**4. Fundamentals of pure component adsorption equilibrium**

Adsorption equilibria information is the most important piece of information in understanding an adsorption process. No matter how many components are present in the system, the adsorption equilibria of pure components are the essential ingredient for the understanding of how many those components can be accommodated by a solid adsorbent. With this information, it can be used in the study of adsorption kinetics of a single component, adsorption equilibria of multicomponent systems, and then adsorption kinetics of multicomponent systems. In this section, we present the fundamentals of pure component equilibria. Various fundamental equations are shown, and to start with the proceeding we will present the most basic theory in adsorption: the Langmuir theory (1918). This theory allows us to understand the monolayer surface adsorption on an ideal surface. By an ideal surface here, we mean that the energy fluctuation on this surface is periodic and the magnitude of this fluctuation is larger than the thermal energy of a molecule (kT), and hence the troughs of the energy fluctuation are acting as the adsorption sites. If the distance between the two neighboring troughs is much larger than the diameter of the adsorbate molecule, the adsorption process is called localised and each adsorbate molecule will occupy one site. Also, the depth of all troughs of the ideal surface are the same, that is the adsorption heat released upon adsorption on each site is the same no matter what the loading is. After the Langmuir theory, we will present the Gibbs thermodynamics approach. This approach treats the adsorbed phase as a single entity, and Gibbs adapted the classical thermodynamics of the bulk phase and applied it to the adsorbed phase. In doing this the concept of volume in the bulk phase is replaced by the area, and the pressure is replaced by the so-called spreading pressure. By assuming some forms of thermal equation of state relating the number of mole of adsorbate, the area and the spreading pressure (analogue of equations of state in the gas phase) and using them in the Gibbs equation, a number of fundamental equations can be derived, such as the linear isotherm, etc (Mohan D., Pittman Jr C.U. 2006).

Following the Gibbs approach, we will show the vacancy solution theory developed by Suwanayuen and Danner in 1980. Basically in this approach the system is assumed to consist of two solutions. One is the gas phase and the other is the adsorbed phase. The

To give the reader a feel about the magnitude of this bombardment rate of molecule, we

This shows a massive amount of collision between gaseous molecules and the surface even at a pressure of 10-3 Torr. A fraction of gas molecules striking the surface will condense and is held by the surface force until these adsorbed molecules evaporate again (see Figure 10.4.). Langmuir (1918) quoted that there is good experimental evidence that this fraction is unity, but for a real surface which is usually far from ideal this fraction could be much less than unity. Allowing for the sticking coefficient a (which accounts for non perfect sticking),

the rate of adsorption in mole adsorbed per unit bare surface area per unit time is:

Fig. 4. Schematic diagram of Langmuir adsorption mechanism on a flat surface

is equal to the rate given by eq. (54) multiplied by the fraction of empty sites, that is:

*<sup>P</sup> <sup>R</sup>*

This is the rate of adsorption on a bare surface. On an occupied surface, when a molecule strikes the portion already occupied with adsorbed species, it will evaporate very quickly, just like a reflection from a mirror. Therefore, the rate of adsorption on an occupied surface

> (1 ) <sup>2</sup> *<sup>a</sup> g*

*MR T* 

(55)

2 *<sup>a</sup>*

*<sup>P</sup> <sup>R</sup>*

*g*

*MR T* 

*g*

*MR T* (53)

(54)

2 *<sup>s</sup>*

tabulate below this rate at three pressures

Table 1. Magnitude of bombardment rate of molecule

*<sup>P</sup> <sup>R</sup>* 

difference between these two phases is the density. One is denser than the other. In the context of this theory, the vacancy solution is composed of adsorbate and vacancies. The latter is an imaginary entity defined as a vacuum space which can be regarded as the solvent of the system. Next, we will discuss one of the recent equations introduced by Nitta and his co-workers. This theory based on statistical thermodynamics has some features similar to the Langmuir theory, and it encompasses the Langmuir equation as a special case. Basically it assumes a localized monolayer adsorption with the allowance that one adsorbate molecule can occupy more than one adsorption site. Interaction among adsorbed molecules is also allowed for in their theory. As a special case, when the number of adsorption sites occupied by one adsorbate molecule is one, their theory is reduced to the Fowler-Guggenheim equation, and further if there is no adsorbate-adsorbate interaction this will reduce to the Langmuir equation. Another model of Nitta and co-workers allowing for the mobility of adsorbed molecules is also presented in this section.

Fig. 3. Surface energy fluctuations

### **4.1 Langmuir equation**

Langmuir (1918) was the first to propose a coherent theory of adsorption onto a flat surface based on a kinetic viewpoint, that is there is a continual process of bombardment of molecules onto the surface and a corresponding evaporation (desorption) of molecules from the surface to maintain zero rate of accumulation at the surface at equilibrium. The assumptions of the Langmuir model are:

### **4.1.1 Surface is homogeneous, that is adsorption energy is constant over all sites 4.1.2 Adsorption on surface is localized, that is adsorbed atoms or molecules are adsorbed at definite, localized sites**

### **4.1.3 Each site can accommodate only one molecule or atom**

The Langmuir theory is based on a kinetic principle, that is the rate of adsorption (which is the striking rate at the surface multiplied by a sticking coefficient, sometimes called the accommodation coefficient) is equal to the rate of desorption from the surface. The rate of striking the surface, in mole per unit time and unit area, obtained from the kinetic theory of gas is:

difference between these two phases is the density. One is denser than the other. In the context of this theory, the vacancy solution is composed of adsorbate and vacancies. The latter is an imaginary entity defined as a vacuum space which can be regarded as the solvent of the system. Next, we will discuss one of the recent equations introduced by Nitta and his co-workers. This theory based on statistical thermodynamics has some features similar to the Langmuir theory, and it encompasses the Langmuir equation as a special case. Basically it assumes a localized monolayer adsorption with the allowance that one adsorbate molecule can occupy more than one adsorption site. Interaction among adsorbed molecules is also allowed for in their theory. As a special case, when the number of adsorption sites occupied by one adsorbate molecule is one, their theory is reduced to the Fowler-Guggenheim equation, and further if there is no adsorbate-adsorbate interaction this will reduce to the Langmuir equation. Another model of Nitta and co-workers allowing for the

Langmuir (1918) was the first to propose a coherent theory of adsorption onto a flat surface based on a kinetic viewpoint, that is there is a continual process of bombardment of molecules onto the surface and a corresponding evaporation (desorption) of molecules from the surface to maintain zero rate of accumulation at the surface at equilibrium. The

**4.1.1 Surface is homogeneous, that is adsorption energy is constant over all sites 4.1.2 Adsorption on surface is localized, that is adsorbed atoms or molecules are** 

The Langmuir theory is based on a kinetic principle, that is the rate of adsorption (which is the striking rate at the surface multiplied by a sticking coefficient, sometimes called the accommodation coefficient) is equal to the rate of desorption from the surface. The rate of striking the surface, in mole per unit time and unit area, obtained from the kinetic theory of

**4.1.3 Each site can accommodate only one molecule or atom** 

mobility of adsorbed molecules is also presented in this section.

