Chemical Reactions on Solid Surfaces

C H A P T E R

8 Chemical Reactions on Solid Surfaces O U T L I N E 8.1 Catalysis

376

8.2 How Does Reaction with Solid Occur?

379

8.3 Adsorption and Desorption 382 8.3.1 Ideal Surfaces and Langmuir Adsorption Isotherm 384 8.3.2 Idealization of Nonideal Surfaces 389 8.3.3 UniLan Adsorption Isotherms 390 8.3.4 Cooperative Adsorption 395 8.3.5 Common Empirical Approximate Isotherms 409 8.3.6 Pore Size and Surface Characterization 412 8.4 LHHW: Surface Reactions with Rate-Controlling Steps

414

8.5 Chemical Reactions on Nonideal Surfaces Based on the Distribution of Interaction Energy 431 8.6 Chemical Reactions on Nonideal Surfaces With the Multilayer Approximation

436

8.7 Kinetics of Reactions on Surfaces Where the Solid is either a Product or Reactant

437

8.8 Decline of Surface Activity: Catalyst Deactivation

439

8.9 Summary

440

Bibliography

444

Problems

445

How do reactions occur on solid surfaces? What is catalysis? In Chapter 7, we learned that enzymes can be employed to produce a specific product from a given substrate. In the process, the enzyme is not consumed. While the enzyme participates in the reaction, it emerges unchanged. This is commonly seen in making wine, cheese, and bread; small amounts of the previous batch are added to make a new batch (in practice for over two millennia). In 1835, Berzelius realized that small amounts of a foreign substance could greatly affect the course of chemical reactions. Later in 1894, Ostwald formally stated that catalysts are substances that accelerate the rate of chemical reactions without being consumed. Catalysts are employed in industry today to selectively speed up desired reactions.

Bioprocess Engineering http://dx.doi.org/10.1016/B978-0-444-63783-3.00008-3

375

# 2017 Elsevier B.V. All rights reserved.

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8. CHEMICAL REACTIONS ON SOLID SURFACES

8.1 CATALYSIS Catalysis is the study of catalytic reactions or reactions involving catalysts. Catalysis can be divided into six categories: gas-phase catalysis, liquid-phase catalysis, acid-base catalysis, organometallic catalysis, autocatalysis, and solid catalysis. 1. Gas-phase catalysis. Lowers the activation energy and thus makes an otherwise inert system react, for example, ðC2 H5 Þ2 O ðAÞ ! C2 H6 + CH4 + CO,  rA ¼ kCA ; Ea ¼ 222 kJ=mol

(8.1)

whereas I2

ðC2 H5 Þ2 OðAÞ


! C2 H6 + CH4 + CO,  rA ¼ kCA CI 2 ; Ea ¼ 142 kJ=mol

(8.2)

2. Liquid-phase catalysis. For example, 2H2 O2 ðAÞ ! 2H2 O + O2 

2+

(8.3)

2+

can be catalyzed by I , Mn , Fe , catalase. 3. Acid-base catalysis. Either acids (protons or hydronium ions) or bases (eg, hydroxide ions), or both can catalyze the reaction. For example, acid-base catalysis is employed in pulping, bleaching, hydrolysis, and transesterification. The reaction rate is increased because of higher pH, or lower pH, or both as shown in Fig. 8.1. 4. Organometallic catalysis. Many homogeneous liquid-phase catalytic chemical processes use organometallic catalysts, for example, the catalytic reaction of a per-organic acid with propylene to form organic acid and propylene oxide (Fig. 8.2). 5. Autocatalysis. Autocatalysis is a special type of molecular catalysis in which one of the products of the reaction acts as a catalyst for the reaction. As a result, the concentration of this product appears in the observed rate law with a positive exponent if it is a catalyst in a usual sense, or with a negative exponent if it is an inhibitor. For example, the gas-phase decomposition of ethylene iodide FIG. 8.1 Effect of pH on the reaction rate for some acid-base catalyzed reactions: (a) protons or hydronium ions are the active catalytic agent, kH is very large and kOH ¼ 0; (b) same as (a) but kH is small; (c) both acid and base catalyze the reaction; (d) only bases or hydroxide ions are the active catalytic agents: kH ¼ 0 and kOH is small; and (e) same as (d) but kOH is large.

1012

a)

k = k0 + kH × 10–pH + kOH × 10pH

1010 b) 108 c) k 106 104 102

d) e)

100 0

2

4

6

8 pH

10

12

14

377

8.1 CATALYSIS

O H

H C H

HO ⎯ O ⎯ R

C CH3

Mo O⎯R

L

HO ⎯ R

L Mo

Mo

O⎯ O⎯ R

O⎯R O

H

H C H

L H

H

C

Mo

CH3

O⎯ R L

O H

H

C CH3

H

H C

C

C CH3

FIG. 8.2 Organometallic catalytic reaction of propylene oxidation with a per-organic acid ROOH. L is a ligand.

1=2

C2 H4 I2 ðAÞ ! C2 H4 + I2 ,  rA ¼ kCA CI2

(8.4)

6. Solid catalysis. The most common and preferred catalysts are solids, as solids can be separated and recovered more easily from the reaction mixture. This is the focus of this chapter. As illustrated in Fig. 8.2, catalysts participate in the reaction to enable the reaction to occur at desired reaction conditions, while emerging unchanged. Active catalytic agents participate in the reaction by forming an active complex with one or more reactant molecule(s). The complex undergoes changes, resulting in the reaction. Therefore an active intermediate (complex) is the key in catalytic reactions. Since catalysts emerge unchanged from the reaction, use of a catalyst that can be easily isolated from the reaction mixture is highly desirable, reducing separation cost. From a different perspective, feedstock for chemical and energy production is desired to be of sustainable origin, eg, plant biomass. Plant biomass is solid under conditions of applications. Thus, the reactions most important in bioprocess engineering applications are those that occur on solid surfaces when either a heterogeneous catalyst is used to promote the rate of reaction (catalysis) or one main reactant is in the solid phase. Solid catalysts are easily recovered and reused for carrying out reactions otherwise involving only fluids (gaseous and/or liquid substances). Solid feedstock and/or products are also common in bioprocesses where renewable biomass is being converted. Reactions occurring on surfaces are remarkably similar kinetically. Catalysts for various reactions are found among a wide variety of metals, metal oxides, and metals on various support materials such as carbon or metal oxides. One property of most catalysts is that they provide a large amount of surface area per unit volume on which the reaction can occur, which normally requires the effective surface to be contained within a porous matrix of some sort. This particular characteristic leads to a number of interesting and important problems arising from the interaction of the rates of transport of mass and energy through such porous matrices, which we shall discuss in detail later.

378

8. CHEMICAL REACTIONS ON SOLID SURFACES

Common catalytic reactions are: 1. Alkylation and dealkylation reactions. Alkylation is the addition of an alkyl group to an organic compound. It is commonly carried out with Friedel-Crafts catalysts, AlCl3 along with trace amount of HCl. For example, AlCl3

CH3 CH2 CH ¼ CH2 + C4 H10



! C8 H18

(8.5)

SiO2–Al2O3, SiO2–MgO, and montmorillonite clay are common dealkylation catalysts. 2. Isomerization reactions. Acid-promoted Al2O3 is a catalyst used in the conversion of normal (straight-chain) hydrocarbons to branched hydrocarbons. Acid and base catalysts are both found suited in most isomerization reactions. 3. Hydrogenation and dehydrogenation. Most ingredients involving hydrogen are d-orbital containing metals Co, Ni, Rh, Ru, Os, Pd, Ir, Pt, and metal oxides MoO2 and Cr2O3. Dehydrogenation is favored at high temperatures (> 600°C) and hydrogenation is favored at low temperatures. For example, CH3 CH ¼ CHCH3 ! CH2 ¼ CHCH ¼ CH2 + H2

(8.6)

is catalyzed by calcium nickel phosphate, Cr2O3, etc. 4. Oxidation reactions. The transition group (IUPAC group 8, 10, 11) elements are commonly used. Ag, Cu, Pt, Fe, Ni, their oxides, and V2O5 and MnO2 are good oxidation catalysts. For example, Ag

2C2 H4 + O2

! 2C2 H4 O V2 O5

2SO2 + O2


! 2SO3 Cu

2C2 H5 OH + O2

! 2CH3 CHO + 2H2 O Ag

2CH3 OH + O2

! 2HCHO + 2H2 O Pt

4NH3 + 5O2

! 4NO + 6H2 O Ni

2C2 H6 + 7O2

! 4CO2 + 6H2 O

(8.7) (8.8) (8.9) (8.10) (8.11) (8.12)

5. Hydration and dehydration (or condensation). This type of reaction is catalyzed by substances that have a strong affinity to water. Al2O3, SiO2–Al2O3 gel, MgO, clay, phosphoric acid and salts are good catalysts. For example, CH2 ¼ CH2 + H2 O ! CH3 CH2 OH

(8.13)

is an interesting reaction. It used to be a cheap way to produce ethanol from petroleumbased ethylene. The reverse of the reaction is more important today, as ethanol can be produced from renewable resources, and ethylene is a valuable monomer for higher alkenes (jet-fuel) and polymer production. 6. Halogenation and dehalogenation. These reactions are commonly catalyzed with an active ingredient of CuCl2, AgCl, or Pd. Halogenation can occur without a catalyst; however, catalysts are used to improve selectivity

379

8.2 HOW DOES REACTION WITH SOLID OCCUR?

Faujasite-type zeolite

ZSM-5 zeolite

7.4Å

6.6Å

12-Member ring

10-Member ring

FIG. 8.3 Typical zeolite (alumina-silica or Al2O3–SiO2) catalyst: 3-D pore formed by a 12 oxygen ring and 1-D pore formed by 10 oxygen ring. More zeolite structures can be found at http://www.iza-structure.org/databases/.

Solid catalysts usually have fine pores such that reactants can be “fixed” or brought tightly together by active centers on the surface for the reaction to occur. Fig. 8.3 shows a schematic of typical zeolite catalysts. Another characteristic of catalytic reactions is that it cannot be assumed, once a reaction is established at a certain rate under given conditions, that the rate will remain constant with the passage of time. Catalysts normally lose some or all of their specific activity for a desired chemical transformation over time of utilization. This effect, normally referred to as deactivation, can come from a number of different sources and is often very important in the analysis and/or design of catalytic processes and reactors.

8.2 HOW DOES REACTION WITH SOLID OCCUR? For chemical reactions involving a solid material and a substance A in the bulk-fluid (liquid- or gas-) phase, the substance A in the bulk-fluid phase needs to be in direct contact with the active ingredient (active catalytic center σ, or reacting group) on the solid material. This process is exemplified by Fig. 8.4. There are numerous examples of bioprocesses or reaction processes involving solid materials. Common processes involving solid materials can be divided into four categories: 1. Fixed bed or packed bed. Examples include coal-fired stoves, wood or wood pellet-fired stoves, ammonia synthesis, and hydrocracking of heavy oil. While the reactions involving fixed beds are commonly gas-solid or liquid-solid systems, hydrocracking of heavy oil is a three-phase system. Heavy oil trickles down the column packed with solid catalyst particles, while light-cracked substances flow up as gas. This type of application is also

380

8. CHEMICAL REACTIONS ON SOLID SURFACES

FIG. 8.4 Typical steps of reaction involving one species in the bulk-fluid phase and one active group in the solid material.

A A

B 1

B

A

7

External 7 mass transfer

1

A B 2

6

2

6 3

Internal diffusion

5

A ⎯4→ B Solid (catalyst) surface

termed a trickle-bed, where the liquid phase is far from saturating the reactor volume and it is in effect a “discontinuous” phase. 2. Settling column. Examples include chemical pulping of woodchips and high consistency bleaching of fibers. Continuous chemical pulping of woodchips is usually conducted in a reactor with woodchips falling downward by gravity in the pulping liquor; the liquor can be either upward or downward flowing. 3. Moving bed. Examples include twin-extruder reactor and pneumatic transport bed reactor. A settling column is also a special type of moving bed, where the solid materials move by gravity. In a pneumatic transport bed, the solid particles are carried upward by a fluid stream. In a twin-extruder reactor, solid material is transported by a screw action from one end to the other while contacting with a fluid phase and reacting. 4. Fluidized bed. Examples include coal gasification, gasification of woody biomass, and oil catalytic cracking. In a fluidized bed, the solid particles are suspended (gravity balanced by inertia and friction) by the upward-flowing fluid. A reaction involving solid material occurs in seven steps as exemplified in Fig. 8.4: 1. Mass transfer of reactant A from bulk-fluid phase (far-field) to the external surface of the solid material. 2. Mass transfer or diffusion of reactant A through the internal void spaces inside the solid material from the external surface to the active centers on the internal surface. 3. Formation of a complex of reactant A with the active center on the solid material, or (chemical) adsorption of A. 4. Reaction on the solid surface, where solid material is actively involved either as a catalyst or a reactant. The reaction transforms a complex of solid with A to a complex of solid with B. 5. Freeing product B from the complex of B with the solid material, or desorption of B. 6. Mass transfer or diffusion of B through the internal void spaces inside the solid material from the reaction site to the external surface of the solid material. 7. Mass transfer of B from the external surface of the solid material to the far-field or bulkfluid phase.

8.2 HOW DOES REACTION WITH SOLID OCCUR?

381

Clearly, steps 1, 2, 6, and 7 are mass transfer steps. This implies that mass transfer is important when reactions involve two or more phases. Step 3 exemplifies the approaching of reactant A to the active center in the solid material. Step 4 is the actual reaction. Step 5 exemplifies the leaving/freeing of product from the active center on the solid surface. Therefore, the key to catalysis and reaction kinetics involving solid materials is steps 3, 4, and 5. Similar to two molecules approaching each other as depicted by Fig. 6.1, as molecule A approaches the solid surface, there is an expulsive force preventing it from getting closer. High interaction potential will incur as molecule A approaches the surface, especially if molecule A is not forming a complex with any group on the solid surface. If a group on the solid surface and molecule A can form an intermediate or complex (like a product, or new species), there is an energy well instead. We name the complex of molecule A with the specific group or center on the solid surface “adsorption.” To illustrate the interaction of adsorbing and nonadsorbing molecules with solid surface, Fig. 8.5 shows a simple tertiary system (A, B, and AB) with the solid surface. For example, when AB is approaching the solid surface, if it does not form a complex with any group on the solid surface, the interaction potential (Gibbs free energy) increases monotonously with decreasing contacting distance (green curve). If AB forms a complex with an active group or center on the solid surface, the interaction potential appears to show an energy well or attractive force, at an intermediate distance of approach. This energy well is for a new species denoted as AB•σ. Further decreasing the distance will cause a sharp increase in the interaction potential (dotted brown line). Similar behavior is shown by either A or B, or A + B. When the interaction potentials of A, B, and AB are combined, as shown by the blue solid curve, the required activation energy for the transformation on the solid surface between the adsorbed A plus adsorbed B and the adsorbed AB can be observed. Therefore, for a reaction to occur on the solid surface, the reactant in the bulk-fluid phase needs to “adsorb” on the active center of the solid surface. A collision of reactant A with the active center on the solid surface causes adsorption. As such, the kinetics of adsorption can be described by collision theory as learned in Chapter 6.

FIG. 8.5

Gibbs free energy

The effect of the distance of a molecule from the solid surface on the Gibbs free energy for the interaction of a tertiary system (A, B, and AB) with a solid surface

No molecule-surface interaction

A+B

AB AB • σ A•σ + B•σ

Distance from surface

382

8. CHEMICAL REACTIONS ON SOLID SURFACES

At this point, it becomes clear that steps 3, 4, and 5 in Fig. 8.4 are key steps in reactions involving a solid material. It is most convenient to consider surface reactions as a series of these three steps, which is in essence a type of chain reaction (as we have seen in Chapter 3) employing the active centers of reaction in a closed sequence. Let us again turn to the simple example of an isomerization reaction, A ! B, now taking place on a surface containing one type of active center, σ. The individual reaction steps are:

+)

A + σ>A  σ

(8.14)

Aσ!Bσ

(8.15)

B  σ>B + σ

(8.16)

A·

B

(8.17)

The first step in this sequence, which we have written as a chemical reaction, represents the adsorption of A on the surface, Eq. (8.14); the second is the reaction of the surface-adsorbed A species, Aσ, to the corresponding B species on the surface, Bσ, Eq. (8.15); and third is the desorption of product B from the surface, Eq. (8.16). The overall reaction, by summing up the three steps, is given by Eq. (8.17), which does not show the involvement of the surface active center. We will have to develop our ideas concerning surface reactions by considering each of three steps individually, much as we did in examining the Lindemann scheme. The development is based mostly on gas/solid systems, which comprise the most common types of heterogeneous catalytic reactions; however, the analysis is also generally valid for liquid/ solid systems. Normally, for gas/solid systems, the expressions for adsorption equilibrium and reaction rate are written in terms of the partial pressures or concentrations of reactant and product species, whereas in liquid/solid systems, only concentrations are employed. This is not the first time that we have considered kinetics in heterogeneous (ie, more than one phase) reaction systems, as enzymes can effectively be considered as “solid.” Still, it is not an easy transition to make to “two-dimensional chemistry,” whereby concentrations are based on area (surface area) rather than volume.

8.3 ADSORPTION AND DESORPTION According to the first step of reaction, Eq. (8.14), given above, an essential feature of catalysis is the adsorption of the reacting species on the active surface prior to reaction. As indicated, this type of adsorption is generally a very specific interaction between surface and adsorbate, which is a chemical reaction in itself called chemisorption. Chemisorption involves a forming chemical bond between the adsorbate and surface active center. Desorption is just the reverse of this, so it is logical to discuss the two together. To begin, let us examine the thermodynamics of adsorption. The Gibbs free energy for the adsorption process is given by ΔG ¼ ΔH  TΔS

(8.18)

8.3 ADSORPTION AND DESORPTION

383

where ΔH is the enthalpy change due to adsorption of the molecule on the solid surface or the heat of adsorption, ΔS is the entropy change from the bulk phase (more random) to the adsorbed phase (more orderly), T is the temperature, and ΔG is the Gibbs free energy change due to adsorption. For the adsorption to occur, ΔG < 0, therefore, ΔH ¼ ΔG + TΔS < 0 or the adsorption is exothermic. This condition holds because ΔS < 0 for the adsorption process, if the adsorption does not involve the breaking apart of the adsorbate molecule (as in dissociative adsorption). For dissociative adsorption, the heat of adsorption can be endothermic, as the entropy change is positive. The kinetics of adsorption follows naturally with the collision theory as we learned in Chapter 6. To begin, let us consider the rate at which A molecules in a homogeneous gas (or liquid) will strike a solid surface. This is a problem we have already solved (Chapter 6). rffiffiffiffiffiffiffiffiffiffiffiffiffi RT (6.21) ZcT ðA, surfaceÞ ¼ NAV CA 2πMA This would give us the maximum possible rate of adsorption (the forward reaction in Eq. 8.14) in any system if every molecule striking the surface was adsorbed. This is independent of whether the adsorption is chemisorption or physisorption, which we will define later. Thus, it is not difficult to imagine the net adsorption-desorption rate is given by r ¼ kad CA Cσv  kdes CSA

(8.19)

r ¼ kA CA θCσ  kA θA Cσ

(8.20)

or where kad ¼ kA is the adsorption rate constant, CA is the concentration of the adsorbate in the bulk-fluid phase, Cσv is the surface concentration of vacant active centers, CSA is the surface concentration of the active centers that are occupied by the adsorbate molecule A, θ is the fraction of the vacant active centers, Cσ is the surface concentration of the total possible available positions or sites or active centers where A can be adsorbed, and θA is the fraction of the total possible available positions that are occupied by A. Clearly, θ¼

Cσv Cσ

(8.21)

CSA Cσ

(8.22)

θA ¼

Eq. (8.19) is analogous to the (elementary) reaction rate for a stoichiometry shown in Eq. (8.14). This is the basis of our discussion on adsorption and desorption. We still need to figure out what Cσ and the two rate constants are. Before we move further, we shall need to define the surface and types of interactions. The adsorption can be physical or chemical. We have alluded to chemisorption at the beginning of this section, in which chemical bonds form between the adsorbate molecule and the active center on the solid surface. Physisorption, on the other hand, does not form a “strong” bond between the adsorbate molecule and the active center on the solid surface. Fig. 8.6 illustrates the differences between physisorption and chemisorption. More qualitative differences are listed in Table 8.1.

