Adsorption of Gases on Heterogeneous Surfaces

By W. Rudzinski and D. H. Everett

All real solid surfaces are heterogeneous to a greater or lesser extent and this book provides a broad yet detailed survey of the present state of gas adsorption. Coverage is comprehensive and extends from basic principles to computer simulation of adsorption. Underlying concepts are clarified and the strengths and weaknesses of the various methods described are discussed.

• Adsorption isotherm equations for various types of heterogeneous solid surfaces
• Methods of determining the nature of surface heterogeneity and porosity from experimental data
• Studies of phase behavior of gases absorbed on heterogeneous solid surfaces
• Computer simulation of adsorption on heterogeneous solid surfaces

• Language: English
• Publisher: Academic Press
• Release date: Dec 2, 2012
• ISBN: 9780080984360

Book preview

1988.

Chapter 1

Basic Principles

Publisher Summary

This chapter provides an overview of some of the basic thermodynamic principles and the adsorption of gases on heterogeneous surfaces. The importance of the isosteric enthalpy of adsorption is that it can be related to enthalpies of adsorption measured calorimetrically. The exact relationship depends on the type of experimental arrangement employed. The interaction energy of an isolated adsorbed molecule with the adsorbent can be described by a potential function U(x, y, z), where x, y, and z are the Cartesian coordinates of the adsorbed molecule. If the oscillations in U are much less than kT (b), the surface may be described as a homogeneous periodic surface. On the other hand, if the oscillations in U greatly exceed kT (c), the local Um are usually called adsorption sites or adsorption. The vibrations of adsorbed molecules are usually assumed to be harmonic, although in real systems, appreciable anharmonicity effects must be present except at very low temperatures.

1.1 Introduction

Before embarking on a discussion of the adsorption of gases on heterogeneous surfaces, it is first necessary to outline the main ideas upon which theories of adsorption on homogeneous surfaces are based. To do this we need a brief preliminary discussion of the thermodynamics of adsorption, followed by a summary of the two main lines of approach to a statistical mechanical theory based on the alternative models of localised adsorption on ‘active sites’ of equal adsorption energy, or of a completely mobile adsorbed layer.

These will act as reference points against which experimental data are to be compared. Failure of these theories to provide an adequate representation of the observed behaviour will be the motivation for questioning the assumption of homogeneity of the surface, an assumption that is common to both theories in their simplest form.

1.2 Thermodynamics of adsorption

We begin by summarising some of the basic thermodynamic relationships which will be employed in this book. Their formulation for adsorption processes is not as straightforward as for bulk phases. It is therefore not surprising that, although, following Gibbs,¹ several aspects of surface thermodynamics had been studied by Butler² and Guggenheim,³ there was until the 1940s no clear thermodynamic treatment of the physical adsorption of gases by solids. In outlining the main features of this problem we follow mainly the work of Hill,⁴–⁷ Everett⁸–¹¹ and Rusanov.¹²

First we must decide how to approach the problem in thermodynamic terms: in particular how to define ‘the system’ precisely. In the most general treatment the system is regarded as comprising an adsorption chamber containing a known amount of adsorbent, into which a known amount of gas (the adsorptive) is admitted and comes into equilibrium at a measured pressure and temperature.¹³ The exchange of work and heat with the surroundings can be measured. While the temperature, pressure, total volume and amounts of adsorbent and adsorptive can be measured, a molecular interpretation involves a knowledge of that part of the system from which the adsorptive is excluded. This implies a knowledge of the volume occupied by the solid which cannot be determined directly under the conditions of the experiment, but must be deduced from, for example, the density of the adsorbent, or its behaviour towards a non-adsorbed gas such as He. If the adsorbent is porous than some uncertainty must remain as to whether its density is equal to the density of the bulk material, and whether molecular sieving effects mean that its excluded volume is different for He and the adsorptive molecule under consideration. These uncertainties must be present in any transition from experimental data to their theoretical interpretation. In practice, for many purposes, however, the errors introduced by conventional methods of estimating the excluded volume do not constitute a major problem.

