes into account the interactions between adsorbent and adsorbate. It is based on the uniformly distributed binding energies up to some maximum energy.
The Temkin isotherm states that heat of the adsorption decreases linearly with the increasing adsorbate coverage [27,28].
(
T e)
equilibrium binding constant i.e. maximum binding energy. Rg is the universal gas constant.General adsorption isotherm (Biosorption model)
The general adsorption isotherm was derived by Liu et al. [29] according to the thermodynamics of adsorption (see the Langmuir case).
Gen equilibrium constant, and nGen is a positive constant. When nGen = 1, the equation reduces to the Langmuir equation and when equilibrium concentration (Ce) in the liquid phase is much lower
than the constant Kads, equation becomes the Freundlich isotherm. The Sips isotherm can also be derived from the general adsorption equation [29].
Dubinin-Radushkevich isotherm
The Dubinin-Radushkevich isotherm is based on the potential theory and assumes a Gaussian energy distribution.
(
2)
m
e q exp BDRε
q = − (22)
where ε is the Polanyi potential given by:
+
=
e
1 1
ln C
ε RT (23)
A constant BDR (mmol2/J2) is related to the mean free energy Eads of the adsorption per molecule when it is transferred to the surface from infinity of the bulk phase.
BDR
E 2
1
ads= (24)
Besides of Eq. (22) a modified form of the Dubinin-Radushkevich isotherm can be found in the literature [13].
Fritz-Schlunder isotherm
Increasing the amount of estimated parameters make isotherm equations more flexible leading generally better fitting results. The following empirical four-parameter isotherm equation was proposed by Fritz and Schlunder [19,30]:
FS FS
e FS
e FS m
e 1 m
n
C K
C K q q
= + (25)
where KFS (L/mmol) corresponds to the Langmuir affinity constant and nFS and mFS describe the heterogeneity of the surface. This kind of models, however, should be carefully used since the physical meaning of the parameters is not clear.
BiLangmuir isotherm (Two-site Langmuir)
The heterogeneity of the adsorption system can be taken into account by assuming two or more different surface active sites that follow a Langmuir type behavior [31]. The BiLangmuir model (two-site Langmuir model) is the simplest four-parameter isotherm incorporating two Langmuir equations:
e BiL2
e BiL2 2 m e BiL1
e BiL1 m1
e 1 1 K C
C K q C K
C K q q
+ +
= + (26)
where qm1 (mmol/g) is the maximum adsorption capacity of the first active site and KBiL1 (L/mmol) the adsorption energy related to that active site. Similarly qm2 and KBiL2 are the corresponding parameters related to the second adsorption site. It should be noted that both of the active sites are homogeneous and can bind only one adsorbate at a time according to the assumptions of the simple Langmuir model.
Other isotherms
Some other empirical three-parameter models, although less commonly used, are for example the Khan [19,32], Koble-Carrigan [19,33], and Jossens [18,34] isotherms. A three-parameter Radke-Prausnitz isotherm [19,35] has thermodynamic basis. Also quite rarely applied four-parameter models are the Weber-van Vliet [18,36], Vieth-Sladek [21,37], and Marczewski-Jaroniec [38]
isotherms.
Isotherm shape
Four different isotherm shapes are commonly observed (Figure 3) [39,40]. The C-type isotherm is a line passing through the origin (Figure 3a). It refers to a system were the ratio between the concentration of the compound in solution and adsorbed on the solid is the same at whole concentration range. In practice, this kind of isotherm can only be obtained for a narrow range of concentrations or low concentrations.
Figure 3. The main types of isotherm curves [modified from refs. 39,40].
The L-type isotherm proposes a progressive saturation of the solid with or without of a strict plateau (Figure 3b). Most of the isotherm equations are of this type. The H-type isotherm is a special case of the L-type isotherm (Figure 3c) with a very high initial slope indicating strong adsorbate–adsorbent interactions [41]. Finally, the S-type isotherm, although quite rarely observed, shows a low adsorption affinity at low adsorbate concentrations and enhanced adsorption after the point (point of inflection, Figure 3d) at which some adsorption has already occurred.