The liquid-vapour phase transition

The liquid-vapour phase transition is a first-order transition. For example, the molar entropy S, being a derivative of te Gibbs potential G, S = —(8G/dT)p, where T is the temperature and p the pressure, has a discontinuity at the boiling point T = Tb. For a description of the liquid-vapour equilibrium, various approximate state equations are employed. Let us consider one of the most common equations of this type, the semiempirical van der Waals equation, written in terms of temperature T, volume V and pressure p or a gas 

In a real liquid-vapour system the following relationship must hold (8p/8V)T 0 (3.15a) 

This inequality, called the condition of mechanical stability, implies that if at a constant temperature the pressure in a system increases, then its volume decreases. It follows from Fig. 37 that along the section BC the van der Waals isotherm does not satisfy the condition (3.15a). This signifies that the isotherms obtaind from equation (3.14) are, at least partially, non-physical, and the corresponding states are physically unattainable. Thus, equation (3.14) is only an approximate state equation. 

The non-physical van der Waals isotherm may be improved using the so-called Maxwell construction. It involves drawing the horizontal section AD, for which (8p/8V)T = 0, joining the two branches of the isotherm, EA and DF, corresponding to the liquid and gaseous phase of a system, respectively. It follows from the condition of equality of chemical potentials at a critical point that the section AD should be thus selected that the areas and S2 be equal. Between the points A and D the system is nonhomo-geneous, i.e. separated into two phases coexisting in equilibrium. The 

it is a degenerate critical point meeting the above conditions (see Section 2.2.2). 

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The shape of the binodal near the critical point is not predicted correctly by the mean-field (Flory-Huggins) theory as demonstrated in Fig. 5.3(a). The difference in the two concentrations coexisting at equilibrium in the two-phase region is called the order parameter. This order parameter is analogous to the order parameter of van der Waals for the liquid-vapour phase transition, that is proportional to the density difference between the two coexisting phases. This order parameter is predicted to vary as a power law of the proximity to the critical point ... 

Fig. 39. The liquid-vapour phase transition on the surface of the cusp catastrophe (A3).

The first important conclusion which can be drawn from the above construction is a structural stability of phenomena described by the van der Waals equation. The conclusion is consistent with an observation that the reduced van der Waals equation (3.18) describes the critical transition for a number of gases. The liquid-vapour phase transition may thus be describred on the surface of the cusp catastrophe, see Fig. 39, in which a projection on the control parameters plane and the isotherm T = const are also shown. 

Many of the considered problems, such as the problem of stability of soap films, the liquid-vapour phase transition, the diffraction phenomena, descriptions of the heartbeat or the nerve impulse transmission, catastrophes described by non-linear recurrent equations have a close relation to chemical problems. 

Figure J.2J. Stale diagram of a one-component system in the coordinates F-V ACD is the binodal of liquid-vapour phase equilibria, Ee and Ff are portions of the binodal of crystal-liquid phase equilibria, CL and DM are portions of the binodal of crystal-vapour phase equilibria (no corresponding surface defined by Equation 1.2-33 is shown in Figure 1.20), BCC is the spinodal of the liquid-vapour phase transition, Kj is the spinodal of the crystal-liquid phase transition, GAD is the straight line of three-phase (vapour-liquid-cryslal) equilibrium at the triple point (Kirilin ct al., 1983 Skripov and Koverda, 1984)...

I. Brovchenko, A. Geiger, A. Oleinikova, Water in nanopores II. The liquid-vapour phase transition near hydrophobic surfaces, J. Phys. Condens. Matt. 16 (2004) S5345-S5370. 

Geometrical analysis of the state equation surfaces of liquid-vapour and crystal-liquid equilibria (Equation 1.2-33, Figures 1.20 and 1.21), analysis of expcrimotilal data and computer simulation results lead to the conclusion of that there is no spinodal of the liquid crystal phase transition while the spinodal of crystal —> liquid transition does exist (Skripov, 1975 Skripov and Koverda, 1984). Consequently, the liquid crystdiization occurs only through the formation of critical nuclei (through the metastable state) and no barrierless transition is possible. 

Surface evaporation is the only liquid/vapour phase transition taking place, whereby all the heat inflow is absorbed by the latent heat of vaporisation. The enthalpy of the vapour carried out of the vessel balances this heat inflow under equilibrium conditions of storage. 

We discuss classical non-ideal liquids before treating solids. The strongly interacting fluid systems of interest are hard spheres characterized by their harsh repulsions, atoms and molecules with dispersion interactions responsible for the liquid-vapour transitions of the rare gases, ionic systems including strong and weak electrolytes, simple and not quite so simple polar fluids like water. The solid phase systems discussed are ferroniagnets and alloys. 

The best-known examples of phase transition are the liquid-vapour transition (evaporation), the solid-liquid transition (melting) and the solid-vapour transition (sublimation). The relationships between the phases, expressed as a function of P, V and T consitute an equation of state that may be represented graphically in the form of a phase diagram. An idealized example, shown in figure 1, is based on the phase relationships of argon [126]. 