Fig. 3. Surface energy fluctuations

assumptions of the Langmuir model are:

**adsorbed at definite, localized sites** 

**4.1 Langmuir equation** 

gas is:

$$R\_s = \frac{P}{\sqrt{2\pi M R\_s T}}\tag{53}$$

To give the reader a feel about the magnitude of this bombardment rate of molecule, we tabulate below this rate at three pressures


Table 1. Magnitude of bombardment rate of molecule

This shows a massive amount of collision between gaseous molecules and the surface even at a pressure of 10-3 Torr. A fraction of gas molecules striking the surface will condense and is held by the surface force until these adsorbed molecules evaporate again (see Figure 10.4.). Langmuir (1918) quoted that there is good experimental evidence that this fraction is unity, but for a real surface which is usually far from ideal this fraction could be much less than unity. Allowing for the sticking coefficient a (which accounts for non perfect sticking), the rate of adsorption in mole adsorbed per unit bare surface area per unit time is:

$$R\_s = \frac{\alpha P}{\sqrt{2\pi M R\_s T}}\tag{54}$$

Fig. 4. Schematic diagram of Langmuir adsorption mechanism on a flat surface

This is the rate of adsorption on a bare surface. On an occupied surface, when a molecule strikes the portion already occupied with adsorbed species, it will evaporate very quickly, just like a reflection from a mirror. Therefore, the rate of adsorption on an occupied surface is equal to the rate given by eq. (54) multiplied by the fraction of empty sites, that is:

$$R\_s = \frac{\alpha P}{\sqrt{2\pi M R\_\odot T}} (1 - \theta) \tag{55}$$

<sup>5</sup> 1/2 1 *b* 5.682 10 ( ) *MT Torr*

The isotherm equation (59) reduces to the Henry law isotherm when the pressure is very low (bP << 1), that is the amount adsorbed increases linearly with pressure, a constraint demanded by statistical thermodynamics. When pressure is sufficiently high, the amount adsorbed reaches the saturation capacity, corresponding to a complete coverage of all adsorption sites with adsorbate molecules, this is called monolayer coverage, 1

When the affinity constant b is larger, the surface is covered more with adsorbate molecule as a result of the stronger affinity of adsorbate molecule towards the surface. Similarly, when the heat of adsorption Q increases, the adsorbed amount increases due to the higher energy barrier that adsorbed molecules have to overcome to evaporate back to the gas phase. Increase in the temperature will decrease the amount adsorbed at a given pressure. This is due to the greater energy acquired by the adsorbed molecule to evaporate. The isotherm equation (59) written in the form of fractional loading is not useful for the data correlation as isotherm data are usually collated in the form of amount adsorbed versus pressure. We now let Cµ be the amount adsorbed in mole per unit mass or volume1, and Cµs be the maximum adsorbed concentration corresponding to complete monolayer coverage, then the Langmuir equation written in terms of the amount adsorbed useful for data

> ( ) 1 () *<sup>s</sup> bT P C C*

Here we use the subscript µ to denote the adsorbed phase, and this will be applied throughout this text. For example, Cµ is the concentration of the adsorbed phase, and Dµ is

<sup>1</sup> This volume is taken as the particle volume minus the void volume where molecules are present in

 

*bT P*

(63)

( ) exp( / ) *<sup>g</sup> bT b Q RT* (64)

behavior of the Langmuir isotherm (Ө versus P) is shown in Fig. 5.

Fig. 5. Behavior of the Langmuir equation

correlation is:

where

free form.

(62)

. The

where Ɵ is the fractional coverage. Here Ra is the number of moles adsorbed per unit area (including covered and uncovered areas) per unit time. The rate of desorption from the surface is equal to the rate, which corresponds to fully covered surface (kd), multiplied by the fractional coverage, that is:

$$R\_d = k\_d \theta = k\_{d\_n} \exp\left(\frac{-E\_d}{R\_s T}\right) \theta \tag{56}$$

where Ed is the activation energy for desorption, which is equal to the heat of adsorption for physically adsorbed species since there is no energy barrier for physical adsorption. The parameter *<sup>d</sup> k* is the rate constant for desorption at infinite temperature. The inverse of this parameter is denoted as

$$\mathbf{r}\_{\rm dev} = \frac{1}{\mathbf{k}\_{\rm dev}}\tag{57}$$

The average residence time of adsorption is defined as:

$$
\pi\_a = \pi\_{d v} e^{\mathbb{E}d/\mathbb{E}T} \tag{58}
$$

This means that the deeper is the potential energy well higher Ed the longer is the average residence time for adsorption. For physical adsorption, this surface residence time is typically ranging between 10-13 to 10-9 sec, while for chemisorption this residence time has a very wide range, ranging from 10-6 (for weak chemisorption) to about 109 for systems such as CO chemisorbed on Ni. Due to the Arrhenius dependence on temperature this average surface residence time changes rapidly with temperature, for example a residence time of 109 at 300K is reduced to only 2 sec at 500K for a system having a desorption energy of 120 kJ/mole. Equating the rates of adsorption and desorption (Equations. 55 and 56), we obtain the following famous Langmuir isotherm written in terms of fractional loading:

$$\theta = \frac{bP}{1 + bP} \tag{59}$$

where

$$b = \frac{a \exp(Q \mid R\_{\sharp}T)}{k\_{\text{des}} \sqrt{2\pi M R\_{\sharp}T}} = b\_{\text{es}} \exp(Q \mid R\_{\sharp}T) \tag{60}$$

Here Q is the heat of adsorption and is equal to the activation energy for desorption, Ed. The parameter b is called the affinity constant or Langmuir constant. It is a measure of how strong an adsorbate molecule is attracted onto a surface. The pre exponential factor *<sup>b</sup>* of the affinity constant is:

$$b\_v = \frac{\alpha}{k\_{\rm{av}} \sqrt{2\pi M R\_\odot T}}\tag{61}$$

which is inversely proportional to the square root of the molecular weight. When P is in Torr, the magnitude of *<sup>b</sup>* for nitrogen is given by Hobson (1965) as:

where Ɵ is the fractional coverage. Here Ra is the number of moles adsorbed per unit area (including covered and uncovered areas) per unit time. The rate of desorption from the surface is equal to the rate, which corresponds to fully covered surface (kd), multiplied by

*dd d*

*<sup>E</sup> Rk k*

where Ed is the activation energy for desorption, which is equal to the heat of adsorption for physically adsorbed species since there is no energy barrier for physical adsorption. The parameter *<sup>d</sup> k* is the rate constant for desorption at infinite temperature. The inverse of this

> d 1 k *<sup>d</sup>*

*E RT* /

*d*

*a d*

This means that the deeper is the potential energy well higher Ed the longer is the average residence time for adsorption. For physical adsorption, this surface residence time is typically ranging between 10-13 to 10-9 sec, while for chemisorption this residence time has a very wide range, ranging from 10-6 (for weak chemisorption) to about 109 for systems such as CO chemisorbed on Ni. Due to the Arrhenius dependence on temperature this average surface residence time changes rapidly with temperature, for example a residence time of 109 at 300K is reduced to only 2 sec at 500K for a system having a desorption energy of 120 kJ/mole. Equating the rates of adsorption and desorption (Equations. 55 and 56), we obtain

> 1 *bP bP*

*g*

strong an adsorbate molecule is attracted onto a surface. The pre exponential factor *<sup>b</sup>*

 

*Q RT <sup>b</sup> b Q RT k MR T*

Here Q is the heat of adsorption and is equal to the activation energy for desorption, Ed. The parameter b is called the affinity constant or Langmuir constant. It is a measure of how

2 *d g*

which is inversely proportional to the square root of the molecular weight. When P is in

for nitrogen is given by Hobson (1965) as:

*k MR T* 

exp( / ) exp( / ) <sup>2</sup>

*g*

(60)

(61)

 

the following famous Langmuir isotherm written in terms of fractional loading:

*d g*

*b*

exp *<sup>d</sup>*

*g*

*R T*

  (56)

(57)

*e* (58)

(59)

of the

the fractional coverage, that is:

parameter is denoted as

where

affinity constant is:

Torr, the magnitude of *<sup>b</sup>*

The average residence time of adsorption is defined as:

$$h\_{\pi} = 5.682 \times 10^{-5} (MT)^{-1/2} Torr^{-1} \tag{62}$$

The isotherm equation (59) reduces to the Henry law isotherm when the pressure is very low (bP << 1), that is the amount adsorbed increases linearly with pressure, a constraint demanded by statistical thermodynamics. When pressure is sufficiently high, the amount adsorbed reaches the saturation capacity, corresponding to a complete coverage of all adsorption sites with adsorbate molecules, this is called monolayer coverage, 1 . The behavior of the Langmuir isotherm (Ө versus P) is shown in Fig. 5.