The statistical mechanical approach relevant to this formulation is that which treats the system as a one-component system in which the vapour is subjected to an external force field due to the adsorbent.¹⁴ No precise distinction is made between ‘adsorbed’ and ‘non-adsorbed’ gas. In the interpretation of the experimental results one needs to know both the volume available to the gas and the area of the solid surface. This leads to the virial formulation that will be discussed in detail in a later chapter.

The alternative is to divide the system into two sub-systems: the solid adsorbent plus the adsorbed molecules (adsorbate), and the equilibrium bulk gas phase of freely moving molecules (adsorptive). The description is appropriate when there is an abrupt transition between a dense adsorbed phase and the non-adsorbed gas. The sub-system adsorbent plus adsorbate can then be treated as a separate thermodynamic system.¹⁵

A third way, which is a variant on the second, is to regard the adsorbate itself as a single system under the influence of an external field emanating from the adsorbent.³,¹⁶-¹⁸ It must be remembered, however, that quantities attributed to the adsorbate in this method are, strictly speaking, mutual properties of the adsorbate and the adsorbent, since it is impossible to separate thermodynamically the influence of adsorption on the properties of the adsorbent.

In the following we limit discussion to a one-component gas and consider the thermodynamics of a system consisting of the adsorbate plus adsorbent.¹⁹ We then consider the case in which the adsorbent is assumed to be inert, i.e. provides only a potential field in which the adsorbate molecules move.

The system contains an amount na of solid adsorbent, plus an amount ns of adsorbed gas at a temperature T enclosed in volume V. This system is in equilibrium with adsorptive at a temperature T and pressure p.

The differential of the energy, U, of such a condensed phase is

(1.2.1)

where S is the entropy of the system (i.e. adsorbate plus adsorbent), μs the chemical potential of the adsorbate, and μa the chemical potential of the adsorbent in the presence of the adsorbate.

For the pure adsorbent, in the absence of adsorbate,

(1.2.2)

where μ⁰,a is the chemical potential of the adsorbent when it is devoid of adsorbed molecules, V⁰,a the volume of the solid under these conditions, and S⁰,a its entropy.

The properties of the system relative to those in the absence of adsorbate can be defined by the differences:

(1.2.3)

Subtracting equation 1.2.2 from equation 1.2.1 we obtain

(1.2.4)

This formulation, in which, in equation 1.2.1, adsorbate and adsorbent are treated symmetrically, constitutes the ‘solution thermodynamic’ approach to the thermodynamics of adsorption.⁵

To make the transition to ‘adsorption thermodynamics’⁶ we assume that the adsorbent is completely inert. In that case Us, Ss, and Vs and their related functions characterise the thermodynamic properties of the ns adsorbed molecules only.

From equation 1.2.4 it follows that

(1.2.5)

φ represents the energy change under the specified conditions accompanying a unit increase in the amount of adsorbent in the same state as that already present, and hence implies a corresponding increase in the solid surface area.

It is reasonable to assume that for an inert adsorbent the surface area As, or the number M of ‘adsorption sites’ on the surface, will be proportional to na. (An exact definition of ‘adsorption site’ will be given later.)

Thus, defining Cα = na/As and CM = na/M,

(1.2.6)

or

(1.2.7)

It will be seen later that equation 1.2.6 is the more convenient form for testing ‘mobile adsorption’, while equation 1.2.7 applies to ‘localised adsorption’. The relationship of these concepts to the molecular kinetic state of the adsorbed molecules will be outlined in Section 1.3. Both ϕ and π may be termed spreading pressures, although their physical interpretation is simple only in the case of mobile adsorption.

For the sake of simplicity we shall for the moment confine our attention to the case of mobile adsorption.

