Because chemisorption is usually a necessary part of a catalytic process, we shall discuss it before treating catalytic reaction rates. The letter S will represent an active site; alone, it will denote a vacant site, with no atom, molecule, or complex adsorbed on it. The combination of S with another letter (e.g., A · S) will mean that one unit of species A will be adsorbed on the site S. Species A can be an atom, molecule, or some other atomic combination, depending on the circumstances. Consequently, the adsorption of A on a site S is represented by
The total molar concentration of active sites per unit mass of catalyst is equal to the number of active sites per unit mass divided by Avogadro’s number and will be labeled Ct (mol/gcat). The molar concentration of vacant sites, Cv, is the number of vacant sites per unit mass of catalyst divided by Avogadro’s number. In the absence of catalyst deactivation, we assume that the total concentration of active sites remains constant. Some further definitions include
Pi = partial pressure of species i in the gas phase, (atm or kPa)
Ci·S = surface concentration of sites occupied by species i, (mol/g cat)
A conceptual model depicting species A and B on two sites is shown in Figure 10-10.
Figure 10-10. Vacant and occupied sites.
For the system shown in Figure 10-10, the total concentration of sites is
This equation is referred to as a site balance.
Now consider the adsorption of a nonreacting gas onto the surface of a catalyst. Adsorption data are frequently reported in the form of adsorption isotherms. Isotherms portray the amount of a gas adsorbed on a solid at different pressures but at one temperature.
First, a model system is proposed, and then the isotherm obtained from the model is compared with the experimental data shown on the curve. If the curve predicted by the model agrees with the experimental data, the model may reasonably describe what is occurring physically in the real system. If the predicted curve does not agree with the experimental data, the model fails to match the physical situation in at least one important characteristic and perhaps more.
To illustrate the difference between molecular adsorption and dissociative adsorption, we will postulate two models for the adsorption of carbon monoxide on metal surfaces. In one model, CO is adsorbed as molecules, CO,
as is the case on nickel
In the other, carbon monoxide is adsorbed as oxygen and carbon atoms instead of molecular CO.
as is the case on iron5
The former is called molecular or nondissociated adsorption (e.g., CO) and the latter is called dissociative adsorption (e.g., C and O). Whether a molecule adsorbs nondissociatively or dissociatively depends on the surface.
The adsorption of carbon monoxide molecules will be considered first. Because the carbon monoxide does not react further after being adsorbed, we need only to consider the adsorption process:
In obtaining a rate law for the rate of adsorption, the reaction in Equation (10-2) can be treated as an elementary reaction. The rate of attachment of the carbon monoxide molecules to the active site on the surface is proportional to the number of collisions that these molecules make with a surface active site per second. In other words, a specific fraction of the molecules that strike the surface become adsorbed. The collision rate is, in turn, directly proportional to the carbon monoxide partial pressure, PCO. Because carbon monoxide molecules adsorb only on vacant sites and not on sites already occupied by other carbon monoxide molecules, the rate of attachment is also directly proportional to the concentration of vacant sites, Cv. Combining these two facts means that the rate of attachment of carbon monoxide molecules to the surface is directly proportional to the product of the partial pressure of CO and the concentration of vacant sites; that is,
Rate of attachment = kAPCOCv
PCO = CCORT
The rate of detachment of molecules from the surface can be a first-order process; that is, the detachment of carbon monoxide molecules from the surface is usually directly proportional to the concentration of sites occupied by the adsorbed molecules (e.g., CCO · S):
Rate of detachment = k–ACCO · S
The net rate of adsorption is equal to the rate of molecular attachment to the surface minus the rate of detachment from the surface. If kA and k–A are the constants of proportionality for the attachment and detachment processes, then
The ratio KA = kA/k–A is the adsorption equilibrium constant. Using KA to rearrange Equation (10-3) gives
The adsorption rate constant, kA, for molecular adsorption is virtually independent of temperature, while the desorption constant, k–A, increases exponentially with increasing temperature. Consequently, the equilibrium adsorption constant KA decreases exponentially with increasing temperature.