384

8. CHEMICAL REACTIONS ON SOLID SURFACES

Physisorption

Ead ΔHad

A

Gibbs free energy

Gibbs free energy

Chemisorption

A ΔHad A•σ

A•σ Distance away from surface

Distance away from surface

FIG. 8.6 Interaction potential between molecule A and the active site σ on the surface. On the right is a typical physisorption where, as molecule A approaches the surface, the Gibbs free energy decreases slowly and finally encounters an energy well that corresponds to the adsorbed A on the surface. On the left is a typical chemisorption where, as molecule A approaches the surface, it first encounters a minor energy well and the stronger adsorption occur after overcoming an energy barrier Ead.

TABLE 8.1 Physisorption and Chemisorption Property

Physisorption

Chemisorption

Heat of adsorption

Usually around 5 kcal/mol, can be as high as 20 kcal/mol, but always greater than zero

Usually >20 kcal/mole, but can be less, even less than zero

Rate of adsorption

Fast with zero activation energy

May be slow with activation energy, but could also be fast with zero activation energy

Rate of desorption

Desorption activation energy equal to heat of adsorption

Desorption activation energy may be larger than heat of adsorption

Temperature range over which adsorption occurs

Close to condensation temperature of adsorbate; however, Kelvin condensation may occur for highly porous catalysts

Occurs over a wide range of temperatures and at temperatures much above the condensation temperature

Specificity

None

Depends on type of surface

Electrical conductivity

Not affected

May have an effect

8.3.1 Ideal Surfaces and Langmuir Adsorption Isotherm Rather than focusing on molecules, which are already understood, we will examine the nature of surfaces. The simplest model of a surface, the ideal surface, is one in which each adsorption site, σ, has the same energy of interaction with the adsorbate molecules, which is not affected by the presence or absence of adsorbate molecules on adjacent adsorbent sites, and which can accommodate only one adsorbate molecule or atom. We might represent the energy contours of each surface qualitatively as shown in Fig. 8.7. Adsorption would occur

8.3 ADSORPTION AND DESORPTION

385

Gibbs Free Energy

FIG. 8.7 A schematic of atomic force (interaction energy) on an energetically homogeneous surface.

G

Distance from a point along a given direction on the surface

when a molecule or atom of adsorbate with the required energy strikes an unoccupied site, and the energy contours (or energy variation when approaching the adsorbent surface) would be unaffected by the extent of adsorption, as depicted by Fig. 8.6 for Gibbs free energy change. These requirements for a reaction (adsorption) to occur look very similar to those we imposed on the bimolecular collision number in order to derive the reactive collision frequency, ZcT(A, B), of Eq. (6.20). If we designate E as the activation energy required for chemisorption, Cσ as the total concentration of sites available for chemisorption, θ as the fraction of free sites available for chemisorption, and θA as the fraction of sites on the surface covered by the adsorbate molecule A, the following analog to Eq. (6.21) may be written as rffiffiffiffiffiffiffiffiffiffiffiffiffi Ead RT (8.23) ZcT ðA, adsÞ ¼ NAV CA Cσ θe RT 2πMA We may also include a term analogous to the steric factor to be used as a measure of the deviation of chemisorption rates from this ideal limit. rffiffiffiffiffiffiffiffiffiffiffiffiffi RT  Ead (8.24) ZcT ðA, adsÞ ¼ NAV CA Cσ θξ e RT 2πMA where ξ is commonly termed the sticking probability. The adsorption rate constant is thus predicted from Eq. (8.24). A potential-energy diagram for the adsorption-desorption process A + σ>A  σ

(8.14)

is shown in Fig. 8.6. As illustrated, the chemisorption is exothermic, which is generally the case. Also, since adsorption results in a more ordered state (similar to solid state) compared to bulk gas or liquid, we can argue in thermodynamic terms that entropy changes for chemisorption are negative. This fact is useful in testing the reasonableness of rate expressions for reactions on surfaces. Eq. (8.14) as we have written implies that Eq. (8.19) is valid, which we will call the Langmuir adsorption rate. Eq. (8.24) reaffirms that this is indeed the case. An important step in the consideration of surface reactions is the equilibrium level of adsorption on a surface. With Eq. (8.14), the rate is shown to be reversible and equilibrated over a sufficiently long time scale, so the rate of adsorption equals the rate of desorption. It is

386

8. CHEMICAL REACTIONS ON SOLID SURFACES

convenient to think of this as a process of dynamic equilibrium, where the net rate of change is zero. Assuming A is the only species adsorbed on the surface or at least on the active site of interest, by setting r ¼ 0 to Eq. (8.19), we have θA kad kad0  Ead Edes kad0 ΔHad RT ¼ ¼ e ¼ e RT ¼ KA θCA kdes kdes0 kdes0

(8.25)

This expression is extraordinarily useful, since it permits us to obtain some information concerning the surface coverage factors, θ and θA, about which we have been rather vague. These factors cannot be measured conveniently, while macroscopic quantities such as CA and the heat of absorption can. Eq. (8.25) provides a link between the two. Solving for the ratio θA/θ, it can be seen that the surface coverage at equilibrium (or at least some function of it), is determined by the temperature of the system and the concentration of adsorbate in the bulk phase. Such an equation for a fixed temperature and varying concentrations expresses the adsorption isotherm for the adsorbate, or, for a fixed concentration and varying temperatures, the adsorption isobar. The heat of adsorption, ΔHad, appears in Eq. (8.25) since in solving for the ratio of surface coverage functions, the difference (Edes–Ead) appears in the exponential; from Fig. 8.7 we see that this is equal in magnitude to the heat of adsorption. Eq. (8.25) is termed the Langmuir isotherm, which we may write in more general notation as ΔHad θA ∘ ¼ KA CA ¼ KA e RT CA θ

(8.26)

where KA¼

ΔHad kad ∘ ¼ KA e RT kdes

(8.27)

The total number of active sites or centers is fixed for a given amount of surfaces. Therefore, an active site balance (just as for active sites in enzymes) leads to θA + θ ¼ 1

(8.28)

if A is the only adsorbate on the particular type of active center. The site balance is not dependent on temperature or concentration. Combining Eqs. (8.26) and (8.28), we obtain θ A¼

KA CA 1 + KA CA

(8.29)

which is the Langmuir isotherm equation for the nondissociative adsorption of a single species on a surface (ie, single molecule on a single-type active site). In many cases of practical importance, the adsorbate molecule will dissociate on adsorption (eg, H2 on many metals), or occupy two adjacent active sites by bonding at two points in the molecule (eg, ethylene on nickel). In such cases, A2 + 2σ>2A  σ

(8.30)

A + 2σ>σ  A  σ

(8.31)

or

8.3 ADSORPTION AND DESORPTION

387

and the corresponding adsorption equilibrium is r ¼ kad θ2 C2σ CA2  kdes θ2A C2σ ¼ 0  2 θA kad ¼ CA θ kdes 2

(8.32) (8.33)

which leads to the isotherm equation θ A¼

ðKA CA2 Þ1=2 1 + ðKA CA2 Þ1=2

(8.34)

If the adsorbate molecule is immobile on the surface, the occupancy of nearest-neighbor active sites should be accounted for. This is not a large refinement, however, and Eq. (8.32) will be employed without reference to the detailed nature of the adsorbate layer. A second modification of practical importance is when more than one adsorbate species is present on the surface competing for the same type of active center (or site). For example, consider the adsorption equilibrium (no surface reaction) A + σ>A  σ

(8.14)

B  σ>B + σ

(8.16)

where A and B are chemisorbed on the same type of surface site, σ; that is, they are competitively adsorbed on the surface. In this case, Eq. (8.26) is applicable to both A and B (with different heats of adsorption) as dictated by the net adsorption rate of zero at equilibrium, ie, θA θB ¼ KA CA and ¼ KB CB . The total active site balance is now θ θ θA + θB + θ ¼ 1

(8.35)

as the active sites are shared by three parts: active sites occupied by A; active sites occupied by B; and free active sites available for adsorption. Therefore, the corresponding isotherm equations for the surface coverages of A and B are θA ¼

KA CA 1 + KA CA + KB CB

(8.36)

θB ¼

KB CB 1 + KA CA + KB CB

(8.37)

A major property of the Langmuir isotherm is that of saturation. In Eq. (8.29), for example, when KACA ≫ 1, θA ! 1, and no further adsorption occurs. This is a result of the surface model, in which each adsorption site can accommodate only one adsorbate molecule. Saturation of the surface, then, corresponds to the occupancy of all sites and is called monolayer coverage. At low concentrations, KACA ≪ 1 and Eq. (8.29) assumes a linear form in CA corresponding to Henry’s law of adsorption. These general features are shown in Fig. 8.8. Experimentally, one measures either the weight or volume of material adsorbed, and the ratio of this quantity at a given partial pressure (for adsorption from gas phase) or concentration to that at saturation can be taken as a direct measure of surface coverage. Mathematically,

388

8. CHEMICAL REACTIONS ON SOLID SURFACES

FIG. 8.8 Monolayer surface coverage, typical Langmuir adsorption isotherm.

Saturation region

1 qA = KACA

qA

1 2

0

0

θ A¼

CA

nAS CSA ¼ nAS1 CSA1

(8.38)

where nAS is the number of moles of A adsorbed on the surface and nAS1 is the maximum number of moles that can be adsorbed on the surface (for multiple species adsorption, without interference from other species). At this point, we observe that the ideal (or Langmuir) adsorption isotherms for a single component and a multicomponent mixture are quite similar. Indeed, in general, the ideal adsorption isotherm or coverage for a particular species j is given by Kj Cj

θj ¼ 1+

Ns X

(8.39)

Km Cm

m¼1

where θj is the fraction of sites that are covered by species j. If there are dissociative species in the mixture, modification of Eq. (8.39) needs to be made for the dissociative species in the same fashion as in Eq. (8.35). The interpretation of data on adsorption in terms of the Langmuir isotherm is most easily accomplished using the procedure previously described for reaction rate data. As pointed out earlier, the total number of active sites is not dependent on the temperature or concentration; the Langmuir isotherm is thus restricted by the total amount of adsorbate that is able to be adsorbed on the surface. Examples of false obedience to the Langmuir isotherm abound. These usually arise when adsorption data have not been obtained over a sufficiently wide range of concentrations or partial pressures (of gas adsorbate); a good test is to see if the values of the saturated adsorbate nmax evaluated from isotherm data at different temperatures are equal, since within the framework of the Langmuir surface model, the saturation capacity should not vary with the temperature. It can be difficult to tell, however, in view of experimental error. Higher temperatures usually result in a lower adsorption isotherm equilibrium constant KA.

8.3 ADSORPTION AND DESORPTION

FIG. 8.9 Variation of adsorption iso-

nAS∞

nAS

389 therm with temperature for Langmuir adsorption.

T1

T2

T3

T4

T1 < T2 < T3 < T4 lnCA

The decrease in KA requires much higher concentration CA in order to achieve the same level of adsorption as depicted by Fig. 8.9, where concentration is plotted on log scale.

8.3.2 Idealization of Nonideal Surfaces It is not hard to imagine that no real surface could have the potential-energy distribution as depicted in Fig. 8.7, nor is it reasonable to expect that adsorbate molecules on the surface would not interact with each other. Somewhat more plausible is the limitation of monolayer adsorption. In any of these events, one would expect to observe some distribution of energies of interaction with the surface to be correlated with surface coverage. Since the strongest interactions would occur on the nearly unoccupied surface, the experimental observation would be that of decreasing heat of chemisorption with increasing coverage. It is not possible to distinguish from this information alone whether energetic inhomogeneity of the surface or adsorbate interactions are the cause, but in most of the practical applications with which we are concerned, this amount of detail is not necessary. In the following discussion, the case of an inhomogeneous surface shall be considered, using the specific example of nondissociative adsorption of a single species for illustration. Consider that the surface may be divided into a number of groups of sites, each characterized by a similar heat of chemisorption, and so capable of being represented by a Langmuir isotherm. That is, ri ¼ kAi CA Cσiv  kAi CSAi

(8.40)

where the subscript i denotes the active center group. The overall adsorption rate is the sum over all the distinct groups. The adsorption rate constants are given by Eai

kAi ¼ kA0 e RT kAi ¼ kA0 e

Eai ΔHi RT

(8.41) (8.42)

where Eai is the activation energy of adsorption for active center group i, and ΔHi is the adsorption heat for the adsorption of adsorbate molecule A on the active center group i. Based

390

8. CHEMICAL REACTIONS ON SOLID SURFACES

on the discussions we had earlier, the adsorption activation energy is relatively small and should not be very different for different active site groups when the surface coverage is not great. Therefore, Eai

kAi ¼ kA0 e RT ¼ kA kAi ¼ kA0 e

Eai ΔHi RT

(8.43) ΔHi

¼ k
A e RT

(8.44)

In this fashion, we can examine a nonideal adsorption by applying idealized models for the heat of adsorption and for how the adsorption occurs among different groups. If the adsorption occurs one group at a time, we effectively have a multilayer adsorption model. The multilayer adsorption can better describe the adsorption process for high surface coverages. On the other hand, if the adsorption occurs simultaneously among all the active centers, we have a distributed adsorption model. At equilibrium, the adsorption rate for each active center group is zero or ri ¼ 0. Thus, KAi CA 1 + KAi CA

(8.45)

ΔHi kAi ¼ K
A e RT kAi

(8.46)

θAi ¼ KAi ¼ and the total coverage is

θA nσ ¼

nT X

θAi nσi

(8.47)

i¼1

where nσi is the number of adsorption sites belonging to group i and can be represented by an appropriate distribution function. It is reasonable to write this distribution function in terms of the heat of chemisorption, ΔHi, which is usually positive, as the interaction between the adsorbate and adsorbent is expected to vary. Generally, ΔHi ¼ fðCσv Þ

(8.48)

ie, the heat of adsorption is a function of the vacant site concentration Cσv still remaining. A higher number of vacant sites will have a higher value of heat of adsorption, while the lower the number of vacant sites remaining, the lower the value of heat of adsorption is expected.

8.3.3 UniLan Adsorption Isotherms With different distribution functions, one would expect different adsorption isotherms. One simplistic distribution function is the uniform distribution as shown in Fig. 8.10, ie, all the available active centers (sites) are distributed linearly along the surface energy rise dnσi ¼ 

nσ dEs RTfT

(8.49)

where Es varies between 0 and fTRT (or more precisely ΔHi between ΔHmin to ΔHmin + fTRT). For a general multiple competitive adsorption with the ideal case given by Eq. (8.39), the fractional coverage by species j is then

391

8.3 ADSORPTION AND DESORPTION

–ΔHmin + fTRT

fTRT

Es

–ΔHi

–ΔHmin

0

0

0

qi or nσI or Cσ

FIG. 8.10 Linear distribution of adsorption interaction energy with vacant site concentration.

1 θj ¼ nσ

ð nσ 0

1 θji dnσi ¼  RTfT

ð0

Es

K
j eRT Cj dEs Ns X Es RTfT
Km Cm eRT 1+

(8.50)

m¼1

where Eq. (8.46) has been applied with Es ¼ ΔHmin  ΔHi. Integration of Eq. (8.50) yields Ns X

K
m Cm efT K
j Cj m¼1 θj ¼ N ln Ns s X X K
m Cm K
m Cm fT 1+ 1+

m¼1

(8.51)

m¼1

For single-species adsorption, Eq. (8.51) is reduced to θA ¼

1 1 + K
A CA efT ln fT 1 + K
A CA

(8.52)

which is also known as the UniLan (uniform distribution Langmuir) model. Example 8.1: Adsorption of Phenol on Activated Carbon From Aqueous Solutions. A Comparison of a Nonideal Isotherm and Ideal Isotherm Table E8.1.1 shows the experimental data collected in an undergraduate lab run for the adsorption of phenol on activated carbon. Examine the suitability of Langmuir isotherm and linear interaction energy distribution nonideal isotherm (UniLan) to describe the experimental data. Solution This example is typical of adsorption experiments where the concentrations of adsorbate (phenol in this example) are measured in the bulk-fluid phase and in the adsorbent. This is

392

8. CHEMICAL REACTIONS ON SOLID SURFACES

TABLE E8.1.1 Equilibrium Concentrations of Phenol in Aqueous Solution and in Activated Carbon Phenol Concentration in Activated Carbon CSA, mg/g

Phenol Concentration in Aqueous Solution CA, mg/L 0

0

1.5

53

2

62.4

5.1

82.8

6.6

105

22.9

137

51.6

172.2

80.2

170.9

different from what we have discussed in Langmuir isotherms and the nonideal isotherms; however, it is clear that θA ¼

CSA CSA1

(8.38)

where CSA1 is the concentration of adsorbate on the adsorbent surface when the concentration of the adsorbate in the bulk-fluid phase is infinitely high, ie, all the active sites on the adsorbent surface are covered by the adsorbate. This quantity is a property of the adsorbent and adsorbate pair, and is not a function of temperature. Substituting Eq. (8.38) into Eq. (8.29), we obtain the Langmuir model CSA ¼ θA CSA1 ¼ CSA1

KA CA 1 + KA CA

(E8.1.1)

Therefore, we can fit the experimental data with Eq. (E8.1.1) to check if the model can reasonably describe the experimental data. For this exercise, a spreadsheet program is convenient to use as illustrated in Table E8.1.2. One can observe from Table E8.1.2 that the Langmuir model fit is consistent with the experimental data; the errors or deviations are consistent with the errors apparent in the experimental data. The fit, however, is better at high coverage than the lower coverage (concentrations). Fig. E8.1 shows the visual quality of the fit. Visual inspection of the data shown in Fig. E8.1 agrees with our assessment based on Table E8.1.2. Furthermore, visual inspection shows that the data are scattered around the model (solid line). Therefore, the Langmuir model is consistent with experimental data within experimental accuracy. We next examine the suitability of the linear adsorption energy distribution nonideal adsorption isotherm, Eq. (8.52). Substituting Eq. (8.28)into Eq. (8.52), we obtain the UniLan model CSA ¼

CSA1 1 + K
A CA e fT ln fT 1 + K
A CA

(E8.1.2)

393

8.3 ADSORPTION AND DESORPTION

TABLE E8.1.2

Least Square Parameter Estimation for the Langmuir Isotherm

Aqueous Solution CA, mg/L

On Activated Carbon C0 SA, mg/g

Eq. (E8.1.1), CSA, mg/g

Error2 (CSA 2 C0 SA)2

0

0

0

0

1.5

53

43.78934

84.8362

2

62.4

53.96358

71.17314

5.1

82.8

93.63607

117.4205

6.6

105

104.9492

0.002578

22.9

137

148.3163

128.0591

51.6

172.2

163.5351

75.07994

80.2

170.9

168.4535 5.985588 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u 8  u CSA, i  C0SA, i t ¼ 8:302814 σL ¼ i¼1 81

The final solution shown is when CSA1 ¼ 178.1 mg-phenol/g-activated carbon and KA ¼ 0.21732 (mg/L)1, which was obtained by minimizing the variance between the langmuir model and the experimental data, σL, while changing CSA1 and KA.

200

CSA, g/kg

150

100

50 Langmuir isotherm UniLan isotherm 0 0

20

40

60

80

100

CA, g/m3

FIG. E8.1 Langmuir isotherm and UniLan isotherm fit to the experimental data of phenol adsorption on activated carbon.

394

8. CHEMICAL REACTIONS ON SOLID SURFACES

The data fit is performed the same way as for the Langmuir adsorption isotherm model and illustrated in Table E8.1.3 for the UniLan adsorption model. The fitted line is also shown in Fig. E8.1 as the dot-dashed line. One can observe both from Fig. E8.1 and Table E8.1.3 that the UniLan nonideal adsorption isotherm describes the experimental data well within the apparent experimental error. Compared to the Langmuir model fit (Table E8.1.2), the UniLan adsorption isotherm fit improved the low coverage region, and the overall fit is slightly better (lower standard deviation or variance). One can conclude that despite their simplicity, both Langmuir adsorption isotherm and the UniLan adsorption isotherm describe the experimental data quite well. Fig. 8.11 shows the change of coverage with bulk-phase concentration as predicted by the Langmuir and UniLan models. One can observe that the general shape (or qualitative behaviors) of the two models are quite similar; however, compared to the ideal Langmuir isotherm, the nonideality introduced to the isotherms by the UniLan model causes the isotherm to bend upward, ie, coverage increases quicker at low bulk-phase concentration. This is due to the nonideal adsorption models we used: the interaction energy level starts from the base line that corresponds to the ideal surfaces (when equating K
j and Kj). Therefore as one would expect, adsorbate molecules prefer to adsorb to high interaction energy sites (with higher ΔHad values) than lower ones (with lower ΔHad values). The level of adsorption increases with nonideality (or introduction of higher interaction energy sites).