We now introduce the following definitions: surface enthalpy, Hs; surface Helmholtz energy, Fs; and surface Gibbs energy, Gs.³,⁸

(1.2.8)

(1.2.9)

(1.2.10)

With these definitions we have

(1.2.11)

(1.2.12)

(1.2.13)

(1.2.14)

Keeping the intensive variables constant, one obtains after integration:

(1.2.15)

(1.2.16)

(1.2.17)

(1.2.18)

Sets of equations similar to equations 1.2.11 to 1.2.18 are obtained for localised adsorption by replacing ϕ dAs and ϕAs by π dM and πM respectively.

The application of standard thermodynamic procedures to the above equations leads to a number of other thermodynamic relations. Thus from equation 1.2.18 we have by differentiation:

(1.2.19)

which on comparison with equation 1.2.14 yields

(1.2.20)

or on introducing the surface concentration, (areal concentration) Γ = ns/As,

(1.2.21)

where vs = Vs/ns and ss = Ss/ns are mean molar quantities.

Since the adsorbed phase is in equilibrium with the bulk phase (μs = μg), we must also have, from the Gibbs-Duhem equation for the bulk phase:

(1.2.22)

here sg and vg are the mean molar quantities for the gas phase. If the gas is ideal, so that vg = RT/p, then at constant temperature,

(1.2.23)

Now vs can usually be neglected in comparison with vg so that, on integration,

(1.2.24)

where p† is the standard pressure, usually set equal to the unit of pressure. This is one form of the Gibbs adsorption isotherm, which can be written in the alternative forms:

(1.2.25)

where θ = bNs/As. Here Ns is the number of molecules in the adsorbed layer, and b is the area occupied by one molecule in the surface.

One way of handling experimental data is through equation 1.2.22 when, neglecting vs in comparison with vg,

(1.2.26)

If the gas is ideal then

(1.2.27)

For an equilibrium process carried out at constant T, ϕ (i.e. constant intensive variables),

(1.2.28)

where hs = Hs/ns is the mean molar enthalpy of the adsorbed molecules, and similarly hg is the mean molar enthalpy of the gas.

Thus

(1.2.29)

Δah is thus the difference between the mean molar enthalpies of the adsorbed species and the same species in the gas phase. To apply this equation it is necessary to convert experimental isotherms, ns as a function of p, into ϕ(p) isotherms using equation 1.2.24. Since this involves integrating the experimental isotherm from low pressures (p → 0), the precision with which Δah can be determined is very sensitive to the accuracy of the low-pressure region of the isotherm.

An alternative procedure which is more convenient in practice is to express μs as a function of T, p, ns and As:

(1.2.30)

Now

(1.2.31)

is a differential molar entropy, to be distinguished from the mean molar entropy Ss/ns.

Furthermore,

(1.2.32)

the differential molar volume of the adsorbate. At constant Γ, we have from equation 1.2.30

(1.2.33)

At equilibrium μs is equal to the chemical potential of the gas:

(1.2.34)

It follows that, at constant Γ,

(1.2.35)

Again neglecting the volume of the adsorbate in comparison with that of the gas, and assuming the gas to be ideal,

(1.2.36)

Since the process of transferring adsorptive from the gas to the adsorbed state is an equilibrium process,

(1.2.37)

Equation 1.2.36 can then be written

(1.2.38)

, the differential enthalpy with reversed sign, the isosteric enthalpy of adsorption and to denote it by qst.²⁰ (It should, strictly, be called the isosteric enthalpy of desorption.) The term ‘isosteric’ derives from the fact that the differential in equation 1.2.38 is taken at constant coverage ns/As of the surface. The usual method of calculating qst from experimental isotherms is to plot ln (p/p†) for a given Γ as a function of 1/T, whence

(1.2.39)

The relationship between Δah follows from

(1.2.40)

Differentiating equations 1.2.21 and 1.2.33 with respect to T at constant p, Γ:

(1.2.41)

