**6. Empirical isotherm equations**

In this section, we present a number of popularly used isotherm equations. We start first with the earliest empirical equation proposed by Freundlich, and then Sips equation which is an extension of the Freundlich equation, modified such that the amount adsorbed in the Sips equation has a finite limit at sufficiently high pressure (or fluid concentration). We then present the two equations which are commonly used to describe well many data of hydrocarbons, carbon oxides on activated carbon and zeolite: Toth and Unilan equations. A recent proposed equation by Keller et al. (1996), which has a form similar to that of Toth, is also discussed. Next, we describe the Dubinin equation for describing micropore filling, which is popular in fitting data of many microporous solids. Finally we present the relatively less used equations in physical adsorption, Jovanovich and Tempkin, the latter of which is more popular in the description of chemisorption systems (Erbil, H.Y.,1997).

### **6.1 Freundlich equation**

The Freundlich equation is one of the earliest empirical equations used to describe equilibria data. The name of this isotherm is due to the fact that it was used extensively by Freundlich (1932) although it was used by many other researchers. This equation takes the following form:

$$\mathcal{C}\_{\mu} = K P^{1/n} \tag{122}$$

where Cµ is the concentration of the adsorbed species, and K and n are generally temperature dependent. The parameter n is usually greater than unity. The larger is this value; the adsorption isotherm becomes more nonlinear as its behavior deviates further away from the linear isotherm. To show the behavior of the amount adsorbed versus pressure (or concentration) we plot (Cµ/Cµ0) versus (P/P0) as shown in Figure 10.8, that is

$$\frac{C\_{\mu}}{C\_{\mu\nu}} = \left(\frac{P}{P\_o}\right)^{1/n} \tag{123}$$

where Po is some reference pressure and Cµ0 is the adsorbed concentration at that reference pressure, 1/ 0 0 *<sup>n</sup> C KP* 

We see from Figure 10.8 that the larger is the value of n, the more nonlinear is the adsorption isotherm, and as n is getting larger than about 10 the adsorption isotherm is approaching a so-called rectangular isotherm (or irreversible isotherm). The term "irreversible isotherm" is normally used because the pressure (or concentration) needs to go

*<sup>g</sup>* ln *<sup>P</sup> A RT*

adsorbate molecule. On the other hand, if the gas phase adsorption potential is greater, then the site will be unoccupied (Fig. 9). Therefore, if the surface has a distribution of surface adsorption potential F(A') with F(A')dA' being the amount adsorbed having adsorption

> *A C F A dA*

potential between A' and A'+dA', the adsorption isotherm equation is simply:

If the density function F(A') takes the form of decaying exponential function

<sup>0</sup> *FA A A* ( ) .exp( / ) 

where Ao is the characteristic adsorption potential, the above integral can be integrated to

1/*<sup>n</sup> C KP* 

0 0 <sup>0</sup> ( ) *RTA <sup>g</sup> K AP* 

1 *R Tg*

The parameter n for most practical systems is greater than unity; thus eq. (131) suggests that the characteristic adsorption energy of surface is greater than the molar thermal energy RgT.

0

/

where the parameter K and the exponent (l/n) are related to the distribution parameters

is less than the adsorption potential A'

( ') '

Fig. 9. Distribution of surface adsorption potential

Ao, and the vapor pressure and temperature as follows:

give the form of the Freundlich equation:

0

(126)

of a site, then that site will be occupied by an

(127)

(128)

,

(129)

(130)

*n A* (131)

*P*

down to an extremely low value before adsorbate molecules would desorb from the surface. The Freundlich equation is very popularly used in the description of adsorption of organics from aqueous streams onto activated carbon. It is also applicable in gas phase systems having heterogeneous surfaces, provided the range of pressure is not too wide as this isotherm equation does not have a proper Henry law behavior at low pressure, and it does not have a finite limit when pressure is sufficiently high. Therefore, it is generally valid in the narrow range of the adsorption data. Parameters of the Freundlich equation can be found by plotting log10 (CM) versus log10 (P)

Fig. 8. Plots of the Freundlich isotherm versus P/Po

$$\log\_{10}(\mathbb{C}\_{\mu}) = \log\_{10} K + \frac{1}{n} \log\_{10} P \tag{124}$$

which yields a straight line with a slope of (1/n) and an intercept of log10(K).

#### **6.1.1 Temperature dependence of K and n**

The parameters K and n of the Freundlich equation (122) are dependent on temperature. Their dependence on temperature is complex, and one should not extrapolate them outside their range of validity. The system of CO adsorption on charcoal has temperaturedependent n such that its inverse is proportional to temperature. This exponent was found to approach unity as the temperature increases. This, however, is taken as a specific trend rather than a general rule. To derive the temperature dependence of K and n, we resort to an approach developed by Urano et al. (1981). They assumed that a solid surface is composed of sites having a distribution in surface adsorption potential, which is defined as:

$$A' = R\_\lg T \ln\left(\frac{P\_\circ}{P}\right) \tag{125}$$

The adsorption potential A' is the work (energy) required to bring molecules in the gas phase of pressure P to a condensed state of vapor pressure Po. This means that sites associated with this potential A will have a potential to condense molecules from the gas phase of pressure P If the adsorption potential of the gas

down to an extremely low value before adsorbate molecules would desorb from the surface. The Freundlich equation is very popularly used in the description of adsorption of organics from aqueous streams onto activated carbon. It is also applicable in gas phase systems having heterogeneous surfaces, provided the range of pressure is not too wide as this isotherm equation does not have a proper Henry law behavior at low pressure, and it does not have a finite limit when pressure is sufficiently high. Therefore, it is generally valid in the narrow range of the adsorption data. Parameters of the Freundlich equation can be

> 10 10 10 <sup>1</sup> log ( ) log log *CKP*

The parameters K and n of the Freundlich equation (122) are dependent on temperature. Their dependence on temperature is complex, and one should not extrapolate them outside their range of validity. The system of CO adsorption on charcoal has temperaturedependent n such that its inverse is proportional to temperature. This exponent was found to approach unity as the temperature increases. This, however, is taken as a specific trend rather than a general rule. To derive the temperature dependence of K and n, we resort to an approach developed by Urano et al. (1981). They assumed that a solid surface is composed

> <sup>0</sup> ' ln *<sup>g</sup> <sup>P</sup> A RT*

phase of pressure P to a condensed state of vapor pressure Po. This means that sites associated with this potential A will have a potential to condense molecules from the gas

*P* 

is the work (energy) required to bring molecules in the gas

which yields a straight line with a slope of (1/n) and an intercept of log10(K).

of sites having a distribution in surface adsorption potential, which is defined as:

*n*

(124)

(125)

found by plotting log10 (CM) versus log10 (P)

Fig. 8. Plots of the Freundlich isotherm versus P/Po

**6.1.1 Temperature dependence of K and n** 

phase of pressure P If the adsorption potential of the gas

The adsorption potential A'

$$A = R\_{\chi} T \ln \left( \frac{P\_0}{P} \right) \tag{126}$$

is less than the adsorption potential A' of a site, then that site will be occupied by an adsorbate molecule. On the other hand, if the gas phase adsorption potential is greater, then the site will be unoccupied (Fig. 9). Therefore, if the surface has a distribution of surface adsorption potential F(A') with F(A')dA' being the amount adsorbed having adsorption potential between A' and A'+dA', the adsorption isotherm equation is simply:

$$\mathbf{C}\_{\mu} = \stackrel{\text{\tiny \phantom{\mu}}}{\int} F(A') dA' \tag{127}$$

Fig. 9. Distribution of surface adsorption potential

If the density function F(A') takes the form of decaying exponential function

$$F(A) = \delta.\exp(-A \mid A\_\circ) \tag{128}$$

where Ao is the characteristic adsorption potential, the above integral can be integrated to give the form of the Freundlich equation:

$$\mathcal{C}\_{\mu} = KP^{1/n} \tag{129}$$

where the parameter K and the exponent (l/n) are related to the distribution parameters , Ao, and the vapor pressure and temperature as follows:

$$K = \left(\mathcal{S}A\_o\right) P\_o^{-\mathcal{R}g^{T/A0}} \tag{130}$$

$$\frac{1}{m} = \frac{R\_{\frac{\pi}{8}}T}{A\_0} \tag{131}$$

The parameter n for most practical systems is greater than unity; thus eq. (131) suggests that the characteristic adsorption energy of surface is greater than the molar thermal energy RgT.

suggesting that the two parameters K and n in the Freundlich equation are not independent. Huang and Cho (1989) have collated a number of experimental data and have observed the linear dependence of ln(K) and (1/n) on temperature. We should, however, be careful about using this as a general rule for extrapolation as the temperature is sufficiently high, the isotherm will become linear, that is n = 1, meaning that 1/n no longer follows the linear temperature dependence as suggested by eq. (131). Thus, eq. (136) has its narrow range of validity, and must be used with extreme care. Using the propane data on activated carbon, we show in Figure 10 that lnK and 1/n are linearly related to each other, as suggested by

Knowing K and n as a function of temperature, we can use the van't Hoff equation

to determine the isosteric heat of adsorption. The result is (Huang and Cho, 1989)

Thus, the isosteric heat is a linear function of the logarithm of the adsorbed amount.

