1.3 CHARACTERIZING ADSORBENTS
1.3.2 Mathematics for liquid phase characterization
In this research, the adsorption isotherm data were calculated and fitted to the three best known isotherm equations, which are Freundlich, Langmuir and Temkin. The key difference between these three models is the variation of heat of adsorption with the surface coverage. The Freundlich model assume logarithmic decrease, the Langmuir model assumes uniformity and the Temkin model assumes a linear decrease [57].
1.3.2.1 Freundlich model calculation
In 1907, H.M.F. Freundlich derived an equation based on empirical considerations. This equation was based on a heterogeneous surface, multi-layer adsorption and non-linear energy distribution for the adsorption sites [30, 58, 59]. For adsorption from solution phase, the Freundlich equation as expressed in Equation 1-6 [18], is an exponential equation, which assumes that as the adsorbate solution concentration increases, the concentration of adsorbate on the adsorbent surface increases. Thus, by using this equation, an infinite amount of adsorption can occur [60].
ππ = ππͺππ/π Equation 1-6
qe = X/M is the amount of adsorbate adsorbed from solution per unit mass of adsorbent (mg/g
or mmol/g)
Ce is the residual concentration of adsorbate at the equilibrium (mg/l or mmol/l)
k is the adsorption capacity constant for multi-layer coverage ((mg/g)(l/mg)1/n)
n is the intensity of adsorption constant/Freundlich adsorption constant that represents the
parameter characterizing quasi-Gaussian energetic heterogeneity of the adsorption surface
The linear form of the Freundlich equation can be obtained by taking logs as shown in Equation 1-7.
e
e k n C
q log 1log
log ο½ ο« Equation 1-7
The benefit of the Freundlich model is that it uses simple expressions. However, it still has a drawback because it does not show a limiting value at high concentration, unlike the Langmuir equation [30, 61]. For the Freundlich equation, if the value of Ce is increasing, the
qe value also increases without limit. Therefore, the Freundlich model is only suitable to fit
data in the range of low to medium concentration.
1.3.2.2 Langmuir model calculation
The Langmuir equation [62] is the simplest and the most widely used expression for physisorption and chemisorption from either a gas or liquid phase. There has been many isotherm equations proposed since Langmuir and many of them are based on this equation. Langmuir proposed an equation in 1918 using three assumption, as follows [18, 30, 47, 62, 63];
ο· Surface is homogenous, where the adsorption energy is constant over all sites. ο· Each adsorption site can be occupied by one adsorbate entity.
ο· There is no transmigration and interaction between adsorbed molecules.
The general adsorption model is normally expressed as shown in Equation 1-8;
e e e bC bC Q q ο« ο½ 1 max Equation 1-8
From the original Langmuir equation, the most well know Langmuir linear form of this equation can be presented in Equations 1-9 and 1-10.
Equation 1-9, derived by Langmuir, produces a linear regression with a plot using Ce qe againstCe. e e e C Q b Q q C max max 1 1 ο« ο½ Equation 1-9
Equation 1-10 was derived by Lineweaver & Burk in 1948, and divided Ce through Equation
1-9. For Equation 1-11, 1 qe was plotted against 1Ceto obtain the constant values of Qmax
and b using the linear regression technique.
e e Q Q bC q max max 1 1 1 ο« ο½ Equation 1-10 where;
Ce is the residual concentration of adsorbate at the equilibrium (mg/l or mmol/l)
qe or X/M is the amount of adsorbate adsorbed from solution per unit mass of adsorbent (mg/g
or mmol/g)
Qmax is the maximum adsorption capacity corresponding to complete monolayer coverage
(mg/g or mmol/g)
b is the Langmuirβs adsorption affinity directly related to the adsorption energy between solid
and adsorbate compound (l/mg or l/mmol).
Note that the linear equation 1-9 and 1-10 were used to fit with the adsorption data in this research because these are the most commonly used in the study of the AC adsorption, respectively [30, 34, 57, 64, 65]. The reason for using both of these linear equations is that by observation from the literature, Equation 1-9 usually produced a better R2 than other model
equations, such as Freundlich and Temkin, while the Equation 1-10 seem to produce a lesser
R2 but with a better fit to the adsorption data as determined by Normalized deviation (P), see
Appendix IV section 12.4.2 [57, 65]. Therefore, by not just assessing the data fit to the equations based upon the R2, it avoids the possibility of a wrong conclusion when compared
Weber and Chakkravorti (1974) defined the essential characteristics of the Langmuir isotherm by expressing it as a dimensionless constant called the separation factor or equilibrium parameter or RL, which is shown in Equation 1-11 [66-68]. RL was used to indicate the shape
of the adsorption isotherm, as illustrated in Table 1-4 and Figure 1-11.
0 1 1 bC RL ο« ο½ Equation 1-11
Where C0 is the initial concentration of adsorbate or the highest concentration of adsorbate
(mg/l or mmol/l), when determining adsorption isotherm by varying solution volume or adsorbate concentration, respectively.
Table 1-4 RL parameter indicates the nature of the adsorption isotherm behaviour [58, 66]
Figure 1-11 Shape of isotherm indicated by separation factor [66]
Further advantages of the Langmuir equation are that it is easy to interpret the parameters and has a uptake limiting aspect. This means if the Ce value becomes very high, the qe will
become closed to Qmax. Qmax is the limit value for qe; this means, no matter how high the Ce
value becomes the qe is still lower than Qmax. The disadvantage is that it only accounts for
monolayer adsorption [61].
value of RL Type of isotherm
RL>1 Unfavourable
RL=1 Linear
0<RL<1 Favourable
1.3.2.3 Temkin model calculation
Very often the decline in the heat of adsorption is more linear than logarithmic and it is this type of phenomena that led to the derivation of the Temkin adsorption isotherms [18]. In 1940, Temkin and Pyzhev derived an isotherm equation based on the consideration of the indirect adsorbate/adsorbate interactions on an adsorption isotherm. The equation was proposed by assuming that the heat of adsorption of all the molecules in the layer decreases linearly in relation to the coverage due to adsorbent-adsorbate interactions [60, 67, 69, 70]. The isotherm is, in fact, derived from Langmuir adsorption isotherms by adding the condition that the heat of adsorption drops linearly with surface coverage and such an effect can arise from repulsive forces on a uniform surface or from surface heterogenity of the surface [18]. The heat of adsorption is characterized by a uniform distribution of binding energies up to a maximum [67, 70].
The original expression, as shown in Equation 1-12 in the linear form, is expressed in Equation 1-13. The plot of qe versus lnCe enables the determination of the constant value for
the Temkin equation.
ο¨
eο©
e e b A C RT AC b RT q ο½ ln( )ο½ ln ο«ln Equation 1-12In the linear form;
) (ln ) (ln ) (ln ) (ln e e e C B A B C b RT A b RT q ο½ ο« ο½ ο« Equation 1-13 where;
Ce is the residual concentration of adsorbate at the equilibrium (mg/l or mmol/l)
qe is the amount of adsorbate adsorbed from solution per unit mass of adsorbent (mg/g or
mmol/g)
A is the equilibrium binding constant corresponding to the maximum binding energy (l/g or
l/mg)
b is the Temkin constant related to the heat of adsorption (J/mol) R is the universal gas constant (8.314 J/mol K)
T is temperature (Kelvin, K)