Substitution in equation 1.2.40 gives⁵,⁸

(1.2.42)

Equations 1.2.29 and 1.2.39 are of the Clausius-Clapeyron type, and in their derivation the volumes vhave been neglected as have deviations of the gas phase from ideality. The latter approximation, which may lead to errors in qst of 1-2%, can be taken care of in practice where higher precision is needed by introducing the second virial coefficient of the gas phase, or at higher pressures by replacing p by p*, the fugacity. On the other hand there is no unambiguous way of introducing the molar volume of the adsorbate: one might set it equal to that of bulk liquid, but this is obviously a crude approximation which is unrealistic at low coverages. This ambiguity is a consequence of the physical unreality of treating the adsorbate as a separate phase, and constitutes a fundamental shortcoming of the method.

The importance of the isosteric enthalpy of adsorption is that it can be related to enthalpies of adsorption measured calorimetrically. The exact relationship depends on the type of experimental arrangement employed. In an isothermal volume calorimeter (Figure 1.1), in which the reversible adsorption of an amount of gas dns is accompanied by an exchange of heat dQs with the surroundings, the differential enthalpy of adsorption is given by:²¹

Figure 1.1 Schematic arrangement of gas adsorption calorimeters. (a) Isothermal heat flux calorimeter: vapour from A is fed through a heat exchanger B to the calorimeter vessel C containing the adsorbent D. The heat flux through the thermopile E to the constant temperature heat jacket is measured (Calvet principle). (b) Adiabatic calorimeter: the calorimetric vessel is thermally isolated by an evacuated space from the jacket whose temperature is programmed to follow that of the calorimeter so that no heat flow occurs between them.

(1.2.43)

In the case of an adiabatic calorimeter, adsorption is accompanied by a rise in temperature and

(1.2.44)

where Cc is the constant pressure heat capacity of the calorimeter and the adsorbent, cgp the molar heat capacity of the gas, and csp the molar heat capacity of the adsorbate at constant pressure and area. The relationship between the isothermal enthalpy and the adiabatic enthalpy changes is²²

(1.2.45)

For many purposes we may characterise an adsorbent system in terms of the quantities ns(p) and qst(ns).

A further observable quantity is the heat capacity of the adsorbate, Csp. This may be measured experimentally in specially designed calorimeters²³ as the difference between the heat capacities of the calorimeter plus adsorbent in the presence and absence of adsorptive. It is related to the temperature dependence of qst:

(1.2.46)

and its interpretation will be discussed later (Chapter 9).

In discussing the thermodynamics of adsorption it is often convenient to direct attention to the enthalpies and entropies of adsorption.(or q. This is defined as the entropy change on going from gas at the standard pressure p† to the adsorbed phase. The steps, reversible compression of the gas from p† to p, and reversible adsorption at p, make two contributions:

(1.2.47)

We note that this is the integrated form of equation 1.2.38 and can be written:

(1.2.48)

1.3 The molecular state of adsorbed species

The interaction energy of an isolated adsorbed molecule with the adsorbent can be described by a potential function U(x, y, z), where x, y and z are the Cartesian coordinates of the adsorbed molecule. In the case of a plane surface, this is taken to be the (x, y)-plane, while the z-axis is normal to the surface.

Surfaces can be characterised by the form of U(x, y, z). The general form of U(z) for a plane surface is shown schematically in Figure 1.2.

Figure 1.2 Adsorption potential as a function of distance, z, of a molecule from a surface. ΔU⁰ is the depth of the potential energy well. ΔUvib0 is the zero point vibrational energy, ΔUvib the vibrational energy at a finite temperature T. The energy of the gas includes a kinetic energy contribution (3/2)RT.

