Liquid-vapour relations

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

The non-consen>ed variable (.t,0 is a broken symmetry variable, it is the instantaneous position of the Gibbs surface, and it is the translational synnnetry in z direction that is broken by the inlioinogeneity due to the liquid-vapour interface. In a more microscopic statistical mechanical approach 121, it is related to the number density fluctuation 3p(x,z,t) as... 

As with all thermodynamic relations, the Kelvin equation may be arrived at along several paths. Since the occurrence of capillary condensation is intimately, bound up with the curvature of a liquid meniscus, it is helpful to start out from the Young-Laplace equation, the relationship between the pressures on opposite sides of a liquid-vapour interface. 

J. G. WOJTASINSKI. J. Chem. Eng. Data, 1963 (July), pp. 381-385. Measurement of total pressures for determining liquid-vapour equilibrium relations of the binary system isobutyraldehyde-n-butyraldehyde. 

The vapour pressure p above a liquid is related to the absolute temperature T by the equation... 

Standard heat pipe is shown on Figure 1. Basic phenomena and equations are related with liquid-vapour interface, heat transport between the outside and the interface ( radial heat transfer), vapour flow and liquid flow. 

Intuitively, surface and interfacial tensions may be expected to be related to a number of physical characteristics of the liquid or the liquid-vapour transition. Two of these are the enthalpy and entropy of evaporation, discussed in sec. 2.9. Other parameters that come to mind are the molar volume V, the isothermal compressibility and the expansion coefficient. The combination of certain powers of such parameters and y sometimes leads to products with interesting properties, like temperature independence or additivity. Severed of such scaling rules have been proposed over the past century, mostly with limited quantitative success. A few of these wUl now be discussed. 

Data from immersion enthalpies into liquids whose vapour is adsorbed according to the Dubinin-Radushkevich (D-R) equation can also be analysed following the approach suggested by Stoeckli and Kraehenbuehl [5]. They showed that the immersion enthalpy of a solid into a given liquid is related to the... 

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 ... 

Any liquid can exist in three thermodynamic states with regard to the phase diagram stable, metastable, and unstable. When it is metastable with respect to its vapour, the so-called superheated liquid persists over the more stable vapour owing to the nucleation barrier related to the cost to create the liquid-vapour interface. Practically speaking, a superheated liquid undergoes any P-T conditions located between the saturation and the spinodal curves (Fig. 1). It should be noted that the term superheating does not refer to a particular... 

The nucleation barrier is easy to formulate considering that the created gas-liquid interface related to bubble nucleation increases the system energy by 27ir (r is the radius of the spherical bubble, and y is the liquid-vapour surface tension), while the formation of the most stable phase provides bulk energy (4/37ir AP, with AP = Pliquid - Pvapour) According to the CNT, the competition between these two opposite effects results in an energy barrier... 

According to the Phase Rule, each different variety of ice constitutes a separate phase and, consequently, it must be possible to obtain not only the ordinary triple point for solid— liquid— vapour which has already been described, but also other triple points at which the other forms of ice exist. Of such forms Bridgman has distinguished no fewer than four, besides ordinary ice, these different forms being designated ice I. (ordinary ice), ice II., ice III., ice V., ice VI. The existence and stability relations of another form, ice IV., discovered by Tammann, do not appear to be definitely settled. 

We have seen that the activity of a gas is related to its effective pressure. For liquids, it is the liquid vapour pressure which is important, and for solutions, the vapour pressure is linked to the concentration of the component of interest. The concentration of a solvent or major component is conveniently measured as the mole fraction, x. For ideal vapours and an ideal solution it may be shown that the chemical potential of a component number 1 is ... 

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. 

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. 

Equations relating the relative adsorption to observable quantities are given in Appendix III. However, the direct measurement of surface excesses using these equations is not always easy. Thus it is extremely difficult to measure T 1 at a liquid/vapour interface, although accurate measurements are possible for liquid/solid interfaces. In the next section wc shall sec how this quantity can be related to the surface tension of the interface and obtainable from experimental measurements of surface tension. 

Even though the graphical correlation works well and is simple to use for binary systems, attempts have continued to find suitable functional forms to describe the properties of homologous mixtures in terms of composition and chain length. McGlashan and co-workers used the zeroth approximation of the lattice theory for molecules of different sizes to account for their measurements of liquid vapour equilibria and enthalpies of mixing of n-alkanes. The relations used were ... 

J. J. H. Haftka, J. R. Parsons and H. A. J. Covers, Supercooled liquid vapour pressures and related thermodynamic properties of polycyclic aromatic hydrocarbons determined by gas chromatography, /. Chromatogr., A, 2006, 1135, 91-100. 

For illustrative purposes, vapour pressure may be portrayed as solubility in air. This parameter is strongly dependent on the ambient temperature if measured at different temperatures, the logarithm of can be linearly related to the reciprocal temperatures (K). For most liquids, vapour pressure ranges between 10 and 4 x 10 Pa at room temperature. It is experimentally accessible using a (mercury) manometer to measure the pressure established in the gas phase above the pure compound at defined temperatures. For volatile chemicals p > 100 Pa), measured data are generally accurate, whereas for low-volatility compounds (p < 100 Pa), the experimental results may scatter by one order of magnitude (Schwarzenbach, Geschwend and Imboden, 1993). 

Dynamic is a key word in our understanding of what is happening in liquid-vapour equilibria. It implies continuous activity and helps explain many everyday phenomena relating to the behaviour of liquids, including the variation of boiling point with atmospheric pressure. 

The operating lines y =f(x) shown in Figure 16.2 are derived from the respective mass balances of the lighter component in the rectification and stripping sections. The equilibrium line / =f(x), however, is derived from the thermodynamic characteristics of the system in terms of the liquid vapour equilibrium. This equilibrium can be then described by the following relation between the... 

---