TABLE E8.1.3 Aqueous Solution CA, mg/L 0

Least Square Parameter Estimation for the UniLan Nonideal Adsorption Isotherm On Activated Carbon C0 SA, mg/g 0

Eq. (E8.1.2), CSA, mg/g 0

Error2 (CSA 2 C0 SA)2 0

1.5

53

52.14

0.7369

2

62.4

60.55

3.4239

5.1

82.8

90.63

61.3314

6.6

105

99.32

32.2165

22.9

137

140.66

13.3634

51.6

172.2

164.42

60.4657

80.2

170.9

175.31 19.4148 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u 8  u CSA, i  C0SA, i t ¼ 5:2229 σL ¼ i¼1 81

 1 The final solution shown is when CSA1 ¼ 209.95 mg-phenol/g-activated carbon, K
A ¼ 0:008528 mg=L and maximum interaction energy distribution range fT ¼ 5.421, and is obtained by minimizing the variance between the nonideal adsorption model and the experimental data, σL, while changing CSA1 and K
A .

395

8.3 ADSORPTION AND DESORPTION

1.0

0.8

qA

0.6

0.4 Langmuir UniLan, fT = 5 UniLan, fT = 1

0.2

0.0 0

2

4

6

8

10

KACA

FIG. 8.11 The adsorption isotherms as predicted by ideal Langmuir adsorption and uniform interaction energy distribution (UniLan).

8.3.4 Cooperative Adsorption Apart from the interaction energy difference or distribution on the adsorption on surfaces, there is an effect of the size of the active center or how close the active centers are as compared with the size of the adsorbate molecule. Fig. 8.12 illustrates the adsorptions of two different types of adsorbate molecules A and B on the same type of active centers σ. The adsorption can occur only in pairs of the active center (σ) and the adsorbate molecule as illustrated, irrespective of the level of interaction energy. The active centers are distributed over the entire adsorbent surface, and are of finite “distances” between adjacent active centers. Because all molecules have finite nonvanishing sizes, some of the active centers cannot be accessed by adsorbate molecules when neighboring site(s) have already been occupied. Therefore, the maximum number (or concentration) of adsorbate molecules that can be accommodated σ Aσ σ σ σ σ B σ σ σ σ σ σ Aσ Aσ A σ σ σ σ σ σ σ σ σ σ B B B B σ σ σ σA σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ

FIG. 8.12 Adsorption or occupation of adsorbate molecules on adsorbent surfaces. Active centers (σ) are distributed over the surface, with finite distances between adjacent active centers. Adsorbate molecules are of finite sizes, needing a finite space in every direction. If the adsorbate molecule (A) is small enough, when paired with the active center (Aσ), the active neighboring centers can still accommodate small adsorbate molecules unhindered; however, when a large adsorbate molecule (B) occupies an active center (Bσ), the immediate adjacent active center(s) are no longer available for other adsorbate molecules to access. For example, the dashed B molecule could not be there if the adjacent molecule(s) are present.

396

8. CHEMICAL REACTIONS ON SOLID SURFACES

on an adsorbent surface is not necessarily identical to the exact number (or concentration) of active centers on the surface. As a result, the maximum or saturation coverages of different types of adsorbate molecules on a given surface can be different, and different from that of the generic active center Cσ. Just as the interaction energy may not be uniform, this steric effect can also cause the adsorption to deviate from ideal interactions. At high surface coverages, the idealization of noninteraction between adsorbate molecules is not a valid assumption. It is well known that adsorbate molecules “pack” very closely on the adsorbent surface, which results in significant adsorbate-adsorbate interactions. In order for a “new” adsorbate molecule to approach the active center when neighboring active centers are occupied, the “new” adsorbate molecule must also inadvertently approach the other adsorbed adsorbate molecules already on the surface. At a minimum, steric interactions exist. The cooperative adsorption model (Liu, 2015) is well suited to describe nonideal adsorption on solid surfaces in general. Fig. 8.13 illustrates the cooperative adsorption idealization. When an adsorbate molecule collides with the solid surface, it slides to the “lowest” energy well when available. When the adjacent “lower” energy wells are occupied, a successful adsorbate molecule collision can result in the adsorption to occur on a “higher” energy well. Therefore, the adsorption is effectively multilayered. While the adsorbate-adsorbent interaction can be chemical (ie, chemical bonds form), the interactions between adsorbate and adsorbate near or in contact with each other is physical. The orderly state of adsorbate molecules on the adsorbent surface is at a lower energy well than that in the bulk-fluid phase, irrespective of whether new chemical bonds formed due to the state change. While the actual physical orientation of the adsorbate molecules on the adsorbent surface is monolayer, the adsorption mechanism (or mathematical model) needs not be. Thus, modeling the adsorbate-adsorbent and adsorbate-adsorbentadsorbate interactions is best accomplished with multiple layers. When steric interaction and adsorption chemical interaction energy distributions are combined, it produces multilayer adsorption kinetics. As shown in Fig. 8.14, the overall interaction energy distribution needs not be monotonous due to the introduction of steric interactions. Comparing Figs. 8.10 and 8.14, cooperative or multilayer adsorption approximates the adsorption interaction energy with piece-wise discrete distribution.

layer i

4 3 2 1

FIG. 8.13

Multilayer surface adsorption model

397

8.3 ADSORPTION AND DESORPTION

–ΔHi Layer 1

0

0

Layer 2

Layer 3

q i or nσi or Cσi

FIG. 8.14 Multilayer surface adsorption model

The cooperative adsorption model so far is applicable to both chemisorptions and physisorptions, as no assumption has been made on the type of interactions. What happens if more than one layer of adsorbate can truly be adsorbed on the surface? This would lead to pure physical adsorptions, since the interaction between the adsorbate molecules and the adsorbent surfaces becomes negligible on layers above the adsorbed molecules. If we summarize to this point, chemisorption is a chemical interaction between the adsorbate and the surface. The heats and activation energies of chemisorption are typical of those of a chemical reaction, and that is exactly what it is: a chemical reaction, albeit three-dimensional (in the bulk-fluid phase) on one side of the arrow, and two-dimensional (on the adsorbent surface) on the other side. The activation energies are such that the species involved have sufficient energy to cross the activation energy barrier at temperature levels that are experimentally accessible and of practical importance. One can imagine that physical adsorption does things a bit differently from chemisorption. The analysis of physical adsorption can be complicated, even more so than for chemisorption, because attractive-repulsive interactions are involved that may be more complex than direct chemisorptive interactions. Nevertheless, for adsorption to occur, the adsorbate molecule has to “collide” with the center where adsorption is to take place. Therefore, Eq. (8.20) applies equally well to physisorptions. The analysis of physical adsorption in general, and that used to approach this particular problem, derives from a classification later summarized by Brunauer (1945). He classified the isotherms for the physical adsorption of gases on surfaces into five general types, as shown in Fig. 8.15. These are the five types of cooperative adsorption. Isotherm I is readily recognizable as a typical Langmuir isotherm. Isotherms II-V show the various complexities of physical adsorption. Early work showed that that isotherm II was typical for the adsorption of nitrogen (at liquid nitrogen temperatures) on a large number of porous adsorbents. The result of that observation led to the derivation of an appropriate analytical theory to describe this type of adsorption in terms of the internal surface area of the adsorbent (Brunauer et al., 1938). From the last initials of the authors comes the name “BET theory.”

398

8. CHEMICAL REACTIONS ON SOLID SURFACES

Type II CSA

0

CSA

CSA

Type I

p°A

pA

0

0

pA

0

p°A

0

pA

0

p°A

Type V

Type IV CSA

0

Type III

CSA

0

pA

p°A

0

0

pA

p°A

FIG. 8.15 Classification of cooperative adsorption isotherms based on the adsorption of a gas on a solid material. The horizontal axes are the partial pressure of the adsorbate gas, and the vertical axes are the amount/concentration of gas adsorbed on the surfaces. p°A is the “maximum” vapor pressure of A or saturation vapor pressure of A. For the adsorption of a liquid, the horizontal axis would be replaced by concentration.

It is reasonable that the multilayer cooperative adsorption model can express all five types of adsorption isotherms (Liu, 2014). 8.3.4.1 Cooperative Adsorption of Single Species We start with a single adsorbate molecule approaching one active site on the surface, k1

σ + A

!

σ  A

(8.53)

k1

which is the usual adsorption stoichiometry. Here k1 is the adsorption rate constant (forward reaction) and k1 is the desorption rate constant (the reverse reaction). If multiple layers of adsorbate can be adsorbed onto one single active site, the adsorption can occur only one layer at a time; ie, the layers of adsorbate are added only one layer at a time on any given active site. The stoichiometry for the last layer on any given active site can be represented by k1

σ  Ai1 + A

!

σ  Ai k1

81  i  N

(8.54)

where N is the total number of layers of adsorbate that can be adsorbed on to the surface, ki denotes the adsorption rate constant and ki denotes the desorption rate constant for the i-th layer.

8.3 ADSORPTION AND DESORPTION

399

Let us focus on each individual active site that already has i–1 layers of adsorbates. Assuming elementary reactions, Eq. (8.54), the adsorption rate for the i-th layer can be represented by ri ¼ ki ½σ  Ai1 CA  ki ½σ  Ai 

81  i  N

(8.55)

where [σAi] is the concentration of active sites that have i-layers of adsorbate molecule A adsorbed and CA is the adsorbate concentration in the bulk-fluid phase. In Liu (2014), the forward (adsorption) rate constants were assumed to be the same except for the first layer. Based on collision theory, there should not be any difference for the adsorbate molecule approaching the different “layers.” Therefore, we assume here that k1 ¼ k 2 ¼ ⋯ ¼ k i ¼ ⋯ ¼ k N ¼ k A

(8.56)

while allowing the reverse (desorption) rate constants to differ due to the different levels of interaction. At equilibrium, the net adsorption rate for each layer is zero, ie, ri ¼ 0, and thus Ki ¼

ki ½σ  Ai  ¼ ki ½σ  Ai1 CA

81  i  N

(8.57)

In theory, the affinity Ki for active site to possess i-layers of adsorbate can be different (ie, Ki’s have different values for different layer). For physisorption, one can imagine that the affinity on the first layer is different from the rest of the layers, as on the first layer, adsorbate molecules interact with the adsorbent (active site), and on the rest of the layers, only adsorbate-adsorbate interactions exist. Therefore, there are two values for the affinity only for physisorption in true multilayer adsorption. In actuality, multilayer adsorption is a result of adsorbate-adsorbate-adsorbent interactions. In most applications, one can imagine a change of affinity from layer to layer, at least for the first two to three layers. The change in affinity may become negligible on the “upper layers.” Therefore, without loss of generality, let us consider K1 ¼ KA

(8.58)

K2 ¼ c1 KA

(8.59)

Ki ¼ c2 KA

83  i  N

(8.60)

In other words, the reverse (or desorption) rate constants are the same on the “top” layers, and differ on the first two layers only, kA KA

(8.61)

kA c1 KA

(8.62)

k1 ¼ k2 ¼ ki ¼

kA c2 K A

83  i  N

(8.63)

To determine the isotherm of adsorption, we start by determining the concentrations of active sites that have adsorbate molecule attached. Eq. (8.57) can be rewritten as

400

8. CHEMICAL REACTIONS ON SOLID SURFACES

½σ  A ¼ KA ½σCA

(8.64)

2 ½σ  A2  ¼ c1 KA ½σ  ACA ¼ c1 KA ½σC2A

(8.65)

i ½σ  Ai  ¼ c1 c2i2 KA ½σCiA

82  i  N

(8.66)

And active site balance leads to Cσ ¼ ½σ +

N X

2 ½σ  Ai  ¼ ½σ + KA ½σCA + c1 KA ½σC2A

i¼1

N X

ðc2 KA CA Þi2

(8.67)

i¼2

leading to ½σ ¼

Cσ N1 2 C2 1  ðc2 KA CA Þ 1 + KA CA + c1 KA A 1  c2 KA CA

½σ  A  ¼

½σ  Ai  ¼

Cσ KA CA N1 2 C2 1  ðc2 KA CA Þ 1 + KA CA + c1 KA A 1  c2 KA CA i i Cσ c1 c2i2 KA CA

N1 2 C2 1  ðc2 KA CA Þ 1 + KA CA + c1 KA A 1  c2 KA CA

82  i  N

(8.68)

(8.69)

(8.70)

where Cσ is the total active site concentration (or the saturation concentration of monolayer coverage). The total adsorbed adsorbate molecules can be computed by " # c1 1  ð1 + N  Nc2 KA CA Þðc2 KA CA ÞN 1+ 1 N X c2 ð1  c2 KA CA Þ2 i½σ  Ai  ¼ Cσ KA CA (8.71) CSA ¼ N1 i¼1 2 C2 1  ðc2 KA CA Þ 1 + KA CA + c1 KA A 1  c2 KA CA When N ¼ 1, the multilayer adsorption isotherm Eq. (8.71) is reduced to CSA ¼

KA CA 1 + KA CA

(8.72)

which is the Langmuir adsorption isotherm, identical to Eq. (8.29). When N ! 1, Eq. (8.71) is reduced to   c1 c1 + 1 ð1  c2 KA CA Þ2 Cσ KA CA c2 c2 CSA ¼ (8.73) 2 C2 1  c2 KA CA 1 + ð1  c2 ÞKA CA + ðc1  c2 ÞKA A Eq. (8.73) is the cooperative adsorption isotherm for single-species adsorption on solid surfaces. It is applicable to both chemisorption and physisorption.

8.3 ADSORPTION AND DESORPTION

401

Eq. (8.73) is applicable for physisorptions where adsorption is not dependent on the interaction between “marked” active centers on the solid surface and the adsorbate molecules. It is similar to the condensation of gaseous molecules onto a solid surface. When gaseous molecules condense on the surface, the top layers exhibit a phase equilibrium between the adsorbed phase (or condensed phase) and the gaseous phase as if the adsorbent was absent, c2 KA CA ¼

pA p0A

(8.74)

where p0A is the vapor pressure of A on the adsorbed phase. Substituting Eq. (8.74) into Eq. (8.73), we obtain   pA pA 2 Cσ 0 c1 + ðc2  c1 Þ 1  0 pA pA (8.75) CSA ¼  2 pA p A 1  0 c22 + c2 ð1  c2 Þ + ðc1  c2 Þ pA0 pA pA p0A Eq. (8.75) is important in that it can be reduced to the well known BET equation. By setting c1 ¼ c2 ¼

1 c

(8.76)

Eq. (8.75) is reduced to pA Cσ c 0 p  A  CSA ¼  pA pA 1  0 1  ð1  cÞ 0 pA pA

(8.77)

which is the BET isotherm. When CA ! 1, Eq. (8.71) is reduced to CSA ¼ NCσ

(8.78)

ie, the amount of adsorbate at saturation is identical to the total number of adsorption layers multiplied by the total available adsorption sites. Therefore, Eq. (8.71) can also be written as " # c1 1  ð1 + N  Nc2 KA CA Þðc2 KA CA ÞN 1+ 1 c2 ð1  c2 KA CA Þ2 CSA1 KA CA (8.79) CSA ¼ N 1  ðc2 KA CA ÞN1 2 2 1 + KA CA + c1 KA CA 1  c2 KA CA To this end, we have derived the most general adsorption isotherm equation for a single species. Example 8.2: Adsorption of Water Vapor on Silica Gel Experimental sorption data given by Knaebel (2009) for water vapor on silica gel at 25°C is shown in Table E8.2.1. Perform data fit to Langmuir isotherm and the cooperative adsorption isotherm. Comment on your solutions.

402

8. CHEMICAL REACTIONS ON SOLID SURFACES

TABLE E8.2.1 Adsorption Isotherm Data for Water Vapor on Silica Gel at 25°C. Relative Humidity, p/p°

Loading, mol/L

0

0

0.0116

1.73

0.0198

2.23

0.0378

3.53

0.06

4.84

0.133

8.68

0.177

11.2

0.18

11.82

0.218

13.67

0.238

15.26

0.241

15.59

0.256

17.08

0.278

17.4

0.279

17.94

0.292

20.03

0.307

20.43

0.35

23.45

0.369

25.28

0.526

30.97

0.539

32.04

0.649

33.03

0.697

33.32

0.758

33.87

0.825

34.11

0.825

34.15

Taken from Knaebel, K.S. 2009. 14 Adsorption. In: Albright, L.F. (Ed.), Albright’s Chemical Engineering Handbook. CRC Press, Taylor & Francis Group, Boca Raton, FL.

Solution We first examine the quality of fit for the Langmuir model, as it is the simplest of all the isotherms with a theoretical basis. Eq. (8.29) can be rearranged to give CSA ¼ CSA1

αpA =p∘A 1 + αpA =p∘A

(E8.2.1)

403

8.3 ADSORPTION AND DESORPTION

Table E8.2.2 shows the results of fitting the Langmuir model, Eq. (E8.2.1) to the experimental data. To show the visual appearance of the fit, Fig. E8.2 shows the experimental data (as circles) and the Langmuir model fit (as a solid line). One can observe that the fit is reasonably good; however, there is noticeable deviation at high surface coverage.

TABLE E8.2.2

Adsorption Isotherm Data for Water Vapor on Silica Gel at 25°C and Ideal Isotherm Fits

Relative humidity, p/p°

Loading, C0 SA, mol/L

Langmuir, Eq. (E8.2.1), CSA, mol/L

Cooperative isotherm, Eq. (8.79)

0

0

0

0

0.0116

1.73

1.029687

1.651855

0.0198

2.23

1.739614

2.500308

0.0378

3.53

3.248241

3.843479

0.06

4.84

5.020141

4.996589

0.133

8.68

10.24104

8.23079

0.177

11.2

13.00431

10.84518

0.18

11.82

13.18352

11.04757

0.218

13.67

15.36051

13.81845

0.238

15.26

16.4412

15.37881

0.241

15.59

16.59966

15.6155

0.256

17.08

17.37825

16.80091

0.278

17.4

18.48034

18.52063

0.279

17.94

18.52935

18.59766

0.292

20.03

19.15819

19.586

0.307

20.43

19.86512

20.69024

0.35

23.45

21.78792

23.5775

0.369

25.28

22.5918

24.70405

0.526

30.97

28.34119

30.79731

0.539

32.04

28.75603

31.10299

0.649

33.03

31.9647

32.97331

0.697

33.32

33.21535

33.5036

0.758

33.87

34.6923

34.01786

0.825

34.11

36.18538

34.43575

0.825

34.15

36.18538

34.43575 Continued

404

8. CHEMICAL REACTIONS ON SOLID SURFACES

TABLE E8.2.2

Adsorption Isotherm Data for Water Vapor on Silica Gel at 25°C and Ideal Isotherm Fits—cont’d

Relative Loading, C0 SA, humidity, p/p° mol/L vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u 24  u CSA, i  C0SA, i t σL ¼ i¼1 24  1

Langmuir, Eq. (E8.2.1), CSA, mol/L

Cooperative isotherm, Eq. (8.79)

1.536587

0.451881

KA

1.2773

19.29

c1

0.000703

c2

2.5608

CSA1

70.52 mol/L

49.0275

N

1

4

The parameters are shown in the bottom rows. Taken from Knaebel, K.S. 2009. 14 Adsorption. In: Albright, L.F. (Ed.), Albright’s Chemical Engineering Handbook. CRC Press, Taylor & Francis Group, Boca Raton, FL.

40 Cooperative, n = 4 Langmuir isotherm

CSA, mol/L

30

20

10

0 0.0

0.2

0.4 0.6 Relative humidity

0.8

1.0

FIG. E8.2 Fit of ideal isotherm models to the water vapor adsorption on silica gel at 25°C. The symbols are experimental data taken from Knaebel (Knaebel, K.S. 2009. 14 Adsorption. In: Albright, L.F. (Ed.), Albright’s Chemical Engineering Handbook. CRC Press, Taylor & Francis Group, Boca Raton, FL.), and the lines are fits to Eq. (8.79).