The dependence of U on x and y may exhibit various characteristics. In the hypothetical case of a perfectly homogeneous surface U(x, y) at a given z⁰, U⁰ is constant (Figure 1.3a). Because of the molecular structure of real surfaces, U will, however, be a function of x and y. If the surface is uniform in the sense that it has a periodic molecular structure, then U(x, y, z⁰) will be a periodic function of the vector i(x, y)²⁵ (Figure 1.3b, c). If the oscillations in U are much less than kT(b), the surface may be described as a homogeneous periodic surface. On the other hand, if the oscillations in U greatly exceed kT (c), then the local Um are usually called adsorption sites, or adsorption centres. Adsorbed molecules are effectively trapped on these sites. The difference between the energy of these minima and the average energy of molecules in the equilibrium bulk state is the energy of site adsorption, and is denoted by ΔU⁰. The energy barrier between sites, which controls lateral diffusion, is called the activation energy for surface diffusion ΔV⁰. If the local minima are characterised by a constant value of Um the surface may be described as a homogeneous site surface. A further possibility is that the periodicity may be characterised by minima having different values of Um (Figure 1.3d): it may be called a heterogeneous periodic surface. Many real surfaces will exhibit defects of various kinds distributed at random on the surface: U(x, y, z⁰) is no longer periodic and the surface is described as a random heterogeneous surface (Figure 1.3e). Finally, a surface may expose different crystal faces, each of which is a homogeneous surface: the distribution of site energies will now be represented by Figure 1.3f. The surface is patchwise heterogeneous or homotattic.²⁶

Figure 1.3 Variation of ΔU⁰ across the surface of an adsorbing solid: (a) perfectly homogeneous surface; (b) homogeneous periodic surface; (c) homogeneous site surface; (d) heterogeneous periodic surface; (e) random heterogeneous surface; (f) patchwise or homotattic heterogeneous surface corresponding to (a).

The kinetic motion of adsorbed molecules is controlled by the energetic topography of the surface. Thus in the case of a completely homogeneous, or homogeneous periodic, surface, for which ΔVkT, adsorbed molecules will be able to move freely in the (x, y)-plane, but if |ΔUkT then their motion in the z-direction is restricted. Adsorbed phases in which the molecules retain two translational degrees of freedom are classified as mobile adsorbed phases. On the other hand, if ΔVkT, lateral translation of the adsorbed molecules is hindered, and the two translational degrees of freedom parallel to the surface are transformed into vibrations in the x- and y-directions. The adsorbed phase is said to be localised. It must be stressed that except in extreme cases, the adsorbed molecules are not completely immobile since diffusion from site to site can still occur when a molecule acquires an excess energy greater than ΔV⁰. From a statistical point of view, however, at any instant the vast majority of molecules are located on or close to an adsorption site, and undergoing vibrations in three directions relative to the site.

The magnitude of ΔU⁰ relative to kT is relevant in determining the appropriate theoretical approach. Thus, if |ΔU⁰|/kT 1, there will be a sharp change in density near the adsorbent surface, and one may be justified in treating the adsorbate as a separate ‘adsorbed phase’. On the other hand, for much smaller values of |ΔU⁰|/kT there will be no rapid change in the density of the adsorbed molecules, but the local concentration will fall off steadily. From a statistical point of view the system has to be treated as a bulk fluid in the presence of an external field due to the presence of the adsorbent.

Since the behaviour of the system depends on ΔV⁰/kT and |ΔU⁰|/kT, a system may be localised at low temperatures but become mobile at high temperatures; similarly, while the model of an adsorbed phase may be acceptable at low temperatures, it may become necessary to adopt a different approach at high temperatures.

There are, of course, intermediate regimes where the molecules undergo restricted translation, or whose concentration near the surface is becoming diffuse. A further complication can arise in the case of heterogeneous surfaces where a wide range of values of |ΔU⁰|/kT may exist so that no simple model can be employed.

For the present purposes we shall begin by treating the adsorbed layer as a separate phase of vibrating molecules in the presence of an external potential field, and in equilibrium with a bulk phase of freely translating molecules. More complicated situations will be dealt with in later chapters.