Fig. 5. Behavior of the Langmuir equation

When the affinity constant b is larger, the surface is covered more with adsorbate molecule as a result of the stronger affinity of adsorbate molecule towards the surface. Similarly, when the heat of adsorption Q increases, the adsorbed amount increases due to the higher energy barrier that adsorbed molecules have to overcome to evaporate back to the gas phase. Increase in the temperature will decrease the amount adsorbed at a given pressure. This is due to the greater energy acquired by the adsorbed molecule to evaporate. The isotherm equation (59) written in the form of fractional loading is not useful for the data correlation as isotherm data are usually collated in the form of amount adsorbed versus pressure. We now let Cµ be the amount adsorbed in mole per unit mass or volume1, and Cµs be the maximum adsorbed concentration corresponding to complete monolayer coverage, then the Langmuir equation written in terms of the amount adsorbed useful for data correlation is:

$$\mathbf{C}\_{\mu} = \mathbf{C}\_{\mu} \frac{b(T)P}{1 + b(T)P} \tag{63}$$

where

$$b(T) = b\_v \exp(Q / R\_\varepsilon T) \tag{64}$$

Here we use the subscript µ to denote the adsorbed phase, and this will be applied throughout this text. For example, Cµ is the concentration of the adsorbed phase, and Dµ is

<sup>1</sup> This volume is taken as the particle volume minus the void volume where molecules are present in free form.

concentration) must be zero. This is to say that the saturation capacity is independent of temperature, and as a result the heat of adsorption is a constant, independent of loading.

The last section dealt with the basic Langmuir theory, one of the earliest theories in the literature to describe adsorption equilibria. One should note that the Langmuir approach is kinetic by nature. Adsorption equilibria can be described quite readily by the thermodynamic approach. What to follow in this section is the approach due to Gibbs. More

In the bulk α-phase containing N components (Fig. 6), the following variables are specified:

N). The upper script is used to denote the phase. With these variables, the total differential

*dA S dT P dV dn*

where Sα is the entropy of the a phase, Pα is the pressure of that phase, nα is the number of

is its chemical potential.

1 *N*

*i*

*i i*

(69)

 

(thermal expansion coefficient of the saturation

<sup>α</sup> (for i = 1, 2, ...,

(71)

details can be found in Yang (1987) and Rudzinski and Everett (1992).

 

the temperature Tα, the volume Vα and the numbers of moles of all species ni

 

Fig. 6. Equilibrium between the phases α and β separated by a plane interface a

 

 

;; *T TP P i i*

Similarly, for the β -phase, we can write a similar equation for the differential Helmholtz

*dA S dT P dV dn*

 

 

If equilibrium exists between the two phases with a plane interface (Fig. 6), we have:

1 *N*

*i*

*i i*

(70)

 

coverage (that is Ɵ —> 1) the parameter

Helmholtz free energy is:

free energy:

molecule of the species *i*, and ( *<sup>i</sup>*

**4.3 Isotherms based on the gibbs approach** 

the diffusion coefficient of the adsorbed phase, Vµ is the volume of the adsorbed phase, etc. The temperature dependence of the affinity constant (e.g. 60) is T-1/2 exp(Q/Rg T). This affinity constant decreases with temperature because the heat of adsorption is positive, that is adsorption is an exothermic process. Since the free energy must decrease for the adsorption to occur and the entropy change is negative because of the decrease in the degree of freedom, therefore

$$
\Delta H = \Delta G + T\Delta S < 0\tag{65}
$$

The negativity of the enthalpy change means that heat is released from the adsorption process. The Langmuir equation can also be derived from the statistical thermodynamics, based on the lattice statistics.

### **4.2 Isosteric heat of adsorption**

One of the basic quantities in adsorption studies is the isosteric heat, which is the ratio of the infinitesimal change in the adsorbate enthalpy to the infinitesimal change in the amount adsorbed. The information of heat released is important in the kinetic studies because when heat is released due to adsorption the released energy is partly absorbed by the solid adsorbent and partly dissipated to the surrounding. The portion absorbed by the solid increases the particle temperature and it is this rise in temperature that slows down the adsorption kinetics because the mass uptake is controlled by the rate of cooling of the particle in the later course of adsorption. Hence the knowledge of this isosteric heat is essential in the study of adsorption kinetics. The isosteric heat may or may not vary with loading. It is calculated from the following thermodynamic van't Hoff equation:

$$\frac{\Delta H}{R\_s T^2} = -\left(\frac{\partial \ln P}{\partial T}\right)\_{c\_\mu} \tag{66}$$

For Langmuir isotherm of the form given in eq. (63), we take the total differentiation of that equation and substitute the result into the above van't Hoff equation to get:

$$\frac{\Delta H}{R\_{\circ}T^{2}} = \frac{Q}{R\_{\circ}T^{2}} + \delta(1+bP) \tag{66}$$

in which we have allowed for the maximum adsorbed concentration (Cµs) to vary with temperature and that dependence is assumed to take the form:

$$\frac{1}{C\_{\mu}} \frac{d\mathbf{C\_{\mu}}}{dT} = -\delta \tag{67}$$

Since (1+bP) = 1/(1-Ɵ), eq. (66) will become:

$$-\Delta H = Q + \frac{\delta R\_{\frac{\nu}{s}} T^2}{1 - \theta} \tag{68}$$

The negativity of the enthalpy change indicates that the adsorption process is an exothermic process. If the maximum adsorbed concentration, C^s, is a function of temperature and it decreases with temperature, the isosteric heat will increase with the loading due to the second term in the RHS of eq. (68). For the isosteric heat to take a finite value at high

the diffusion coefficient of the adsorbed phase, Vµ is the volume of the adsorbed phase, etc. The temperature dependence of the affinity constant (e.g. 60) is T-1/2 exp(Q/Rg T). This affinity constant decreases with temperature because the heat of adsorption is positive, that is adsorption is an exothermic process. Since the free energy must decrease for the adsorption to occur and the entropy change is negative because of the decrease in the degree

 0 *H G TS* (65) The negativity of the enthalpy change means that heat is released from the adsorption process. The Langmuir equation can also be derived from the statistical thermodynamics,

One of the basic quantities in adsorption studies is the isosteric heat, which is the ratio of the infinitesimal change in the adsorbate enthalpy to the infinitesimal change in the amount adsorbed. The information of heat released is important in the kinetic studies because when heat is released due to adsorption the released energy is partly absorbed by the solid adsorbent and partly dissipated to the surrounding. The portion absorbed by the solid increases the particle temperature and it is this rise in temperature that slows down the adsorption kinetics because the mass uptake is controlled by the rate of cooling of the particle in the later course of adsorption. Hence the knowledge of this isosteric heat is essential in the study of adsorption kinetics. The isosteric heat may or may not vary with

loading. It is calculated from the following thermodynamic van't Hoff equation:

2

*g g*

*RT RT*

equation and substitute the result into the above van't Hoff equation to get:

temperature and that dependence is assumed to take the form:

Since (1+bP) = 1/(1-Ɵ), eq. (66) will become:

ln

(66)

(66)

(68)

(67)

*g C H P RT T*

For Langmuir isotherm of the form given in eq. (63), we take the total differentiation of that

2 2 (1 )

 

> 1 *R Tg*

The negativity of the enthalpy change indicates that the adsorption process is an exothermic process. If the maximum adsorbed concentration, C^s, is a function of temperature and it decreases with temperature, the isosteric heat will increase with the loading due to the second term in the RHS of eq. (68). For the isosteric heat to take a finite value at high

2

*H Q bP*

in which we have allowed for the maximum adsorbed concentration (Cµs) to vary with

1 *<sup>s</sup> s*

*H Q*

*dC C dT* 

of freedom, therefore

based on the lattice statistics.

**4.2 Isosteric heat of adsorption** 

coverage (that is Ɵ —> 1) the parameter (thermal expansion coefficient of the saturation concentration) must be zero. This is to say that the saturation capacity is independent of temperature, and as a result the heat of adsorption is a constant, independent of loading.