Recognizing the problem of the continuing increase in the adsorbed amount with an increase in pressure (concentration) in the Freundlich equation, Sips (1948) proposed an equation similar in form to the Freundlich equation, but it has a finite limit when the

*g*

*<sup>P</sup> H RT*

<sup>2</sup> ln

*T*

0 0 0 0 ln( ) ln *Rg H A AAC A*

> 1/ 1/

*bP*

*n s n*

( ) 1( )

*bP C C*

  *C*

(137)

(138)

(139)

eq.(136).

**6.2 Heat of adsorption** 

**6.3 Sips equation (langmuir-freundlich)** 

Fig. 11. Plots of the Sips equation versus bP

pressure is sufficiently high.

Provided that the parameters 5 and Ao of the distribution function are constant, the parameter l/n is a linear function of temperature, that is nRT is a constant, as experimentally observed for adsorption of CO in charcoal for the high temperature range (Rudzinski and Everett, 1992). To find the temperature dependence of the parameter K, we need to know the temperature dependence of the vapor pressure, which is assumed to follow the Clapeyron equation:

$$
\ln P\_0 = \alpha - \frac{\beta}{T} \tag{132}
$$

Taking the logarithm of K in eq. (131) and using the Clapeyron equation (132), we get the following equation for the temperature dependence of lnK:

$$\ln K = \left[ \ln(\mathcal{S}A\_o) + \frac{\mathcal{J}\mathcal{R}\_s}{A\_o} \right] - \frac{\alpha \mathcal{R}\_s T}{A\_o} \tag{133}$$

This equation states that the logarithm of K is a linear function of temperature, and it decreases with temperature. Thus the functional form to describe the temperature dependence of K is

$$K = K\_o \exp(-\frac{aR\_sT}{A\_o})\tag{134}$$

and hence the explicit temperature dependence form of the Freundlich equation is:

ln ln( )

*K A*

$$\mathbf{C}\_{\mu} = \mathbf{K}\_{0} \exp\left(-\frac{a R\_{\text{g}} T}{A\_{0}}\right) \mathbf{P}^{\text{g}\_{\text{g}} \text{T} / 40} \tag{135}$$

(136)

Since lnCµ and 1/n are linear in terms of temperature, we can eliminate the temperature and obtain the following relationship between lnK and n:

0

*Rg*

*A n*

Fig. 10. Plot of ln(K) versus 1/n for propane adsorption on activated carbon

Provided that the parameters 5 and Ao of the distribution function are constant, the parameter l/n is a linear function of temperature, that is nRT is a constant, as experimentally observed for adsorption of CO in charcoal for the high temperature range (Rudzinski and Everett, 1992). To find the temperature dependence of the parameter K, we need to know the temperature dependence of the vapor pressure, which is assumed to follow the

<sup>0</sup> ln *P*

following equation for the temperature dependence of lnK:

*T* 

0 0

*A A*

0 exp( ) *R Tg*

/

0

*A* 

0

0

*A n*

*Rg*

 

Since lnCµ and 1/n are linear in terms of temperature, we can eliminate the temperature and

0

 

*R RT g g*

 

Taking the logarithm of K in eq. (131) and using the Clapeyron equation (132), we get the

0

This equation states that the logarithm of K is a linear function of temperature, and it decreases with temperature. Thus the functional form to describe the temperature

 

0

and hence the explicit temperature dependence form of the Freundlich equation is:

exp *R Tg RTA <sup>g</sup> CK P A*

*K K*

0

ln ln( )

Fig. 10. Plot of ln(K) versus 1/n for propane adsorption on activated carbon

*K A*

obtain the following relationship between lnK and n:

ln ln( )

*K A*

(132)

(134)

(133)

(135)

(136)

Clapeyron equation:

dependence of K is

suggesting that the two parameters K and n in the Freundlich equation are not independent. Huang and Cho (1989) have collated a number of experimental data and have observed the linear dependence of ln(K) and (1/n) on temperature. We should, however, be careful about using this as a general rule for extrapolation as the temperature is sufficiently high, the isotherm will become linear, that is n = 1, meaning that 1/n no longer follows the linear temperature dependence as suggested by eq. (131). Thus, eq. (136) has its narrow range of validity, and must be used with extreme care. Using the propane data on activated carbon, we show in Figure 10 that lnK and 1/n are linearly related to each other, as suggested by eq.(136).

### **6.2 Heat of adsorption**

Knowing K and n as a function of temperature, we can use the van't Hoff equation

$$
\Delta H = -R\_{\rm s} T^2 \left( \frac{\partial \ln P}{\partial T} \right)\_{\rm c\_{\mu}} \tag{137}
$$

to determine the isosteric heat of adsorption. The result is (Huang and Cho, 1989)

$$
\Delta H = -\left[\ln(\sigma A\_{\text{o}}) + \frac{R\_{\text{g}}\mathcal{J}}{A\_{\text{o}}}\right]A\_{\text{o}} + A\_{\text{o}}\ln C\_{\mu} \tag{138}
$$

Thus, the isosteric heat is a linear function of the logarithm of the adsorbed amount.

### **6.3 Sips equation (langmuir-freundlich)**

Recognizing the problem of the continuing increase in the adsorbed amount with an increase in pressure (concentration) in the Freundlich equation, Sips (1948) proposed an equation similar in form to the Freundlich equation, but it has a finite limit when the pressure is sufficiently high.

$$\mathbf{C}\_{\mu} = \mathbf{C}\_{\mu} \frac{\left(bP\right)^{1/n}}{1 + \left(bP\right)^{1/n}} \tag{139}$$

Fig. 11. Plots of the Sips equation versus bP

Table 4. Optimal parameters for the Sips equation in fitting propane data on activated

**6.3.1 The temperature dependence of the sips equation** 

temperature dependence of the Sips equation

The parameter n is greater than unity, suggesting some degree of heterogeneity of this propane/ activated carbon system. The larger is this parameter, the higher is the degree of heterogeneity. However, this information does not point to what is the source of the heterogeneity, whether it be the solid structural property, the solid energetically property or the sorbet property. We note from the above table that the parameter n decreases with temperature, suggesting that the system is "apparently" less heterogeneous as temperature

For useful description of adsorption equilibrium data at various temperatures, it is important to have the temperature dependence form of an isotherm equation. The

*bP C C*

*Q QT b b <sup>b</sup> RT RT T* 

> 0 1 1 <sup>1</sup> *<sup>T</sup> nn T*

either taken as constant or it can take the following temperature dependence:

,0

 

exp[ (1 )] *S S <sup>T</sup> CC x*

parameter. This choice of this temperature-dependent form is arbitrary. This temperature

*<sup>S</sup>*,0 is the saturation capacity at the reference temperature To, and x is a constant

 

for the affinity constant b and the exponent n may take the following form:

1/ 1/

*bP*

*n s n*

0

0

0

(143)

*T*

0

(140)

(142)

(141)

( ) 1( )

0

Here *b* is the adsorption affinity constant at infinite temperature, b0 is that at some reference temperature To is the parameter n at the same reference temperature and a is a constant parameter. The temperature dependence of the affinity constant b is taken from the of the Langmuir equation. Unlike Q in the Langmuir equation, where it is the isosteric heat, invariant with the surface loading, the parameter Q in the Sips equation is only the measure of the adsorption heat. The temperature-dependent form of the exponent n is empirical and such form in eq. (142) is chosen because of its simplicity. The saturation capacity can be

exp exp ( 1) *g g*

carbon

increases.

