Equilibrium gas pressure

Fugacity, like other thermodynamics properties, is a defined quantity that does not need to have physical significance, but it is nice that it does relate to physical quantities. Under some conditions, it becomes (within experimental error) the equilibrium gas pressure (vapor pressure) above a condensed phase. It is this property that makes fugacity especially useful. We will now define fugacity, see how to calculate it, and see how it is related to vapor pressure. We will then define a related quantity known as the activity and describe the properties of fugacity and activity, especially in solution. 

Adsorption experiments are conducted at constant temperature, and an empirical or theoretical representation of the amount adsorbed as a function of the equilibrium gas pressure is called an adsorption isotherm. Adsorption isotherms are studied for a variety of reasons, some of which focus on the adsorbate while others are more concerned with the solid adsorbent. In Chapter 7 we saw that adsorbed molecules can be described as existing in an assortment of two-dimensional states. Although the discussion in that chapter was concerned with adsorption at liquid surfaces, there is no reason to doubt that similar two-dimensional states describe adsorption at solid surfaces also. Adsorption also provides some information about solid surfaces. The total area accessible to adsorption for a unit mass of solid —the specific area Asp — is the most widely encountered result determined from adsorption studies. The energy of adsorbate-adsorbent interaction is also of considerable interest, as we see below. 

Although UHV is required for LEED measurement, there is considerable interest in applying this technique to surfaces that carry adsorbed species. In view of our discussion of adsorption equilibrium above in the chapter, there is a difficulty here since adsorbed molecules imply an equilibrium gas phase. One way around this problem is to study surfaces at a sufficiently low coverage that the equilibrium gas pressure is compatible with the LEED technique. When higher pressures are desired, the surface is first equilibrated, then the excess gas is pumped out, and the surfaces before and after adsorption are compared through LEED. Chemisorption is better suited for study by this method than physical adsorption because the adsorbed layer remains intact when the equilibrium gas is removed. 

In this method, a graph of the number of moles of gas adsorbed per gram of solid, at constant temperature, against the equilibrium gas pressure is called an adsorption isotherm. A point must be chosen on this isotherm corresponding to the completion of the adsorbed monolayer in order to calculate Sw [29]. 

FIGURE 5.3 Standard free energy of reaction as a function of temperature —equilibrium gas pressure above oxide/carbonate or oxide/hydroxide. Data from Kingery et al. [3], with additions by T. A. Ring. 

We now cite the types of experimental data in the literature, by which an analysis of surface adsorption effects is carried out. One common experiment involves measuring adsorption isotherms. By weighing or by volumetric techniques one determines as a function of equilibrium gas pressure the amount of gas held on a given surface at a specified temperature. Usually this quantity varies sigmoidally with rising pressure P, as sketched in Fig. 5.2.1 for a variety of temperatures 7). By standard methods that rely on the Brunauer, Emmett, Teller isotherm equa-tion one can determine the point on the isotherms at which monolayer coverage of the surface is complete it is usually is located fairly close to the knee of the isotherm. From the cross sectional area of the adsorbate molecules and from the amount needed for monolayer coverage one may then ascertain more or less quantitatively the surface area of the adsorbent. As-... 

This relation shows how the two-dimensional pressure may be determined through measurements of F at a sequence of equilibrium gas pressures F at a fixed temperature, either by graphical integration, or analytically, by curve fitting procedures. 

In principle, knowing the molar entropy of the perfect gas (Section 1.17), and by measuring the change of equilibrium gas pressure as a function of temperature, one can determine the molar entropy of the adsorbed phase. The problem here is that the experiment has to be carried out at constant 0, a problematic task. Methods for circumventing this difficulty are shown below. Meanwhile, for completeness, we observe that at equilibrium the chemical potentials of the gas and adsorbate must match then Hg — Hs — T(Sg — Ss), so that we obtain the alternative formulation... 

A second type of adsorption is called chemisorption. In this case, the adsorption energy is comparable to the chemical bond energies and adsorbate molecules have the tendency to be localized at particular sites even though surface diffusion or some molecular mobility may still occur. Due to the chemical nature of the interactions between the gas and the solid surface, the equilibrium gas pressure in the adsorption system can be extremely low. This enables one to study the adsorbent-adsorbate system under high vacuum using diffraction and spectroscopic techniques for the identification of the actual species presented on the surface and the determination of their packing and chemical state. 

In Secs. II and III we shall summarize the present state of the theory of physical adsorption, starting with the simplest case of essentially independent molecules on a surface (very low equilibrium gas pressure) and then considering monolayer adsorption with interactions, and finally multilayer adsorption. In Sec. IV, the thermodynamics of adsorption will be discussed. 

In these approximations, as well as in higher ones, one finds that when w < 0 (attraction) there exists a critical temperature below which a first order phase change will be observed—a sudden condensation, as the equilibrium gas pressure is increased, from a dilute localized monolayer to a relatively condensed localized monolayer. For a plane square surface lattice of sites, the Bragg-Williams approximation gives — w/kTc = 1 and the quasi-chemical approximation — w/kTc = 1.386. 

Below the three-dimensional critical temperature of the adsorbate, multilayer adsorption and eventually bulk condensation (on a nonporous adsorbent) will occur as the equilibrium gas pressure p approaches the vapor pressure p0 of liquid adsorbate. From a theoretical point of view this problem is extremely complicated. 

In a recent paper Barrer and Robins (76) have introduced an important refinement (in the direction of the Wheeler-Ono treatment) in the above picture by use of the van der Waals equation as equation of state not only for the gas and liquid but also for molecules in the film. The chemical potential must have the same value at all distances z from the surface, and this value is determined by the temperature T and equilibrium gas pressure p far from the surface (z = >). This chemical potential at any point z1, according to Barrer and Robins, is made up of two terms (1) the chemical potential of the usual van der Waals bulk fluid, appropriate to T and the local density p(z ) and (2) the potential field of the solid, U(z ), falling off as 2 s (Eq. 2). Since p (= constant) and the potential field of the solid are known, one can solve for the density p at each z. This gives a film density p(z) which is not constant going out from the surface and not the density of bulk liquid. The adsorption isotherm is found by computing the surface excess (amount adsorbed) from p z) for each p. Isotherm types I to V (1,76) may be computed. 

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