Vapor Coexistence Curve and the Critical Point

Coles in The Optics of Thermotropic Liquid Crystals, S. Elston and R. Sambles (eds.), pp. 57-84, esp. Table 4.2, Taylor Francis, London (1998). 

Collinss and M. Hird, Introduction to Liquid Crystals Chemistry and Physics, Taylor Francis, London (1997). 

Oswald and P. Pieranski, Nematic and Cholesteric Liquid Crystals Concepts and Physical Properties Illustrated by Experiments, Taylor Francis, London (2005). 

Liquid Crystals Fundamentals, World Science Publ., Singapore (2002). 

Liquid-Vapor Coexistence Curve and the Critical Point 

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Exp. 16 Liquid-Vapor Coexistence Curve and the Critical Point 229... 

Here is the vapor pressure of pure liquid solute at the same temperature and total pressure as the solution. If the pressure is too low for pure B to exist as a liquid at this temperature, we can with little error replace with the saturation vapor pressure of liquid B at the same temperature, because the effect of total pressure on the vapor pressure of a liquid is usually negligible (Sec. 12.8.1). If the temperature is above the critical temperature of pure B, we can estimate a hypothetical vapor pressure by extrapolating the liquid-vapor coexistence curve beyond the critical point. 

Figure 7 The vapor pressure and liquid-vapor coexistence curves (including estimated critical points) for methanethiol from simulation and experiment. The square symbols are obtained using the quantum-mechanical-based potential in simulation, the triangles from using the OPLS-UA potential, and the solid line shows the smoothed experimental data. The filled symbols are measured and estimated critical points (ref 22)...

In P-r space, we see only two remarkable features the vapor pressure curve, indicating the conditions under which the vapor and liquid coexist, and the critical point, at which the distinction between vapor and liquid disappears. We indicate in this figure the critical isotherm 7 = Tc and the critical isobar P = Pc. If the liquid is heated at a constant pressure exceeding the critical pressure, it expands and reaches a vapor-like state without undergoing a phase transition. Andrews and Van der Waals called this phenomenon the continuity of states. 

The calculated phase diagram for pentadecanoic acid using the Karaborni and Toxvaerd model has been reported in a previous publication [29]. The results can be summarized as follows (i) the model yields coexistence between a liquid and a vapor phase (ii) the liquid phase of the model monolayer is substantially denser than the LE phase of the real monolayer, and the critical point seems to be shifted to higher temperatures and (iii) the coexistence curve obtained from simulations can best be fitted with a scaling exponent of 0.32, supporting a three-dimensional character of the finite-size model system. Thus the Karaborni and Toxvaerd model yields merely qualitative agreement for the G-LE coexistence with experiments on the same systems. 

Fig. 6.9. Prewetting phase diagram for mercury showing liquid-vapor coexistence curve and diameter together with line of prewetting transitions extending from to the prewetting critical point, The dashed extension of the prewetting line indicates loci of maximum two-dimensional compressibility in the supercritical range above Inset, Prewetting layer thicknesses estimated with a slab model as described in text.

The liquid-vapor coexistence curve of the 10-mer as obtained from simulation and theory is shown in Fig. 23. Both the RHNC and MSA versions overestimate the critical temperature as obtained from simulation by about 15%. Of course, this is expected for any classical theory. On the other hand, far away from the critical point, results from both versions of the theory yield good agreement with simulation. The MSA version is somewhat more convenient, however, because it allows us to calculate the coexistence at low temperatures with no additional cost, while it becomes rather problematic to calculate the coexistence for the RHNC version below... 

It has been proposed to define a reduced temperature Tr for a solution of a single electrolyte as the ratio of kgT to the work required to separate a contact +- ion pair, and the reduced density pr as the fraction of the space occupied by the ions. (M+ ) The principal feature on the Tr,pr corresponding states diagram is a coexistence curve for two phases, with an upper critical point as for the liquid-vapor equilibrium of a simple fluid, but with a markedly lower reduced temperature at the critical point than for a simple fluid (with the corresponding definition of the reduced temperature, i.e. the ratio of kjjT to the work required to separate a van der Waals pair.) In the case of a plasma, an ionic fluid without a solvent, the coexistence curve is for the liquid-vapor equilibrium, while for solutions it corresponds to two solution phases of different concentrations in equilibrium. Some non-aqueous solutions are known which do unmix to form two liquid phases of slightly different concentrations. While no examples in aqueous solution are known, the corresponding... 

Figure 2.9 Phase diagram for C02, showing solid-gas (S + G, sublimation ), solid-liquid (S + L, fusion ), and liquid-gas (L + G, vaporization ) coexistence lines as PT boundaries of stable solid, liquid, or gaseous phases. The triple point (triangle), critical point (x), and selected 280K isotherm of Fig. 2.8 (circle) are marked for identification. Note that the fusion curve tilts slightly forward (with slope 75 atm K-1) and that the sublimation and vaporization curves meet with slightly discontinuous slopes (angle < 180°) at the triple point. The dotted and dashed half-circle shows two possible paths between a liquid (cross-hair square) and a gas (cross-hair circle) state, one discontinuous (dashed) crossing the coexistence line, the other continuous (dotted) encircling the critical point (see text).

Figure 7.1 Schematic phase diagram of water (not to scale), showing phase boundaries (heavy solid lines), triple point (triangle), critical point (circle-x), and a representative point (circle, dotted lines) at 25°C on the liquid-vapor coexistence curve.

A key role in this debate was played by experiments by Bischoff and Rosenbauer [153], who reported accurate data on isothermal vapor-liquid coexistence curves as a function of pressure near the critical line of NaCl + H20. Far from the critical point of pure water, one expects the compositions... 

Studies of liquid-vapor coexistence are, generally, best addressed in the framework of an open ensemble thus the state variables here comprise both the particle coordinates r and the particle number N. A path with the appropriate credentials can be constructed by identifying pairs of values of the chemical potential p and the temperature T which trace out some rough approximation to the coexistence curve in the p—7 plane, but extend into the one-phase region beyond the critical point. Once again there is some circularity here to which we shall return. Making the relevant variables explicit, the sampling distribution [Eq. (26)] takes the form... 

The vapor-liquid equilibrium curve terminates at the critical temperature and critical pressure (7c nnd Pc). Above and to the right of the critical point, two separate phases never coexist. 

J.J. De Pablo, T.M. Prausnitz, H.J. Strauch and P.T. Cummings, Molecular simulation of water along the liquid-vapor coexistence curve from 25 C to the critical point, J. Chem. Phys., 93, (1990) 7355-7359. 

The critical point of a binary mixture occurs where the nose of a loop in Fig. 10.3 is tangent to the envelope curve. Put another way, the envelope curve is the critical locus. One can verify tliis by considering two closely adjacent loops and noting what happens to the point of intersection as their separation becomes infinitesimal. Figure 10.3 illustrates that the location of the critical point on the nose of the loop varies with composition. For a pure species the critical point is the highest temperature and highest pressure at which vapor and liquid phases can coexist, but for a mixture it is in general neither. Therefore under certain conditions a condensation process occurs as the result of a reduction in pressure. 

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