### **4.3 Isotherms based on the gibbs approach**

The last section dealt with the basic Langmuir theory, one of the earliest theories in the literature to describe adsorption equilibria. One should note that the Langmuir approach is kinetic by nature. Adsorption equilibria can be described quite readily by the thermodynamic approach. What to follow in this section is the approach due to Gibbs. More details can be found in Yang (1987) and Rudzinski and Everett (1992).

In the bulk α-phase containing N components (Fig. 6), the following variables are specified: the temperature Tα, the volume Vα and the numbers of moles of all species ni <sup>α</sup> (for i = 1, 2, ..., N). The upper script is used to denote the phase. With these variables, the total differential Helmholtz free energy is:

$$dA = -S^{a}dT^{a} - P^{a}dV^{a} + \sum\_{i=1}^{N} \mu\_{i}^{a} dn\_{i}^{a} \tag{69}$$

where Sα is the entropy of the a phase, Pα is the pressure of that phase, nα is the number of molecule of the species *i*, and ( *<sup>i</sup>* is its chemical potential.

Fig. 6. Equilibrium between the phases α and β separated by a plane interface a 

Similarly, for the β -phase, we can write a similar equation for the differential Helmholtz free energy:

$$dA = -S^{\rho}dT^{\rho} - P^{\rho}dV^{\rho} + \sum\_{i=1}^{N} \mu\_{i}^{\rho} dn\_{i}^{\rho} \tag{70}$$

If equilibrium exists between the two phases with a plane interface (Fig. 6), we have:

$$T^{\
u} = T^{\rho}; P^{\
u} = P^{\rho}; \mu\_i^{\
u} = \mu\_i^{\rho} \tag{71}$$

*S dT Ad n d*

the Gibbs adsorption isotherm equation is:

For pure component systems (N = 1), we have:

ideal, i.e.

P.W., 1998).

**5.1 Linear isotherm** 

the Gibbs equation , we get:

equation of state to finally get:

1

*i i*

 

*i*

Adsorption equilibria experiments are usually carried out at constant temperature, therefore

1

where we have dropped the superscript a for clarity. At equilibrium, the chemical potential of the adsorbed phase is equal to that of the gas phase, which is assumed to be

0

ln *<sup>g</sup> T d n R T dP A* 

This equation is the fundamental equation relating gas pressure, spreading pressure and amount adsorbed. It is very useful in that if the equation of state relating the spreading pressure and the number of mole on the adsorbed phase is provided, the isotherm expressed as the number of mole adsorbed in terms of pressure can be obtained (Atkins,

For an ideal surface at infinite dilution, the equation of state relating the spreading pressure

an analogue of the ideal gas law (i.e. diluted systems), that is the spreading pressure is linear with the number of molecules on a surface of area A. Substituting this equation of state into

> ln *<sup>T</sup> d d P*

Integrating this equation at constant T, we obtain *n =* C(T)P, where C(T) is some function of temperature. This equation means that at equilibrium the spreading pressure in the adsorbed phase is linearly proportional to the pressure in the gas phase. The spreading pressure is not, however, useful in the correlation of adsorption equilibrium data. To relate the amount adsorbed in the adsorbed phase in terms of the gas phase pressure, we use the

and the number of mole on the surface has the following form:

 

Substituting eq.(82) into eq.(81), the following Gibbs isotherm equation is derived:

*i Ad n d*

*Ad nd* 

*N*

<sup>0</sup> *<sup>N</sup> i i*

(79)

(80)

*ggg RT P* ln (82)

(83)

*A nR T <sup>g</sup>* (84)

(85)

0 (81)

 

0

 

that is equality in temperature, pressure and chemical potential is necessary and sufficient for equilibrium for a plane interface.

$$A^{a} = -P\;V^{a} + \sum\_{i=1}^{N} \mu\_{i}^{a} n\_{i}^{a} \tag{72}$$

Differentiating eq. (72) and subtracting the result from eq. (69) will give the following Gibbs-Duhem equation:

$$-V^{a}dP - S^{a}dT + \sum\_{i=1}^{N} \mu\_{i}^{a} n\_{i}^{a} = 0\tag{73}$$

for the bulk α phase. As a special case of constant temperature and pressure, the Gibbs-Duhem's relation is reduced to:

$$\sum\_{i=1}^{N} \mu\_i^a n\_i^a = 0 \tag{74}$$

Similarly, the Gibbs-Duhem equation for the (β-phase at constant temperature and pressure is:

$$\sum\_{i=1}^{N} \mu\_i^{\rho} n\_i^{\rho} = 0 \tag{75}$$

### **5. Thermodynamics of the surface phase**

We now can develop a similar thermodynamic treatment for the surface phase , which is the interface between the phases α and β, and is in equilibrium with these two phases. When the adsorbed phase is treated as a two dimensional surface, fundamental equations in classical thermodynamics can still be applied. Applying the same procedure to surface free energy, we will obtain the Gibbs adsorption equation. This is done as follows. The total differentiation of the surface free energy takes the form similar to eq. (69) with PαdVβ being replaced by πdA :

$$dA^{\sigma} = -S^{\sigma}dT - \pi dA + \sum\_{i=1}^{N} \mu\_i dn\_i^{\sigma} \tag{76}$$

where the surface chemical potentials µi have the same values as those of the two joining phases, *π* is the spreading pressure, playing the same role as pressure in the bulk phase. Integrating eq. (76) with constant T, *π* and µi yields:

$$A'' = -\pi dA + \sum\_{i=1}^{N} \mu\_i n\_i'' \tag{77}$$

which is an analogue of eq. (72). Differentiation of this equation yields:

$$
\Delta dA^{\sigma} = -\pi dA - A d\pi + \sum\_{i=1}^{N} \mu d\_i n\_i^{\sigma} + \sum\_{i=1}^{N} n\_i^{s} d\_i \mu\_i \tag{78}
$$

Subtracting eq. (72) from eq. (70), we have the Gibbs equation for a planar surface:

that is equality in temperature, pressure and chemical potential is necessary and sufficient

Differentiating eq. (72) and subtracting the result from eq. (69) will give the following Gibbs-

*V dP S dT n*

for the bulk α phase. As a special case of constant temperature and pressure, the Gibbs-

<sup>0</sup> *<sup>N</sup> i i*

*n* 

Similarly, the Gibbs-Duhem equation for the (β-phase at constant temperature and pressure

*i i*

*n* 

the interface between the phases α and β, and is in equilibrium with these two phases. When the adsorbed phase is treated as a two dimensional surface, fundamental equations in classical thermodynamics can still be applied. Applying the same procedure to surface free energy, we will obtain the Gibbs adsorption equation. This is done as follows. The total differentiation of the surface free energy takes the form similar to eq. (69) with PαdVβ being

*dA S dT dA dn*

where the surface chemical potentials µi have the same values as those of the two joining phases, *π* is the spreading pressure, playing the same role as pressure in the bulk phase.

which is an analogue of eq. (72). Differentiation of this equation yields:

 *dA dA Ad d n n d*

Subtracting eq. (72) from eq. (70), we have the Gibbs equation for a planar surface:

0

1 *N*

*i i i A dA n*

> 1 1 *N N*

*i i*

*ii i i i*

 

(78)

 

 

*i*

1 *N*

 

*i i*

 

(76)

(77)

 

1

1

*i*

We now can develop a similar thermodynamic treatment for the surface phase

 *N*

*i*

1 *N*

*i A PV n*

1

*i*

*N*

*i i*

0

(73)

(74)

(75)

, which is

(72)

 

*i i*

 

for equilibrium for a plane interface.

