Escaping tendency

One of the simplest cases of phase behavior modeling is that of soHd—fluid equilibria for crystalline soHds, in which the solubility of the fluid in the sohd phase is negligible. Thermodynamic models are based on the principle that the fugacities (escaping tendencies) of component are equal for all phases at equilibrium under constant temperature and pressure (51). The soHd-phase fugacity,, can be represented by the following expression at temperature T ... 

The activity coefficient y can be defined as the escaping tendency of a component relative to Raonlt s law in vapor-liqnid eqnihbrinm (see Sec. 4 in this handbook or Null, Phase Equilibrium in Process Design, Wiley-Interscience, 1970). 

A negative deviation reduces the activity of the solute in the solvent, which enhances the liqnid-hqnid partition ratio but also leads to maximnm-boihng-point azeotropes. Among other classes of solvents shown in Table 15-4 that suppress the escaping tendency of a ketone are classes 1 and 2, i.e., phenohcs and acids. 

In Chapter 5, we considered systems in which composition becomes a variable, and defined and described the chemical potential. We showed that the chemical potential provides the condition for spontaneity or equilibrium. It is the potential that drives the flow of mass in a chemical process, A useful quantity related to the chemical potential is the fugacity. It can also be thought of as a measure of the flow of mass in a chemical process, and can be used to determine the point of equilibrium. It is often known as the escaping tendency since it can be used to describe the ease with which mass flows from one phase to another, particularly the flow from a solid or liquid phase to a gas phase. 

The activities of the various components 1,2,3. .. of an ideal solution are, according to the definition of an ideal solution, equal to their mole fractions Ni, N2,. . . . The activity, for present purposes, may be taken as the ratio of the partial pressure Pi of the constituent in the solution to the vapor pressure P of the pure constituent i in the liquid state at the same temperature. Although few solutions conform even approximately to ideal behavior at all concentrations, it may be shown that the activity of the solvent must converge to its mole fraction Ni as the concentration of the solute(s) is made sufficiently small. According to the most elementary considerations, at sufficiently high dilutions the activity 2 of the solute must become proportional to its mole fraction, provided merely that it does not dissociate in solution. In other words, the escaping tendency of the solute must be proportional to the number of solute particles present in the solution, if the solution is sufficiently dilute. This assertion is equally plausible for monomeric and polymeric solutes, although the... 

The vapor pressure of a polymer is, of course, far too small to measure We may, nevertheless, insist that such a vapor pressure exists, however small it may be. Or we may resort to the use of the escaping tendency, or fugacity, in place of the partial vapor pressure in the development given above, in accordance with usual thermodynamic procedures applied to the treatment of solutions. The treatment given here is in no way restricted to volatile solutes. 

Thus, equilibrium is achieved when the escaping tendency from the vapor and liquid phases for Component i are equal. The vapor-phase fugacity coefficient, fj, can be defined by the expression ... 

For a given solid material, a progressive reduction of particle size corresponds to increases in the surface/volume ratio and the escaping tendency of the molecules until the nature of the surface dominates the properties of the material. Two related thermodynamic consequences of this effect are an increase of solubility in any solvent and an increase of vapor pressure as the size of the particle is reduced. For a spherical particle of radius r, thermodynamic arguments lead to the Thomson-Freundlich equation [19] ... 

G. N. Lewis proposed the term escaping tendency to give a strong kinetic-molecular flavor to the concept of the chemical potential. Let us consider two solutions of iodine, in water and carbon tetrachloride, which have reached equilibrium with each other at a flxed pressure and temperature (Fig. 9.2). In this system at equilibrium, let us carry out a transfer of an inflnitesimal quantity of iodine from the water phase to the carbon tetrachloride phase. On the basis of Equation (9.17), we can say that... 

Thus, we may say that the escaping tendency of the iodine is greater in the water than in the carbon tetrachloride phase, and the chemical potentials in the two phases describe the spontaneous direction of transport from one phase to the other. 

The concept of escaping tendency also can be applied to the chemical reaction in Equation (9.47). At equilibrium, from Equation (9.54), we can say that the sum of the escaping tendencies of the reactants is equal to the sum of the escaping tendencies of the products. 

An advantage of the fugacity over the chemical potential as a measure of escaping tendency is that an absolute value of the fugacity can be calculated, whereas an absolute value of the chemical potential cannot be calculated. 

The condition of Equation (13.7) can be met only if p,j = p,n, which is the condition of transfer equilibrium between phases. Or, to put the argument differently, if the chemical potentials (escaping tendencies) of a substance in two phases differ, spontaneous transfer will occur from the phase of higher chemical potential to the phase of lower chemical potential, with a decrease in the Gibbs function of the system, until the chemical potentials are equal (see Section 10.5). For each component present in aU p phases, (p 1) equations of the form of Equation (13.7) provide constraints at transfer equilibrium. Furthermore, an equation of the form of Equation (13.7) can be written for each one of the C components in the system in transfer equUibrium between any two phases. Thus, C(p — 1) independent relationships among the chemical potentials can be written. As chemical potentials are functions of the mole fractions at constant temperamre and pressure, C(p — 1) relationships exist among the mole fractions. If we sum the independent relationships for temperature. 

If a quantity of a solute A is distributed between two immiscible solvents, for example I2 between carbon tetrachloride and water, then at equUibrium the chemical potentials or escaping tendencies of the solute are the same in both phases thus, for A(in solvent a) = A(in solvent b)... 

The chemical potential and the escaping tendency of the pure solvent are always greater than the chemical potential and the escaping tendency of the solvent in the solution. 

If the solute in solution A is in equUibrium with that in solution B, its escaping tendency is the same in both solvents. Consequently, its chemical potential p,2 at equihbiium also must be identical in both solvents. Nevertheless, the solute will have different activities in solution A and B since [Equation (16.1)]... 

Fugacity. Accdg to Hackh s (Ref 1), it is the escaping tendency in a heterogeneous mixture, by which. a chemical equilibrium responds to altered conditions. In a dilute soln obeying the gas laws, the fugacity equals the osmotic pressure. In other solns it is the value of the pressure for which these equations are still valid... 

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