8.3 ADSORPTION AND DESORPTION

405

We now examine the quality fit can be achieved from the more general multilayer isotherm. Eq. (8.79) can be rearranged to give 2 3   N 0 pA 0 pA c 1  1 + N  Nc α α 2 2 7 p∘A p∘A c1 6 6 7 1+ 6  17  2 5 c2 4 0 pA 1  c2 α ∘ pA pA (E8.2.2) CSA ¼ CSA11 α0 ∘  0 N1 pA pA  1  c α ∘ 2 p 2 pA 0 p A 1 + α0 ∘ + c1 α pA∘ pA A pA 1  c2 α 0 ∘ pA Table E8.2.2 shows the results of fitting the general multilayer isotherm model, Eq. (E8.2.2) to the experimental data. The parameters obtained are: N ¼ 4; α0 ¼ 19.29; c1 ¼ 0.000703; c2 ¼ 2.5608; CSA11 ¼ 9.0275 and the variance is small; σL ¼ 0.451881. One can observe from Fig. E8.2 that the Langmuir isotherm fits reasonably well, but systematic deviation is visible. There is noticeable deviation at high surface coverage. The solid line or the cooperative multilayer isotherm fits the data very well, even with the detailed changes in localized regions, compared to the Langmuir fit. Further observations from Table E8.2.2 include that the Langmuir isotherm fit to the data is poor and the cooperative isotherm model Eq. (8.79) fits the data quite well. In this case, the cooperative adsorption model indicates that the adsorption can be expressed as four-layer adsorption, which is intuitively reasonable for adsorptions whose interactions are dominated by adsorbateadsorbent interactions. The adsorption rates have been established for single-solute (or species) multilayer adsorption in general, as Eqs. (8.55), (8.56), and (8.61) through (8.63). To describe the kinetic behavior of adsorption in batch processes, one needs to consider both mass transfer and adsorption. Fig. 8.4 illustrates the adsorption process. Figs. 8.16 through 8.18 show the quality of the cooperative adsorption isotherm and kinetic behaviors by Liu (2015). Fig. 8.16 shows that the adsorption isotherm data of Koubaissy et al. FIG. 8.16 Adsorption isotherm of phenol on HBEA zeolite at 20°C. Reprinted with permission from Liu, S. 2015. Cooperative adsorption on solid surfaces. J. Colloid Interface Sci. 450, 224–238.

40

CSA, mg/g

30

20 Data of Koubaissy et al. (2012) 3-Layer adsorption Langmuir fit Langmuir fit with data near saturation

10

0 0.0

0.2

0.4

0.6

0.8 CA, g/L

1.0

1.2

1.4

1.6

8. CHEMICAL REACTIONS ON SOLID SURFACES

40

40

30

30 Data of Koubaissy et al. (2012) ds = 0.6–0.7 mm

20

20

dS = 0.2–0.4 mm

CSA, mg/L

CSA, mg/L

406

dS = 0.05–0.07 mm kp0 = 0.000978; kc = 0.608

10

10

kp0 = 0.001158; kc = 0.748 kp0 = 0.001476; kc = 0.816 0 0

10

20

30

40

50

Time, min

0 100 150 200 250 300 350 Time, min

FIG. 8.17 Adsorption dynamics of phenol on HBEA zeolite at 20°C with different zeolite particle-size suspensions. Reprinted with permission from Liu, S. 2015. Cooperative adsorption on solid surfaces. J. Colloid Interface Sci. 450, 224–238.

1.0

1.0

0.8

0.8

Layer 1

0.6

qi

qi

0.6

0.4

0.4

Layer 2

0.2

0.2

Layer 3

0.0

0.0 0

2

4

6 Time, min

8

10

100 Time, min

FIG. 8.18

Surface coverage as a function of time for phenol on HBEA zeolite at 20°C. Reprinted with permission from Liu, S. 2015. Cooperative adsorption on solid surfaces. J. Colloid Interface Sci. 450, 224–238.

(2012) for phenol on HBEA zeolite at 20°C fit well with Eq. (8.79) when Cσ ¼ 14.018 mg/g, KA ¼ 3.376 L/g, c1 ¼ 0.1663, c2 ¼ 5.2964, and N ¼ 3. Fig. 8.17 shows the dynamic behavior of phenol adsorption on HEBA zeolite as given by Koubaissy et al. (2012) for three different sorbent particle sizes. One can observe that as particle sizes decrease, the adsorption rate increases. The data fit well to the multilayer adsorption kinetic model of the same isotherm parameters: CSA11 ¼ 14.018 mg/g, KA ¼ 3.376 L/g, c1 ¼ 0.1663, c2 ¼ 5.2964, and N ¼ 3. Only three additional parameters are needed for the

407

8.3 ADSORPTION AND DESORPTION

TABLE 8.2 Mass Transfer Parameters for the Multilayer Adsorption Kinetic Model Fit in Fig. 8.17. Isotherm parameters are: CAS11 ¼ 14.018 mg/g, KA ¼ 3.376 L/g, c1 ¼ 0.1663, c2 ¼ 5.2964, and N ¼ 3. The adsorbate rate constant is: kA ¼ 0.0178 min1 HEBA Zeolite Particle Size, mm

kc, min21

kp0, min21

Standard Deviation of Fit, σ

Correlation Coefficient, R2

0.6–0.7

0.608

0.000978

1.6104 mg/L

0.9753

0.2–0.4

0.748

0.001158

1.0219 mg/L

0.9899

0.05–0.07

0.816

0.001476

0.5765 mg/L

0.9965

dynamic behavior: kA, kc, and kp0 of which the last two parameters are mass transfer or particle size-dependent. One can observe that the lines fit the data reasonably well. The parameters are shown in Table 8.2. One can observe from Fig. 8.17 that the adsorption occurs very rapidly in the beginning, and then slows down, a typical behavior of multilayer adsorptions. Fig. 8.18 shows the coverage of each layer for the case of HEBA zeolite with particle sizes between 0.05–0.07 mm. One can observe that the first layer is quickly covered to a large extent, while the top layer, layer 3, is covered very slowly. 8.3.4.2 Cooperative Competitive Adsorption The cooperative adsorption model in Section8.3.4.1 can be extended readily to competitive multiple species adsorptions. Assume that the interactions among adsorbate species are negligible; the stoichiometry can be written based on Eq. (8.53), as k1j

σ + Aj

!

σ  Aj

(8.80)

k1j

where the subscript j denotes for the species j. When multiple layers of adsorbate molecules are adsorbed onto one single active site, the stoichiometry can be written as kij

σ  L1 ⋯Li1 + Aj

!

σ  L1 ⋯Li1 Aj 81  i  N

(8.81)

kij

where Li stands for (adsorbed) layer i, and L can be any adsorbate species. Thus, σL1⋯Li1Aj in reaction (Eq. 8.81) can also be represented by σL1⋯Li; however, σL1⋯Li is more general than σL1⋯Li1Aj. The net rate of adsorption for species j on to the i-th layer is given by

rji ¼ kji ½σ  L1 ⋯Li1 CAj  kji σ  L1 ⋯Li1 Aj 81  i  N (8.82) where [σL1⋯Li] is the concentration of active sites that have i-layers of adsorbate molecules adsorbed, [σL1⋯Li1Aj] is the concentration of active sites that have i-layers of adsorbate molecules with species j being adsorbed on the last layer, and CAj is the adsorbate Aj concentration in the bulk-fluid phase. At equilibrium, the net adsorption rate is zero and thus the affinity of the species j on the i-th layer is defined by

408

8. CHEMICAL REACTIONS ON SOLID SURFACES



σ  L1 ⋯Li1 Aj kji Kji ¼ ¼ kji ½σ  L1 ⋯Li1 CAj

81  i  N

(8.83)

Assuming the affinities change only in the first three layers and are not affected by other adsorbate species, we let Kj1 ¼ Kj

(8.84)

Kj2 ¼ c1 Kj

(8.85)

Kji ¼ c2 Kj

83  i  N

(8.86)

The adsorption on the i-th layer is given by ½σ  L1 ⋯Li1 Li  ¼

NS NS X X

Cσj σ  L1 ⋯Li1 Aj ¼ ½σ  L1 ⋯Li1  Kji CAj 81  i  N C j¼1 j¼1 σ

(8.87)

and the adsorption of species j on each layer is given by

σ  L1 ⋯Li1 Aj Liþ1 ⋯ ¼

Cσj Kji CAj NS X

½σ  L1 ⋯Li1 Li Liþ1 ⋯ 81  i  N

(8.88)

Cσm Kmi CAm

m¼1

Thus the total adsorbed species j can be computed by CSj ¼ Cσj

N X Kj Cj Kj Cj ½ σ  L  þ C i½σ  L1 ⋯Li  1 σj NS NS X X i¼2 Cσl Kl Cl Cσl Kl Cl l¼1

(8.89)

l¼1

Let
c2 ¼

NS X cσj j¼1

Eq. (8.89) leads to

CSj ¼ Cσj Kj Cj

cσ

Kj Cj

" # c1 1  ð1 + N  Nc2
cÞ
cN 1+ 1 c2 ð1  c2
cÞ2 1 +
c +
c1
c2

1  ðc2
cÞN1 1  c2
c

(8.90)

(8.91)

which is the isotherm equation for competitive cooperative adsorptions. When NS ¼ 1, Eq. (8.91) reduces to Eq. (8.79). Therefore, we have obtained the general competitive multilayer adsorption isotherm expression, Eq. (8.91).

8.3 ADSORPTION AND DESORPTION

409

8.3.5 Common Empirical Approximate Isotherms While our discussions in ideal (Langmuir) and nonideal adsorption isotherms have led to more general adsorption isotherms that could be applicable to a variety of surfaces, we shall make a detour and mention a couple of simple empirical isotherms. These isotherms were thought of as accounting for nonideality of the adsorbent surfaces. They are approximations to a variety of ideal and nonideal isotherms. As such, their utilities are more restrictive than the above isotherms. The reason we introduce these isotherms is that there are a few classic applications that have utilized these simple empirical isotherms. In the low coverage region well before the coverage levels off as the bulk concentration is increased, the Langmuir isotherm (Eq. 8.29), as well as the generalized logarithm coverage Eq. (8.52), can be approximated by θA ¼ cCm A

(8.92)

In addition, one can note that dissociative adsorption (Eq. 8.34) can be approximated by Eq. (8.52) as well in the low-coverage region. This versatile approximation, Eq. (8.92), is called the Freundlich isotherm. In the intermediate coverage region, K
A CA << 1 << K
A CA efT Eq. (8.52) may be approximated by

  ln K
A CA efT θA  fT

(8.93)

which is the Temkin isotherm. Eq. (8.93) correlates experimental data better in the midrange of surface coverage and is best known for the adsorption of hydrogen and nitrogen in ammonia synthesis reactions. To show the closeness of the Freundlich isotherm and Temkin isotherm to the UniLan isotherm, Fig. 8.19 shows the UniLan isotherm with Emax ¼ 5 RT together with Temkin isotherm with identical parameters. One can observe that Temkin approximation, Eq. (8.93), is reasonably close to the original UniLan isotherm if KACA 2 (0.01, 1), which is less accurate than the Freundlich isotherm, θA ¼ 0.9128(KACA)0.2636; however, when the Temkin isotherm  is regarded as an empirical model, its correlation in KACA 2 (0.1, 10), θA ¼ 10:8041 ln 0:14212K
A CA e10:804 , showed a reasonable approximation to the UniLan isotherm in this region. Therefore, both Freundlich and Temkin isotherms can be applied to correlate adsorption isotherms. Example 8.3: Temkin and Freundlich Isotherms Recorrelate the adsorption data in Example 8.1, Table E8.1.1, with Temkin and Freundlich isotherm models. Compare the quality of the fits with that of the Langmuir isotherm fit. Solution We first look at the power-law model (Freundlich isotherm). Substituting Eq. (8.38) into Eq. (8.92), we obtain CSA ¼ c0 Cm A

(8.94)

410

8. CHEMICAL REACTIONS ON SOLID SURFACES

1.4 UniLan Temkin Freundlich, for KACA = [0, 1) Temkin, correlation for KACA = (0.1, 10)

1.2 1.0

qA

0.8 0.6 0.4 0.2 0.0 0.01

0.1

1

10

KACA

FIG. 8.19 Approximation of a UniLan isotherm (fT ¼ 5) by Temkin and Freundlich isotherms. The Freundlich isotherm correlation is for KACA 2 (0, 1): θA ¼ 0.9128(KACA)0.2636, and the Temkin isotherm correlation is for KACA 2 (0.1, 10): θA ¼ 10:8041 ln 0:14212K
A CA e10:804 .

Therefore, we can fit the experimental data with Eq. (8.94) to check if the model can reasonably describe the experimental data. Since the power-law model is not valid at high coverages, we perform the least square fits on a spreadsheet program either with the highest coverage point removed or kept. The solutions are illustrated in Tables E8.3.1 and E8.3.2. TABLE E8.3.1

Least Square Parameter Estimation for the Freundlich Isotherm

Aqueous Solution CA, mg/L 0

On Activated Carbon C0 SA, mg/g 0

Eq. (8.94), CSA, mg/g 0

Error2 (CSA 2 C0 SA)2 0

1.5

53

62.33299

87.10465

2

62.4

67.38618

24.86198

5.1

82.8

86.8411

16.33046

6.6

105

93.12478

141.0209

22.9

137

130.4549

42.83818

51.6

172.2

162.5764

92.61345

80.2

170.9

183.2113 151.5683 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u 8  u CSA, i  C0SA, i t ¼ 8:914979 σL ¼ i¼1 81

The final solution shown is when c0 ¼ 55.85 (mg/L)m  mg-phenol/g-activated carbon and m ¼ 0.271, which were obtained by minimizing the variance between the Freundlich model and the experimental data, σL, while changing c0 and m.

411

8.3 ADSORPTION AND DESORPTION

TABLE E8.3.2

Least Square Parameter Estimation for the Freundlich Isotherm With the Highest Coverage Data Point not Accounted for in Data-Fitting

Aqueous Solution CA, mg/L 0

On Activated Carbon C0 SA, mg/g

Eq. (8.94), CSA, mg/g

0

0

1.5

53

59.11624

2

62.4

64.56151

5.1

82.8

85.99851

Error2 (CSA2C0 SA)2 0 37.40836 4.672121 10.23046

6.6

105

93.0651

142.4419

22.9

137

136.2292

51.6

172.2

80.2

170.9

174.7157 6.328751 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u 7  u CSA, i  C0SA, i t ¼ 5:79764 σL ¼ i¼1 71

0.594159

The final solution shown is when c0 ¼ 52.21 (mg/L)m  mg-phenol/g-activated carbon and m ¼ 0.306, which were obtained by minimizing the variance between the Freundlich model and the experimental data, σL.

One can observe from Table 9.3.1 that the Freundlich model fit is not as good as the Langmuir fit as shown in Example 9.1. The deviation of the model with the experimental data is higher at the high coverage region. Fig. E8.3.1 shows the visual quality of the fit. Visual inspection of the data shown in Fig. E8.3.1 reinforces our assessment based on Table E8.3.1.

200

CAs, g/kg

150

100

50 Langmuir isotherm Freundlich isotherm Freundlich isotherm, last point removed 0 0

20

40

60

80

100

CA, g/m3

FIG. E8.3.1 Freundlich isotherm model fit as compared with the Langmuir isotherm fit for the adsorption of phenol on activated carbon.

412

8. CHEMICAL REACTIONS ON SOLID SURFACES

One can observe from Table E8.3.2 and Fig. E8.3.1 that the Freundlich model fit with the highest coverage data removed has improved quality of fit. Within the region of fit, the Freundlich isotherm fit and the Langmuir isotherm fit are of similar quality, ie, both within the apparent experimental data error. We next examine the Temkin isotherm fit to the same data. Substituting Eq. (8.38) into Eq. (8.93), we obtain (8.95) CSA ¼ fTm ln ðKAT CA Þ ΔHad + Emax Emax RT ∘ RT CAs1 and KAT ¼ K
A e RT ¼ KA e . Since the Temkin isotherm is only Emax valid in the intermediate coverage region, we remove the zero coverage data from consideration. Table E8.3.3 shows the least square fit using the Temkin isotherm to the experimental data. Visual representation of the fit as compared with Langmuir isotherm fit is shown in Fig. E8.3.2. One can observe from Table E8.3.3 and Fig. E8.3.2 that the Temkin model fit is consistent with the experimental data. Within the region of fit, the Temkin isotherm fit and the Langmuir isotherm fit are of similar quality, both within the apparent experimental data error.

where fTm ¼

8.3.6 Pore Size and Surface Characterization As we can conclude from the above discussions, the amount of adsorbate that can be accommodated is directly proportional to the available surface sites, or the surface area. To TABLE E8.3.3 Least square Parametric Estimation for the Temkin Isotherm Aqueous Solution, CA, mg/L 0

On Activated Carbon, C0 SA, mg/g

Eq. (8.95), CSA, mg/g

Error2 (CSA2C0 SA)2

0

1.5

53

52.49601

0.254001

2

62.4

61.53407

0.749841

5.1

82.8

90.94313

66.31056 35.48232

6.6

105

99.0433

22.9

137

138.1279

51.6

172.2

163.6504

80.2

170.9

1.272193 73.09481

177.5053 43.63024 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 8  2 u u CSA, i  C0SA, i t ¼ 6:066217 σL ¼ i¼2 71

The final solution shown is when fTm ¼ 31.42 mg-phenol/g-activated carbon and KAT ¼ 3.545 L/mg, obtained by minimizing the variance between the Temkin model and the experimental data, σL, while changing fTm and KAT.

413

8.3 ADSORPTION AND DESORPTION

200

CAs, g/kg

150

100

50 Langmuir isotherm Temkin isotherm 0 0

20

40

60

80

100

CA, g/m3

FIG. E8.3.2 Temkin isotherm model fit as compared with the Langmuir isotherm fit for the adsorption of phenol on activated carbon.

achieve high specific surface area (surface area per mass of solid adsorbent material), solid catalysts are commonly manufactured to be porous. Therefore, the structural characteristics of solid (catalyst) materials are important. Pore volume, pore size, and specific pore surface area are important parameters that dependent on the catalyst preparation procedures. These parameters are commonly determined experimentally. Complete coverage via physisorption is a preferred method of characterizing the surface area of catalyst. In this case, the BET isotherm, Eq. (8.77), can be applied to determine the surface area as nσ1 or monolayer coverage from experimental measurements on the total amount of molecules adsorbed as a function of pressure. The total surface area can be computed via aT ¼ nσ1 NAv aσ

(8.96)

where NAv is the Avogadro’s number, 6.0231023 mol1, and aσ is the projected area of an adsorbate molecule. Table 8.3 shows the parameters of commonly used adsorbate molecules. Pore volume and pore size are traditionally measured by mercury penetration. Because mercury is a nonwetting liquid (ie, contact angle φ > 90°), it is not capable of penetrating pores ˚ (or 1.5106 m) in diameter at atmospheric conditions. At high pressmaller than 150 000 A sure, mercury can be forced into fine pores as the surface tension σγ is overcome by external pressure p. The pore diameter DP and the pressure p at breakthrough is related by the Washburn equation or DP ¼ 

4σγ cos ϕ p

(8.97)

414

8. CHEMICAL REACTIONS ON SOLID SURFACES

TABLE 8.3 Properties of Commonly Used Adsorbates for Catalyst Surface Characterization Adsorbate N2

Adsorption Temperature, K 77.4

Saturation Vapor Pressure p°, MPa

Projected Molecular Area aσ, 10220 m2

0.10132

16.2

Kr

77.4

3.45610

Ar

77.4

0.03333

4

19.5 14.6

3

C6H6

293.2

9.87910

40

CO2

195.2

0.10132

19.5

CH3OH

293.2

0.01280

25

By measuring the amount of mercury penetration as the pressure is increased, we can determine the pore size distributions as well as the pore volume.