Duhem's relation is reduced to:

**5. Thermodynamics of the surface phase** 

Integrating eq. (76) with constant T, *π* and µi yields:

Duhem equation:

replaced by πdA :

is:

$$-S^{\sigma}dT - Ad\pi + \sum\_{i=1}^{N} n\_i^{\delta} d\mu\_i = 0\tag{79}$$

Adsorption equilibria experiments are usually carried out at constant temperature, therefore the Gibbs adsorption isotherm equation is:

$$-Ad\pi + \sum\_{i=1}^{N} n\_i^{\delta} d\mu\_i = 0\tag{80}$$

For pure component systems (N = 1), we have:

$$-Ad\pi + nd\mu = 0\tag{81}$$

where we have dropped the superscript a for clarity. At equilibrium, the chemical potential of the adsorbed phase is equal to that of the gas phase, which is assumed to be ideal, i.e.

$$
\mu = \mu\_s = \mu\_s^\diamond + R\_s T \ln P \tag{82}
$$

Substituting eq.(82) into eq.(81), the following Gibbs isotherm equation is derived:

$$\left(\frac{d\pi}{d\ln P}\right)\_r = \frac{n}{A} R\_s T \tag{83}$$

This equation is the fundamental equation relating gas pressure, spreading pressure and amount adsorbed. It is very useful in that if the equation of state relating the spreading pressure and the number of mole on the adsorbed phase is provided, the isotherm expressed as the number of mole adsorbed in terms of pressure can be obtained (Atkins, P.W., 1998).

### **5.1 Linear isotherm**

For an ideal surface at infinite dilution, the equation of state relating the spreading pressure and the number of mole on the surface has the following form:

$$
\pi A = mR\_gT\tag{84}
$$

an analogue of the ideal gas law (i.e. diluted systems), that is the spreading pressure is linear with the number of molecules on a surface of area A. Substituting this equation of state into the Gibbs equation , we get:

$$
\pi = \left(\frac{d\pi}{d\ln P}\right)\_r \tag{85}
$$

Integrating this equation at constant T, we obtain *n =* C(T)P, where C(T) is some function of temperature. This equation means that at equilibrium the spreading pressure in the adsorbed phase is linearly proportional to the pressure in the gas phase. The spreading pressure is not, however, useful in the correlation of adsorption equilibrium data. To relate the amount adsorbed in the adsorbed phase in terms of the gas phase pressure, we use the equation of state to finally get:

$$\frac{m}{A} = K(T)P\tag{86}$$

<sup>2</sup> ln (1 )

( ) exp( ) 1 1

where the affinity constant b(T) is a function of temperature, which can take the following

( ) exp( )

Eq. (97) is known as the Volmer equation, a fundamental equation to describe the adsorption on surfaces where the mobility of adsorbed molecules is allowed, but no interaction is allowed among the adsorbed molecules. The factor exp(Ɵ/(l- Ɵ)) in eq. (96) accounts for the mobility of the adsorbate molecules. If we arrange eq. (96) as follows:

> .exp( ) 1 1 *b P*

the Volmer equation is similar to the Langmuir isotherm equation with the apparent affinity

.exp( ) <sup>1</sup> *app*

The difference between the Volmer equation and the Langmuir equation is that while the affinity constant remains constant in the case of Langmuir mechanism, the "apparent" affinity constant in the case of Volmer mechanism decreases with loading. This means that the rate of increase in loading with pressure is much lower in the case of Volmer compared

It is now seen that the Gibbs isotherm equation (83) is very general, and with any proper choice of the equation of state describing the surface phase an isotherm equation relating the amount on the surface and the gas phase pressure can be obtained as we have shown in the last two examples. The next logical choice for the equation of state of the adsorbate is an equation which allows for the co-volume term and the attractive force term. In this theme

<sup>2</sup> <sup>0</sup> ( ) *<sup>g</sup>*

*R T*

(100)

*a* 

 

*<sup>Q</sup> bT b*

*<sup>d</sup> <sup>P</sup>*

Written in terms of the fractional loading, Ɵ, eq. (93) becomes:

Carrying out the integration, we finally get the following equation:

form:

as

to that in the case of Langmuir.

the following van der Waals equation can be used:

**5.3 Hill-deboer isotherm** 

*bT P*

*b b*

00 0 ( /) ( /) *A An A An*

*g*

 

(94)

(95)

(96)

*R T* (97)

(98)

(99)

where

$$K(T) = \frac{C(T)}{R\_g T} \tag{87}$$

The parameter K(T) is called the Henry constant. The isotherm obtained for the diluted system is a linear isotherm, as one would anticipate from such condition of infinite dilution.

#### **5.2 Volmer isotherm**

We have seen in the last section that when the system is dilute (that is the equation of state follows eq. 84), the isotherm is linear because each adsorbed molecule acts independently from other adsorbed molecules. Now let us consider the case where we allow for the finite size of adsorbed molecules. The equation of state for a surface takes the following form:

$$
\pi(A - A\_o) = nR\_gT\tag{88}
$$

where A0 is the minimum area occupied by n molecules. The Gibbs equation (83) can be written in terms of the area per unit molecule as follows:

$$\left(\frac{\partial \pi}{\partial \ln P}\right)\_{\mathrm{r}} = \frac{R\_{\mathrm{g}}T}{\mathcal{S}}\tag{89}$$

where the variable is the area per unit molecule of adsorbate

$$
\delta = \frac{A}{n} \tag{90}
$$

Integrating equation (89) at constant temperature, we have:

$$\ln P = \frac{1}{R\_{\text{g}}T} \int \delta d\pi \tag{91}$$

We rewrite the equation of state in terms of the new variable and get:

$$
\hbar \pi (\mathcal{S} - \mathcal{S}\_o) = \mathcal{R}\_g T \tag{92}
$$

Substituting the spreading pressure from the equation of state into the integral form of the Gibbs equation (91), we get:

$$\ln P = -\int \frac{\delta d\delta}{(\delta - \delta\_o)^2} \tag{93}$$

But the fractional loading is simply the minimum area occupied by n molecules divided by the area occupied by the same number of molecules, that is

( ) *<sup>n</sup> KTP*

( ) ( ) *g*

The parameter K(T) is called the Henry constant. The isotherm obtained for the diluted system is a linear isotherm, as one would anticipate from such condition of infinite dilution.

We have seen in the last section that when the system is dilute (that is the equation of state follows eq. 84), the isotherm is linear because each adsorbed molecule acts independently from other adsorbed molecules. Now let us consider the case where we allow for the finite size of adsorbed molecules. The equation of state for a surface takes

> <sup>0</sup> ( )

> > ln

is the area per unit molecule of adsorbate

<sup>1</sup> ln

<sup>0</sup> ( ) 

ln

 *R Tg* 

Substituting the spreading pressure from the equation of state into the integral form of the

*<sup>d</sup> <sup>P</sup>* 

But the fractional loading is simply the minimum area occupied by n molecules divided by

*g P d R T*

 

> 2 0

( )

 

*P* 

written in terms of the area per unit molecule as follows:

Integrating equation (89) at constant temperature, we have:

We rewrite the equation of state in terms of the new variable

the area occupied by the same number of molecules, that is

where A0 is the minimum area occupied by n molecules. The Gibbs equation (83) can be

*T*

*A n* 

*g*

*R T*

*C T K T*

where

**5.2 Volmer isotherm** 

the following form:

where the variable

Gibbs equation (91), we get:

*<sup>A</sup>* (86)

*R T* (87)

*A A nR T <sup>g</sup>* (88)

(89)

(90)

and get:

(92)

(93)

(91)

$$\theta = \frac{A\_o}{A} = \frac{\{A\_o \,/\, n\}}{\{A \,/\, n\}} = \frac{\delta\_o}{\delta} \tag{94}$$

Written in terms of the fractional loading, Ɵ, eq. (93) becomes:

$$\ln P = \int \frac{d\theta}{\theta(1-\theta)^2} \tag{95}$$

Carrying out the integration, we finally get the following equation:

$$b(T)P = \frac{\theta}{1-\theta} \exp(\frac{\theta}{1-\theta})\tag{96}$$

where the affinity constant b(T) is a function of temperature, which can take the following form:

$$b(T) = b\_v \exp(\frac{Q}{R\_s T})\tag{97}$$

Eq. (97) is known as the Volmer equation, a fundamental equation to describe the adsorption on surfaces where the mobility of adsorbed molecules is allowed, but no interaction is allowed among the adsorbed molecules. The factor exp(Ɵ/(l- Ɵ)) in eq. (96) accounts for the mobility of the adsorbate molecules. If we arrange eq. (96) as follows:

$$\frac{\theta}{1-\theta} = b.\exp(-\frac{\theta}{1-\theta})P\tag{98}$$

the Volmer equation is similar to the Langmuir isotherm equation with the apparent affinity as

$$b \underset{\text{app}}{\text{app}} = b . \text{exp}(-\frac{\theta}{1-\theta}) \tag{99}$$

The difference between the Volmer equation and the Langmuir equation is that while the affinity constant remains constant in the case of Langmuir mechanism, the "apparent" affinity constant in the case of Volmer mechanism decreases with loading. This means that the rate of increase in loading with pressure is much lower in the case of Volmer compared to that in the case of Langmuir.