Here *C*

In form this equation resembles that of Langmuir equation. The difference between this equation and the Langmuir equation is the additional parameter "n" in the Sips equation. If this parameter n is unity, we recover the Langmuir equation applicable for ideal surfaces. Hence the parameter n could be regarded as the parameter characterizing the system heterogeneity. The system heterogeneity could stem from the solid or the adsorbate or a combination of both. The parameter n is usually greater than unity, and therefore the larger is this parameter the more heterogeneous is the system. Figure 11 shows the behavior of the Sips equation with n being the varying parameter. Its behavior is the same as that of the Freundlich equation except that the Sips equation possesses a finite saturation limit when the pressure is sufficiently high. However, it still shares the same disadvantage with the Freundlich isotherm in that neither of them have the right behavior at low pressure, that is they don't give the correct Henry law limit. The isotherm equation (139) is sometimes called the Langmuir-Freundlich equation in the literature because it has the combined form of Langmuir and Freundlich equations.

To show the good utility of this empirical equation in fitting data, we take the same adsorption data of propane onto activated carbon used earlier in the testing of the Freundlich equation. The following Figure (Figure 10.12) shows the degree of good fit between the Sips equation and the data. The fit is excellent and it is fairly widely used to describe data of many hydrocarbons on activated carbon with good success. For each temperature, the fitting between the Sips equation and experimental data is carried out with MatLab nonlinear optimization outline, and the optimal parameters from the fit are tabulated in the following table. A code ISOFIT1 provided with this book is used for this optimization, and students are encouraged to use this code to exercise on their own adsorption data.

Fig. 12. Fitting of the propane/activated carbon data with the Sips equation (symbol -data; line:fitted equation)

The optimal parameters from the fitting of the Sips equation with the experimental data are tabulated in Table 4.

In form this equation resembles that of Langmuir equation. The difference between this equation and the Langmuir equation is the additional parameter "n" in the Sips equation. If this parameter n is unity, we recover the Langmuir equation applicable for ideal surfaces. Hence the parameter n could be regarded as the parameter characterizing the system heterogeneity. The system heterogeneity could stem from the solid or the adsorbate or a combination of both. The parameter n is usually greater than unity, and therefore the larger is this parameter the more heterogeneous is the system. Figure 11 shows the behavior of the Sips equation with n being the varying parameter. Its behavior is the same as that of the Freundlich equation except that the Sips equation possesses a finite saturation limit when the pressure is sufficiently high. However, it still shares the same disadvantage with the Freundlich isotherm in that neither of them have the right behavior at low pressure, that is they don't give the correct Henry law limit. The isotherm equation (139) is sometimes called the Langmuir-Freundlich equation in the literature because it has the combined form of

To show the good utility of this empirical equation in fitting data, we take the same adsorption data of propane onto activated carbon used earlier in the testing of the Freundlich equation. The following Figure (Figure 10.12) shows the degree of good fit between the Sips equation and the data. The fit is excellent and it is fairly widely used to describe data of many hydrocarbons on activated carbon with good success. For each temperature, the fitting between the Sips equation and experimental data is carried out with MatLab nonlinear optimization outline, and the optimal parameters from the fit are tabulated in the following table. A code ISOFIT1 provided with this book is used for this optimization, and students are encouraged to use this code to exercise on their own

Fig. 12. Fitting of the propane/activated carbon data with the Sips equation (symbol -data;

The optimal parameters from the fitting of the Sips equation with the experimental data are

Langmuir and Freundlich equations.

adsorption data.

line:fitted equation)

tabulated in Table 4.


Table 4. Optimal parameters for the Sips equation in fitting propane data on activated carbon

The parameter n is greater than unity, suggesting some degree of heterogeneity of this propane/ activated carbon system. The larger is this parameter, the higher is the degree of heterogeneity. However, this information does not point to what is the source of the heterogeneity, whether it be the solid structural property, the solid energetically property or the sorbet property. We note from the above table that the parameter n decreases with temperature, suggesting that the system is "apparently" less heterogeneous as temperature increases.

### **6.3.1 The temperature dependence of the sips equation**

For useful description of adsorption equilibrium data at various temperatures, it is important to have the temperature dependence form of an isotherm equation. The temperature dependence of the Sips equation

$$\mathbf{C}\_{\mu} = \mathbf{C}\_{\mu} \frac{\{bP\}^{1/n}}{\mathbf{1} + \{bP\}^{1/n}} \tag{140}$$

for the affinity constant b and the exponent n may take the following form:

$$b = b\_n \exp\left(\frac{Q}{R\_s T}\right) = b\_0 \exp\left[\frac{Q}{R\_s T\_0} (\frac{T\_o}{T}) - 1\right] \tag{141}$$

$$\frac{1}{n} = \frac{1}{n\_o} + \alpha \left(1 - \frac{T\_o}{T}\right) \tag{142}$$

Here *b* is the adsorption affinity constant at infinite temperature, b0 is that at some reference temperature To is the parameter n at the same reference temperature and a is a constant parameter. The temperature dependence of the affinity constant b is taken from the of the Langmuir equation. Unlike Q in the Langmuir equation, where it is the isosteric heat, invariant with the surface loading, the parameter Q in the Sips equation is only the measure of the adsorption heat. The temperature-dependent form of the exponent n is empirical and such form in eq. (142) is chosen because of its simplicity. The saturation capacity can be either taken as constant or it can take the following temperature dependence:

$$\mathbf{C}\_{\mu s} = \mathbf{C}\_{\mu s, 0} \exp[\mathbf{x}(1 - \frac{T}{T\_o})] \tag{143}$$

Here *C<sup>S</sup>*,0 is the saturation capacity at the reference temperature To, and x is a constant parameter. This choice of this temperature-dependent form is arbitrary. This temperature

pressures, the Toth equation is recommended as the first choice of isotherm equation for fitting data of many adsorbates such as hydrocarbons, carbon oxides, hydrogen sulfide, and alcohols on activated carbon as well as zeolites. Sips equation presented in the last section is also recommended but when the behavior in the Henry law region is needed, the Toth

Like the other equations described so far, the temperature dependence of equilibrium parameters in the Toth equation is required for the purpose of extrapolation or interpolation of equilibrium at other temperatures as well as the purpose of calculating isosteric heat. The parameters b and t are temperature dependent, with the parameter b taking the usual form

0

where *b* is the affinity at infinite temperature, b0 is that at some reference temperature To and Q is a measure of the heat of adsorption. The parameter t and the maximum adsorption

*Q QT b b <sup>b</sup> RT RT T* 

capacity can take the following empirical functional form of temperature dependence

<sup>0</sup> <sup>1</sup> *<sup>T</sup> t t*

,0

 

exp[ (1 )] *S S <sup>T</sup> CC x*

The temperature dependence of the parameter t does not have any sound theoretical footing; however, we would expect that as the temperature increases this parameter will

Keller and his co-workers (1996) proposed a new isotherm equation, which is very similar in form to the original Toth equation. The differences between their equation and that of Toth

1/ 1( ) *S m*

*bP*

*<sup>m</sup> P P* 

 

a. the exponent a is a function of pressure instead of constant as in the case of Toth

*bP C C*

1 1

 

 

b. the saturation capacities of different species are different

where the parameter αm takes the following equation:

exp exp ( 1) *g g*

0

(146)

(147)

(145)

(148)

(149)

0

0

0

*T*

*T*

equation is the better choice.

of the adsorption affinity that is

approach unity.

are that:

**6.5 Keller, staudt and toth's equation** 

The form of Keller et al.'s equation is:

**6.4.1 Temperature dependence of the toth equation** 

dependence form of the Sips equation (142) can be used to fit adsorption equilibrium data of various temperatures simultaneously to yield the parameter b0, *C<sup>S</sup>*,0 , Q/RT0, ratio and α.