8.4 LHHW: SURFACE REACTIONS WITH RATE-CONTROLLING STEPS At the beginning of this chapter, it was stated that reaction sequences involving surface steps (for example, Eq. 8.17), could be visualized as a type of chain reaction. This is indeed so and it naturally leads to the utility of pseudosteady-state hypothesis (PSSH) derivations for kinetic rate simplifications; however, also associated with the development of the theory of surface reaction kinetics has been the concept of the rate-limiting or rate-controlling step. This presents a rather different view of sequential steps than pure chain reaction theory, since if a single step controls the rate of reaction, then all other steps must be at (rapid) equilibrium. This is a result that is not a consequence of the general PSSH. As we have learned that both PSSH and rapid equilibrium steps lead to similar asymptotic rate expressions, there are no clear advantages gained by using one or the other for a complicated reaction network. The Langmuir isotherm is generally applied in surface reaction analyses, which was formalized by Hinshelwood, Hougen, and Watson. The Langmuir adsorption isotherm-based solid catalysis is thus known today as being characterized by the Langmuir-Hinshelwood-Hougen-Watson (LHHW) kinetics. In fact, pursuing the example in Eq. (8.17) a bit further in this regard, if the surface reaction is rate-limiting, we can express the net rate of reaction directly in terms of the surface species concentrations, nσθA and nσθB: r ¼ ks nσ θA  ks nσ θB

(8.98)

where kS and kS are rate constants for the forward and reverse surface reaction steps of Eq. (8.15). This represents a considerable simplification from the normal chain reaction

8.4 LHHW: SURFACE REACTIONS WITH RATE-CONTROLLING STEPS

415

analysis, because the first and third steps (Eq. 8.14) and (Eq. 8.16) are at equilibrium and the surface species concentrations are entirely determined by their adsorption/desorption equilibrium on the surface. Substitution of Eq. (8.15) into Eq. (8.98) gives the rate of reaction in terms of the bulk-fluid phase concentrations of the reacting species. r ¼ ks nσ

KA ðCA  CB =KC Þ 1 + KA CA + KB CB

(8.99)

where KC ¼

K A ks KB ks

(8.100)

is the equilibrium constant. In fact, the value of nσ may be a somewhat elusive quantity, so that in practice, this is often absorbed into the rate constant as r¼k

CA  CB =KC 1 + KA CA + KB CB

(8.101)

The rates of catalytic reactions are usually expressed in terms of unit mass or unit total surface (not external surface) of the catalyst. This is understood because the active sites are proportional to the total surface area. As a second example of surface-reaction-rate control, consider the slightly more complicated bimolecular reaction on one single type of active site:

+) = overall

A + σ>A  σ

(8.102a)

B + σ>B  σ

(8.102b)

Aσ+Bσ!Cσ+Dσ

(8.102c)

C  σ>C + σ

(8.102d)

D  σ>D + σ

(8.102e)

A+B

C+D

(8.102f)

Let us assume that the surface reaction (Eq. 8.102c) is the rate-limiting step, whereas the adsorption-desorption steps are fast equilibrium steps (FESs). The adsorption step (Eq. 8.102a) is a fast equilibrium step, that is 0 ¼ raddes, A ¼ kA nσ θCA  kA nσ θA

(8.103)

which yields after rearrangement θA ¼ KA θCA

(8.104)

where KA ¼

kA kA

(8.105)

416

8. CHEMICAL REACTIONS ON SOLID SURFACES

is the Langmuir isotherm constant for species A on the catalyst surface. Similarly, one can write for B, C, and D (8.106) θB ¼ KB θCB θC ¼ KC θCC

(8.107)

θD ¼ KD θCD

(8.108)

There are four species adsorbed on the same type of active sites. Site balance leads to the summation of the fraction of vacant (ie, available sites) and the fractions of the sites occupied by A, B, C, and D being unity θ + θA + θB + θ C + θ D ¼ 1

(8.109)

Substituting Eqs. (8.104) and (8.106) through (8.108) into Eq. (8.109), we obtain after rearrangement θ¼

1 KA CA + KB CB + KC CC + KD CD

(8.110)

Since the surface reaction (Eq. 8.119c) is the rate-limiting step, the overall reaction rate for Eq. (8.119f) is identical to that for Eq. (8.119c) as the stoichiometric coefficients are the same for A, B, C, and D. That is, r ¼ rs ¼ ks n2σ θA θB

(8.111)

Substituting Eqs. (8.110), (8.104) and (8.106) into Eq. (8.111), we obtain r¼

ks n2σ KA KB CA CB ðKA CA + KB CB + KC CC + KD CD Þ2

(8.112)

It is clear from Eq. (8.111) that when a surface reaction step is rate-controlling, one needs only information concerning adsorption equilibria of reactant and product species in order to write the appropriate rate equation. Table 8.4 shows a collection of the simple LHHW rate expressions. As shown in Table 8.4, in each of these reactions, the active centers or sites form a closed sequence in the overall reaction. According to this view, when the stoichiometry between reactants and products is not balanced, the concentration of vacant active centers also enters the rate equation, Reaction 2 in Table 8.4. The bimolecular reaction involving only one single active center is shown as Case 7 in Table 8.4, and one extreme where one of the reactants is nonadsorbed is sometimes referred to as a Rideal or Eley-Rideal simplification. It is encountered primarily in interpretation of catalytic hydrogenation kinetics. In general, when a reactant or product is not adsorbed on the surface, this particular species is not to appear in the denominator of the rate expression if the surface reaction is the rate-limiting step. The rate-controlling steps other than the surface reaction on ideal surfaces are also shown in Table 8.4. If, for example, the rate of adsorption of a reactant species on the surface is slow compared to other steps, it is no longer correct to suppose that its surface concentration is determined by adsorption equilibrium. Rather, there must be chemical equilibrium among

417

8.4 LHHW: SURFACE REACTIONS WITH RATE-CONTROLLING STEPS

TABLE 8.4

Some Examples of LHHW Rate Expressions

Overall Reaction

Mechanism/Controlling Step

1. Isomerization: A > B

A + σ>A  σ A  σ>B  σ B  σ>B + σ

2. Decomposition (double site): A>B + C

A + σ>A  σ A  σ + σ>B  σ + C  σ B  σ>B + σ C  σ>C + σ

3. Decomposition (dual site): A>B + C

A + σ1 >A  σ1 A  σ1 + σ2 >B  σ1 + C  σ2

Rate Expression r ¼ kA Cσ

CA  CB =Keq 1 K C 1 + KB CB + Keq A B

r ¼ ks K A C σ r ¼ kB Cσ

Keq CA  CB 1 + KA CA + Keq KB CA

r ¼ kA Cσ

CA  CB CC =Keq 1 C C + K C + K C 1 + KA Keq B C B B C C

r ¼ ks KA C2σ

CA  CB CC =Keq ð1 + KA CA + KB CB + KC CC Þ2

r ¼ kB Cσ

Keq CA  CB CC KB Keq CA + ð1 + KA CA + KC CC ÞCC

r ¼ kC Cσ

Keq CA  CB CC KC Keq CA + ð1 + KA CA + KB CB ÞCB

r ¼ kA Cσ1

CA  CB CC =Keq 1 C C + K C 1 + KA Keq B C B B CA  CB CC =Keq ð1 + KA CA + KB CB Þð1 + KC CC Þ

B  σ1 >B + σ1

r ¼ ks KA Cσ1 Cσ2

C  σ2 >C + σ2

Keq CA  CB CC  r ¼ kB Cσ1 KB Keq CA + ð1 + KA CA ÞCC ð1 + KC CC r ¼ kC Cσ2

4. Bimolecular (double site): A + B>C + D

CA  CB =Keq 1 + KA CA + KB CB

A + σ>A  σ B + σ>B  σ A  σ + B  σ>C  σ + D  σ

r ¼ kA C σ

CA CB  CC CD =Keq 1 C C + ð1 + K C + K C + K C ÞC KA Keq C D B B C C D D B

r ¼ kB Cσ

CA CB  CC CD =Keq 1 C C + ð1 + K C + K C + K C ÞC KB Keq C D A A C C D D A

C  σ>C + σ D  σ>D + σ

Keq CA  CB CC KC Keq CA + CB

r ¼ ks KA KB C2σ

CA CB  CC CD =Keq ð1 + KA CA + KB CB + KC CC + KD CD Þ2

r ¼ kC Cσ

Keq CA CB  CC CD Keq KC CA CB + ð1 + KA CA + KB CB + KD CD ÞCC

r ¼ kD C σ

Keq CA CB  CC CD Keq KD CA CB + ð1 + KA CA + KB CB + KC CC ÞCD Continued

418

8. CHEMICAL REACTIONS ON SOLID SURFACES

TABLE 8.4 Some Examples of LHHW Rate Expressions—cont’d Overall Reaction 5. Bimolecular (dissociative): 1 A + B>C + D 2 2

Mechanism/Controlling Step A2 + 2 σ>2 A  σ B + σ>B  σ

Rate Expression r ¼ kA2 C2σ h

2 CA2 C2B  C2C C2D =Keq 1=2

1 C C + ð1 + K C + K C + K C ÞC KA2 Keq C D B B C C D D B

A  σ + B  σ>C  σ + D  σ C  σ>C + σ

i2

1=2

r ¼ kB C σ

1 C C KB Keq C D

D  σ>D + σ

CA2 CB  CC CD =Keq  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  + 1 + KA2 CA2 + KC CC + KD CD CA 1=2

1=2

r ¼ ks KA2 KB C2σ 

CA2 CB  CC CD =Keq pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 + KA2 CA2 + KB CB + KC CC + KD CD 1=2

r ¼ kC C σ

Keq CA2 CB  CC CD  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1=2 Keq KC CA2 CB + 1 + KA2 CA2 + KB CB + KD CD CC 1=2

r ¼ kD Cσ 6. Bimolecular (dual site): A + B>C + D

A + σ1 >A  σ1 B + σ2 >B  σ2 A  σ1 + B  σ2 >C  σ1 + D  σ2 C  σ1 >C + σ1 D  σ2 >D + σ2

7. Bimolecular (single site): A + B>C

A + σ>A  σ B + σ>B  σ B + A  σ>A  σ  B

CA CB  CC CD =Keq 1 C C + ð1 + K C ÞC KA Keq C D C C B

r ¼ kB Cσ2

CA CB  CC CD =Keq 1 C C + ð1 + K C ÞC KB Keq C D D D A

r ¼ ks KA KB Cσ1 Cσ2

Keq CA CB  CC CD Keq KC CA CB + ð1 + KA CA ÞCC

r ¼ kD Cσ2

Keq CA CB  CC CD Keq KD CA CB + ð1 + KB CB ÞCD

r ¼ kA Cσ

CA CB  CC =Keq 1 C + ð1 + K C + K C ÞC KA Keq C B B C C B

r ¼ kB C σ

CA CB  CC =Keq 1 C + ð1 + K C + K C ÞC KB Keq C A A C C A

r ¼ kAB Cσ

Keq CA CB  CC 1 K 1 C 1 + KA CA + KB CB + KC CC + KAB eq C

r ¼ ks KAB Cσ

A + σ>A  σ A  σ + B>C  σ C  σ>C + σ

CA CB  CC CD =Keq ð1 + KA CA + KC CC Þð1 + KB CB + KD CD Þ

r ¼ kC Cσ1

C  σ>C + σ

8. Bimolecular (Eley-Rideal): A + B>C (B not adsorbing)

Keq CA2 CB  CC CD  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  + 1 + KA2 CA2 + KB CB + KC CC CD

r ¼ kA Cσ1

A + B  σ>A  σ  B A  σ  B>C  σ

1=2 Keq KD CA2 CB

CA CB  CC =Keq 1 + KA CA + KB CB + KC CC + KAB CA CB

r ¼ kC C σ

Keq CA CB  CC 1 + Keq KC CA CB + KA CA + KB CB + KAB CA CB

r ¼ kA Cσ

CA CB  CC =Keq 1 C + ð1 + K C ÞC KA Keq C C C B

r ¼ ks KA Cσ r ¼ kC C σ

CA CB  CC =Keq 1 + KA CA + KC CC

Keq CA CB  CC 1 + KA CA + Keq KC CA CB

419

8.4 LHHW: SURFACE REACTIONS WITH RATE-CONTROLLING STEPS

all species on the surface, and the surface concentration of the species involved in the ratelimiting step is determined by this equilibrium. The actual concentration of the rate-limiting adsorbate consequently will not appear in the adsorption terms (denominator) of the rate equation, but is replaced by that concentration corresponding to the surface concentration level established by the equilibrium of other steps. Again consider the isomerization example of Eq. (8.1), this time in which the rate of adsorption of A is controlling. From Eq. (8.20), we may write the net rate of reaction as the difference between rates of adsorption and desorption of A (8.20) r ¼ kA CA θCσ  kA θA Cσ in which kA (or kad), and kA (or kdes) are adsorption and desorption rate constants EA

(ie, kA ¼ k∘A e RT ) and θA is the fraction of surface coverage by A as determined by the equilibrium of all other steps. Since the surface reaction (Eq. 8.15) is a rapid equilibrium step, we have 0 ¼ rs ¼ ks θA Cσ  ks θB Cσ

(8.113)

which leads to Ks ¼

ks θB ¼ ks θA

(8.114)

θB Ks

(8.115)

or θA ¼

The desorption of B (Eq. 8.16) is also a rapid equilibrium step, 0 ¼ rB, des-ad ¼ kB θB Cσ  kB CB θCσ

(8.116)

which gives θB ¼

kB θCB ¼ KB CB θ kB

(8.117)

The total active site balance leads to θ + θA + θB ¼ 1

(8.118)

Substituting Eqs. (8.115) and (8.117) into Eq. (8.118), we obtain after rearrangement θ¼

1 1 + KB CB + KS1 KB CB

(8.119)

and the fraction of sites occupied by A and B are given by θ B ¼ KB CB θ ¼

KB CB 1 + KB CB + KS1 KB CB

(8.120)

θA ¼ KS1 θB ¼

KS1 KB CB 1 + KB CB + KS1 KB CB

(8.121)

420

8. CHEMICAL REACTIONS ON SOLID SURFACES

Substituting Eqs. (8.119) and (8.121) into Eq. (8.113), we obtain r ¼ Cσ

1 kA CA  kA KS1 KB CB CA  k1 A kA KS KB CB ¼ kA C σ 1 1 1 + KB CB + KS KB CB 1 + KB CB + KS KB CB

(8.122)

At equilibrium, r ¼ 0, we have Keq ¼

CB,eq 1 ¼ 1 CA,eq kA kA KS1 KB

(8.123)

kA kA

(8.124)

KS KA KB

(8.125)

Also noting that KA ¼ we have Keq ¼ Eq. (8.122) can be rewritten as r ¼ kA Cσ

1 CA  Keq CB 1 K C 1 + KB CB + Keq A B

(8.126)

Eq. (8.126) can also be written as r ¼ kA Cσ

1 CA  Keq CB

1 + KB CB + KA CA

(8.127)

by noting that 1 CA ¼ Keq CB

(8.128)

as the equilibrium constant is not dependent on the nature of the surfaces. CA* is the virtual or equilibrium concentration of A (in the bulk-fluid phase with surface coverage). This is derivable because the concentration of A on the surface is in equilibrium with other components (or species) in the system, as the rest of the steps that govern the overall reaction are fast equilibrium steps. One can observe from Table 8.4 that it holds true for all other cases. Comparing Eq. (8.127) with Eq. (8.122), we note that θ¼

1 1 + KB CB + KA CA

(8.129)

KA CA 1 + KB CB + KA CA

(8.130)

and θA ¼

Thus, it resembles the Langmuir isotherm including species A, although the adsorption of A is not in equilibrium. The nonequilibrium of A in the isotherm is accommodated by the replacement of CA with the virtual concentration of A that is in equilibrium with other species

8.4 LHHW: SURFACE REACTIONS WITH RATE-CONTROLLING STEPS

421

in the system. Therefore, the virtual concentration may be applied to simplify the derivation for adsorption-controlled processes or provide a check on the final expression. Example 8.4: The reaction network of isomerization on a solid catalyst follows the following scheme

+) = overall

A + σ>A  σ

(E8.4.1)

A  σ>B  σ

(E8.4.2)

B  σ>B + σ

(E8.4.3)

A

(E8.4.4)

B

Discuss the relevance of LHHW kinetics based on detailed kinetics as the reaction is performed in a batch reactor. Treat the reaction network via (a) microkinetics (ie, full solutions); (b) using LHHW approximations, ie, rate-limiting step assumptions; and (c) PSSH on intermediates. Solution This is the simplest case of surface catalyzed reactions shown in Table 8.4. As we have discussed in detail the derivation of LHHW kinetic expressions, we now go back to the detailed reaction network analysis before turning to the LHHW simplifications (assumptions). (a) Microkinetics (full solutions) As the reaction rate expression follows stoichiometry for elementary steps, reactions E8.4.1–E8.4.3 r1 ¼ kA CA θCσ  kA θA Cσ

(E8.4.5)

r2 ¼ kS θA Cσ  kS θB Cσ

(E8.4.6)

r3 ¼ kB θB Cσ  kB CB θCσ

(E8.4.7)

Mole balances of all the species in the batch reactor lead to dCA ¼ kA ðCA θ  θA =KA ÞCσ dt

(E8.4.8)

dCB ¼ kB ðθB =KB  θCB ÞCσ dt

(E8.4.9)

dθA ¼ kA ðθCA  θA =KA Þ  kS ðθA  θB KA =KB =KC Þ dt

(E8.4.10)

dθB ¼ kS ðθA  θB KA =KB =KC Þ  kB ðθB =KB  θCB Þ dt

(E8.4.11)

θ ¼ 1  θA  θB

(E8.4.12)

where KA ¼

kA kA

(E8.4.13)

422

8. CHEMICAL REACTIONS ON SOLID SURFACES

kB kB

(E8.4.14)

kS KA  kS KB

(E8.4.15)

KB ¼ KC ¼

KA and KB are adsorption equilibrium constants of A and B, respectively, and KC is the equilibrium constant. The five Eqs. (E8.4.8–E8.4.12) can be solved simultaneously, together with the initial conditions (in the reactor at time 0) CA ¼ CA0 , Cσ , CB ¼ 0, θ ¼ 1, θA ¼ 0, θB ¼ 0 when t ¼ 0 We shall use an integrator to show how this problem can be solved on a computer. In this case, we use ODexLims to integrate Eqs. (E8.4.8–E8.4.11). Let y1 ¼ CA , y2 ¼ CB , y3 ¼ qA , y4 ¼ qB , and the independent variable x ¼ t: y1 ¼ CA , y2 ¼ CB , y3 ¼ qA , y4 ¼ qB , and the independent variablex ¼ t: The kernel functions can then be input into the visual basic module as shown in Fig. E8.4.1. The setup and solution on an Excel worksheet for one case (one set of parameters) is shown in Fig. E8.4.2. Some of the results for selected values of the parameters are shown in Figs. E8.4.3–E8.4.5. We first examine a case where the surface reaction rate is limiting. For this case, we scale the time with the surface reaction rate constant. The solution when CA0 ¼ 10 Cσ, kA Cσ ¼ 10 kS, KA ¼ 1/Cσ, KC ¼ 4, KB ¼ 1/Cσ and kB Cσ ¼ 10 kS are shown in Fig. E8.4.3. One can observe that in the batch operation, the concentration of the reactant in the bulk-fluid phase experiences a quick drop before product formation is observed (Fig. E8.4.3A and C). The decrease is simply due to the adsorption of reactants on the catalyst surface, and thus is dependent on the

FIG. E8.4.1

Visual basic module showing the integral kernels for Example 8.4.

FIG. E8.4.2

Excel worksheet for Example 8.4.

10

1.0

9

0.9

8

0.7

6

0.6

5

0.5

qj

Cj / Cσ

0.8

CA

7

0.4

4 3

0.2

1

0.1

0

0.0 0

1

2

3

(A)

qB

0.3

CB

2

4

5

6

7

8

9

q 0

10

3

4

5

6

7

8

9

10

kSt 0.9

CA

8

0.8 0.7

6

0.6

5

0.5

qj

7

4

0.4

3

0.3

2

qA

qB

0.2

CB

1

(C)

2

1.0

9

0 0.01

1

(B)

kSt 10

Cj / Cσ

qA

0.1

q

0.1 1

kSt

10

100

0.0 0.01

(D)

0.1

1

10

100

kSt

FIG. E8.4.3 Variations of concentrations in the bulk phase (A, C), and surface coverages (B, D) as functions of time when kA Cσ ¼ 10 kS; KA ¼ 1/Cσ; KC ¼ 4; KB ¼ 1/Cσ; and kB Cσ ¼ 10 kS.

424

8. CHEMICAL REACTIONS ON SOLID SURFACES

amount of catalyst or total amount of adsorption sites. Since the product B is also adsorbed on the surface, the coverage of A increases sharply at the start, reaches a maximum, and then decreases, as the product B is being formed and adsorbed on the surface. The coverage of B increases with time, and as more product B is formed, it also desorbs to the bulk phase and causes the concentration B in the bulk phase to increase. In the second case, we examine the situation where the adsorption of A and surface reaction are the same, but the desorption rate of B is fast. The results are shown in Fig. E8.4.4. One can observe that there is a drop in the concentration of A initially, but the drop is more gradual or at a much slower speed that in Fig. E8.4.3. θA reaches its maximum more gradually and not as obviously when comparing Fig. E8.4.4C with Fig. E8.4.3B. After θA reaches its maximum, the vacant site fraction becomes “steady” at about kSt ¼ 0.3 (Fig. E8.4.4D). We then examine the case when adsorption of A is the rate-limiting step as shown in Fig. E8.4.5. In this case, the decrease in CA initially is not as apparent as the previous two cases. The increase in the concentration of B in the bulk phase shows a “delay.” Therefore, the effect of adsorption is still important. The effects of a catalyst and adsorption on the concentrations of reactants and products are important if there is a significant amount of catalyst being added for batch systems. Therefore, when we perform LHHW analysis, we need to take adsorption into consideration.

