Heterogeneous phase equilibria equilibrium conditions

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

This is an equation which fixes the relation existing between the number of phases (/ ), the number of components ( i), and the variance, or number of degrees of f reedom (F), of a heterogeneous system in equilibrium, subject to certain conditions which are usually satisfied in practice. The rule states that... 

A great many electrolytes have only limited solubility, which can be very low. If a solid electrolyte is added to a pure solvent in an amount greater than corresponds to its solubility, a heterogeneous system is formed in which equilibrium is established between the electrolyte ions in solution and in the solid phase. At constant temperature, this equilibrium can be described by the thermodynamic condition for equality of the chemical potentials of ions in the liquid and solid phases (under these conditions, cations and anions enter and leave the solid phase simultaneously, fulfilling the electroneutrality condition). In the liquid phase, the chemical potential of the ion is a function of its activity, while it is constant in the solid phase. If the formula unit of the electrolyte considered consists of v+ cations and v anions, then... 

Phase diagrams show coexistent phases in equilibrium. We have seen in Chapter 1 that the conditions for equilibrium in a heterogeneous closed system at constant pressure and temperature can be expressed in terms of the chemical potential of the components of the phases in equilibrium ... 

The equilibrium conditions for systems with curved interfaces [3] are in part identical to those defined earlier for heterogeneous phase equilibria where surface effects where negligible ... 

Equation (6.27) is the Laplace equation, or Young-Laplace equation, which defines the equilibrium condition for the pressure difference over a curved surface. In Section 6.2 we will examine the consequences of surface or interface curvature for some important heterogeneous phase equilibria. 

Liquid phase hydrogenation catalyzed by Pd/C is a heterogeneous reaction occurring at the interface between the solid catalyst and the liquid. In our one-pot process, the hydrogenation was initiated after aldehyde A and the Schiff s base reached equilibrium conditions (A B). There are three catalytic reactions A => D, B => C, and C => E, that occur simultaneously on the catalyst surface. Selectivity and catalytic activity are influenced by the ability to transfer reactants to the active sites and the optimum hydrogen-to-reactant surface coverage. The Langmuir-Hinshelwood kinetic approach is coupled with the quasi-equilibrium and the two-step cycle concepts to model the reaction scheme (1,2,3). Both A and B are adsorbed initially on the surface of the catalyst. Expressions for the elementary surface reactions may be written as follows ... 

An accurate evaluation of kxa is complicated by the heterogeneous nature and poor definition of contaminant/soil systems. Some success has been achieved in modeling mass transfer from a separate contaminant phase. During degradation these nonaqueous phase liquids (NAPLs) often dissolve under conditions where phase equilibrium is not achieved and dissolution is proportional to k a. Experimental determinations and correlations for k-p depend on interfacial area of the NAPL and liquid velocity at the interface (Geller Hunt, 1993). For adsorbed contaminants, kxa varies with soil composition and structure, concentration and age of contamination, and therefore with time. For example, slurry reactor tests indicate that the rate of naphthalene mass transfer decreases with time, with media size, and with aging of the tar prior to testing (Luthy et al., 1994). 

The calculation of the equilibrium conversion of heterogeneous reactions is in most cases much more complicated then in the case of homogeneous reactions, because the calculations involve in general the solution of the conditions for chemical equilibrium and the conditions for phase equilibrium. In the following a relatively simple example is given. 

If a heterogeneous system consists of two phases, a and / , and we treat the phase boundary conceptually as a separate (interface) phase with thickness A b, we can derive the equilibrium condition as before and obtain... 

The condition of equilibrium is also applicable to changes of state that involve heterogenous reactions, and the same methods used for homogenous reactions to obtain expressions of the equilibrium constant are used for heterogenous reactions. One difference is that in many heterogenous reactions one or more of the substances taking part in the change of state is a pure phase at equilibrium. In such cases the standard state of the substance is chosen as the pure phase at the experimental temperature and pressure. The chemical potential of the pure substance in its standard state still appears in Y.k vkPk but the activity of the substance is unity and its activity does not appear in the expression for the equilibrium constant. 

Equation (12.2) is applicable to every phase in a heterogenous system. Because of the identity of this equation with Equation (4.12) with the exclusion of other work terms, the conditions of equilibrium must be the same as those developed in Chapter 5 in a heterogenous system without restrictions, the temperature of every phase must be the same, the pressure of every phase must be the same, and the chemical potential of a species must be the same in every phase in which the species exists. For phase equilibrium, then,... 