### **6.4 Toth equation**

The previous two equations have their limitations. The Freundlich equation is not valid at low and high end of the pressure range, and the Sips equation is not valid at the low end as they both do not possess the correct Henry law type behavior. One of the empirical equations that is popularly used and satisfies the two end limits is the Toth equation. This equation describes well many systems with sub-monolayer coverage, and it has the following form:

$$\mathbf{C}\_{\mu} = \mathbf{C}\_{\mu^{\rm s}} \frac{bP}{\left[1 + (bP)'\right]^{1/\gamma}} \tag{144}$$

Here t is a parameter which is usually less than unity. The parameters b and t are specific for adsorbate-adsorbent pairs. When t = 1, the Toth isotherm reduces to the famous Langmuir equation; hence like the Sips equation the parameter t is said to characterize the system heterogeneity. If it is deviated further away from unity, the system is said to be more heterogeneous. The effect of the Toth parameter t is shown in Figure10-13, where we plot the fractional loading (Cµ/Cµs) versus bP with t as the varying parameter. Again we note that the more the parameter t deviates from unity, the more heterogeneous is the system. The Toth equation has correct limits when P approaches either zero or infinity.

Fig. 13. Plot of the fractional loading versus bP for the Toth equation

Being the three-parameter model, the Toth equation can describe well many adsorption data. We apply this isotherm equation to fit the isotherm data of propane on activated carbon. The extracted optimal parameters are: Cµs=33.56 mmole/g , b=0.069 (kPa)-1, t=0.233 The parameter t takes a value of 0.233 (well deviated from unity) indicates a strong degree of heterogeneity of the system. Several hundred sets of data for hydrocarbons on Nuxit-al charcoal obtained by Szepesy and Illes (Valenzuela and Myers, 1989) can be described well by this equation. Because of its simplicity in form and its correct behavior at low and high

dependence form of the Sips equation (142) can be used to fit adsorption equilibrium data of

The previous two equations have their limitations. The Freundlich equation is not valid at low and high end of the pressure range, and the Sips equation is not valid at the low end as they both do not possess the correct Henry law type behavior. One of the empirical equations that is popularly used and satisfies the two end limits is the Toth equation. This equation describes well many systems with sub-monolayer coverage, and it has the

1/

1( ) *S t <sup>t</sup>*

Here t is a parameter which is usually less than unity. The parameters b and t are specific for adsorbate-adsorbent pairs. When t = 1, the Toth isotherm reduces to the famous Langmuir equation; hence like the Sips equation the parameter t is said to characterize the system heterogeneity. If it is deviated further away from unity, the system is said to be more heterogeneous. The effect of the Toth parameter t is shown in Figure10-13, where we plot the fractional loading (Cµ/Cµs) versus bP with t as the varying parameter. Again we note that the more the parameter t deviates from unity, the more heterogeneous is the system.

*bP*

*bP C C*

 

The Toth equation has correct limits when P approaches either zero or infinity.

Fig. 13. Plot of the fractional loading versus bP for the Toth equation

Being the three-parameter model, the Toth equation can describe well many adsorption data. We apply this isotherm equation to fit the isotherm data of propane on activated carbon. The extracted optimal parameters are: Cµs=33.56 mmole/g , b=0.069 (kPa)-1, t=0.233 The parameter t takes a value of 0.233 (well deviated from unity) indicates a strong degree of heterogeneity of the system. Several hundred sets of data for hydrocarbons on Nuxit-al charcoal obtained by Szepesy and Illes (Valenzuela and Myers, 1989) can be described well by this equation. Because of its simplicity in form and its correct behavior at low and high

*<sup>S</sup>*,0 , Q/RT0, ratio and α.

(144)

various temperatures simultaneously to yield the parameter b0, *C*

**6.4 Toth equation** 

following form:

pressures, the Toth equation is recommended as the first choice of isotherm equation for fitting data of many adsorbates such as hydrocarbons, carbon oxides, hydrogen sulfide, and alcohols on activated carbon as well as zeolites. Sips equation presented in the last section is also recommended but when the behavior in the Henry law region is needed, the Toth equation is the better choice.

### **6.4.1 Temperature dependence of the toth equation**

Like the other equations described so far, the temperature dependence of equilibrium parameters in the Toth equation is required for the purpose of extrapolation or interpolation of equilibrium at other temperatures as well as the purpose of calculating isosteric heat. The parameters b and t are temperature dependent, with the parameter b taking the usual form of the adsorption affinity that is

$$b = b\_n \exp\left(\frac{Q}{R\_s T}\right) = b\_0 \exp\left[\frac{Q}{R\_s T\_o} (\frac{T\_o}{T} - 1)\right] \tag{145}$$

where *b* is the affinity at infinite temperature, b0 is that at some reference temperature To and Q is a measure of the heat of adsorption. The parameter t and the maximum adsorption capacity can take the following empirical functional form of temperature dependence

$$t = t\_0 + \alpha \left(1 - \frac{T\_o}{T}\right) \tag{146}$$

$$\mathbf{C}\_{\mu\mathbf{s}} = \mathbf{C}\_{\mu\mathbf{s},\phi} \exp[\mathbf{x}(1 - \frac{T}{T\_o})] \tag{147}$$

The temperature dependence of the parameter t does not have any sound theoretical footing; however, we would expect that as the temperature increases this parameter will approach unity.

### **6.5 Keller, staudt and toth's equation**

Keller and his co-workers (1996) proposed a new isotherm equation, which is very similar in form to the original Toth equation. The differences between their equation and that of Toth are that:


The form of Keller et al.'s equation is:

$$\mathbf{C}\_{\mu} = \mathbf{C}\_{\mu^{\rm s}} \alpha\_{\boldsymbol{m}} \frac{bP}{\left[1 + (bP)^{\boldsymbol{a}}\right]^{1/a}} \tag{148}$$

$$\alpha = \frac{1 + \alpha\_w \beta P}{1 + \beta P} \tag{149}$$

where the parameter αm takes the following equation:

$$\frac{\alpha\_{\boldsymbol{w}}}{\alpha\_{\boldsymbol{w}}^{\*}} = \left(\frac{\boldsymbol{r}}{\boldsymbol{r}^{\*}}\right)^{-D} \tag{150}$$

where Po is the vapor pressure. The premise of their derivation is the functional form V(A)

where the logarithm of the amount adsorbed is linearly proportional to the square of the adsorption potential. Eq. (155) is known as the Dubinin-Radushkevich (DR) equation.

> 1 exp ln ( ) *<sup>g</sup> <sup>P</sup> V V R T*

where Eo is called the solid characteristic energy towards a reference adsorbate. Benzene has been used widely as the reference adsorbate. The parameter β is a constant which is a function of the adsorptive only. It has been found by Dubinin and Timofeev (1946) that this parameter is proportional to the liquid molar volume. Fig. 14 shows plots of the DR equation versus the reduced pressure with E/RT as the varying parameter (Foo K.Y.,

We see that as the characteristic energy increases the adsorption is stronger as the solid has stronger energy of interaction with adsorbate. One observation in that equation is that the slope of the adsorption isotherm at zero loading is not finite, a violation of the thermodynamic requirement Eq. (156) when written in terms of amount adsorbed (mole/g)

1 *S g* exp ln *<sup>P</sup> C C R T*

2 0 0

*E P*

2

(157)

2

0 0

*E P*

<sup>0</sup> ln ln *V V BA* (155)

(156)

2

which is independent of temperature. They chose the following functional form:

0 2

Writing this equation explicitly in terms of pressure, we have:

Fig. 14. Plots of the DR equation versus the reduced pressure

Where the maximum adsorption capacity is:

 

Hameed B.H., 2009).

is:

Here r is the molecular radius, and D is the fractal dimension of sorbent surface. The saturation parameter*C<sup>S</sup>* , the affinity constant b, and the parameter (3 have the following temperature dependence):

$$\mathbf{C}\_{\mu s} = \mathbf{C}\_{\mu s, \boldsymbol{\rho}} \exp[\mathbf{x}(1 - \frac{T}{T\_o})] \tag{151}$$

$$b = b\_o \exp\left[\frac{Q\_i}{R\_g T\_0} (\frac{T\_o}{T} - 1)\right] \tag{152}$$

$$\beta = \beta\_o \exp\left[\frac{Q\_z}{R\_g T\_o}(\frac{T\_o}{T} - 1)\right] \tag{153}$$