Because carbon monoxide is the only material adsorbed on the catalyst, the site balance gives
At equilibrium, the net rate of adsorption equals zero, i.e., rAD ≡ 0. Setting the left-hand side of Equation (10-4) equal to zero and solving for the concentration of CO adsorbed on the surface, we get
Using Equation (10-5) to give Cv in terms of CCO · S and the total number of sites Ct, we can solve for the equilibrium value of CCO · S in terms of constants and the pressure of carbon monoxide:
CCO · S = KACυPCO = KAPCO(Ct – CCO · S)
Rearranging gives us
This equation thus gives the equilibrium concentration of carbon monoxide adsorbed on the surface, CCO · S, as a function of the partial pressure of carbon monoxide, and is an equation for the adsorption isotherm. This particular type of isotherm equation is called a Langmuir isotherm.6 Figure 10-11(a) shows the Langmuir isotherm for the amount of CO adsorbed per unit mass of catalyst as a function of the partial pressure of CO.
Figure 10-11. Langmuir isotherms for (a) molecular adsorption (b) dissociative adsorption of CO.
One method of checking whether a model (e.g., molecular adsorption versus dissociative adsorption) predicts the behavior of the experimental data is to linearize the model’s equation and then plot the indicated variables against one another. For example, Equation (10-7) may be arranged in the form
and the linearity of a plot of PCO/CCO · S as a function of PCO will determine if the data conform to a Langmuir single-site isotherm.
Next, we derive the isotherm for carbon monoxide disassociating into separate atoms as it adsorbs on the surface, i.e.,
When the carbon monoxide molecule dissociates upon adsorption, it is referred to as the dissociative adsorption of carbon monoxide. As in the case of molecular adsorption, the rate of adsorption is proportional to the pressure of carbon monoxide in the system because this rate is governed by the number of gaseous collisions with the surface. For a molecule to dissociate as it adsorbs, however, two adjacent vacant active sites are required, rather than the single site needed when a substance adsorbs in its molecular form. The probability of two vacant sites occurring adjacent to one another is proportional to the square of the concentration of vacant sites. These two observations mean that the rate of adsorption is proportional to the product of the carbon monoxide partial pressure and the square of the vacant-site concentration, .
For desorption to occur, two occupied sites must be adjacent, meaning that the rate of desorption is proportional to the product of the occupied-site concentration, (C · S) × (O · S). The net rate of adsorption can then be expressed as
Factoring out kA, the equation for dissociative adsorption is
where
For dissociative adsorption, both kA and k–A increase exponentially with increasing temperature, while KA decreases with increasing temperature.
At equilibrium, rAD ≡ 0, and
For CC · S = CO · S,
Substituting for CC · S and CO · S in a site balance Equation (10-1),
Ct = Cυ + CO · S + CC · S = Cυ + (KCOPCO)½CυKCOPCO½Cυ = Cυ(1 + 2(KCOPCO)½)
Solving for Cυ
Cυ = Ct/(1 + 2(KCOPCO)½)
This value may be substituted into Equation (10-10) to give an expression that can be solved for the equilibrium value of CO · S. The resulting equation for the isotherm shown in Figure 10-11(b) is
Taking the inverse of both sides of the equation, then multiplying through by (PCO)½, yields
If dissociative adsorption is the correct model, a plot of versus should be linear with slope (2/Ct).
When more than one substance is present, the adsorption isotherm equations are somewhat more complex. The principles are the same, though, and the isotherm equations are easily derived. It is left as an exercise to show that the adsorption isotherm of A in the presence of another adsorbate B is given by the relationship
When the adsorption of both A and B are first-order processes, the desorptions are also first order, and both A and B are adsorbed as molecules. The derivations of other Langmuir isotherms are relatively easy.
In obtaining the Langmuir isotherm equations, several aspects of the adsorption system were presupposed in the derivations. The most important of these, and the one that has been subject to the greatest doubt, is that a uniform surface is assumed. In other words, any active site has the same attraction for an impinging molecule as does any other active site. Isotherms different from the Langmuir isotherm, such as the Freundlich isotherm, may be derived based on various assumptions concerning the adsorption system, including different types of nonuniform surfaces.
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