The Metastable Region

Let us now consider points 2 and 10 that satisfy the stability criterion. In addition, since areas I and II are not equal, they do not represent two phases at equilibrium. Which of the two states will the fluid find itself in at r 325 K and P = PqI 

The only way to answer this question is to determine in which of the two states the fluid assumes the lowest Gibbs free energy. We proceed, therefore, to calculate the molar Gibbs free energy of propane as a function of pressure, using the vdW EoS. 

We can thus calculate /t relative to its value at some reference state only choosing, arbitrarily, P = 1.013 bar n = 0.0. (This does not present any problems, however, since we are interested in comparing n values, not in absolute ones.) To facilitate the integration in Eq. 12.9.1, we use  

We proceed next to point 4 and the associated ones, 6 and 12. Point 6 is rejected again, but now /i(12) /i(4), and point 12 represents the stable state for propane (liquid). 

Let us now consider the question Since point 4 does not violate the stability criterion, can propane exist in this state The answer is yes. If the conipression is done very slowly, without any vibrations, the system can find itself in point 4. The slightest tapping, however, will shift the system to point 12. 

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For initial post-quench states in the metastable region between the classical spinodal and coexistence curves,... 

It is, however, possible to calculate the tensile strength of a liquid by extrapolation of an equation of state for the fluid into the metastable region of negative pressure. Burgess and Everett in their comprehensive test of the tensile strength hypothesis, plot the theoretical curves of T /T against zjp, calculated from the equations of state of van der Waals, Guggenheim, and Berthelot (Fig. 3.24) (7], and are the critical temperature and critical... 

Sihcate solutions of equivalent composition may exhibit different physical properties and chemical reactivities because of differences in the distributions of polymer sihcate species. This effect is keenly observed in commercial alkah sihcate solutions with compositions that he in the metastable region near the solubihty limit of amorphous sihca. Experimental studies have shown that the precipitation boundaries of sodium sihcate solutions expand as a function of time, depending on the concentration of metal salts (29,58). Apparently, the high viscosity of concentrated alkah sihcate solutions contributes to the slow approach to equihbrium. 

Some guidelines have been provided for defining the metastable region. If the seed crystals dissolve when added to the metastable solution, this implies that saturation conditions have not been reached. If the addition of the seed leads to the formation of an oil dispersion, it may be concluded that supersaturation has been realized (Anderson, 2000). 

Fig. 4.8 The Frumkin adsorption isotherm. The value of the interaction coefficient a is indicated at each curve. The transition in the metastable region is indicated by a dashed...

The decomposition of a solution with composition outside the spinodal region but within the metastable region can be analyzed in a similar way. Let us assume that a sample with composition in this region is cooled to low temperatures. Small fluctuations in composition now initially lead to an increase in the Gibbs energy and the separation of the original homogeneous solution must occur by nucleation of a new phase. The formation of this phase is thermally activated. Two solutions with different composition appear, but in this case the composition of the nucleated phase is well defined at all times and only the relative amount of the two phases varies with time. 

If hexane is used as the low molecular weight liquid, the desired phase separation is observed when precursor mixtures containing 6-15 wt % hexane are cured isothermally at 40 °C. Further discussion of the phase separation behavior requires more detailed consideration of the schematic phase diagram, as presented in Fig. 17, which resembles the real phase diagram shown in Fig. 13. Experimentally it is found, that no phase separation occurs with hexane concentrations equal to or lower than 5 wt %. Hence the critical amount for phase separation, (j)p, is given by the intercept of the binodal line and the imaginary value of Hence no phase separation occurs if is reached before the metastable region is entered. 

Based on Eqs. (42) and (43), the development of a narrow or bimodal size distribution can be qualitatively explained without the detailed knowledge of the real phase diagram nor the exact dependency of the diffusion constant as a function of time. The final morphology depends mainly on the extent of reaction at which the metastable region is entered and the difference between ( )p and ( )o, as discussed below. 

At higher concentrations of hexane, the metastable region is entered at lower conversions, hence lower viscosity. As the nucleation starts at the point d the diffusion constant is high, leading to a very fast growth of the separated domains immediately after the start of the phase separation. This allows the system to reach the equilibrium concentration after a rather short period. Consequently, the driving force for the phase separation, given by becomes nearly ze-... 

In this section, a brief description of the necessary experiments to identify the kinetic parameters of a seeded naphthalene-toluene batch crystallization system is presented. Details about the experimental apparatus and procedure are given by Witkowski (12). Operating conditions are selected so that the supersaturation level is kept within the metastable region to prevent homogeneous nucleation. To enhance the probability of secondary nucleation, sieved naphthalene seed particles are introduced into the system at time zero. 

Limiting Supersaturation for Nucleation. Like sucrose, D-fructose solutions can tolerate a high degree of supersaturation without nucleating, even in the presence of seed crystals. This is the metastable region on the Miers supersolubility diagram (9). 

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