Here the subscript 0 denotes for properties at some reference temperature T0. The Keller et al.'s equation contains more parameters than the empirical equations discussed so far. Fitting the Keller et equation with the isotherm data of propane on activated carbon at three temperatures 283, 303 and 333 K, we found the fit is reasonably good, comparable to the good fit observed with Sips and Toth equations. The optimally fitted parameters are:


Table 5. The parameters for Keller, Staudt and Toth's Equation

#### **6.6 Dubinin-radushkevich equation**

The empirical equations dealt with so far, Freundlich, Sips, Toth, Unilan and Keller et al., are applicable to supercritical as well as subcritical vapors. In this section we present briefly a semi-empirical equation which was developed originally by Dubinin and his co-workers for sub critical vapors in microporous solids, where the adsorption process follows a pore filling mechanism. Hobson and co-workers and Earnshaw and Hobson (1968) analysed the data of argon on Corning glass in terms of the Polanyi potential theory. They proposed an equation relating the amount adsorbed in equivalent liquid volume (V) to the adsorption potential

$$A = R\_\circ T \ln(\frac{P\_o}{P}) \tag{154}$$

*r r*

*D*

*<sup>S</sup>* , the affinity constant b, and the parameter (3 have the following

0

*T*

(150)

(151)

(152)

(153)

 

\* \*

Here r is the molecular radius, and D is the fractal dimension of sorbent surface. The

exp[ (1 )] *S S <sup>T</sup> CC x*

1 0

2 0

0 exp ( 1) *g Q T RT T*

Here the subscript 0 denotes for properties at some reference temperature T0. The Keller et al.'s equation contains more parameters than the empirical equations discussed so far. Fitting the Keller et equation with the isotherm data of propane on activated carbon at three temperatures 283, 303 and 333 K, we found the fit is reasonably good, comparable to the

The empirical equations dealt with so far, Freundlich, Sips, Toth, Unilan and Keller et al., are applicable to supercritical as well as subcritical vapors. In this section we present briefly a semi-empirical equation which was developed originally by Dubinin and his co-workers for sub critical vapors in microporous solids, where the adsorption process follows a pore filling mechanism. Hobson and co-workers and Earnshaw and Hobson (1968) analysed the data of argon on Corning glass in terms of the Polanyi potential theory. They proposed an equation relating the amount adsorbed in equivalent liquid volume (V) to the adsorption potential

ln( ) *<sup>o</sup>*

*<sup>P</sup>* (154)

*g <sup>P</sup> A RT*

0 exp ( 1) *g*

*RT T* 

*m m*

,0

*Q T b b*

 

0

0

good fit observed with Sips and Toth equations. The optimally fitted parameters are:

 

Table 5. The parameters for Keller, Staudt and Toth's Equation

**6.6 Dubinin-radushkevich equation** 

saturation parameter*C*

temperature dependence):

where Po is the vapor pressure. The premise of their derivation is the functional form V(A) which is independent of temperature. They chose the following functional form:

$$
\ln V = \ln V\_0 B A^2 \tag{155}
$$

where the logarithm of the amount adsorbed is linearly proportional to the square of the adsorption potential. Eq. (155) is known as the Dubinin-Radushkevich (DR) equation. Writing this equation explicitly in terms of pressure, we have:

$$V = V\_0 \exp\left[-\frac{1}{\left(\mathcal{J}E\_0\right)^2} \left(R\_s T \ln \frac{P}{P\_0}\right)^2\right] \tag{156}$$

where Eo is called the solid characteristic energy towards a reference adsorbate. Benzene has been used widely as the reference adsorbate. The parameter β is a constant which is a function of the adsorptive only. It has been found by Dubinin and Timofeev (1946) that this parameter is proportional to the liquid molar volume. Fig. 14 shows plots of the DR equation versus the reduced pressure with E/RT as the varying parameter (Foo K.Y., Hameed B.H., 2009).

Fig. 14. Plots of the DR equation versus the reduced pressure

We see that as the characteristic energy increases the adsorption is stronger as the solid has stronger energy of interaction with adsorbate. One observation in that equation is that the slope of the adsorption isotherm at zero loading is not finite, a violation of the thermodynamic requirement Eq. (156) when written in terms of amount adsorbed (mole/g) is:

$$\mathbf{C}\_{\mu} = \mathbf{C}\_{\mu \mathbf{s}} \exp\left[ -\frac{1}{\left(\mathcal{J}\mathcal{E}\_{\mathbf{0}}\right)^{2}} \left(\mathcal{R}\_{\mathbf{s}} T \ln \frac{P}{P\_{\mathbf{0}}}\right)^{2} \right] \tag{157}$$

Where the maximum adsorption capacity is:

$$C\_{\mu\text{S}} = \frac{\mathcal{W}\_{\text{o}}}{V\_{\text{M}}(T)} \tag{158}$$

By fitting the equilibria data of all three temperatures simultaneously using the ISOFIT1 program, we obtain the following optimally fitted parameters: W0 = 0.45 cc/g, E = 20,000 Joule/mole Even though only one value of the characteristic energy was used in the fitting of the three temperature data, the fit is very good as shown in Fig. 15, demonstrating the good utility of this equation in describing data of sub-critical vapors in microporous solids.

Of lesser use in physical adsorption is the Jovanovich equation. It is applicable to mobile and localized adsorption (Hazlitt et al, 1979). Although it is not as popular as the other

1 exp *<sup>P</sup>*

1 *bP CC e*

exp( / ) *<sup>g</sup> b b Q RT* (161)

reduces to the Henry's law at low pressure. At high pressure, it reaches the saturation limit. The Jovanovich equation has a slower approach toward the saturation than that of the

Another empirical equation is the Temkin equation proposed originally by Slygin and Frumkin (1935) to describe adsorption of hydrogen on platinum electrodes in acidic

solutions (chemisorption systems). The equation is (Rudzinski and Everett, 1992):

0

 

(159)

*<sup>S</sup>* . Thus, this equation

(160)

*vP C cP* ( ) ln( . ) (162)

*a P*

empirical equations proposed so far, it is nevertheless a useful empirical equation:

 *S*

At low loading, the above equation will become ( ) *C C b P HP*

Fig. 15. Fitting the benzene/ activated carbon data with the DR equation

**6.7 Jovanovich equation** 

where

Langmuir equation.

**6.8 Temkin equation** 

or written in terms of the amount adsorbed:

The parameter W0 is the micropore volume and VM is the liquid molar volume. Here we have assumed that the state of adsorbed molecule in micropores behaves like liquid. Dubinin-Radushkevich equation (157) is very widely used to describe adsorption isotherm of sub-critical vapors in microporous solids such as activated carbon and zeolite. One debatable point in such equation is the assumption of liquid-like adsorbed phase as one could argue that due to the small confinement of micropore adsorbed molecules experience stronger interaction forces with the micropore walls, the state of adsorbed molecule could be between liquid and solid. The best utility of the Dubinin-Radushkevich equation lies in the fact that the temperature dependence of such equation is manifested in the adsorption potential A, defined as in eq. (154), that is if one plots adsorption data of different temperatures as the logarithm of the amount adsorbed versus the square of adsorption potential, all the data should lie on the same curve, which is known as the characteristic curve. The slope of such curve is the inverse of the square of the characteristic energy E = βE0. To show the utility of the DR equation, we fit eq. (157) to the adsorption data of benzene on activated carbon at three different temperatures, 283, 303 and 333 K. The data are tabulated in Table 10.6 and presented graphically in Figure 10.15.


Table 6. Adsorption data of benzene on activated carbon

The vapor pressure and the liquid molar volume of benzene are given in the following table.


