Stability diffusional

In this chapter we develop the stability criteria for both pure substances and for mixtures. Since we have three kinds of equilibria, we have three kinds of stabilities thermal stability, mechanical stability, and diffusional stability. If the proposed state of a single phase violates any of these criteria, then the phase might spontaneously split into two or more phases. Therefore, violations of stability criteria contribute to the wealth of phase behavior observed in Nature. In this chapter we introduce some of the phase behavior that results from instabilities, but the subject is an extensive one, so the descriptions of observable phase behavior are continued in the next chapter. 

In 8.1 we derive the thermal and mechanical stability criteria for closed systems, and in 8.2 we apply those criteria to pure substances. In pure substances only thermal and mechanical instabilities are possible diffusional instabilities never occur because pure substances caimot exhibit concentration gradients. Then in 8.3 we derive the diffusional stability criteria for open systems, and in 8.4 we apply those... 

The thermal and mechanical stability criteria (8.1.23) and (8.1.31) apply both to pure fluids and to mixtures however, for homogeneous mixtures, those criteria are not sufficient to identify stable systems because, in addition to energy and volume fluctuations, mixtures have concentration fluctuations. These fluctuations occur in localized regions of a system when material spontaneously aggregates and redisperses. If such fluctuations are not to disturb a system s stability, then the mixture must satisfy a set of conditions known as the material or diffusional stability criteria. These criteria are derived in a manner similar to that given in 8.1.2 for (8.1.23) and (8.1.31), so we only sketch the procedure here. 

This is the criterion for material or diffusional stability for a binary mixture to be differentially stable, the mixture must have Q > 0, Kt- > 0, and (at fixed T and P) the chemical potential of component 1 must always increase in response to any increase in Nj. This means that if an isothermal-isobaric plot of the chemical potential (or fugacity) passes through an extremum with Xj, then the mixture is unstable for some Xj-values. The result (8.3.13) confirms (3.7.29) in which we claimed that the chemical potential of a pure component is always greater than its value in any mixture at the same T and P. 

The criterion (8.3.13) implies that if a mixture is mechanically unstable kj < 0), then it is also diffusionally unstable, just as (8.1.30) implies that if a fluid is thermally unstable (Q < 0), then it is also mechanically unstable. But a fluid may be diffusion-ally unstable while remaining mechanically and thermally stable. In fact, whenever a stable mixture is driven into an unstable region of its phase diagram, the diffusional stability limit is always violated before the mechanical or thermal limits are violated, because higher-order terms approach zero before lower-order terms [3]. This can be seen in Figure 8.11. This means that the diffusional stability criterion (8.3.13) is a stronger test for thermodynamic stability than the mechanical criterion and (as noted in 8.1.2) the mechanical criterion, in turn, is a stronger test than the thermal criterion. 

The middle envelope is the spinodal the set of states that separate metastable states from unstable states. Recall from 8.3 that one-phase mixtures become diffusionally unstable before becoming mechanically unstable. Therefore, the mixture spinodal is the locus of points at which the diffusional stability criterion (8.3.14) is first violated that is, it is the locus of points having... 

Note that the parameter A is dimensionless and depends only on temperature. Although simple. Porter s equation can reproduce states that violate the diffusional stability criterion, thereby giving rise to liquid-liquid or solid-solid equilibria. Whether or not such violations occur depends on the parameter A. To identify the stability bound on A, we substitute Porter s equation (8.4.32) into (8.4.31), and find... 

Porter s equation also provides an estimate for the spinodal, which separates unstable states from metastable ones. In terms of g, the spinodal of a binary occurs when the diffusional stability criterion is first violated, that is, when... 

Figure 8 1 g ix) and its second mole-fraction derivative computed from Porter s equation for the binary mixtures in Figure 8.20. At 60°C the diffusional stability criterion is satisfied at all compositions and the mixture is a stable single phase. However at 30°C, states between C and D violate the diffusional stability criterion and the mixture splits into two phases C and D lie on the spinodal. Filled circles at 30°C correspond to states of the same labels in Figure 8.20.

If the pressure is reduced to 30 bar, below the mechanical critical point, the loop in the fugacity becomes more pronounced. Here violates the diffusional stability criterion dfildxi > 0 for stability) over only a small range of Xj but the mechanical stability criterion (k > 0) is also violated at states between the extrema in fp Finally, at the lowest pressure (10 bar), the loop in has completely closed, dividing into two parts a vapor part that is linear and obeys the ideal-gas law, and a fluid part that includes stable and metastable liquid states at small x values plus mechanically unstable fluid states at higher x values. The broken horizontal lines in Figure 10.1 are vapor-liquid tie lines, computed by solving the phi-phi equations (10.1.3) simultaneously for both components. 

Since neither an ideal gas nor an ideal solution can violate the diffusional stability criteria derived in Chapter 8, Tabitha the Untutored maintains that RaoulFs law must be nonsense. Do you agree If so, then how do you explain that ideal gases can exist in VLB with ideal liquid solutions If you do not agree, then how do you explain phase separation without instabilities ... 

Prigoginc and Defay (1954) have shown the validity of Equations 46 or 47 to cau.se the validity of Equations 39 and 40, henr , the diffusional stability condition (any of Equations 46, 47, 53-56) is a necessary and sufficient stability condition of the ono-phase state of multicomponent systems (including the metastable state). 

I he equation obtained should be followed by inequalities II -13 (which arc identical to the diffusional stability conditions from Subsection 1.1.2) to allow the solution stability condition (the decrea.se in the probability of composition fluctuations as the system deviates from equilibrium) to be true. 

The extent of monomer diffusion can be reduced by the incorporation of a costabiUser. Costabiliser (104, 314, 354) must have two properties low molecular weight (to maximise the mixing free energy contribution to the droplet diffusional stability) and low... 

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