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...