Table 7. Vapor pressure and liquid molar volume of benzene

The parameter W0 is the micropore volume and VM is the liquid molar volume. Here we have assumed that the state of adsorbed molecule in micropores behaves like liquid. Dubinin-Radushkevich equation (157) is very widely used to describe adsorption isotherm of sub-critical vapors in microporous solids such as activated carbon and zeolite. One debatable point in such equation is the assumption of liquid-like adsorbed phase as one could argue that due to the small confinement of micropore adsorbed molecules experience stronger interaction forces with the micropore walls, the state of adsorbed molecule could be between liquid and solid. The best utility of the Dubinin-Radushkevich equation lies in the fact that the temperature dependence of such equation is manifested in the adsorption potential A, defined as in eq. (154), that is if one plots adsorption data of different temperatures as the logarithm of the amount adsorbed versus the square of adsorption potential, all the data should lie on the same curve, which is known as the characteristic curve. The slope of such curve is the inverse of the square of the characteristic energy E = βE0. To show the utility of the DR equation, we fit eq. (157) to the adsorption data of benzene on activated carbon at three different temperatures, 283, 303 and 333 K. The data

are tabulated in Table 10.6 and presented graphically in Figure 10.15.

Table 6. Adsorption data of benzene on activated carbon

Table 7. Vapor pressure and liquid molar volume of benzene

The vapor pressure and the liquid molar volume of benzene are given in the following table.

0 ( ) *<sup>S</sup> M <sup>W</sup> <sup>C</sup> V T*

(158)

Fig. 15. Fitting the benzene/ activated carbon data with the DR equation

By fitting the equilibria data of all three temperatures simultaneously using the ISOFIT1 program, we obtain the following optimally fitted parameters: W0 = 0.45 cc/g, E = 20,000 Joule/mole Even though only one value of the characteristic energy was used in the fitting of the three temperature data, the fit is very good as shown in Fig. 15, demonstrating the good utility of this equation in describing data of sub-critical vapors in microporous solids.

### **6.7 Jovanovich equation**

Of lesser use in physical adsorption is the Jovanovich equation. It is applicable to mobile and localized adsorption (Hazlitt et al, 1979). Although it is not as popular as the other empirical equations proposed so far, it is nevertheless a useful empirical equation:

$$1 - \theta = \exp\left[-a\left(\frac{P}{P\_0}\right)\right] \tag{159}$$

or written in terms of the amount adsorbed:

$$\mathbf{C}\_{\mu} = \mathbf{C}\_{\mu \delta} \left[ \mathbf{1} - e^{-\mathbf{b}^{\rho}} \right] \tag{160}$$

where

$$b = b\_{\text{\tiny in\tag{20}}} \exp(\mathbf{Q} \mid \mathbf{R}\_{\text{\tiny j}} \mathbf{T}) \tag{161}$$

At low loading, the above equation will become ( ) *C C b P HP <sup>S</sup>* . Thus, this equation reduces to the Henry's law at low pressure. At high pressure, it reaches the saturation limit. The Jovanovich equation has a slower approach toward the saturation than that of the Langmuir equation.

#### **6.8 Temkin equation**

Another empirical equation is the Temkin equation proposed originally by Slygin and Frumkin (1935) to describe adsorption of hydrogen on platinum electrodes in acidic solutions (chemisorption systems). The equation is (Rudzinski and Everett, 1992):

$$\upsilon(P) = \mathbb{C}\ln(c.P) \tag{162}$$

where a1, b1 and *E1* are constant, independent of the amount adsorbed. Here E1 is the interaction energy between the solid and molecule of the first layer, which is expected to be higher than the heat of vaporization. Similarly, the rate of adsorption onto the first layer

2 0 22 exp

*i i ii*

The total area of the solid is the sum of all individual areas, that is

*<sup>E</sup> a Ps b s*

The same form of equation then can be applied to the next layer, and in general for the i-th

<sup>1</sup> exp *<sup>i</sup>*

*<sup>E</sup> a Ps b s R T* 

<sup>0</sup> *<sup>i</sup> <sup>i</sup> S s*

Therefore, the volume of gas adsorbed on surface covering by one layer of molecules is the fraction occupied by one layer of molecules multiplied by the monolayer coverage Vm:

> 1 *m <sup>s</sup> V V*

2

The volume of gas adsorbed on the section of the surface which has two layers of molecules

The factor of 2 in the above equation is because there are two layers of molecules occupying a surface area of s2 (Fig. 16). Similarly, the volume of gas adsorbed on the section of the

> *i m is V V*

> > 0

*i*

 

Hence, the total volume of gas adsorbed at a given pressure is the sum of all these volumes:

*i s <sup>V</sup> V is V <sup>S</sup> <sup>s</sup>*

To explicitly obtain the amount of gas adsorbed as a function of pressure, we have to express Si in terms of the gas pressure. To proceed with this, we need to make a further assumption beside the assumptions made so far about the ideality of layers (so that Langmuir kinetics could be applied). One of the assumptions is that the heat of adsorption of the second and subsequent layers is the same and equal to the heat of liquefaction, EL

1

*S*

2

*S*

*i*

0

 

. *<sup>i</sup> m i i m*

.

*i*

*S*

2 *m <sup>s</sup> V V*

2

(164)

(165)

(166)

(166)

(167)

(168)

(169)

*g*

*g*

*R T*

must be the same as the rate of evaporation from the second layer, that is:

layer, we can write

is:

surface having "i" layers is:

where C and c are constants specific to the adsorbate-adsorbent pairs. Under some conditions, the Temkin isotherm can be shown to be a special case of the Unilan equation (162).

#### **6.9 BET<sup>2</sup> isotherm**

All the empirical equations dealt with are for adsorption with "monolayer" coverage, with the exception of the Freundlich isotherm, which does not have a finite saturation capacity and the DR equation, which is applicable for micropore volume filling. In the adsorption of sub-critical adsorbate, molecules first adsorb onto the solid surface as a layering process, and when the pressure is sufficiently high (about 0.1 of the relative pressure) multiple layers are formed. Brunauer, Emmett and Teller are the first to develop a theory to account for this multilayer adsorption, and the range of validity of this theory is approximately between 0.05 and 0.35 times the vapor pressure. In this section we will discuss this important theory and its various versions modified by a number of workers since the publication of the BET theory in 1938. Despite the many versions, the BET equation still remains the most important equation for the characterization of mesoporous solids, mainly due to its simplicity. The BET theory was first developed by Brunauer et al. (1938) for *a flat* surface (no curvature) and there is *no limit* in the number of layers which can be accommodated on the surface. This theory made use of the same assumptions as those used in the Langmuir theory, that is the surface is energetically homogeneous (adsorption energy does not change with the progress of adsorption in the same layer) and there is no interaction among adsorbed molecules. Let S0, S1, S2 and Sn be the surface areas covered by no layer, one layer, two layers and n layers of adsorbate molecules, respectively (Fig. 16).

Fig. 16. Multiple layering in BET theory

The concept of kinetics of adsorption and desorption proposed by Langmuir is applied to this multiple layering process, that is the rate of adsorption on any layer is equal to the rate of desorption from that layer. For the first layer, the rates of adsorption onto the free surface and desorption from the first layer are equal to each other:

$$a\_i P s\_0 = b\_i s\_i \exp\left(\frac{-E\_i}{R\_s T}\right) \tag{163}$$

<sup>2</sup> Brunauer, Emmett and Teller

where C and c are constants specific to the adsorbate-adsorbent pairs. Under some conditions, the Temkin isotherm can be shown to be a special case of the Unilan equation

All the empirical equations dealt with are for adsorption with "monolayer" coverage, with the exception of the Freundlich isotherm, which does not have a finite saturation capacity and the DR equation, which is applicable for micropore volume filling. In the adsorption of sub-critical adsorbate, molecules first adsorb onto the solid surface as a layering process, and when the pressure is sufficiently high (about 0.1 of the relative pressure) multiple layers are formed. Brunauer, Emmett and Teller are the first to develop a theory to account for this multilayer adsorption, and the range of validity of this theory is approximately between 0.05 and 0.35 times the vapor pressure. In this section we will discuss this important theory and its various versions modified by a number of workers since the publication of the BET theory in 1938. Despite the many versions, the BET equation still remains the most important equation for the characterization of mesoporous solids, mainly due to its simplicity. The BET theory was first developed by Brunauer et al. (1938) for *a flat* surface (no curvature) and there is *no limit* in the number of layers which can be accommodated on the surface. This theory made use of the same assumptions as those used in the Langmuir theory, that is the surface is energetically homogeneous (adsorption energy does not change with the progress of adsorption in the same layer) and there is no interaction among adsorbed molecules. Let S0, S1, S2 and Sn be the surface areas covered by no layer, one layer,

The concept of kinetics of adsorption and desorption proposed by Langmuir is applied to this multiple layering process, that is the rate of adsorption on any layer is equal to the rate of desorption from that layer. For the first layer, the rates of adsorption onto the free surface

1 0 11 exp

*<sup>E</sup> a Ps b s*

1

(163)

*g*

*R T*

 

two layers and n layers of adsorbate molecules, respectively (Fig. 16).

