Equilibrium volume phase transition diagram

Figure /. Equilibrium volume phase transition diagram for the gel The polymer volume fraction (f> is represented as a function of Flory s interaction parameterX-...

Fig. 6. Equilibrium volume of ionized AJ-isopropylacrylamide (NIPA) gels in water show discontinuous volume phase transition in response to temperature. The values shown in the diagram indicate the amount of ionizable group (sodium acrylate) incorporated in 700 mM NIPA...

Figure 3.10. Phase diagrams of attractive monodisperse dispersions. Uc is the contact pair potential and (j) is the particle volume fraction. For udk T = 0, the only accessible one-phase transition is the hard sphere transition. If Uc/hgT 0, two distinct scenarios are possible according to the value of the ratio (range of the pair potential over particle radius). For < 0.3 (a), only fluid-solid equilibrium is predicted. For % > 0.3 (b), in addition to fluid-solid equilibrium, a fluid-fluid (liquid-gas) coexistence is predicted with a critical point (C) and a triple point (T).

At particular critical points (Tq, Pc) on the phase diagram of a substance, two phases can be found in thermodynamic equilibrium. Therefore, upon application of a pressure or a temperature gradient, a transformation occurs from one phase into the other. This is a phase transition, in many aspects similar to a transformation implying the change of aggregation state. However, the extent of the changes in a solid to solid transformation is much smaller. For example, latent heat or latent volumes associated with the transformations are quite small, sometimes even difficult to detect. 

The order of a transition can be illustrated for a fixed-stoichiometry system with the familiar P-T diagram for solid, liquid, and vapor phases in Fig. 17.2. The curves in Fig. 17.2 are sets of P and T at which the molar volume, V, has two distinct equilibrium values—the discontinuous change in molar volume as the system s equilibrium environment crosses a curve indicates that the phase transition is first order. Critical points where the change in the order parameter goes to zero (e.g., at the end of the vapor-liquid coexistence curve) are second-order transitions. 

Other work with volcanic magmas has demonstrated the significance of the crystallites which appear as the molten material cools. A sharp transition in the Arrhenius plot of log of viscosity versus reciprocal absolute temperature was observed as the volume of crystallites approached 30% of the total volume. This transition has been observed in coal ash systems. The composition of the solid phases or crystallites can be interpreted by comparison with phase equilibrium diagrams of related simpler systems. 

The bulk phase diagrams of pure hydrocarbons and mixtures are well known from the experiments. In the work by Sage et al. [3], the bubble point pressures of methane + n-butane mixtures are determined experimentally from the discontinuity of isothermal compressibility of constant-composition mixture at the point of phase transition. The composition of vapor phase is determined in that work from the residual specific volume of gas. Later experiments employ phase recirculation techniques [4] to achieve vapor-Uquid equilibrium [5, 6], and the phase compositions are analyzed by more advanced methods such as gas chromatography. 

Fig. 38. Isothermal sections at 25°C of (a) intra-lanthanide and (b) intra-actinide generalized binary phase diagrams, showing equilibrium phase boundaries [with estimated hysteresis for (a)] as full hnes (Benedict et al. 1986). The broken line in (a) indicates the interpolated boundary for the volume collapse transition of the lanthanides. The atomic radius of Ce at room temperature as a function of pressure is shown in (c) (Franceschi and Olcese 1969), with the Kondo-volume collapse transition at about 7 kbar. This transition can be traced to negative pressures by alloying (Lawrence et al. 1984), as seen in (d) via the temperature dependence of the resistance.

At T = 0 K, where any transformation of a pure substance tends to be isoen-tropic, phase stability can be related to the enthalpy and a phase transition occurs at those points in the phase diagram where two phases have equal enthalpy. Erom the computational point of view, it is possible to explore a range of crystalline volumes by isometric lattice deformations and obtain the corresponding values of pressure and, consequently, of enthalpy. It is intended that nuclei are allowed to relax to their equilibrium geometry after... 

Figure 3 Phase transition observed in emulsifier-free dispersions of polystyrene latex particles. (A) Schematic representation of sedimentation equilibrium (particle diameter 0.26 ionic strength 8 mol m ) with distinct regions, which are (a) transparent, (b) turbid, (c) milky-white, (d) diffusely coloured, and (e) iridescent. (B) Phase diagram (particle diameter 0.17 ftm) with regions of order (O), disorder (D), and coexistence (C). The volume fraction is plotted against the electrolyte concentration c on a logarithmic scale (Redrawn from Refs. 41 and 45).

As an example, let the system contain a fixed amount n of a pure substance divided into liquid and gas phases, at a temperature and pressure at which these phases can coexist in equilibrium. When heat is transferred into the system at this T and p, some of the liquid vaporizes by a liquid-gas phase transition and V increases withdrawal of heat at this T and p causes gas to condense and V to decrease. The molar volumes and other intensive properties of the individual liquid and gas phases remain constant during these changes at constant T and p. On the pressure-volume phase diagram of Fig. 8.9 on page 208, the volume changes correspond to movement of the system point to the right or left along the tie line AB. 

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