10

10

9

9

CA

7

7

6

6

5 4 3

5 4 3

CB

2

2

CB

1

1

0

0 0.0

0

1

2

3

4

5

6

7

8

9

10

kSt

(A)

1.0

0.9

0.9

0.8

0.8

0.7

0.7

qA

0.4

kSt

0.6

0.8

1.0

0.6

0.8

1.0

qA

qj

0.6 0.5

0.5 0.4 0.3 0.2

qB

0.4

q

0.2

0.3

0.1

0.1

0.0

0.0 0

(C)

0.2

(B)

1.0

0.6

qj

CA

8

Cj / Cσ

Cj / Cσ

8

1

2

3

4

5

kSt

6

7

8

9

q qB 0.0

10

(D)

0.2

0.4

kSt

FIG. E8.4.4 Variations of concentrations in the bulk phase (A, B) and surface coverages (C, D) as functions of time when kA Cσ ¼ kS; KA ¼ 1/Cσ; KC ¼ 4; KB ¼ 1/Cσ; and kB Cσ ¼ 10 kS.

425

8.4 LHHW: SURFACE REACTIONS WITH RATE-CONTROLLING STEPS 10

1.0

9 8

0.8

7

0.7

6

0.6

5

0.5

qj

Cj / Cσ

0.9

CA

4

0.4

3

0.3

2

(A)

qA

0.2

1 0 0.01

q

CB 0.1

qB

0.1

1

10

100

kSt

0.0 0.01

0.1

1

(B)

10

100

kSt

Variations of concentrations in the bulk phase (A) and surface coverages (B) when kA Cσ ¼ 0.1 kS; KA ¼ 1/Cσ; KC ¼ 4; KB ¼ 1/Cσ; and kB Cσ ¼ kS.

FIG. E8.4.5

(b) LHHW approximation We next use the LHHW assumption to examine one case: the surface rate-limiting case to see how LHHW approximates the adsorption steps we have discussed in this example. If the surface reaction is the rate-limiting step, we have the overall rate for reaction (E8.4.4) r ¼ r2 ¼ kS Cσ θA  kS Cσ θB

(E8.4.16)

The other two steps, Eq. (E8.4.3) and Eq. (E8.4.5) are in equilibrium, 0 ¼ r1 ¼ kA CA θCσ  kA θA Cσ

(E8.4.17)

0 ¼ r3 ¼ kB θB Cσ  kB CB θCσ

(E8.4.18)

which can be rearranged to give θA ¼

kA CA θ ¼ KA CA θ kA

(E8.4.19)

kB CB θ ¼ KB CB θ kB

(E8.4.20)

and θB ¼ Total active site balance 1 ¼ θ + θA + θB ¼ θ + KA CA θ + KB CB θ

(E8.4.21)

Thus, θ¼

1 1 + KA CA + KB CB

(E8.4.22)

Substituting Eq. (E8.4.22) into Eqs. (E8.4.19), (E8.4.20), and then (E8.4.16), we obtain θA ¼

KA CA 1 + KA CA + KB CB

(E8.4.23)

426

8. CHEMICAL REACTIONS ON SOLID SURFACES

θB ¼

KB CB 1 + KA CA + KB CB

r ¼ kS Cσ KA

CA  CB =KC 1 + KA CA + KB CB

(E8.4.24) (E8.4.25)

At time t ¼ 0 in the batch reactor, the reaction has not started yet; however, the LHHW expression (Eq. E8.4.25) requires that adsorption and desorption are already in equilibrium. While this requirement is not of issue for reactions carried out in flow reactors (after the transient period) or when the amount of catalyst is negligible, it becomes important when a large amount of catalyst is employed in a batch reactor. For example, the concentration of A can be obtained via mole balance, CAT0 ¼ CA0 + θA0 Cσ ¼ CA0 +

KA CA0 Cσ 1 + KA CA0 + KB CB0

(E8.4.26)

The concentrations of A and B in the bulk-fluid phase charged into the batch reactor are CAT0 and CBT0 ¼ 0, assuming only reactant A was loaded. Since there is no B present in the initial reaction mixture, CB0 ¼ 0. From Eq. (E8.4.26), we can solved for the effective concentration of A in the batch reactor as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 1 2 + 4K1 C CAT0  Cσ  KA + CAT0  Cσ  KA AT0 A CA0 ¼ (E8.4.27) 2 That is to say that the concentration of A in the bulk-fluid phase at t ¼ 0 is smaller than that in the original bulk-fluid phase before contacts with the solid catalyst. Mole balances on A and B in the reactor lead to dCA CA  CB =KC ¼ r ¼ kS Cσ KA dt 1 + KA CA + KB CB

(E8.4.28)

dCB CA  CB =KC ¼ r ¼ kS Cσ KA dt 1 + KA CA + KB CB

(E8.4.29)

The solutions from LHHW model can be obtained by solving Eqs. (E8.4.28) and (E8.4.29) with initial conditions given by CA0 (Eq. E8.4.27) and CB0 ¼ 0 at t ¼ 0. Subsequently, the fractional coverages can be obtained from Eqs. (E8.4.23) and (E8.4.24). One should note that the initial conditions set for the surface rate-limiting LHHW model must have taken the adsorption equilibria into consideration. Instead of having four equations to solve, we now have only two equations to solve. The solutions to one case consistent with the approximation to the first case discussed earlier are shown in Fig. E8.4.6. For comparison purposes, we have plotted the LHHW approximations as dashed lines and the original solutions with individual rates as solid lines. Fig. E8.4.6A and B shows the change of bulk-phase concentrations with reaction time. One can observe that LHHW approximation agrees with the full rate description reasonably well at the shape of curves. In this case, where the adsorption of A and desorption of B are one-tenth of the rate of surface reaction, the effects of adsorption and desorption are still observable, but the simplification is reasonable. Wider differences in rates would make the approximation a better choice.

427

8.4 LHHW: SURFACE REACTIONS WITH RATE-CONTROLLING STEPS 10

10

9

9

CA

8

7

7

6

6

Cj / Cσ

Cj / Cσ

8

5 4

A

5 4 3

3

2

2 0 0.01

0.1

1

10

B

0 0.0

100

kSt

(A)

C

1

CB

1

0.2

0.4

(B)

0.6

0.8

1.0

0.6

0.8

1.0

kSt

1.0

1.0

0.8

C

qA

qA

0.8

0.6

qj

qj

0.6

0.4

0.4

0.2

0.0 0.01

(C)

0.2

q

qB 0.1

1

10

q

0.0 0.0

100

qB 0.2

0.4

kSt

(D)

kSt

FIG. E8.4.6 Variations of concentrations in the bulk phase and fractional surface coverages as functions of time for KA ¼ 1/Cσ; KC ¼ 4; KB ¼ 1/Cσ. The dashed lines are the predictions from LHHW kinetics assuming the surface reaction is the rate-limiting step, whereas the solid lines are for kA Cσ ¼ 10 kS and kB Cσ ¼ 10 kS.

The fractional coverages on the catalyst surface are computed via Eqs. (E8.4.23) and (E8.4.24) after the bulk-phase concentrations were obtained. Figs. E8.4.6C and D show the variations of surface coverage as a function of reaction time. One can observe that the fractional surface coverage as predicted by the LHHW approximation is quite similar to the full rate descriptions. LHHW approximation becomes suitable once the fraction of vacant sites becomes steady (nearly constant). One can observe from Fig. E8.4.6A that while the variation of bulk concentrations for the LHHW approximation looks similar to the full solutions, there is a shift log t (to the right) that could make the agreement closer. This is due to the fact that the finite rates of adsorption of A and desorption of B contributes to the decline of the overall rate used by the LHHW approximation. As a result, the LHHW approximations over-predicted the reaction rate, leading to a quicker change in the bulk-phase concentrations. Overall, the shape of the curve (or how the

A

J 1 = r1 A⋅σ r1 = rnet, Ad-A

J2 = r2 r2 = rS

B⋅σ

J3 = r3 r3= rnet, Des-B

B

FIG. E8.4.7 A schematic of the reaction pathway showing the rates and fluxes between each adjacent intermediate or substance for the isomerization of A to B carried out on a solid catalyst.

428

8. CHEMICAL REACTIONS ON SOLID SURFACES

concentrations change with time) is remarkably similar. Therefore, if one were to correlate the experimental data, the difference between the quality of full solutions and the quality of fit from LHHW approximations would not be noticeable, as different rate parameters would be used. In data analysis/parametric estimation, the prediction curves are shifted left or right to create a better match. (c) Pseudosteady-state hypothesis We next examine the approximation by PSSH based on the same set of parameters. PSSH assumes that the rate of change of intermediates is zero. In this case, 0 ¼ rσ  A ¼ r1  r2 ¼ kA ðθCA  θA =KA ÞCσ  kS ðθA  θB KA =KB =KC ÞCσ

(E8.4.30)

0 ¼ rσ  B ¼ r2  r3 ¼ kS ðθA  θB KA =KB =KC ÞCσ  kB ðθB =KB  θCB ÞCσ

(E8.4.31)

which yield θA ðkA =KA + kS Þ + θB kS KA =KB =KC + θkA CA ¼ 0

(E8.4.32)

θA kS  θB ðkS KA =KB =KC + kB =KB Þ + θkB CB ¼ 0

(E8.4.33)

Eqs. (E8.4.32) and (E8.4.33) can be solved to give  1 1  1 1 KC kB + KA kS CA + KC1 k1 A CB θA ¼ KA θ 1 k1 + k1 KC1 k1 + K B A S A  1  k + K1 k1 CB + k1 CA θB ¼ A 1 1A S 1 1 B1 KB θ KC kB + KA kS + kA

(E8.4.34)

(E8.4.35)

Total site balance leads to 1 ¼ θ + θA + θB  1 1   1  1 1 1 1 KC kB + KA kS CA + KC1 k1 kA + K A kS CB + k1 B CA A CB ¼θ+ KA θ + KB θ 1 1 1 1 1 1 1 1 1 KC kB + KA kS + kA KC kB + KA kS + k1 A

(E8.4.36)

Thus, θ¼

1   1 1  1 1 1 1 KA KC1 k1 + k + k K + KC kA KA + k1 C B A B B A KB + KA kS KB CB S 1+ 1 1 1 KC1 k1 B + KA kS + kA 

(E8.4.37)

Substituting Eq. (E8.4.37) into Eqs. (E8.4.34) and (E8.4.35), we obtain the fractional coverages on the catalyst surface. The overall rate of reaction r ¼ rA ¼ r1 ¼ kA CA qCs  kA qA Cs

(E8.4.38)

r ¼ kA ðCA q  qA =KA ÞCs

(E8.4.39)

That is,

Substituting the fractional coverage of A and the fraction of vacant sites, Eq. (E8.4.40) renders

8.4 LHHW: SURFACE REACTIONS WITH RATE-CONTROLLING STEPS

r ¼ kA

 1 1   1 1 KC kB + KA kS CA + KC1 k1 CA  KC1 CB A CB θC CA  ¼ θCσ σ 1 1 1 1 1 1 KC1 k1 KC1 k1 B + KA kS + kA B + KA kS + kA

429 (E8.4.40)

or CA  KC1 CB C 1 1 1 σ KC1 k1 B + KAkS + kA   r¼ 1 1 1 1 1 1 1 KA KC1 k1 B + kS + kB KB CA + KC kA KA + kA KB + KA kS KB CB 1+ 1 1 1 KC1 k1 B + KA kS + kA

(E8.4.41)

which is the rate expression for the first case shown in Table 8.4; however, the rate expression is derived from PSSH, not from a rate-limiting step assumption. The expression (E8.4.41) is quite similar to the three rate expressions based on different-rate-limiting assumptions, although the rate constants are different from any one of them. In fact, one can reduce 1 Eq. (E8.4.41) to the three expressions in Table 8.4, Case 1 by assuming k1 S ¼ 0 and kB ¼ 0 if the 1 ¼ 0 and k ¼ 0 if the surface reaction is rate-limiting; and adsorption of A is rate-limiting; k1 B A 1 k1 ¼ 0 and k ¼ 0 if the desorption of B is rate-limiting. Therefore, PSSH is more general and A S carries less approximations/simplifications than the LHHW approximations. If one were to use Eq. (E8.4.41) to correlate experimental data, one would not be able to distinguish whether it were a PSSH model or a LHHW model with a rate-limiting step, as the parameters would need to be lumped together. The appeal of the PSSH approach is that it may be able to approximate the reaction rate in general (ie, all the fluxes are considered), without imposing a particular step as the rate-limiting step as illustrated in Fig. E8.4.7. This approximation is particularly useful when only the beginning (reactants) and end (products) are of concern. Based on Fig. E8.4.7, one can think of the reaction network in analogy to electric conduction: (1) the fluxes must be equal at any given point; and (2) the total resistance is the summation of all the resistors in series. If one were to look carefully, these statements could be detected as both are embedded in Eq. (E8.4.41). At this point, one may look back at the full solutions as illustrated in Figs. E8.4.3–E8.4.5, especially with Figs. E8.4.3B and D, E8.4.4C and D, and E8.4.5B, the concentrations of intermediates (in this case, the fractional coverage of A and the fractional coverage of B) are hardly constant or at “steady state” in any reasonably wide region where we would like to have the solutions be meaningful. Therefore, the assumption is rather strong. How do the solutions actually measure up with the full solutions? To apply the PSSH expression (E8.4.40) or (E8.4.41), we must first ensure that the reaction mixture is already in pseudosteady state to minimize the error in the solution. This error can be negligible if the amount of catalyst is negligible or for steady flow reactors where the steady state is already reached. Let us consider again the case where no B is present in the reaction mixture at the start of the reaction. The concentrations of A and B charged into the batch reactor are CAT0 and CBT0 ¼ 0. Since there is no B present in the initial reaction mixture, CB0 ¼ 0. Overall mole balance at the onset of the reaction leads to CAT0 ¼ CA0 + θA0 Cσ + θB0 Cσ

(E8.4.42)

430

8. CHEMICAL REACTIONS ON SOLID SURFACES

where θA0 and θB0 satisfy Eqs. (E8.4.34) and (E8.4.35). Since CB0 ¼ 0, substituting Eqs. (E8.4.34) and (E8.4.35) into Eq. (E8.4.42), we obtain   1 1 KA KC1 k1 B + kS + kB KB Cσ CA0   (E8.4.43) CAT0 ¼ CA0 + 1 1 1 k1 + k1 + K K1 k1 + k1 + k1 K C KC kB + KA A C B B A0 B A S S This quadratic equation can be solved to give qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CAT0  Cσ  a + ðCAT0  Cσ  aÞ2 + 4aCAT0 CA0 ¼ 2

(E8.4.44)

where a¼

1 1 1 KC1 k1 B + KA kS + kA 1 1 1 1 KA KC kB + kS + kB KB

(E8.4.45)

Fig. E8.4.8 shows the comparison between the full solutions and those from the PSSH treatment for the particular case of kA ¼ 10 kS and kB ¼ 10 kS where we have also shown the LHHW solutions. One can observe that bulk-phase concentration solutions based on PSSH are surprisingly close to the full solutions (kSt > 0.02) and are better than the LHHW solutions as shown in Fig. E8.4.8. Although we have used the argument that all the intermediates remain at steady state (pseudosteady state), which does not appear to be the case for the solutions (of surface coverages, Fig. E8.4.8B), yet the computed fractional coverages agree quite well with the full solutions. There is an apparent shift of the dashed curves to the left, but to a smaller degree than that in Fig. E8.4.6. The PSSH approximation is usually better than the FES. Fig. E8.4.9 shows the comparison between the full solutions and those from the PSSH treatment for the case where the rates of reaction are the same for the adsorption and surface reaction when kA ¼ kS and kB ¼ 10 kS. This is a case where we did not show any solutions from the equilibrium step assumptions. One

1.0

10 9

CA

8

0.8

qA 0.6

6

qj

Cj / Cσ

7

5

0.4

4

qB

3 0.2

2

0 0.01

(A)

q

CB

1 0.1

1

kSt

10

100

0.0 0.01

(B)

0.1

1

10

100

kSt

FIG. E8.4.8 Variations of bulk-phase concentrations and fractional surface coverage with time for kA Cσ ¼ 10 kS; kB

Cσ ¼ 10 kS; KA ¼ 1/Cσ; KC ¼ 4; KB ¼ 1/Cσ. The dashed lines are the predictions from PSSH kinetics, whereas the solid lines are from the full solutions.

431

8.5 CHEMICAL REACTIONS ON NONIDEAL SURFACES 10

1.0

9

CA

8

0.8

qA

6

0.6

5

qj

Cj / Cσ

7

4

0.4

3 2

0 0.01

(A)

0.1

1

kS t

q

0.2

CB

1

10

100

0.0 0.01

(B)

qB 0.1

1

10

100

kSt

Variations of bulk-phase concentrations (A) and fractional coverages (B) with time for kA Cσ ¼ kS; kB Cσ ¼ 10 kS; KA ¼ 1/Cσ; KC ¼ 4; KB ¼ 1/Cσ. The dashed lines are the predictions from PSSH kinetics, whereas the solid lines are from the full solutions.

FIG. E8.4.9

can observe that the agreement between the full solutions and the PSSH solutions is rather good for the bulk-phase concentrations (kSt > 0.1), despite the fact that two steps together are rate-controlling steps. Therefore, the PSSH approximations are better suited as kinetic models than the equilibrium step assumptions when true kinetic constants are employed; however, the rate expressions are quite similar to those of LHHW or equilibrium step assumptions. When utilized to correlate experimental data, one would not be able to distinguish the two treatments.

8.5 CHEMICAL REACTIONS ON NONIDEAL SURFACES BASED ON THE DISTRIBUTION OF INTERACTION ENERGY The catalytic reaction rate theoretical development has thus far involved only the theory for ideal surfaces, quite directly in the treatment of surface-reaction rate-determining steps and a little more implicitly using the chain reaction analysis. Yet we have made a special effort in the discussion of adsorption and desorption to point out that ideal surfaces are rare; in fact, when one is concerned with the applications of catalysis in reaction engineering, it is probably fair to say that ideal surfaces are never involved. Nonetheless, LHHW rate forms or modifications of it are widely used and accepted, particularly in chemical engineering practice, for the correlation of rates of catalytic reactions. This is so even though it has been shown in many instances that the adsorption equilibrium constants appearing in the denominator of the kinetic expressions do not agree with adsorption constants obtained in adsorption experiments. There are many reasons for this, but the voluminous illustrations of Table 8.4 show that such equations are of a very flexible mathematical form, with separate product and summation terms in numerator and denominator, and are richly endowed with constants that become adjustable parameters when correlating rate data. The empirical Freundlich isotherm is a favorable isotherm for kinetic studies, although there is no rigorous theoretical basis behind it. Although the Freundlich isotherm is not as good as the Langmuir or other isotherms that have a theoretical basis, it is accurate enough in low

432

8. CHEMICAL REACTIONS ON SOLID SURFACES

coverage regions as it is an approximation of a variety of isotherms. The success of the Freundlich isotherm suggests immediately that simple power-law forms might be applied successfully to surface reaction kinetics; however, this is just empirical. One other isotherm of interest is the Temkin isotherm. The Temkin isotherm was derived above based on the fact that available active centers are linearly distributed with adsorption heat, ie, Eq. (8.49) dnσi ¼ 

nσ dEs RTfT

(8.49)

We know that adsorption occurs first on highly energetic sites (sites of higher ΔHad values). Therefore, Eq. (8.49) is equivalent to dθ ¼

1 dEs RTfT

(8.131)

where θ is the available (vacant) active site fraction. Integrating Eq. (8.131), we obtain Es ¼ RTfT θ

(8.132)

which can be rewritten for adsorption of a single species Es ¼ RTfT ð1  θA Þ

(8.133)

ie, the adsorbed molecules exhibit an adsorption heat fluctuation (Es) linearly related to the site coverage. 0 0  Es ¼ ΔHad  RTfT ð1  θA Þ ΔHad ðθA Þ ¼ ΔHad

(8.134)

Therefore, one can imagine that the activation energy of adsorption should be linearly related to the site coverage as well, Ead ðθA Þ ¼ E0ad + Eα θA

(8.135)

where Ead(θA) is the activation energy of adsorption for A when the fractional surface coverage of A is θA, It is expected that at lower coverage, the activation energy is lower. The activation energy increases as the coverage increases. E0ad is the activation energy of adsorption for A when no A has been adsorbed on the surface, and Eα is a positive scaling constant. Since (see Eq. 8.25) ΔHad ¼ Ead  Edes

(8.136)

Substituting Eq. (8.135) into Eq. (8.136), we have 0 Edes ðθA Þ ¼ Ead  ΔHad ðθA Þ ¼ E0ad  ΔHad + RTfT  ðRTfT  Eα ÞθA

(8.137)

We now derive the activation energies for adsorption and desorption as a function of the surface coverage. Since we expect the desorption activation energy to decrease with increasing coverage, ie, ðRTfT  Eα Þ < 0

(8.138)

