Systems, heterogeneous, equilibrium

The phase rule is a mathematical expression that describes the behavior of chemical systems in equilibrium. A chemical system is any combination of chemical substances. The substances exist as gas, liquid, or solid phases. The phase rule applies only to systems, called heterogeneous systems, in which two or more distinct phases are in equilibrium. A system cannot contain more than one gas phase, but can contain any number of liquid and solid phases. An alloy of copper and nickel, for example, contains two solid phases. The rule makes possible the simple correlation of very large quantities of physical data and limited prediction of the behavior of chemical systems. It is used particularly in alloy preparation, in chemical engineering, and in geology. 

Chemical reactions obey the rules of chemical kinetics (see Chapter 2) and chemical thermodynamics, if they occur slowly and do not exhibit a significant heat of reaction in the homogeneous system (microkinetics). Thermodynamics, as reviewed in Chapter 3, has an essential role in the scale-up of reactors. It shows the form that rate equations must take in the limiting case where a reaction has attained equilibrium. Consistency is required thermodynamically before a rate equation achieves success over tlie entire range of conversion. Generally, chemical reactions do not depend on the theory of similarity rules. However, most industrial reactions occur under heterogeneous systems (e.g., liquid/solid, gas/solid, liquid/gas, and liquid/liquid), thereby generating enormous heat of reaction. Therefore, mass and heat transfer processes (macrokinetics) that are scale-dependent often accompany the chemical reaction. The path of such chemical reactions will be... 

Because the expression for Kn is simpler than that for Kb it is the equilibrium constant of choice for the heterogeneous system. 

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... 

By the variance, or number of degrees of freedom of the system, we mean the number of independent variables which must be arbitrarily fixed before the state of equilibrium is completely determined. According to the number of these, we have avariant, univariant, bivariant, trivariant,. . . systems. Thus, a completely heterogeneous system is univariant, because its equilibrium is completely specified by fixing a single variable— the temperature. But a salt solution requires two variables— temperature and composition—to be fixed before the equilibrium is determined, since the vapour-pressure depends on both. 

All points on the two tangents HRi, HR2, to the curve of solutions represent heterogeneous systems composed of solid hydrate in contact with solutions. If the curve between Ri and R2 is convex the heterogeneous systems are stable, and inversely. At a given temperature and pressure the hydrate can be in equilibrium with two liquid phases of different composition, one containing relatively more, the other relatively less, salt than the hydrate. With rise of temperature the form of the curve and the altitude of H change ... 

Chemical equilibria with reactants and products that are all in the same phase are called homogeneous equilibria. Equilibria C, D, and E are homogeneous. Equilibria in systems having more than one phase are called heterogeneous equilibria. Equilibrium F is heterogeneous so too is the equilibrium between water vapor and liquid water in a closed system ... 

The system, therefore, is at equilibrium at a given temperature when the partial pressure of carbon dioxide present has the required fixed value. This result is confirmed by experiment which shows that there is a certain fixed dissociation pressure of carbon dioxide for each temperature. The same conclusion can be deduced from the application of phase rule. In this case, there are two components occurring in three phases hence F=2-3 + 2 = l, or the system has one degree of freedom. It may thus legitimately be concluded that the assumption made in applying the law of mass action to a heterogeneous system is justified, and hence that in such systems the active mass of a solid is constant. 

The phase rule as has been pointed out in the preceding paragraph deals with the behavior of heterogeneous systems at equilibria. It essentially includes three special terms. These are (i) number of phases in the system (P) (ii) the number of components for the system (C) and (iii) the number of degrees of freedom available to the system (F). A system for the present purpose could be any substance or combination of substances, which is set apart from its surroundings or other substances, such that its equilibrium state may be studied. The simplest way to express the rule in the form of an equation combining the three terms is as follows ... 

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... 

Electron Transfer Far From Equilibrium. We have shown how the Marcus Theory of electron transfer provides a quantitative means of analysis of outer-sphere mechanisms in both homogeneous and heterogeneous systems. It is particularly useful for predicting electron transfer rates near the equilibrium potential,... 

For sub-critical isotherms (T < Tc), the parts of the isotherm where (dp/dV)T < 0 become unphysical, since this implies that the thermodynamic system has negative compressibility. At the particular reduced volumes where (dp/dV)T =0, (spinodal points that correspond to those discussed for solutions in the previous section. This breakdown of the van der Waals equation of state can be bypassed by allowing the system to become heterogeneous at equilibrium. The two phases formed at T[Pg.141]

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 ... 

Two additional points about Equation (8) need to be discussed here. Equation (8) contains mj in the denominator. Thus the solution concentrations must be known before the first increment dE, is taken and none of them can be zero. In practice this means that the set of nonlinear equations (mass action and balance equations) describing the fluid phase in its initial unperturbed equilibrium state must be solved once. Further, Equation (8) does not completely describe a heterogeneous system at partial equilibrium. 

The following Sample Problem shows how to find the equilibrium expression for a reaction. In this chapter, you will use equilibrium expressions for homogeneous reactions (mostly reactions between gases). In Chapters 8 and 9, you will learn how to use equilibrium expressions for heterogeneous systems. 

The coexistence of various solids at equilibrium in a heterogeneous system may be opportunely described by chemical reactions involving the major components of the solid phases. If we observe, for instance, the coexistence of pyroxene, pla-gioclase, and quartz, we can write an equation involving sodic terms ... 

Degree of Saturation of Condensed Phases and Affinity to Equilibrium in Heterogeneous Systems... 

This parameter has the magnitude of an energy (i.e., kJ/mole, kcal/mole) and represents the energy driving toward equilibrium in a chemical potential field— i.e., the higher A becomes, the more the solid phase of interest is unstable in solution. In a heterogeneous system at equilibrium, based on the principle of... 

Figure 8.30 Degree of saturation of various phases as a function of T in a heterogeneous system. A = complete equilibrium B = disequilibrium C = dispersion of saturation curves as a result of boiling. Reprinted from M. Reed and N. Spycher, Geochimica et Cosmochimica Acta, 48, 1479-1492, copyright 1984, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

It is most important to know in this connection the compressibility of the substances concerned, at various temperatures, and in both the liquid and the crystalline state, with its dependent constants such as change of. melting-point with pressure, and effect of pressure upon solubility. Other important data are the existence of new pol3miorphic forms of substances the effect of pressure upon rigidity and its related elastic moduli the effect of pressure upon diathermancy, thermal conductivity, specific heat capacity, and magnetic susceptibility and the effect of pressure in modif dng equilibrium in homogeneous as well as heterogeneous systems. 

Partition Coefficient A constant ratio that occurs when a heterogeneous system of two phases is in equilibrium the ratio of concentrations (or strictly activities) of the same molecular species in the two phases is constant at constant temperature. 

In alcohols, water is found to be partially displaced, and it is possible to have the water/salt ratio in a saturated solution considerably less than that in the hydrated solid in equilibrium with it (4> 10). As with pyridine, an alcohol in a heterogeneous system will sometimes displace water well enough to make the difference between extraction and nonextraction of a metal-ion value (29). Another way in which the differences between strong water-competitors such as alcohol and weaker competitors such as the ketones and ethers are manifested is when liquid such as CC14, with no solvent power of its own for the salt, is mixed with the oxygenated solvent. With thorium nitrate as the test salt, it is seen (Fig. 2) that, whereas addition... 

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