The thermodjmamlcs presented in the previous subsection is easily extended to account for the coexistence of different phases in pores. In particular, the liquid-vapour transition is relevant here. We shall briefly consider three aspects of this topic. 

Phase transitions are central in many industrial problems, for instance, in distillation, absorption, condensation, manufacture of liquid natural gas, and multiphase flow. We shall use the liquid/vapour transition to illustrate the basic hypotheses and usefulness of NET for surfaces, a relatively new application. For applications to homogeneous phases, we refer the reader to the basic literature see Section 2,... 

The obtained distributions of the tetrahedricity measure were used for estimation of the concentration C of the four-coordinated tetrahedrally ordered water molecules. Temperature dependence of this concentration along the liquid-vapour coexistence curve is shown in the upper panel of Fig.5. There is only slight increase of C upon cooling from the liquid-vapour critical temperature to about 350 K (due to the temperature mismatch of ST2 water and real water, about 30 to 35° lower temperature should be expected for real water). The drastic increase of C is evident at lower temperatures, when approaching the liquid-liquid phase transition. At 7 = 270 K, concentrations of the tetrahedrally ordered four-coordinated water molecules in two coexisting phases was found to be about 28% and 46.5%. Such step increase of C is related to a step decrease of density from 0.97 to 0.91 g/cm ... 

Figure 5 Temperature dependence of the concentration C of the tetrahedrally ordered four-coordinated water molecules (upper panel) and of the liquid water density (lower panel) along the liquid-vapour coexistence curve. Vertical dashed line indicates the temperature of the liquid-liquid transition. Dotted lines indicate the densities and concentrations of the coexisting phases. Stars indicate percolation transition of the tetrahedrally ordered four-coordinated molecules.

As we have mentioned in the Introduction, the location of the critical point of the lowest density liquid-liquid transition of real water is unknown and both scenarios (critical point at positive or at negative pressure) can qualitatively explain water anomalies. Recent simulation studies of confined water show the way, how to locate the liquid-liquid critical point of water. Confinement in hydrophobic pores shifts the temperature of the liquid-liquid transition to lower temperatures (at the same pressure), whereas effect of confinement in hydrophilic pores is opposite. If the liquid-liquid critical point in real water is located at positive pressure, in hydrophobic pores it may be shifted to negative pressures. Alternatively, if the liquid-liquid critical point in real water is located at negative pressure, it may be shifted to positive pressures by confinement in hydrophilic pores. Interestingly, that it may be possible in both cases to place the liquid-liquid critical point at the liquid-vapour coexistence curve by tuning the pore hydrophilicity. We expect, that the experiments with confined supercooled water should finally answer the questions, concerning existence of the liquid-liquid phase transition in supercoleed water and its location. 

The similarity of Fig. 6 (model 2 of the phase transition in the liquid-vapour system) and 10 (self-ignition of the H2/02 mixture, example 6.1) may suggest an analogy between catastrophes occurring in these systems such an analogy actually takes place. 

In this chapter we shall show how the observed phenomena may be explained by means of elementary catastrophe theory. In principle, the discussion will be confined to examination of non-chemical systems. However, some of the discussed problems, such as a stability of soap films, a phase transition in the liquid-vapour system, diffraction phenomena or even non-linear recurrent equations, are closely related to chemical problems. This topic will be dealt with in some detail in the last section. The discussion of catastrophes (static and dynamic) occurring in chemical systems is postponed to Chapters 5, 6 these will be preceded by Chapter 4, where the elements of chemical kinetics necessary for our purposes will be discussed. 

A classical theory of phase transitions may be formulated by means of elementary catastrophe theory. We shall describe some notions of the theory of phase transitions in terms of elementary catastrophe theory. Next, we shall describe examples of application of catastrophe theory to the description of the liquid-vapour equilibrium. 

From that point of view it seems natural that special consideration should be given to the composite membranes in which one component consists of liquid crystal. The transitions from crystalline to mesophase state are connected with mobility increase thus influencing the transport properties of these composite systems. However, it was only recently that a discontinuous jump of permeability to liquids, gases and vapours in the vicinity of transition temperature of the liquid crystal phase has been discovered. Membranes of this type form a new class of composites which deserve special consideration due to their particular properties. 

Nichita et al calculated the wax precipitation from hydrocarbon mixtures using a cubic equation of state (see Chapter 4) to describe the vapour and the liquid lumping into pseudo-components to simplify the phase equilibrium calculation. However, the information lost in this procedure effected the location of the predicted solid phase transition. This issue was avoided by an inverse lumping procedure, in which the equilibrium constants of the original system are related to some quantities evaluated from lumped fluid flash results. The method was tested for two synthetic and one real mixture yielding good agreement between calculated and experimental results. 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

Figure A2.5.1. Schematic phase diagram (pressure p versus temperature 7) for a typical one-component substance. The full lines mark the transitions from one phase to another (g, gas liquid s, solid). The liquid-gas line (the vapour pressure curve) ends at a critical point (c). The dotted line is a constant pressure line. The dashed lines represent metastable extensions of the stable phases.

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