Partition function adsorbed molecule

The statistical thermodynamic treatment of the BET theory has the advantage that it provides a satisfactory basis for further refinement of the theory by, say, allowing for adsorbate-adsorbate interactions or the effects of surface heterogeneity. By making the assumptions outlined above, Steele (1974) has shown that the problems of evaluating the grand partition function for the adsorbed phase could be readily solved. In this manner, he arrived at an isotherm equation, which has the same mathematical form as Equation (4.32). The parameter C is now defined as the ratio of the molecular partition functions for molecules in the first layer and the liquid state. 

For physical adsorption of apolar molecules this is probably not a bad approximation. The usual picture is that the adsorbate molecules are piled on top of each other, but the model is somewhat less restrictive. The pile does not have to be straight. In fact, the adsorbate-adsorbate enthalpy is commonly identified as the heat of condensation, which does not correspond to a pair interaction but interaction with more molecules. With this interpretation in mind, tlie necessity of piling molecules on top of each other disappears (molecules can even bridge neighboring sites) but at the expense of losing the connection with the premises of the theory (no lateral interaction) and the rigorous notion of adsorbate-adsorbate pair with the premises of the theory (no lateral interaction). At any rate, mathematically the assumption implies that if we denote the molecular partition functions of molecules in contact with the surface by Qj, those in layer 2, 3,. .. are all Identical, say Q2. Hence q(l) = qj, q[2] = q q. q 3) = q q, ... etc. Substitution in [1.5.38] yields... 

The following derivation is modified from that of Fowler and Guggenheim [10,11]. The adsorbed molecules are considered to differ from gaseous ones in that their potential energy and local partition function (see Section XVI-4A) have been modified and that, instead of possessing normal translational motion, they are confined to localized sites without any interactions between adjacent molecules but with an adsorption energy Q. 

All molecules in the second and subsequent layers are assumed to behave similarly to a liquid, in particular to have the same partition fimction. This is assumed to be different to the partition function (A2.2) of molecules adsorbed into the first layer. 

In the present study we try to obtain the isotherm equation in the form of a sum of the three terms Langmuir s, Henry s and multilayer adsorption, because it is the most convenient and is easily physically interpreted but, using more a realistic assumption. Namely, we take the partition functions as in the case of the isotherm of d Arcy and Watt [20], but assume that the value of V for the multilayer adsorption appearing in the (5) is equal to the sum of the number of adsorbed water molecules on the Langmuir s and Henry s sites ... 

Note that a diatomic molecule in the gas phase has only one vibration, but as soon as it adsorbs on the surface it acquires several more modes, some of which may have quite low frequencies. The total partition function of vibration then becomes the product of the individual partition functions ... 

In the following we consider a surface with adsorbed atoms or molecules that react. We will leave out the details of the internal coordinates of these adsorbed species, but note that their partition functions can be found using the schemes presented above. Let us assume that species A reacts with B to form an adsorbed product AB via an activated complex AB ... 

Hence, the properties of the molecularly adsorbed N2 cancel as soon as we take kj Ki together, which is the relevant term in the formation of atomic nitrogen. Similarly, but on a much larger scale, partition functions cancel in the term hi Eqs. (51) and (52). Returning to Eq. (59), the factor of two arises because the rate describes the number of nitrogen atoms, whereas the transition state refers to the molecule, which dissociates into two atoms. 

Under ivhat conditions are the partition functions for translation, rotation, and vibration of an adsorbed molecule (a) dose to unity, (b) moderate, and (c) large ... 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

One degree of freedom of the adsorbed molecule serves as the reaction coordinate. For desorption, the reaction coordinate is the vibration of the molecule with respect to the substrate. In the transition state this vibration is highly excited and the chance that the adsorption bond breaks is given by the factor kT/h. All other degrees of freedom of the excited molecule are in equilibrium with those of the molecule in the ground state and are accounted for by their partition functions. 

Note that gtrans is given per degree of freedom, implying that the total translational partition function for an adsorbed molecule is given by (Qtrms)2- Also, the total partition functions for vibration and rotation are the products of terms for each individual vibration and rotation, respectively. Table 2.2 gives values for the partition functions for adsorbed atoms and molecules at 500 K. Vibrational partition functions are usually close to one, but rotational and translational partition functions have larger values. 

We use this knowledge to derive preexponential factors from (2-20) for a few desorption pathways (see Fig. 2.15). The simplest case arises if the partition functions Q and Q in (2-20) are about equal. This corresponds to a transition state that resembles the ground state of the adsorbed molecule. In order to compare (2-20) with the Arrhenius expression (2-15) we need to apply the definition of the activation energy ... 

Figure 2.15 Microscopic pictures of the desorption of atoms and molecules via mobile and immobile transition states. If the transition state resembles the ground state, we expect a prefactor of desorption on the order of 1013 s. If the adsorbates are mobile in the transition state, the prefactor goes up by one or two orders of magnitude. In the case of desorbing molecules, free rotation in the transition state increases the prefactor even further. The prefactors are roughly characteristic of atoms such as C, N and O and molecules such as N2, CO, NO and 02. See also the partition functions in Table 2.2 and the prefactors for CO desorption in Table 2.3.