Fig. 16. Multiple layering in BET theory

<sup>2</sup> Brunauer, Emmett and Teller

and desorption from the first layer are equal to each other:

(162).

**6.9 BET<sup>2</sup>**

 **isotherm** 

where a1, b1 and *E1* are constant, independent of the amount adsorbed. Here E1 is the interaction energy between the solid and molecule of the first layer, which is expected to be higher than the heat of vaporization. Similarly, the rate of adsorption onto the first layer must be the same as the rate of evaporation from the second layer, that is:

$$a\_2 P s\_0 = b\_2 s\_2 \exp\left(\frac{-E\_2}{R\_\varepsilon T}\right) \tag{164}$$

The same form of equation then can be applied to the next layer, and in general for the i-th layer, we can write

$$a\_i P s\_{i-1} = b\_i s\_i \exp\left(\frac{-E\_i}{R\_s T}\right) \tag{165}$$

The total area of the solid is the sum of all individual areas, that is

$$S = \sum\_{i=0}^{n} s\_i \tag{166}$$

Therefore, the volume of gas adsorbed on surface covering by one layer of molecules is the fraction occupied by one layer of molecules multiplied by the monolayer coverage Vm:

$$V\_i = V\_m \left(\frac{s\_i}{S}\right) \tag{166}$$

The volume of gas adsorbed on the section of the surface which has two layers of molecules is:

$$V\_2 = V\_w \left(\frac{\mathfrak{L}s\_2}{S}\right) \tag{167}$$

The factor of 2 in the above equation is because there are two layers of molecules occupying a surface area of s2 (Fig. 16). Similarly, the volume of gas adsorbed on the section of the surface having "i" layers is:

$$V\_i = V\_m \left(\frac{\text{is}\_i}{\text{S}}\right) \tag{168}$$

Hence, the total volume of gas adsorbed at a given pressure is the sum of all these volumes:

$$V = \frac{V\_m}{S} \sum\_{i=0}^{n} i.s\_i = V\_m \frac{\sum\_{i=0}^{n} i.s\_i}{\sum\_{i=0}^{n} s\_i} \tag{169}$$

To explicitly obtain the amount of gas adsorbed as a function of pressure, we have to express Si in terms of the gas pressure. To proceed with this, we need to make a further assumption beside the assumptions made so far about the ideality of layers (so that Langmuir kinetics could be applied). One of the assumptions is that the heat of adsorption of the second and subsequent layers is the same and equal to the heat of liquefaction, EL

; <sup>1</sup> (1 )

(1 )(1 ) *<sup>m</sup>*

Eq. (179) can only be used if we can relate x in terms of pressure and other known quantities. This is done as follows. Since this model allows for infinite layers on top of a flat surface, the amount adsorbed must be infinity when the gas phase pressure is equal to the vapor pressure, that is P = Po occurs when x = 1; thus the variable x is the ratio of the

> 0 *P x*

With this definition, eq. (179) will become what is now known as the famous BET equation

0 0 ( )(1 ( 1)( / ) *<sup>m</sup>*

Fig. 17 shows plots of the BET equation (181) versus the reduced pressure with C being the varying parameter. The larger is the value of C, the sooner will the multilayer form and the

*V P P C PP*

*V CP*

Fig. 17. Plots of the BET equation versus the reduced pressure (C = 10,50, 100)

pressure, the constant g and the heat of liquefaction:

Equating eqs.(180) and (176), we obtain the following relationship between the vapor

*x x*

*x x*

1 1

eq. (174) can be simplified to yield the following form written in terms of C and x:

 

*i i*

*i i*

*x ix*

*V Cx V x x Cx* 2

(178)

(179)

(181)

*<sup>P</sup>* (180)

By using the following formulas (Abramowitz and Stegun, 1962)

pressure to the vapor pressure at the adsorption temperature:

convexity of the isotherm increases toward the low pressure range.

containing two fitting parameters, C and Vm:

$$E\_2 = E\_3 = \dots = E\_i = \dots = E\_L \tag{170}$$

The other assumption is that the ratio of the rate constants of the second and higher layers is equal to each other, that is:

$$\frac{b\_2}{a\_2} = \frac{b\_3}{a\_3} = \dots = \frac{b\_i}{a\_i} = \text{g} \tag{171}$$

where the ratio g is assumed constant. This ratio is related to the vapor pressure of the adsorbate. With these two additional assumptions, one can solve the surface coverage that contains one layer of molecule (s,) in terms of s0 and pressure as follows:

$$s\_{\circ} = \frac{a\_{\circ}}{b\_{\circ}} = P s\_{\circ} \exp(\varepsilon\_{\circ}) \tag{172}$$

where ε, is the reduced energy of adsorption of the first layer, defined as

$$
\varepsilon\_1 = \frac{E\_1}{R\_g T} \tag{173}
$$

Similarly the surface coverage of the section containing i layers of molecules is:

$$s\_i = \frac{a\_i}{b\_1} s\_o \text{g.exp}(\varepsilon\_1 - \varepsilon\_2) \left[ \left( \frac{P}{g} \right) \exp \varepsilon\_1 \right]^i \tag{174}$$

for i = 2, 3, ..., where EL is the reduced heat of liquefaction

$$
\varepsilon\_{\perp} = \frac{E\_{\perp}}{R\_{\text{g}}T} \tag{173}
$$

Substituting these surface coverage into the total amount of gas adsorbed (eq. 169), we obtain:

$$\frac{V}{V\_w} = \frac{\text{Cs}\_o \sum\_{i=0}^{n} i.x^i}{s\_o(1 + \text{C} \sum\_{i=1}^{n} x^i)}\tag{174}$$

where the parameter C and the variable x are defined as follows:

$$y = \frac{a\_1}{b\_1} P \exp \varepsilon\_i \tag{175}$$

$$\mathbf{x} = \frac{P}{\mathcal{S}} \exp \mathbf{z}\_{\perp} \tag{176}$$

$$C = \frac{y}{x} = \frac{a\_1 \mathcal{g}}{b\_1} e^{(\epsilon \mathbf{1} - \epsilon \mathbf{L})} \tag{177}$$

The other assumption is that the ratio of the rate constants of the second and higher layers is

where the ratio g is assumed constant. This ratio is related to the vapor pressure of the adsorbate. With these two additional assumptions, one can solve the surface coverage that

> exp( ) *<sup>a</sup> s Ps <sup>b</sup>*

> > 1

*g E R T*

.exp( ) exp

*L*

*g E R T*

Substituting these surface coverage into the total amount of gas adsorbed (eq. 169), we

0 0

*Cs i x <sup>V</sup> <sup>V</sup> s Cx*

> 1 1

*<sup>a</sup> y P <sup>b</sup>*

*i <sup>i</sup> <sup>m</sup>*

 

1

(1 )

*i*

exp *<sup>i</sup>*

exp *<sup>L</sup> P x g*

1 1 <sup>1</sup> *<sup>L</sup> y ag C e x b*

 

.