0 < Eα < RTfT

(8.139)

we have

8.5 CHEMICAL REACTIONS ON NONIDEAL SURFACES

433

This forms the basis for studying kinetics involving nonideal surfaces when steric interactions and size effects are negligible. One example of kinetics involving the application of the nonideal surface theory is to the ammonia synthesis from nitrogen and hydrogen. The synthesis of ammonia has changed the world’s food production (Smil, 2001). The Haber-Bosch process has remained the dominant process in generating ammonia from nitrogen separated from air and hydrogen. Over a Fe catalyst or a promoted Ru/C catalyst

overall

N2 + 2σ>2N  σ

(8.140a)

H2 + 2σ>2H  σ

(8.140b)

H  σ + N  σ>NH  σ + σ

(8.140c)

H  σ + NH  σ>NH2  σ + σ

(8.140d)

H  σ + NH2  σ>NH3  σ + σ

(8.140e)

NH3  σ>NH3 + σ

(8.140f)

N2 + 3H2

2NH3

(8.140g)

The adsorption of nitrogen was found to be the rate-limiting step; all the other steps are rapid equilibrium steps. As such, the overall rate of reaction is determined by the adsorption of nitrogen (Eq. 8.140a). The rate is given by r ¼ rN2 ¼ kN2 C2σ θ2 pN2  kN2 C2σ θ2N

(8.141)

noting that Ead

kN2 ¼ k0N2 e RT ¼ k0N2 e

E0ad + Eα θN RT

Edes

kN2 ¼ k0N2 e RT ¼ k0N2 e

E0ad

¼ k0N2 e RT e

Eα θ N RT

0 E0ad + RTfT ΔHad ðRTfT Eα ÞθN RT RT e

(8.142) (8.143)

Combining the constants together, Eq. (8.143) is reduced to r ¼ k^N2 θ2 pN2 e

Eα θN RT

 k^N2 θ2N e

ðRTfT Eα ÞθN RT

(8.144)

Temkin isotherm, which was derived from the linear distribution of adsorption heat, is given by θN 

 1 ln K
N2 pN2 efT fT

(8.145)

434

8. CHEMICAL REACTIONS ON SOLID SURFACES

Since nitrogen adsorption is the rate-limiting step and the surface coverage of nitrogen is in equilibrium with hydrogen and ammonia via other surface reaction steps, the pressure in the adsorption isotherm, pN2 , is not the actual partial pressure of nitrogen. This virtual partial pressure of N2 is evaluated by the partial pressures of hydrogen and ammonia by KP ¼

p2NH3

(8.146)

pN2 p3H2

Substituting Eq. (8.146) into Eq. (8.145), we obtain p2 1 θN ¼ ln K N2 NH33 efT fT K P pH 2

! (8.147)

Since the dependence of fractional coverage is dominated by the exponential terms, one can neglect the change in the power-law terms of the fractional coverage in Eq. (8.144) while applying the Temkin approximation of θN. Thus, Eq. (8.144) is approximated by r ¼ k^N2 pN2 K
N2

p2NH3 KP p3H2

!  Eα  1 efT

RT fT

 k^N2 K
N2

p2NH3 KP p3H2

!RTfT Eα  1 RT

efT

fT

(8.148)

Noting that at thermodynamic equilibrium the reaction rate becomes zero, we can combine the constant terms together for Eq. (8.148) to give r ¼ kpN2

p3H2 p2NH3

!m

k p2NH3  KP p3H2

!1m (8.149)

  Eα KP fT m ^ and k ¼ kN2
e . Eq. (8.149) is known as the Temkin equation. where m ¼ RTfT K N2 This illustrates that the linear energy distribution with the Temkin simplification also leads to power-law kinetic expressions. The power-law appearance after the Temkin simplification is attractive as it is analogous to homogenous reactions. Because of the approximation made in the Temkin model that the coverage is in the intermediate region and the variation of the power-law terms of the coverage function is negligible compared to the exponential variations, the kinetic expressions derived are not expected to be valid in as wide a range as one would expect from LHHW expressions. In most cases, the expressions derived from this approach are merely comparable to LHHW expressions. Table 8.5 shows a comparison of some examples where both LHHW and power-law form correlations have been made. Table 8.6 shows some of the rate expressions based on the UniLan adsorption isotherms when the surface reaction is limiting. One can observe that when UniLan is employed to derive the rate expressions, the reaction rate expressions are quite similar to those of the LHHW expressions.

435

8.5 CHEMICAL REACTIONS ON NONIDEAL SURFACES

TABLE 8.5

Comparison of LHHW Rate Expressions With Power-Law Expressions

Reaction and Data Source SO2 oxidation: 1 SO2 + O2 >SO3 2 (Lewis and Ries, 1927;

LHHW r¼h

 1=2 k pSO2 pO2  pSO3 =KP 1 + ðKO2 pO2 Þ1=2 + KSO2 pSO2

i2

Uyehara and Watson, 1943)

15.4% deviation over 12 experiments on variation of partial pressures of SO2 and SO3

Hydrogenation of codimer (C):

r¼

H2 + C>P

20.9% deviation at 200°C 19.6% deviation at 275°C 19.4% deviation at 325°C

(Tschernitz et al., 1946) Phosgene synthesis: CO + Cl2 >COCl2 (Potter and Baron, 1951)

r¼

13.3% deviation over 12 experiments on variation of partial pressures of SO2 and SO3

kpH2 pC ð1 + KH2 pH2 + KC pC + KP pP Þ2

r ¼ kðpH2 pC Þ1=2 19.6% deviation at 200°C 32.9% deviation at 275°C 21.4% deviation at 325°C

kpCO pCl2

1=2

2

ð1 + KCl2 pCl2 + KCOCl2 pCOCl2 Þ

3.4% deviation at 5.6% deviation at 2.6% deviation at 7.0% deviation at Toluene alkylation by methanol: T + M>X + W (Sotelo, et al., 1993)

Power-Law  1=2 1=2 1=2 r ¼ k pSO2 pSO3  pSO3 pO2 =KP

30.6°C 42.7°C 52.5°C 64.0°C

kðpM pT  pX pW =KP Þ pW + KM pM + KX pX pW 2.07% deviation r¼

r ¼ kpCO pCl2

13.0% deviation at 30.6°C 9.1% deviation at 42.7°C 13.9% deviation at 52.5°C 3.0% deviation at 64.0°C r ¼ kpT pM 4.65% deviation

Eley-Rideal model (Fraenkel, 1990)

TABLE 8.6

Some Examples of Rate Expressions With UniLan Adsorption Where Surface Reaction Is the Rate-Limiting Step

Overall Reaction

Rate Expression

1) Isomerization: A>B

β ¼ K
A CA + K
B CB

2) Bimolecular: A + B>C + D

β ¼ K
A CA + K
B CB + K
C CC + K
D CD

r ¼ ks K
A Cσ

CA  CB =Keq 1 + βefT ln fT β 1+β

r ¼ ks K
A K
B C2σ 1

CA CB 

1

3) Bimolecular (dissociative): 1 A2 + B>C + D 2

β ¼ K
2A2 C2A2 + K
B CB + K
C CC + K
D CD

4) Bimolecular (Eley-Rideal): A + B>C

β ¼ K
A CA + K
C CC

fT2 β2 1

1 2

r ¼ ks KA2 KB C2σ

CC CD Keq

C2A2 CB 

ln 2

CC CD Keq

fT2 β2

1 + βefT 1+β

ln 2

1 + βefT 1+β

CA CB  CC =Keq 1 + βefT ln r ¼ ks K
A Cσ fT β 1+β

436

8. CHEMICAL REACTIONS ON SOLID SURFACES

8.6 CHEMICAL REACTIONS ON NONIDEAL SURFACES WITH THE MULTILAYER APPROXIMATION In the previous section, we have learned how the Temkin approximation can be applied to perform kinetic analysis to render surface catalyzed reaction rate to a power-law rate expression. The UniLan adsorption isotherms can be applied to derive kinetic models for chemical reactions on nonideal surfaces. Besides interaction energy distribution modeling, multilayer adsorption can also be applied to model nonideal surfaces. In this section, we apply the multilayer adsorption model to reactions occurring on the surfaces. When surface reaction is limiting, the overall rate expression is dependent only on the concentrations of the reactants on the surface. For example, the reaction network of isomerization on a solid catalyst follows the following scheme A + σ>A  σ

(8.150)

A  σ>B  σ

(8.151)

B  σ>B + σ

(8.152)

+) = overall

A

B

(8.153)

Assume that only step, Eq. (8.151), is elementary as the adsorption steps are occurring on nonideal surfaces. The reaction rate expression for the overall reaction is then r ¼ kS CSA  kS CSB

(8.154)

The surface coverage of species A and B can be obtained from Eq. (8.91) for competitive cooperative adsorptions. To be consistent, we assume Keq ¼

kS KA CσA kS KB CσB

(8.155)

and f0 ¼ 1 + c + c1 c2 f1 ¼ 1  c¼ f2 ¼

1  ðc2 cÞN1 1  c2 c

c1 c1 1  ð1 + N  Nc2 cÞðc2 cÞN + c2 c2 ð1  c2 cÞ2

X Cσj Cσ

Kj Cj ¼

CσA KA CA + CσB KB CB Cσ

  2 + NðN + 1ÞðN  2Þðc2 cÞN1 2 + Nð3N  5Þðc2 cÞN1 1  ðN  1Þ N 2 + 4N + 6  6ð1  c2 cÞ 18 6ð1  c2 cÞ2 N1 N1 1 + ð3N  4Þðc2 cÞ 1  ðc2 cÞ  + 3 3ð1  c2 cÞ ð1  c2 cÞ4

(8.156) (8.157)

(8.158)

(8.159)

437

8.7 KINETICS OF REACTIONS ON SURFACES WHERE THE SOLID IS EITHER A PRODUCT OR REACTANT

TABLE 8.7

Some Examples of Rate Expressions With Multilayer Adsorption Where Surface Reaction Is the Rate-Limiting Step

Overall Reaction

Rate Expression   f1 CB r ¼ kS CσA KA CA  f0 Keq

1) Isomerization: A>B

c¼

CσA KA CA + CσB KB CB Cσ

2) Bimolecular: A + B>C + D

c¼

CσA KA CA + CσB KB CB + CσC KC CC + CσD KD CD Cσ

1 1 3) Bimolecular 2 2 (dissociative): c ¼ CσA KA2 CA2 + CσB KB CB + CσC KC CC + CσD KD CD Cσ 1 A2 + B>C + D 2

4) Bimolecular (Eley-Rideal): A + B>C

c¼

CσA KA CA + CσC KC CC Cσ

0 1 ! f2 C C f r ¼ CσA CσB KA KB @kS2 1 + kS1 c1 2 A CA CB  C D 2 f 0 Cσ Keq f0 0 1 ! f2 C C f 1=2 1=2 r ¼ CσA CσB KA KB @kS2 1 + kS1 c1 2 A CA CB C D 2 2 Cσ f0 Keq f02

r ¼ kS CσA KA

  f1 CC CA CB  f0 Keq

We obtain the rate expression for reaction of Eq. (8.153) on nonideal surfaces as modeled by competitive cooperative adsorption f1 X CB r ¼ kS CσA KA CA f0 Keq

(8.160)

Some examples of the reaction rates where the surface reaction is the limiting step and when multilayer adsorption is employed to model the nonideality of the surfaces are listed in Table 8.7.

8.7 KINETICS OF REACTIONS ON SURFACES WHERE THE SOLID IS EITHER A PRODUCT OR REACTANT In the previous section, we have learned that with solid catalysis, the surface active centers are not generated or consumed during reaction. When a solid is one of the reactants, eg, dissolution reactions, combustion reactions, or one of the products, such as vapor deposition, the active centers on the surface change. The change of surface active centers could lead to changes in the kinetic analysis. In this section, we shall use reactions on woody biomass to focus our discussions. Renewable biomass has increasingly become the chemical and energy source for commodity and chemical industry. Reactions involving biomass are usually heterogeneous with the biomass being solid. Pulp and paper is the earliest application, and still remains relevant. Based on surface reaction theory, Liu (2008) applied LHHW kinetics to bleaching, pulping and extraction reactions involving wood and fibers. A series of studies followed resulting in simplistic kinetic relationships. A brief review of the kinetics of “acid-hydrolysis” and hot-water extraction is shown here as an example of reactions involving surfaces. There are multitudes of chemical components in wood that participate in the reactions. At high temperatures, water molecules can be activated and then directly attack carbohydrates, resulting in thermal-hydrodepolymerization. In the case of acid hydrolysis, hydrogen ions act as a

438

8. CHEMICAL REACTIONS ON SOLID SURFACES

catalyst to soften the glycosidic bonds and induce their depolymerization. The glycosidic bond breakage is enhanced when the breakaway oligomers are attracted away from the solid surface. High temperature and high electrolyte content favor the oligomers dissolving in the liquid phase. As an example, we shall examine the hot-water extraction or hydrolysis of wood in the following manner. Under autocatalytic conditions, the hydrogen ion concentration is initially very low; however, acetyl groups are present in wood as they are associated with extractives, lignin and hemicellulose. Hydration of the acetyl groups lead to the acidification of the liquid and thus the formation of hydrogen ions. According to these considerations, extraction-hydrolysis involving solid woodchips can be represented by a simplistic model H2 O>H2 O∗

(8.161)

R-Xn OHðsÞ + H2 O∗ >R-Xn OH  H2 O∗ ðsÞ

(8.162)

R-Xn OH  H2 O∗ ðsÞ ! R-Xm OHðsÞ + HXS OHðaqÞ

(8.163)

R-POAc ðsÞ + H2 O∗ ðaqÞ>R-POAc  H2 OðsÞ

(8.164)

R-POAc  H2 OðsÞ>R-POHðsÞ + HOAcðaqÞ

(8.165)

R-POAc  H2 OðsÞ>R-OHðsÞ + HPOAcðaqÞ

(8.166)

HOAcðaqÞ>H + ðaqÞ + OAcðaqÞ

(8.167)

R-Xn OHðsÞ + H + ðaqÞ>R-Xn OH  H + ðsÞ

(8.168)

R-Xn OH  H + ðsÞ + H2 OðaqÞ ! R-Xm OH  H + ðsÞ + HXs OHðaqÞ

(8.169)

HXn OHðaqÞ + H + ðaqÞ>HXn OH  H + ðaqÞ

(8.170)

HXn OH  H ðaqÞ + H2 OðaqÞ ! HXm OH  H ðaqÞ + HXs OHðaqÞ

(8.171)

+

+

where m + s ¼ n, R- denotes the cellulose and/or lignin bonding with the fibers, P represents a segment/subunit of hemicellulose or lignin, Xn represents an n-xylopolymer middle group, ie, (–O–C5H8O3–)n, whereas HXnOH is an n-xylooligomer, and HOAc represents an acetic acid molecule, where Ac ¼ CH3CO. Eq. (8.168) represents the adsorption of hydrogen ions onto the woody biomass surface, and Eq. (8.169) represents the surface reaction whereby one xylo-oligomer is cleaved from the woody biomass. One should note that the actual entity participating in the extraction reaction/hydrolysis reactions can be either H+ or H3O+. In fact, one may write H + + H2 O∗ >H3 O +

(8.172)

R-Xn OHðsÞ + H3 O + >R-Xn OH  H3 O + ðsÞ

(8.173)

R-Xn OH  H3 O + ðsÞ ! R-Xm OH  H + ðsÞ + HXS OHðaqÞ

(8.174)

R-POAcðsÞ + H3 O + ðaqÞ>R-POAc  H3 O + ðsÞ

(8.175)

and

8.8 DECLINE OF SURFACE ACTIVITY: CATALYST DEACTIVATION

439

R-POAc  H3 O + ðsÞ>R-POH  H + ðsÞ + HOAcðaqÞ

(8.176)

R-POAc  H3 O + ðsÞ>R-OHðsÞ + HPOAc  H + ðaqÞ

(8.177)

HXn OHðaqÞ + H3 O + ðaqÞ>HXn OH  H3 O + ðaqÞ

(8.178)

HXn OH  H3 O + ðaqÞ ! HXm OH  H + ðaqÞ + HXs OHðaqÞ

(8.179)

Kinetically, the steps as shown in Eqs. (8.161–8.179) can represent the extraction/hydrolysis reactions. The monomeric sugar can further dehydrate in the presence of protons to products such as furfural, humic acid, Levulinic acid, etc. HX1 OH  H + ðaqÞ ! H + + H2 O + furfural, and other dehydration products

(8.180)

Besides the extraction reaction, the polysaccharides can condense back into woody biomass R-Xn OH  H + ðsÞ + HXS OHðaqÞ ! R-Xn + s OH  H + ðsÞ + H2 OðaqÞ

(8.181)

R-Xn OHðsÞ + HXs OH  H + ðaqÞ ! R-Xn + s OH  H + ðsÞ + H2 OðaqÞ

(8.182)

or

The dehydration polymerization of the xylo-oligomers also occurs HXn OH  H + ðaqÞ + HXm OHðaqÞ ! HXn + m OH  H + ðaqÞ + H2 OðaqÞ

(8.183)

The polymerization reactions Eqs. (8.181–8.183) can have a significant effect on the extraction and/or hydrolysis process.

8.8 DECLINE OF SURFACE ACTIVITY: CATALYST DEACTIVATION One particular aspect of solid catalysis not encountered in reactions that are not catalytic is a progressive decrease in the activity of the surface with its time of utilization. The reasons for this are numerous, but we will divide them into three general categories. Poisoning: loss of activity due to strong chemisorption of a chemical impurity on the active sites of the surface, denying their access for reactant molecules. This should not be confused with inhibition as expressed by adsorption terms in the denominator of LHHW rate expressions. Coking or fouling: loss of activity, normally in reactions involving hydrocarbons or carbonbased substrates, due to reactant or product degradation, producing a carbonaceous residue on the surface blocking reactants from accessing the active centers on the catalyst surface. Sintering: loss of activity due to a decrease in active surface per volume of catalyst, normally the result of excessively high temperatures. A vast effort has been expended over the years in investigating these types of deactivation as they are encountered in catalytic reactions and catalysts of technological importance. The

440

8. CHEMICAL REACTIONS ON SOLID SURFACES

nσ

0

FIG. 8.20

0

t

Variation of number of active centers with the service time of a catalyst.

uninitiated are often amazed at the fact that many reaction-system process designs are dictated by the existence of catalyst deactivation, as are process operation and optimization strategies. In some cases, the deactivation behavior is so pronounced as to make detailed studies of intrinsic kinetics of secondary importance. In the catalytic reaction rate expressions we have seen so far, the number of active centers is the key factor or activity of the catalyst. When viewed in mathematical terms, catalyst deactivation reduces the number of active centers. This is illustrated in Fig. 8.20. The loss of active centers can be due to a number of reasons as outlined above. From this point on, the mechanism of catalyst deactivation can be modeled again via reaction kinetics. Therefore, the kinetics we have been discussing so far can be applied to catalyst deactivation as well.

8.9 SUMMARY Reactions occurring on solid surfaces start when reactants or catalysts from the fluid phase collide and associate with active centers (or sites) on the surface. A + σ>A  σ

(8.14)

A2 + 2σ>2A  σ

(8.30)

r ¼ kad θCσ CA  kdes θA Cσ

(8.19)

or for dissociative adsorption

which is governed by

or in terms of dissociative adsorption r ¼ kad θ2 C2σ CA2  kdes θ2A C2σ

(8.32)

These equations form the bases for the analysis of reactive fluid-surface interactions. The adsorption isotherms are derived from Eq. (8.19) and/or (8.32) together with the total active center balance when adsorption is at equilibrium, or the net adsorption rate is zero.

441

8.9 SUMMARY

Isotherms directly obtained from Eqs. (8.19) or (8.32) are termed Langmuir isotherms, as uniform surface activity is assumed, so that all the rate constants are not functions of the coverage, θA ¼ and

KA CA 1 + KA CA

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KA2 CA2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi θA ¼ 1 + KA2 CA2

(8.29)

(8.34)

Since dissociative adsorption is a special case of adsorption where the concentration is replaced by the square root of concentration as noted from Eqs. (8.34) and (8.29), we shall now focus on nondissociative adsorption. KA ¼

ΔHad kad 0  RT ¼ KA e kdes

(8.27)

For the adsorption of multiple component mixture at the same active center, the Langmuir adsorption isotherm is given by θj ¼

Kj Cj Ns X 1+ Km Cm

(8.39)

m¼1

Nonideal surfaces can be modeled by a distribution of interaction energy (adsorption heat ΔHad, and/or Ead and Edes). If the available adsorption sites are linearly distributed about the excess interaction energy between 0 and RTfT, then the adsorption isotherm becomes Ns X

K
m Cm efT K
j Cj m¼1 θj ¼ N ln Ns s X X
Km Cm K
m Cm fT 1+ 1+

m¼1

(8.51)

m¼1

For single-species adsorption, Eq. (8.51) is reduced to θA ¼

1 1 + K
A CA efT ln fT 1 + K
A CA

(8.52)

which is also known as the UniLan (uniform distribution Langmuir) model. In the intermediate coverage region, K
A CA << 1 << K
A CA efT Eq. (8.52) may be approximated by

  ln K
A CA efT θA  fT

which is the Temkin isotherm.