A cell may be considered as a heterogenous system at equilibrium with restrictions. In most cells the pressure on each phase is the same and a change of pressure of the system would cause the same change of pressure on all phases. However, it is possible to construct a cell so that the various phases may have different pressures. Then the pressures of some phases may be held constant while the pressures of other phases are changed. In such cases some of the derivatives of the chemical potentials in Equation (12.86) would be zero unless matter would have to be transported across the boundary between phases in order to maintain the equilibrium conditions with a change of pressure. 

In Eqs. (3) and (4), the vapor phase mole fractions y and the liquid phase mole fractions %i have to fulfil the vapor-liquid-liquid equilibrium conditions in case of heterogeneous liquid mixtures, and the vapor-liquid equilibrium conditions in homogeneous mixtures. For the latter case, the reaction term 0 in Eq. (5) simplifies to... 

The hypothesis of the latter is always the instantaneous equilibrium at the contact surface of heterogeneous phases but the fulfillment of this condition is not to be accepted forthwith, particularly in the ease of many organic processes which —for instance, the reduction of nitre-bodies—are able to give a whole series of intermediate phases up to the final equilibrium. 

In systems where heterogeneous chemical equilibria prevail, both chemical and phase equilibrium conditions must be simultaneously satisfied. In practice, this means that the chemical equilibrium condition—Eq. (51) in the discrete description, and Eq. (60) in the continuous one—must be satisfied in one phase, and the phase equilibrium condition [/( or fi(x) to be the same in all phases] must be satisfied this clearly guarantees that the chemical equilibrium condition is automatically satisfied in all phases. 

A homogeneous reaction is one that involves only one phase. A heterogeneous reaction involves more than one phase, and reaction usually occurs at, or very near the interface between the phases. An irreversible reaction is one that proceeds in only one direction and continues in that direction until the reactants Types of reactions are exhausted. A reversible reaction, on the other hand, can proceed in either direction, depending on the concentrations of reactants and products relative to the corresponding equilibrium concentrations. An irreversible reaction behaves as if no equilibrium condition exists. StrictW speaking, no chemical reaction is completely irreversible, but in very many reactions the equilibrium point lies so far to the right that they are treated as irreversible reactions. 

The thermodynamic description of the formation of mlcroporous systems by means of the phase diagrams, eis illustrated in Figures 1 to 3, is based on the assumption of thermodynamic equilibrium. It predicts under what conditions of temperature and composition a system will separate into two phases and the ratio of the two phases in the heterogeneous mixture. As related to the membrane formation procedure, the thermodynamic description predicts the overall porosity that will be obtained at specified states. However, no information is provided about the pore sizes, which are determined by the spatial distribution of the two phases. Equilibrium thermodynamics is not able to offer any explanation about structural variations within the membrane cross-section that is, whether the membrane has a symmetric or asymmetric structure or a dense skin at the surface. These... 

Upon substitution of p from the defining equation of fugacity. Equation (4.302), in the phase-equilibrium equation. Equation (4.111) et seq., canceling out and RT, and exponentiating, we obtain the fugacity equality condition of heterogeneous phase equilibrium. 

For the subject of this book, Langmuir s (1933) extension of the phase rule for adsorption under equilibrium and non-equilibrium conditions is placed at the beginning of the treatment of surface thermodynamics. In the study of heterogeneous equilibrium by Gibbs methods, the term phase is used for a homogeneous part of a system without regarding for quantity or form. 

The total number of ways of distributing the species is independent of the order in which the species are attached. In transition state theory applied to heterogeneous catalysis the coverage of activated complexes is neglected. Consequently, interactions between a particular type of adsorbed species and activated complexe are also neglected. Equilibrium condition for adsorption again implies equality of the chemical potential in the gas-phase and the adsorbed phase ... 

If a mixture consisting of one or more components possesses uniform physical and chemical properties throughout, it is said to be a single-phase, homogeneous system. If, however, a system consists of one or more parts that have different properties and are set apart from each other by bounding surfaces, so that the phases are mechanically separable, the system is heterogeneous. When equilibrium exists between the distinct parts of the system, this condition is known as heterogeneous equilibrium. 

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