The vibrations of adsorbed molecules are usually assumed to be harmonic, although in real systems appreciable anharmonicity effects must be present except at very low temperatures. The energy levels of a one-dimensional oscillator, εn, are

(1.3.1)

where v is the frequency and h is Planck’s constant. The partition function of a one-dimensional oscillator is

(1.3.2)

At high temperatures, T hv/k and

(1.3.3)

while at low temperatures, T hv/k, so that

(1.3.4)

where hv/2 is the zero point (ground state) energy of the harmonic oscillator.

Thus for a simple monatomic adsorbate and mobile monolayer adsorption the molecular partition function of an isolated adsorbed molecule of mass m, qs, will be

(1.3.5)

where af is the ‘free surface area’ or the area of the surface available for translation: the term af/Λ² arises from the two translations of the molecules parallel to the surface. qz is the partition function for vibration at a frequency vz normal to the surface, while qs(i) is the partition function for electronic and nuclear degrees of freedom.

In the case of localised adsorption,

(1.3.6)

where qx, qy refer to vibrations along the x- and y-axes respectively.

If we are dealing with more complicated molecules qs must be multiplied by the partition function qs(e) for internal vibrations and rotational degrees of freedom of the molecule. The transition from bulk to the adsorbed phase is often accompanied by the transformation of some rotational degrees of freedom into the corresponding vibrations (librations). In principle, transitions from internal vibrations to rotations might occur on adsorption, although there is so far no experimental evidence for this.

1.4 Theories of monolayer adsorption on homogeneous surfaces

We now summarise some well-known results of the theory of adsorption on homogeneous surfaces.¹⁹,²⁷ Because problems associated with lateral interactions and surface heterogeneity can very quickly complicate the picture, we begin by developing only the simplest theories of localised and mobile adsorption.

1.4.1 Localised adsorption

The theories of localised adsorption are mainly based on lattice statistics, and constitute a group of ‘lattice gas’ theories based on the Ising model. Excellent detailed reviews of these theories are to be found in the monograph by Clark¹⁹ and in the review by Domb.²⁸

Following Fowler and Guggenheim,²⁹ we consider a lattice of M sites, in which every site has on average c nearest-neighbour adsorption sites. Let N denote the number of molecules adsorbed on this lattice at a temperature T. If the N adsorbed molecules are distributed among the M sites in a particular configuration in which N11 pairs are on adjoining sites, the potential energy of interaction between them is (N11 w), where w is the energy of interaction between molecules on adjacent sites. The canonical partition function of the system will be Q(T, M, N), since the system is characterised by these variables. One possible way of writing the function Q(T, M, N) is as follows:

(1.4.1)

where g(M, N, N11) is the number of configurations containing N11 adjoining pairs.

One may also introduce a new variable, N01, which is the number of adjoining pairs of sites in which one is empty and the other occupied. For an arbitrary lattice of coordination number c the following relationships hold:

(1.4.2)

(1.4.3)

Equation 1.4.1 can be written in the alternative form

(1.4.4)

where g(M, N, N01) denotes the number of configurations with N01 pairs of half-occupied adjacent pairs. The problem now is to find the explicit form for g(M, N, N01). No exact general solution is known, and hence only approximate solutions are available at present.

The simplest approximate solution that retains some of the basic features of the model is the Bragg-Williams³⁰ approximation. In this the configurational degeneracy and the average nearest-neighbour interaction energy are both handled on the basis of a random distribution of molecules among the adsorption sites, an assumption which is strictly applicable only in the limit T → ∞. With this assumption, in each configuration, every adsorbed molecule has on average (cN/M) nearest neighbours on the surface. This enables equation 1.4.1 to be written in the form:

(1.4.5)

where

(1.4.6)

In the above the factor of 1/2 is introduced to avoid counting each pair twice.

The thermodynamic properties of the system can be derived from the canonical partition function by applying the basic