(8.93)

442

8. CHEMICAL REACTIONS ON SOLID SURFACES

Nonideal surfaces are characterized by multiple levels of interaction energies with adsorbates. The multiple levels of interactions can also be modeled with multilayer adsorptions as the adsorbate-adsorbate molecular interactions and steric interactions can be introduced via multilayer adsorption concepts. True multilayer adsorption is common in physisorptions. Assuming that (1) on the first layer, the interaction is only between adsorbate and the adsorbent surface, and this interaction is uniform; (2) the interactions among adsorbate-adsorbate and adsorbate-adsorbent are uniform and identical for the third layer and above; we obtain for competitive N-layer adsorptions, " # c1 1  ð1 + N  Nc2 cÞðc2 cÞN 1+ 1 c2 ð1  c2 cÞ2 (8.91) CSj ¼ Cσj Kj Cj N1 2 1  ðc2 cÞ 1 + c + c1 c 1  c2 c where c
¼

NS X Cσj Kj Cj j¼1


c1 ¼

NS X

Cσ c1j Kj Cj

j¼1

c¼

NS X Cσj Kj Cj j¼1

Cσ

(8.90)

For single-species cooperative adsorption, Eq. (8.91) is reduced to

CSA ¼ Cσ KA CA

" # c1 1  ð1 + N  Nc2 KA CA Þðc2 KA CA ÞN 1+ 1 c2 ð1  c2 KA CA Þ2 N1 2 C2 1  ðc2 KA CA Þ 1 + KA CA + c1 KA A 1  c2 KA CA

(8.71)

or

CSA ¼

CSA1 KA CA N

" # c1 1  ð1 + N  Nc2 KA CA Þðc2 KA CA ÞN 1+ 1 c2 ð1  c2 KA CA Þ2 N1 2 C2 1  ðc2 KA CA Þ 1 + KA CA + c1 KA A 1  c2 KA CA

(8.79)

When N ¼ 1, Eq. (8.79) is reduced to the Langmuir adsorption isotherm, Eq. (8.29). If the adsorption is truly physisorption, the adsorbed adsorbate may be considered to be equivalent to “liquefied” or “solidified” adsorbate. When N ! 1, Eq. (8.71) is reduced to

8.9 SUMMARY

  c1 c1 + 1 ð1  c2 KA CA Þ2 Cσ KA CA c2 c2 CSA ¼ 2 C2 1  c2 KA CA 1 + ð1  c2 ÞKA CA + ðc1  c2 ÞKA A

443

(8.73)

Eq. (8.73) is the cooperative adsorption isotherm for single-species adsorption on solid surfaces. It is applicable to both chemisorptions and physisorptions. Eq. (8.73) is applicable for physisorptions where adsorption occurs independently of the interaction between “marked” active centers on the solid surface and the adsorbate molecules. It is similar to the condensation of gaseous molecules onto a solid surface. When gaseous molecules condense on a surface, the top layers exhibit a phase equilibrium between the adsorbed phase (or condensed phase) and the gaseous phase as if the adsorbent was absent, c2 KA CA ¼

pA p0A

(8.74)

where p0A is the vapor pressure of A on the adsorbed phase. Substituting Eq. (8.74) into Eq. (8.73), we obtain   pA pA 2 Cσ 0 c1 + ðc2  c1 Þ 1  0 pA pA (8.75) CSA ¼  2 pA p 1  0 c22 + c2 ð1  c2 Þ A + ðc1  c2 Þ pA0 pA pA p0A Eq. (8.75) is important in that it can be reduced to the well known BET equation. By setting c1 ¼ c2 ¼

1 c

(8.76)

Eq. (8.75) is reduced to pA Cσ c 0 p  A  CSA ¼  pA pA 1  0 1  ð1  cÞ 0 pA pA

(8.77)

which is the BET isotherm. LHHW kinetics is derived from the Langmuir adsorption isotherm. Table 8.4 shows some examples of rates with different rate-controlling steps. The terms in the denominator of the rate can indicate the type of reaction and/or controlling step. If the concentration of one species did not appear in the denominator, (1) either the rate-controlling step is the adsorptiondesorption of the species; or (2) the particular species is not adsorbed onto the surface. If the square root of a concentration appears, the adsorption of that particular species is dissociative. If an adsorption-desorption step is rate-controlling, the concentration of the particular species does not appear in the denominator; instead its virtual concentration (equilibrium concentration with other species) appears in its place. The rate-control step has two active

444 sites involved if

8. CHEMICAL REACTIONS ON SOLID SURFACES

1 2

(same type of active sites) or

1 ð1 + KA CA + ⋯Þð1 + KB CB + ⋯Þ

ð1 + KA CA + ⋯Þ (two types of active sites) appears. In kinetic analysis, PSSH on all the intermediates) is a more general and better approximation than equilibrium step assumptions. PSSH can be traced back to the overall effect of all the fluxes on the reaction network or pathway; however, the final rate expressions are quite similar when the rate constants are lumped together. Therefore, there is no difference on the quality of fit to experimental data one may obtain. The difference between the actual (or full solution) and either one of the approximation methods (PSSH or LHHW) is noticeable if the initial time period (at the onset of adsorption and reaction) is considered. After the onset or the start of reaction, the difference between the approximation and full solution is negligible. Therefore, equilibrium step (or the opposite, rate-limiting step) assumptions are well suited for kinetic analyses or simplifications of complex problems. The rate expression can also appear to be of power-law form if the Freundlich isotherm or the Temkin approximation were applied to UniLan isotherms. The accuracy of LHHW rate expressions can be as accurate as power-law forms even for nonideal surfaces. This is largely due to the fact that the power-law rate forms are approximate and only valid in narrow ranges of concentrations (intermediate). When a wide range of concentration is examined, LHHW rate expressions do not show larger deviations because of their better agreement at low surface coverages than either the Temkin or the Freundlich isotherms. To obtain rate expressions valid at low coverages as well as at high surface coverages, nonideal isotherms must be employed without the Freundlich and/or the Temkin approximations. UniLan isotherms consider the distribution of available adsorption sites to be linearly dependent on the interaction energy between adsorbate molecules and adsorbent. When a UniLan isotherm is employed to describe nonideal surfaces, some of the reaction rate expressions are shown in Table 8.6. These relations are valid in wide ranges of concentrations and thus are of similar quality as LHHW expressions for ideal surfaces. The multilayer adsorption model can also be applied to describe reactions on nonideal surfaces as steric interactions and interaction potential differences can be effectively modeled by multilayers. Table 8.7 shows some examples of the rate expressions for nonideal surface reactions. Rate expressions for noncatalytic reactions involving solids can be derived in the same manner as LHHW. The participation of the surface active centers in actual reactions can affect the site balance, either active center renewal or depletion can occur. For catalytic surfaces, the activity can decrease with an increasing duration of service. The surface activity decline can be treated the same way as the concentration of a reacting component in the mixture.

Bibliography Brunauer, S., 1945. The Adsorption of Gases and Vapors. Princeton University Press, Princeton, NJ. Brunauer, S., Emmet, P.H., Teller, R., 1938. Adsorption of gases in multi-molecular layers. J. Am. Chem. Soc. 60, 309–316. Butt, J.B., Petersen, E.E., 1989. Activation, Deactivation and Poisoning of Catalysts. Academic Press, San Diego, CA. ¨ berpru¨fung des Desaktivierungsverhaltens eines Christoph, R., Baerns, M., 1986. Simulation und experimentelle U Nickel/Al2O3-Tra¨gerkatalysators fu¨r die Methanisierung von Rest-CO in H2. Chem-Ing-Tech. 58, 494.

PROBLEMS

445

Fogler, H.S., 1999. Elements of Chemical Reaction Engineering, third ed. Prentice Hall PTR, Inc., Upper Saddle River, NJ. Fraenkel, D., 1990. Role of external surface sites in shape selective catalysis over zeolites. Ind. Eng. Chem. Res. 29, 1814–1821. Hinshelwood, C.N., Burk, R.E., 1925. The thermal decomposition of ammonia upon various surfaces. J. Chem. Soc. Trans. 127, 1105–1117. Klusacek, K., 1984. Numerical simulation of a catalytic reaction dynamics in a kinetic region. Collect. Czech. Chem. Commun. 49, 170. Knaebel, K.S., 2009. 14 Adsorption. In: Albright, L.F. (Ed.), Albright’s Chemical Engineering Handbook. CRC Press, Taylor & Francis Group, Boca Raton, FL. Koubaissy, B., Toufaily, J., El-Murr, M., Daou, T.J., Hafez, H., Joly, G., Magnoux, P., Hamieh, T., 2012. Adsorption kinetics and equilibrium of phenol drifts on three zeolites. Central Eur. J. Eng. 2 (3), 435–444. Lewis, W.K., Ries, E.D., 1927. Influence of reaction rate on operating conditions in contact sulfuric acid manufacture. II. Ind. Eng. Chem. 19, 830. Liu, S., 2008. A kinetic model on autocatalytic reactions in Woody biomass hydrolysis. J. Biobased Mater. Bioenerg. 2, 135–147. Liu, S., 2013. Chemical reactions on surfaces during Woody biomass hydrolysis. J. Bioproc. Eng. Biorefinery 2 (2), 125–142. Liu, S., 2014. A visit on the kinetics of surface adsorption. J. Bioproc. Eng. Biorefinery 3 (2), 100–114. Liu, S., 2015. Cooperative adsorption on solid surfaces. J. Colloid Interface Sci. 450, 224–238. Masel, R.I., 1996. Principles of Adsorption and Reaction on Solid Surfaces. Wiley & Sons, New York. Potter, C., Baron, S., 1951. Kinetics of the catalytic formation of phosgene. Chem. Eng. Prog. 47, 473. Saterfield, C.N., 1991. Heterogeneous Catalysis in Industrial Practices, second ed. McGraw-Hill, New York. Smil, V., 2001. Enriching the Earth: Frith Haber Carl Bosch and the Transformation of World Food Production. MIT Press, Cambridge, MA. Somojai, G.A., 1994. Introduction to Surface Chemistry and Catalysis. John Wiley & Sons, New York. Sotelo, J.L., et al., 1993. Kinetics of toluene alkylation with methanol over Mg-modified ZSM-5. Ind. Eng. Chem. Res. 32, 2548–2554. Topchieva, K.V., Yun-Pun, R., Smirnova, I.V., 1957. 81 function of surface compounds in the study of catalytic dehydration of alcohols over aluminum oxide and silica-alumina catalysts. Advan. Catal. 9, 799. Tschernitz, J.L., et al., 1946. Determination of the kinetics mechanism of a catalytic reaction. Trans. Am. Inst. Chem. Eng. 42, 883. Uyehara, O.A., Watson, K.M., 1943. Oxidation of sulfur dioxide. Ind. Eng. Chem. 35, 541. White, M.G., 1990. Heterogeneous Catalysis. Prentice Hall, Upper Saddle River, NJ.

PROBLEMS 8.1. One way of modeling multilayer adsorption is to employ the available site distribution function. Assume that there are n + 1 layers, with equal fractions of available sites in each layer, and that the excess heat of adsorption decreases each layer by an equal fixed amount. This assumption leads to     K1, j Ki1, j Emax Emax 1=n ; c ¼ ¼ exp ¼ exp c¼ Kn + 1, j RT Ki, j nRT Derive the adsorption isotherm expression similar to the way the UniLan isotherm is obtained.

446

8. CHEMICAL REACTIONS ON SOLID SURFACES

Equilibrium Pressure, mmHg T 5 0ºC

3

Volume Adsorbed, cm /g

T 5 25ºC

T 5 100ºC

2

0.12

0.31

5.8

3.5

0.21

0.54

10.3

5

0.31

0.8

14

7

0.48

1.2

21

10

0.8

2.0

32

15

1.7

4.3

56

20

3.1

7.8

91

30

7.7

40

15

19

210

38

380

8.2. When deriving multilayer adsorption isotherms, Eqs. (8.110) and (8.112), we have assumed steric interactions: desorption can occur only if the adsorbed molecule is not covered by the layer above it. Rework P8.1 with n ¼ 2 (or three layers), following the multilayer adsorption derivation with the additional steric interaction implied. 8.3. The following data were recorded for the adsorption of a gas on a solid (a) Plot the three isotherms on a log-log scale. (b) The heat of adsorption can be related to the equilibrium pressure at a constant level d ln P ΔH ¼  2 .Determine of adsorption through the Clausius-Clapeyron equation, dT RT the heat of adsorption and plot the heat of adsorption as a function of surface coverage (or level of adsorption). (c) Is the Langmuir adsorption isotherm a good assumption? Why? How about the Temkin isotherm? (d) Examine the quality of the adsorption isotherm fit with UniLan. (e) Examine the quality of the adsorption isotherm fit with the multilayer model. 8.4. The mechanism proposed by Topchieva et al. (1957) for ethanol dehydration over Al2O3, is OH OC2H5

1. C2H5OH +

2.

Al

Al

OC2H5

OH

Al

Al

+ H2O

+ C2H4

OH The designated active site,

, is consumed in the first step and regenerated in

Al the second, thus forming a closed sequence. Derive a rate equation for the rate of dehydration of alcohol assuming that:

447

PROBLEMS

(a) Both reactions occur at comparable rates. (b) Reaction 1 is much faster than Reaction 2, and Reaction 2 is irreversible. 8.5. Methanol synthesis from syn gas 2H2 + CO>CH3 OH

(P8.5.1)

May be described by the following mechanism H2 + σ>H2  σ

(P8.5.2)

CO + σ>CO  σ

(P8.5.3)

CO  σ + H2  σ>HCOH  σ + σ

(P8.5.4)

HCOH  σ + H2  σ>CH3 OH  σ + σ

(P8.5.5)

Derive a rate expression if the surface is ideal and the reaction (Eq. P8.5.4) is ratelimiting. 8.6. The synthesis of ammonia from nitrogen and hydrogen over a Ru/C catalyst is found to follow the following steps (P8.6.1) N2 + σ>N2  σ

overall

N2  σ + σ>2N  σ

(P8.6.2)

H2 + 2σ>2H  σ

(P8.6.3)

H  σ + N  σ>NH  σ + σ

(P8.6.4)

H  σ + NH  σ>NH2  σ + σ

(P8.6.5)

H  σ + NH2  σ>NH3  σ + σ

(P8.6.6)

NH3  σ>NH3 + σ

(P8.6.7)

N2 + 3H2

(P8.6.8)

2NH3

The adsorption of nitrogen, step (Eq. P8.6.1), is found to be the rate-limiting step. Derive a rate expression that is consistent with the mechanism based on an ideal surface. 8.7. Christoph and Baerns (1986) conducted extensive studies on the methanation of CO CO + 3H2 ! CH4 + H2 O

(P8.7.1)

on a supported Ni catalyst in the presence of a large excess of H2 (H2/CO > 25) and temperature of about 500 K. A correlation proposed that fit their kinetic data is given by rCO ¼ 

2 aC0:5 CO Cσ

d + bC0:5 CO

2

(P8.7.2)

448

8. CHEMICAL REACTIONS ON SOLID SURFACES

and the catalyst decay follows (P8.7.3) rσ ¼ kd CCO Cσ Suggest a mechanism for the methanation that is consistent with this rate expression and a mechanism for the decay of the catalyst activity. 8.8. (a) For the decomposition of NH3 (A) on Pt (as a catalyst), what is the form of the rate law, according to the Langmuir-Hinshelwood model, if NH3 (the reactant) is weakly adsorbed and H2 (the product) is strongly adsorbed on Pt? Explain briefly. Assume N2 does not affect the rate. (b) Do the following experimental results, obtained by Hinshelwood and Burk (1925) in a constant-volume batch reactor at 1411 K, support the form used in (a)? t, s

0

10

60

120

240

360

720

P, kPa

26.7

30.4

34.1

36.3

38.5

40.0

42.7

P is total pressure, and only NH3 is present initially. Justify your answer quantitatively, for example, by using the experimental data in conjunction with the form given in (a). Use partial pressure as a measure of concentrations. 8.9. Dehydration of alcohol over an alumina-silica catalyst can be employed to produce alkenes. The dehydration of n-butanol was investigated by J.F. Maurer (PhD thesis, University of Michigan). The data shown in Fig. P8.9 were obtained at 750 °F in a modified differential reactor. The feed consisted of pure butanol. a) Suggest a mechanism and rate-controlling step that is consistent with the data. b) Evaluate the rate law parameters. 0.8

r'A0, mol/h g-Catalyst

0.6

0.4

0.2

0 0

50

100 PA0, atm

150

200

FIG. P8.9 Dehydration rate of butanol (A) as a function of butanol feed pressure.

449

PROBLEMS

8.10. The catalytic dehydration of methanol (M) to form dimethyl ether (DME) and water (W) was carried out over an ion exchange catalyst (Klusacek, 1984). The packed bed was initially filled with nitrogen gas (N) and at t ¼ 0, a feed of pure methanol vapor entered the reactor at 413 K, 100 kPa, and 0.2  106 m3/s. The following partial pressure data were recorded at the exit of the differential reactor containing 1.0 g of catalyst in a 4.5  106 m3 of reactor. t, s

0

10

50

100

150

200

300

PN, kPa

100

50

10

2

0

0

0

PM, kPa

0

2

15

23

25

26

26

PW, kPa

0

10

15

30

35

37

37

PDME, kPa

0

38

60

45

40

37

37

8.11. In the decomposition of N2O on Pt, if N2O is weakly adsorbed and O2 is moderately adsorbed, what form of rate law would be expected based on a LHHW mechanism? Explain briefly. 8.12. Rate laws for the decomposition of PH3 (A) on the surface of Mo (as a catalyst) in the temperature range 843-918 K are as follows Pressure pA, kPa

Rate Law

0

rA ¼ kpA 3

8  10

rA ¼ kpA/(a + bpA)

2

rA ¼ constant

2.6  10

Interpret these results in terms of an LHHW mechanism. 8.13. Consider the following overall reaction 1 A2 + B ! C + D 2 The mechanism has the following features: (i) Molecules of A2 dissociatively adsorb on two type 1 sites (one A atom on each site). (ii) Molecules of B adsorb on type 2 sites. (iii) The rate-determining step is the surface reaction between adsorbed A and adsorbed B. (iv) Molecules of C adsorb on type 1 sites and molecules of D adsorb on type 2 sites Derive an LHHW type rate expression that is consistent with this mechanism. 8.14. The solid-catalyzed exothermic reaction catalyst


! A2 + 2B












2D is virtually irreversible at low temperatures and low product concentrations, and the rate law is given by

450

8. CHEMICAL REACTIONS ON SOLID SURFACES 1=2

kCA2 CB r ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 1 + KA2 CA2 + KB CB (a) How manypactive ffiffiffiffiffiffiffiffi sites are involved in the rate-limiting step? (b) Why does CA2 appear in the denominator of the rate law? How many species are adsorbed on the catalyst surface? (c) Suggest a rate law that is valid for a wide range of temperatures and concentrations. (d) Suggest a mechanism for the rate law you have suggested. Identify the ratelimiting step. (e) Show that your mechanism is consistent with the rate law. 8.15. The synthesis of ammonia from nitrogen and hydrogen over a Ru/C catalyst is found to follow the following steps

overall

N2 + σ>N2  σ

(P8.15.1)

N2  σ + σ>2N  σ

(P8.15.2)

H2 + 2σ>2H  σ

(P8.15.3)

H  σ + N  σ>NH  σ + σ

(P8.15.4)

H  σ + NH  σ>NH2  σ + σ

(P8.15.5)

H  σ + NH2  σ>NH3  σ + σ

(P8.15.6)

NH3  σ>NH3 + σ

(P8.15.7)

N2 + 3H2

2NH3

(P8.15.8)

The surface reaction step (Eq. P8.15.4) is found to be the rate-limiting step. Derive a rate expression that is consistent with the mechanism based on nonideal surfaces with UniLan isotherm. 8.16. The rate law for the hydrogenation (H) of ethylene (E) to form ethane (A) over a cobaltmolybdenum catalyst is found to be r0A

  1 + K1 PE ¼ kPE PH ln 1 + K2 PE

Suggest a mechanism and rate-limiting step consistent with the rate law. Describe the feature of the catalysts surface.