We first treat the most general case where all sites are different. The grand partition function of a single adsorbent molecule is... 

The molecular approach, adopted throughout this book, starts from the statistical mechanical formulation of the problem. The interaction free energies are identified as correlation functions in the probability sense. As such, there is no reason to assume that these correlations are either short-range or additive. The main difference between direct and indirect correlations is that the former depend only on the interactions between the ligands. The latter depend on the maimer in which ligands affect the partition function of the adsorbent molecule (and, in general, of the solvent as well). The argument is essentially the same as that for the difference between the intermolecular potential and the potential of the mean force in liquids. 

The statistical mechanical approach starts from more fundamental ingredients, namely, the molecular properties of all the molecules involved in the binding process. The central quantity of this approach is the partition function (PF) for the entire macroscopic system. In particular, for binding systems in which the adsorbent molecules are independent, the partition function may be expressed as a product of partition functions, each pertaining to a single adsorbent molecule. The latter function has the general form... 

They showed that b0 may be expressed statistically in terms of the partition function of the molecule in the adsorbed state Sa less the Boltzmann factor, cx/ST, the partition function of the molecules in the... 

Barrer (3) makes similar calculations for the entropies of occlusion of substances by zeolites and reaches the conclusion that the adsorbed material is devoid of translational freedom. However, he uses a volume, area or length of unity when considering the partition function for translation of the adsorbed molecules in the cases where they are assumed to be capable of translation in three, two or one dimensions. His entropies are given for the standard state of 6 = 0.5, and the volume, area or length associated with the space available to the adsorbed molecules should be of molecular dimensions, v = 125 X 10-24 cc., a = 25 X 10-16 cm.2 and l = 5 X 10-8 cm. When these values are introduced into his calculations the entropies in column four of Table II of his paper come much closer together, as is shown in Table I. The experimental values for different substances range from zero to —7 cals./deg. mole or entropy units, and so further examination is required in each case to decide... 

The Langmuir isotherm can be derived from a statistical mechanical point of view. Thus, for the reaction M + Agas Aads, equilibrium is established when the chemical potential on both phases is the same, i.e., pgas = p,ads. The partition function for the adsorbed molecules as a system is given by... 

The surface is assumed to consist of M adsorption sites. Suppose we consider the case in which TV of the sites are occupied that is, TV molecules are adsorbed. To write the partition function Q for the surface molecules, we must ask how these molecules differ from those in the gas phase (superscript g). Some of the internal degrees of freedom may be modified by the adsorption ((/, ,), but the most notable difference will be in the translational degrees of freedom. From three equivalent translational degrees of freedom, the adsorbed molecule goes to two highly restrained translational degrees of freedom (remember the adsorption is localized) and one vibrational degree of freedom normal to the surface(s) ... 

Combining Equations (22) and (23) gives the following expression for the partition function of the adsorbed molecules ... 

The quantity K defined by Equation (31) may easily be expanded somewhat further. First, we write the two-dimensional partition function by analogy with its three-dimensional counterpart, Equation (16). To do this, replace V by the area accessible to the adsorbed molecule a and the exponent by 2/2 (= 1) in place of 3/2 since two rather than three degrees of freedom are involved. Therefore we obtain... 

Let qs be the molecular partition function for an adsorbed species. Consider the adsorption of N molecules on some portion of the surface containing a total of M possible adsorption sites. The system partition function Qs of the collection of N adsorbed species is... 

The partition function for a system of Na adsorbed and Nf free polymer molecules is given by... 

In all other cases to has the dimensions but not the meaning of a reciprocal frequency (193). The time of adsorption can be calculated by means of statistical mechanics from the partition functions of the gaseous and the adsorbed molecule (193). The equilibrium condition for the adsorption may be written as... 

The usual assumption is made that equals unity, where /and fa are the partition functions for the activated complex and adsorbed molecule respectively. 

Here E ( y1 ) stands for the single-particle contribution to the total energy, allowing for molecule interaction with the surface <2 is the heat released in adsorption of molecules z on the /Lh site Fj the internal partition function for the z th molecules adsorbed on the /Lh site F j the internal partition function for the zth molecule in the gas phase the dissociation degree of the z th molecule, and zz the Henry local constant for adsorption of the zth molecule on the /Lh site. Lateral interaction is modeled by E2k([ylj ), and gj (r) allows for interaction between the z th and /Lh particles adsorbed on the /th and gth sites spaced r apart. In the lattice gas model, separations are conveniently measured in coordination-sphere numbers, 1 < r < R. For a homogeneous surface, molecular parameters zz and ej(r) are independent of the site nature, while for heterogeneous, they may depend on it. 

Assuming that the adsorbed molecules have lost their three translational degrees of freedom, we calculate S from Eq. (32) of Chapter 5, taking the energy term as zero. The partition function for the molecules on the surface is given by Eqs. (71) and (72) ... 

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