#### **5.3 Hill-deboer isotherm**

It is now seen that the Gibbs isotherm equation (83) is very general, and with any proper choice of the equation of state describing the surface phase an isotherm equation relating the amount on the surface and the gas phase pressure can be obtained as we have shown in the last two examples. The next logical choice for the equation of state of the adsorbate is an equation which allows for the co-volume term and the attractive force term. In this theme the following van der Waals equation can be used:

$$\left(\pi + \frac{a}{\delta^2}\right)(\delta - \delta\_0) = R\_\text{g}T \tag{100}$$

which involves only measurable quantities. Here P0 is the vapor pressure. This equation can describe isotherm of type II shown in Figure 10.7. The classification of types of isotherm will be discussed in detail in next section. But for the purpose of discussion of the Harkins-Jura equation, we explain type II briefly here. Type II isotherm is the type which exhibits a similar behavior to Langmuir isotherm when the pressure is low, and when the pressure is further increased the amount adsorbed will increase in an exponential fashion. Rearranging the Harkins-Jura equation (105) into the form of adsorbed amount versus the reduced

> / <sup>1</sup> 1 ln(1 / )

1 <sup>1</sup> 1 ln(1 / )

*B*

from which we can see that the only parameter which controls the degree of curvature of the

To investigate the degree of curvature of the Harkins-Jura equation (108), we study its

2 2 5/2

To find the inflexion point, we set the second derivative to zero and obtain the reduced

For the Harkins-Jura equation to describe the Type II isotherm, it must have an inflexion point occurring at the reduced pressure between 0 and 1, that is the restriction on the parameter B between 3/2>B>0. The restriction of positive B is due to the fact that if B is negative, eq. (105) does not always give a real solution. With the restriction on B as shown in eq. (109), the minimum reduced pressure at which the inflexion point occurs is (by putting B

<sup>3</sup> exp[ ( )] <sup>2</sup>

3 1 [1 ln(1 / ) <sup>1</sup> <sup>2</sup> ( ) 2 [1 1 / ln(1 / )] *<sup>x</sup> d V B B dx V Bx B x*

*x*

*x*

max lim*VV CB* / (107)

(109)

*x B* (110)

(106)

(108)

*C B <sup>V</sup>*

max

*V V*

Thus, the Harkins-Jura isotherm equation can be written as

2

pressure at which the isotherm curve has an inflexion point

max

inf,

*B*

where x is the reduced pressure (x=P/P0), We see that when the pressure approaches the vapor pressure, the adsorbed amount reaches a maximum concentration given below:

pressure, we have:

isotherm is the parameter B.

second derivative:

to zero in eq. 110):

**5.5 Characteristics of isotherm** 

With this equation of state, the isotherm equation obtained is:

$$bpP = \frac{\theta}{1-\theta} \exp(\frac{\theta}{1-\theta}) \exp(-c\theta) \tag{101}$$

where

$$bpP = b\_w \exp(\frac{Q}{R\_gT}), c = \frac{2a}{R\_gT\mathcal{S}\_o} = \frac{zw}{R\_gT} \tag{102}$$

where z is the coordination number (usually taken as 4 or 6 depending on the packing of molecules), and w is the interaction energy between adsorbed molecules. A positive w means attraction between adsorbed species and a negative value means repulsion that is the apparent affinity is increased with loading when there is attraction between adsorbed species, and it is decreased with loading when there is repulsion among the adsorbed species. The equation as given in eq. (101) is known as the Hill-de Boer equation, which describes the case where we have mobile adsorption and lateral interaction among adsorbed molecules. When there is no interaction between adsorbed molecules (that is w = 0), this Hill-de Boer equation will reduce to the Volmer equation obtained. The first exponential term in the RHS of eq. (101) describes the mobility of adsorbed molecules, and when this term is removed we will have the case of localized adsorption with lateral interaction among adsorbed molecules, that is:

$$bpP = \frac{\theta}{1-\theta} \exp(-c\theta) \tag{103}$$

This equation is known in the literature as the Fowler-Guggenheim equation, or the quasi approximation isotherm. This equation can also be derived from the statistical thermodynamics. Due to the lateral interaction term exp(-cƟ), the Fowler-Guggenheim equation and the Hill-de Boer equation exhibit a very interesting behavior. This behavior is the two dimensional condensation when the lateral interaction between adsorbed molecules is sufficiently strong (Adam, N.K., 1968).

### **5.4 Harkins-jura isotherm**

We have addressed the various adsorption isotherm equations derived from the Gibbs fundamental equation. Those equations (Volmer, Fowler-Guggenheim and Hill de Boer) are for monolayer coverage situation. The Gibbs equation, however, can be used to derive equations which are applicable in multilayer adsorption as well. Here we show such application to derive the Harkins-Jura equation for multilayer adsorption. Analogous to monolayer films on liquids, Harkins and Jura (1943) proposed the following equation of state:

$$
\pi = b - a\delta \tag{104}
$$

where a and b are constants. Substituting this equation of state into the Gibbs equation (67) yields the following adsorption equation:

$$\ln\left(\frac{P}{P\_o}\right) = B - \frac{C}{V^2} \tag{105}$$

exp( )exp( ) 1 1 *bP c*

  0

*g gg*

*RT RT RT* 

(101)

(102)

(103)

(104)

(105)

 

<sup>2</sup> exp( ),

*Q a zw bP b c*

where z is the coordination number (usually taken as 4 or 6 depending on the packing of molecules), and w is the interaction energy between adsorbed molecules. A positive w means attraction between adsorbed species and a negative value means repulsion that is the apparent affinity is increased with loading when there is attraction between adsorbed species, and it is decreased with loading when there is repulsion among the adsorbed species. The equation as given in eq. (101) is known as the Hill-de Boer equation, which describes the case where we have mobile adsorption and lateral interaction among adsorbed molecules. When there is no interaction between adsorbed molecules (that is w = 0), this Hill-de Boer equation will reduce to the Volmer equation obtained. The first exponential term in the RHS of eq. (101) describes the mobility of adsorbed molecules, and when this term is removed we will have the case of localized adsorption with lateral interaction among

> exp( ) <sup>1</sup> *bP c*

This equation is known in the literature as the Fowler-Guggenheim equation, or the quasi approximation isotherm. This equation can also be derived from the statistical thermodynamics. Due to the lateral interaction term exp(-cƟ), the Fowler-Guggenheim equation and the Hill-de Boer equation exhibit a very interesting behavior. This behavior is the two dimensional condensation when the lateral interaction between adsorbed molecules

We have addressed the various adsorption isotherm equations derived from the Gibbs fundamental equation. Those equations (Volmer, Fowler-Guggenheim and Hill de Boer) are for monolayer coverage situation. The Gibbs equation, however, can be used to derive equations which are applicable in multilayer adsorption as well. Here we show such application to derive the Harkins-Jura equation for multilayer adsorption. Analogous to monolayer films on liquids, Harkins and Jura (1943) proposed the following equation of