*i*

... *<sup>i</sup> i* *g*

2 3 2 3

1 1 01 1

1

0 12

*L*

0

where the parameter C and the variable x are defined as follows:

*i L a P s sg b g* 

Similarly the surface coverage of the section containing i layers of molecules is:

contains one layer of molecule (s,) in terms of s0 and pressure as follows:

where ε, is the reduced energy of adsorption of the first layer, defined as

1

1

for i = 2, 3, ..., where EL is the reduced heat of liquefaction

*bb b*

equal to each other, that is:

obtain:

2 3 ... ... *EE E E i L* (170)

*aa a* (171)

(173)

(173)

*i*

 

(172)

(174)

(174)

(175)

(176)

(177)

By using the following formulas (Abramowitz and Stegun, 1962)

$$\sum\_{i=1}^{n} \mathbf{x}^{i} = \frac{\mathbf{x}}{1 - \mathbf{x}} ; \sum\_{i=1}^{n} \mathbf{i} \mathbf{x}^{i} = \frac{\mathbf{x}}{(1 - \mathbf{x})^{2}} \tag{178}$$

eq. (174) can be simplified to yield the following form written in terms of C and x:

$$\frac{V}{V\_w} = \frac{\text{Cx}}{(1-\text{x})(1-\text{x}+\text{Cx})} \tag{179}$$

Eq. (179) can only be used if we can relate x in terms of pressure and other known quantities. This is done as follows. Since this model allows for infinite layers on top of a flat surface, the amount adsorbed must be infinity when the gas phase pressure is equal to the vapor pressure, that is P = Po occurs when x = 1; thus the variable x is the ratio of the pressure to the vapor pressure at the adsorption temperature:

$$\infty = \frac{P}{P\_0} \tag{180}$$

With this definition, eq. (179) will become what is now known as the famous BET equation containing two fitting parameters, C and Vm:

$$\frac{V}{V\_m} = \frac{CP}{(P\_o - P)(1 + (C - 1)(P \; / \; P\_o)}\tag{181}$$

Fig. 17 shows plots of the BET equation (181) versus the reduced pressure with C being the varying parameter. The larger is the value of C, the sooner will the multilayer form and the convexity of the isotherm increases toward the low pressure range.

Fig. 17. Plots of the BET equation versus the reduced pressure (C = 10,50, 100)

Equating eqs.(180) and (176), we obtain the following relationship between the vapor pressure, the constant g and the heat of liquefaction:

$$P\_0 = \text{g.exp}\left(-\frac{E\_\iota}{R\_sT}\right) \tag{182}$$

Fig. 19. BDDT classification of five isotherm shapes

Fig. 20. Plots of the BET equation when C < 1

Within a narrow range of temperature, the vapor pressure follows the Clausius-Clapeyron equation, that is

$$P\_0 = \alpha. \exp\left(-\frac{E\_\iota}{R\_s T}\right) \tag{183}$$

Comparing this equation with eq.(182), we see that the parameter g is simply the preexponential factor in the Clausius-Clapeyron vapor pressure equation. It is reminded that the parameter g is the ratio of the rate constant for desorption to that for adsorption of the second and subsequent layers, suggesting that these layers condense and evaporate similar to the bulk liquid phase. The pre-exponential factor of the constant C (eq.177)

$$\frac{a\_i g}{b\_i} = \frac{a\_i b\_j}{b\_i a\_j}; \text{forj} > 1\tag{184}$$

can be either greater or smaller than unity (Brunauer et al., 1967), and it is often assumed as unity without any theoretical justification. In setting this factor to be unity, we have assumed that the ratio of the rate constants for adsorption to desorption of the first layer is the same as that for the subsequent layers at infinite temperature. Also by assuming this factor to be unity, we can calculate the interaction energy between the first layer and the solid from the knowledge of C (obtained by fitting of the isotherm equation 3.3-18 with experimental data) The interaction energy between solid and adsorbate molecule in the first layer is always greater than the heat of adsorption; thus the constant C is a large number (usually greater than 100).

### **7. BDDT (Brunauer, Deming, Denting, Teller) classification**

The theory of BET was developed to describe the multilayer adsorption. Adsorption in real solids has given rise to isotherms exhibiting many different shapes. However, five isotherm shapes were identified (Brunauer et al., 1940) and are shown in Fig.19. The following five systems typify the five classes of isotherm.

**Type 1**: Adsorption of oxygen on charcoal at -183 °C

**Type 2:** Adsorption of nitrogen on iron catalysts at -195°C (many solids fall into this type).

**Type 3:** Adsorption of bromine on silica gel at 79°C, water on glass

**Type 4:** Adsorption of benzene on ferric oxide gel at 50°C

**Type 5:** Adsorption of water on charcoal at 100°C

Type I isotherm is the Langmuir isotherm type (monolayer coverage), typical of adsorption in microporous solids, such as adsorption of oxygen in charcoal. Type II typifies the BET adsorption mechanism. Type III is the type typical of water adsorption on charcoal where the adsorption is not favorable at low pressure because of the nonpolar (hydrophobic) nature of the charcoal surface. At sufficiently high pressures, the adsorption is due to the capillary condensation in mesopores. Type IV and type V are the same as types II and III with the exception that they have finite limit as *P P* 0 due to the finite pore volume of porous solids.

<sup>0</sup> .exp *<sup>L</sup>*

Within a narrow range of temperature, the vapor pressure follows the Clausius-Clapeyron

<sup>0</sup> .exp *<sup>L</sup>*

Comparing this equation with eq.(182), we see that the parameter g is simply the preexponential factor in the Clausius-Clapeyron vapor pressure equation. It is reminded that the parameter g is the ratio of the rate constant for desorption to that for adsorption of the second and subsequent layers, suggesting that these layers condense and evaporate similar

> *j j*

can be either greater or smaller than unity (Brunauer et al., 1967), and it is often assumed as unity without any theoretical justification. In setting this factor to be unity, we have assumed that the ratio of the rate constants for adsorption to desorption of the first layer is the same as that for the subsequent layers at infinite temperature. Also by assuming this factor to be unity, we can calculate the interaction energy between the first layer and the solid from the knowledge of C (obtained by fitting of the isotherm equation 3.3-18 with experimental data) The interaction energy between solid and adsorbate molecule in the first layer is always greater than the heat of adsorption; thus the constant C is a large number

The theory of BET was developed to describe the multilayer adsorption. Adsorption in real solids has given rise to isotherms exhibiting many different shapes. However, five isotherm shapes were identified (Brunauer et al., 1940) and are shown in Fig.19. The following five

**Type 2:** Adsorption of nitrogen on iron catalysts at -195°C (many solids fall into this

Type I isotherm is the Langmuir isotherm type (monolayer coverage), typical of adsorption in microporous solids, such as adsorption of oxygen in charcoal. Type II typifies the BET adsorption mechanism. Type III is the type typical of water adsorption on charcoal where the adsorption is not favorable at low pressure because of the nonpolar (hydrophobic) nature of the charcoal surface. At sufficiently high pressures, the adsorption is due to the capillary condensation in mesopores. Type IV and type V are the same as types II and III with the exception that they have finite limit as *P P* 0 due to the finite pore volume of

 

*<sup>E</sup> <sup>P</sup>*

to the bulk liquid phase. The pre-exponential factor of the constant C (eq.177)

**7. BDDT (Brunauer, Deming, Denting, Teller) classification** 

**Type 1**: Adsorption of oxygen on charcoal at -183 °C

**Type 5:** Adsorption of water on charcoal at 100°C

**Type 4:** Adsorption of benzene on ferric oxide gel at 50°C

**Type 3:** Adsorption of bromine on silica gel at 79°C, water on glass

1 1 1 1

*a g a b*

equation, that is

(usually greater than 100).

type).

porous solids.

systems typify the five classes of isotherm.

*<sup>E</sup> P g R T* 

*g*

*g*

; 1

*R T*

(182)

(183)

*forj b ba* (184)

Fig. 19. BDDT classification of five isotherm shapes

Fig. 20. Plots of the BET equation when C < 1

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The BET equation developed originally by Brunauer et al. (1938) is able to describe type I to type III. The type III isotherm can be produced from the BET equation when the forces between adsorbate and adsorbent are smaller than that between adsorbate molecules in the liquid state (i.e. E, < EL). Fig. 20 shows such plots for the cases of C = 0.1 and 0.9 to illustrate type III isotherm.

The BET equation does not cover the last two types (IV and V) because one of the assumptions of the BET theory is the allowance for infinite layers of molecules to build up on top of the surface. To consider the last two types, we have to limit the number of layers which can be formed above a solid surface. (Foo K.Y., Hameed B.H., 2009), (Moradi O. , et al. 2003). (Hirschfelder, and et al. 1954).