> *b a*

0 ln *P C <sup>B</sup> P V*

 

where a and b are constants. Substituting this equation of state into the Gibbs equation (67)

2

With this equation of state, the isotherm equation obtained is:

where

adsorbed molecules, that is:

**5.4 Harkins-jura isotherm** 

state:

is sufficiently strong (Adam, N.K., 1968).

yields the following adsorption equation:

which involves only measurable quantities. Here P0 is the vapor pressure. This equation can describe isotherm of type II shown in Figure 10.7. The classification of types of isotherm will be discussed in detail in next section. But for the purpose of discussion of the Harkins-Jura equation, we explain type II briefly here. Type II isotherm is the type which exhibits a similar behavior to Langmuir isotherm when the pressure is low, and when the pressure is further increased the amount adsorbed will increase in an exponential fashion. Rearranging the Harkins-Jura equation (105) into the form of adsorbed amount versus the reduced pressure, we have:

$$V = \frac{\sqrt{C/B}}{\sqrt{1 + \frac{1}{B} \ln(1/x)}}\tag{106}$$

where x is the reduced pressure (x=P/P0), We see that when the pressure approaches the vapor pressure, the adsorbed amount reaches a maximum concentration given below:

$$\text{Min}\,V = V\_{\text{max}} = \sqrt{\text{C} / B} \tag{107}$$

Thus, the Harkins-Jura isotherm equation can be written as

$$\frac{V}{V\_{\text{max}}} = \frac{1}{\sqrt{1 + \frac{1}{B} \ln(1/\chi)}}\tag{108}$$

from which we can see that the only parameter which controls the degree of curvature of the isotherm is the parameter B.

### **5.5 Characteristics of isotherm**

To investigate the degree of curvature of the Harkins-Jura equation (108), we study its second derivative:

$$\frac{d^2}{d\mathbf{x}^2} (\frac{V}{V\_{\text{max}}}) = \frac{1}{2B\mathbf{x}^2} \frac{\left(\frac{3}{2B} - \left[1 + \frac{1}{B}\ln(1/\mathbf{x})\right]\right)}{\left[1 + 1/B\ln(1/\mathbf{x})\right]^{5/2}}\tag{109}$$

To find the inflexion point, we set the second derivative to zero and obtain the reduced pressure at which the isotherm curve has an inflexion point

$$\propto\_{\text{in}\_{\text{in}\_{\text{}}}} = \exp[-(\frac{\mathfrak{J}}{2} - B)] \tag{110}$$

For the Harkins-Jura equation to describe the Type II isotherm, it must have an inflexion point occurring at the reduced pressure between 0 and 1, that is the restriction on the parameter B between 3/2>B>0. The restriction of positive B is due to the fact that if B is negative, eq. (105) does not always give a real solution. With the restriction on B as shown in eq. (109), the minimum reduced pressure at which the inflexion point occurs is (by putting B to zero in eq. 110):

**Equation of state Isotherm Name** 

*bP*

 *cw bP*

*bP c*

*bP c*

 

> 

<sup>2</sup> exp( )exp( ) 1 1

 

 -

 

 

 

*RT*

Henry law

Fowler-Guggenheim

Langmuir

Volmer

Hill-deBoer


(113)

(114)

*s g*

(115)

 

*bP*

exp( ) 1 1

exp( )exp( ) 1 1

exp( )exp( ) 1 1

equation is popularly used (Dubinin M. M., Radushkevich L.V. 1947)

*bP*

*bP*

Numbers of fundamental approaches have been taken to derive the necessary adsorption isotherm. If the adsorbed fluid is assumed to behave like a two dimensional non-ideal fluid, then the Equation of State developed for three dimensional fluids can be applied to two dimensional fluids with a proper change of variables. The 2D- equation of state (2D.EOS) adsorption isotherm equations are not popularly used in the description of data, but they have an advantage of easily extending to multicomponent mixtures by using a proper mixing rule for the adsorption parameters. For 3D fluids, the following 3 parameter EOS

2 2 *<sup>g</sup>*

 <sup>2</sup> 2 2 1 1 *<sup>g</sup>*

> <sup>2</sup> 2 2 1

*<sup>a</sup> b RT*

*p bd dR T*

Adopting the above form, we can write the following equation for the 2D-EOS as follows:

*p v b RT*

where p is the pressure, v is the volume per unit mole, a and b are parameters of the fluid and α and β represent numerical values. Different values obtained of α and β, give different forms of equation of state. For example, when α = β = 0, we recover the famous van der Waals equation. Written in terms of molar density d (mole/volume), the 3D-EOS will

 

*a*

*v bv b* 

*ad*

*s*

 

*s s*

*b b* 

 *bd b d*

1

 <sup>1</sup>

0

Table 2. Isotherm Equations derived from the Gibbs Equation

 

 exp( ) <sup>1</sup>

0

*R T*

*R T*

**5.7 Equation 2D of state adsorption isotherm** 

 

<sup>2</sup> <sup>0</sup> *<sup>g</sup>*

<sup>3</sup> <sup>0</sup> *<sup>g</sup>*

0 2 *g <sup>a</sup> R T* 

 

 *R Tg* 

*R Tg* ln

<sup>0</sup> ( ) 

*a* 

*a* 

become:

 *R Tg* 

0 0 ln( ) <sup>2</sup> *<sup>g</sup> cw R T*

> 

 

 

$$\mathbf{x}\_{\text{int}} = \exp[- (\frac{\beta}{2})] \approx 0.22\tag{111}$$

Fig. 7 shows typical plots of the Harkins-Jura equation.

Fig. 7. Plots of the Harkins-Jura equation versus the reduced pressure with B = 0.01

Jura and Harkins claimed that this is the simplest equation found so far for describing adsorption from sub-monolayer to multilayer regions, and it is valid over more than twice the pressure range of any two-constant adsorption isotherms. They showed that for TiO2 in the form of anatase, their isotherm agrees with the data at both lower and higher values of pressure than the commonly used BET equation. Harkins and Jura (1943) have shown that a plot of ln(P/P0) versus 1/v2 would yield a straight line with a slope of *C*. The square root of this constant is proportional to the surface area of the solid. They gave the following formula:

$$S\_s = 4.06\sqrt{C} \tag{112}$$

where v is the gas volume at STP adsorbed per unit g, and S has the unit of m2/g. They also suggested that if the plot of ln(P/P0) versus 1/v2 exhibits two straight lines, the one at lower pressure range should be chosen for the area calculation as this is the one in which there exists a transition from a monolayer to a polylayer.

### **5.6 Other isotherms from gibbs equation**

We see that many isotherm equations (linear, Volmer, Hill-deBoer, Harkins-Jura) can be derived from the generic Gibbs equation. Other equations of state relating the spreading pressure to the surface concentration can also be used, and thence isotherm equations can be obtained. The following table (Table 2) lists some of the fundamental isotherm equations from a number of equations of state (Ross and Olivier, 1964; Adamson, 1984).

Since there are many fundamental equations which can be derived from various equations of state, we will limit ourselves to a few basic equations such as the Henry law equation, the Volmer, the Fowler-Guggenheim, and the Hill-de Boer equation. Usage of more complex fundamental equations other than those just mentioned needs justification for doing so.

<sup>3</sup> exp[ ( )] 0.22 <sup>2</sup>

*x* (111)

4.06 *S C <sup>g</sup>* (112)

inf,

Fig. 7. Plots of the Harkins-Jura equation versus the reduced pressure with B = 0.01

Jura and Harkins claimed that this is the simplest equation found so far for describing adsorption from sub-monolayer to multilayer regions, and it is valid over more than twice the pressure range of any two-constant adsorption isotherms. They showed that for TiO2 in the form of anatase, their isotherm agrees with the data at both lower and higher values of pressure than the commonly used BET equation. Harkins and Jura (1943) have shown that a plot of ln(P/P0) versus 1/v2 would yield a straight line with a slope of *C*. The square root of this constant is proportional to the surface area of the solid. They gave the following

where v is the gas volume at STP adsorbed per unit g, and S has the unit of m2/g. They also suggested that if the plot of ln(P/P0) versus 1/v2 exhibits two straight lines, the one at lower pressure range should be chosen for the area calculation as this is the one in which there

We see that many isotherm equations (linear, Volmer, Hill-deBoer, Harkins-Jura) can be derived from the generic Gibbs equation. Other equations of state relating the spreading pressure to the surface concentration can also be used, and thence isotherm equations can be obtained. The following table (Table 2) lists some of the fundamental isotherm equations

Since there are many fundamental equations which can be derived from various equations of state, we will limit ourselves to a few basic equations such as the Henry law equation, the Volmer, the Fowler-Guggenheim, and the Hill-de Boer equation. Usage of more complex fundamental equations other than those just mentioned needs justification for doing so.

from a number of equations of state (Ross and Olivier, 1964; Adamson, 1984).

Fig. 7 shows typical plots of the Harkins-Jura equation.

exists a transition from a monolayer to a polylayer.

**5.6 Other isotherms from gibbs equation** 

formula:


Table 2. Isotherm Equations derived from the Gibbs Equation

### **5.7 Equation 2D of state adsorption isotherm**

Numbers of fundamental approaches have been taken to derive the necessary adsorption isotherm. If the adsorbed fluid is assumed to behave like a two dimensional non-ideal fluid, then the Equation of State developed for three dimensional fluids can be applied to two dimensional fluids with a proper change of variables. The 2D- equation of state (2D.EOS) adsorption isotherm equations are not popularly used in the description of data, but they have an advantage of easily extending to multicomponent mixtures by using a proper mixing rule for the adsorption parameters. For 3D fluids, the following 3 parameter EOS equation is popularly used (Dubinin M. M., Radushkevich L.V. 1947)

$$\left(p + \frac{a}{v^2 + \alpha bv + \beta b^2}\right)(v - b) = R\_g T \tag{113}$$

where p is the pressure, v is the volume per unit mole, a and b are parameters of the fluid and α and β represent numerical values. Different values obtained of α and β, give different forms of equation of state. For example, when α = β = 0, we recover the famous van der Waals equation. Written in terms of molar density d (mole/volume), the 3D-EOS will become:

$$\left(p + \frac{ad^2}{1 + abd + \beta b^2 d^2}\right) \left(1 - bd\right) = dR\_s T \tag{114}$$

Adopting the above form, we can write the following equation for the 2D-EOS as follows:

$$\left(\pi + \frac{a\_\ast \delta^2}{1 + a b\_\ast \delta + \beta b\_\ast^2 \delta^2}\right) \left(1 - b\_\ast \delta\right) = \delta R\_\ast T \tag{115}$$

1 1 ln ( ) *<sup>P</sup>*

> 

*g d P d R T*

Eq. (121) is the adsorption isotherm equation relating the surface density *a* (mole/m2) in terms of the gas phase pressure. The applicability of this isotherm equation rests on the ability of the 2D-EOS (eq. 121) to describe the state of the adsorbed molecule. Discussions on the usage of the above equation in the fitting of experimental data are discussed in

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).

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:

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

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

1/

*n*

0

where Po is some reference pressure and Cµ0 is the adsorbed concentration at that reference

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

*o C P C P* 

*T*

(121)

(122)

(123)

Integrating the above equation, we get

**6. Empirical isotherm equations** 

Zhou et al. (1994).

**6.1 Freundlich equation** 

pressure, 1/

0 0 *<sup>n</sup> C KP* 

where *n* is the spreading pressure, *a* is the surface density (mole/area) and the parameters as and bs are the 2D analogs of a and b of the 3D-EOS. Written in terms of the surface concentration (mole/mass), the above equation becomes:

$$\left(A\pi + \frac{a\_\ast w^2}{1 + \alpha b\_\ast w + \beta b\_\ast^2 w^2}\right) \left(1 - b\_\ast w\right) = w R\_\ast T \tag{116}$$

where A is the specific area (m2/g). To provide an EOS to properly fit the experimental data, Zhou et al. (1994) suggested the following form containing one additional parameter

$$\left(A\pi + \frac{a\_\ast w^2}{1 + a b\_\ast w + \beta b\_\ast^2 w^2}\right) \left(1 - (b\_\ast w)^\ast\right) = w R\_\ast T \tag{117}$$

This general equation reduces to special equations when the parameters α, β and m take some specific values. The following table shows various special cases deduced from the above equation.


Table 3. The various special cases deduced from equations

To fit many experimental data, Zhou et al. (1994) have found that m has to be less than 1/2*.*  They suggested a value of 1/3 for m to reduce the number of parameters in the 2D-EOS equation (117). At equilibrium, the chemical potential of the adsorbed phase is the same as that of the gas phase that is

$$
\mu\_{\mu} = \mu\_{\g} = \mu\_{\g}^{0} + R\_{\g}T \ln P \tag{118}
$$

The chemical potential of the adsorbed phase is related to the spreading pressure according to the Gibbs thermodynamics equation rewritten here for clarity:

$$-Ad\pi + nd\mu = 0\tag{119}$$

Thus

$$\mathcal{S}\left(\frac{d\pi}{d\ln P}\right) = \frac{n}{A}\mathcal{R}\_\circ T = \mathcal{S}\mathcal{R}\_\circ T\tag{119}$$

But the spreading pressure is a function of a as governed by the equation of state (115). We write

$$d\ln P = \frac{1}{R\_{\circ}T} \frac{1}{\delta} (\frac{\partial \pi}{\partial \delta})\_{r} d\delta \tag{120}$$

where *n* is the spreading pressure, *a* is the surface density (mole/area) and the parameters as and bs are the 2D analogs of a and b of the 3D-EOS. Written in terms of the surface

*a w <sup>A</sup> b w wR T*

where A is the specific area (m2/g). To provide an EOS to properly fit the experimental data,

*a w <sup>A</sup> b w wR T*

This general equation reduces to special equations when the parameters α, β and m take some specific values. The following table shows various special cases deduced from the

To fit many experimental data, Zhou et al. (1994) have found that m has to be less than 1/2*.*  They suggested a value of 1/3 for m to reduce the number of parameters in the 2D-EOS equation (117). At equilibrium, the chemical potential of the adsorbed phase is the same as

0

The chemical potential of the adsorbed phase is related to the spreading pressure according

ln *g g d n RT RT*

But the spreading pressure is a function of a as governed by the equation of state (115). We

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

   

*dP A* 

 <sup>2</sup> 2 2 1

<sup>2</sup>

*s g*

*s g*

*aggg RT P* ln (118)

(119)

(120)

(119)

(116)

(117)

concentration (mole/mass), the above equation becomes:

Table 3. The various special cases deduced from equations

to the Gibbs thermodynamics equation rewritten here for clarity:

0 *Ad nd*

above equation.

that of the gas phase that is

Thus

write

1

*s*

*s s*

*bw b w*

Zhou et al. (1994) suggested the following form containing one additional parameter

*s s*

*bw b w*

 

2 2 1( ) <sup>1</sup> *s m*

 

Integrating the above equation, we get

$$\int^{\rho} d\ln P = \frac{1}{R\_{\rm s}T} \int^{\delta} \frac{1}{\delta} (\frac{\partial \pi}{\partial \delta})\_{\rm r} d\delta \tag{121}$$

Eq. (121) is the adsorption isotherm equation relating the surface density *a* (mole/m2) in terms of the gas phase pressure. The applicability of this isotherm equation rests on the ability of the 2D-EOS (eq. 121) to describe the state of the adsorbed molecule. Discussions on the usage of the above equation in the fitting of experimental data are discussed in Zhou et al. (1994).
