Diameter of the coexistence curve

An unexpected conclusion from this fonuulation, shown in various degrees of generality in 1970-71, is that for systems that lack tlie synunetry of simple lattice models the slope of the diameter of the coexistence curve... 

Two recent papers report observations of a divergence of the diameter of the coexistence curve of the pure fluid SF , although no analysis is given. Gopal et al. have reported similar divergences in the systems CSj + CH3NO2 and cyclohexane + acetic anhydride. They fit their data over a range 3 x 10" <... 

The anomaly in the diameter of the coexistence curve characterized by P is weak and difficult to observe. By combining Equations 8.20 and 8.22, an equation for the coexistence vapor and liquid densities can be obtained as ... 

Fig.l. Coexistence curve of acetonitrile + water densities p (left) and static dielectric constants (right) at the transition temperature ( k, ) and in the one-phase region ( ). Diameter of the coexistence curve ( , ). Composition x mole% of water. 

Figure 13 Coexistence curves corresponding to the first layering transition of water in cylindrical pores of various radii Rp and with smooth hydrophilic surface (t/o =-4.62 kcal/mol). Open circles densities of the coexisting phases. Closed circles diameter of the coexistence curve.

If the small temis in p- and higher are ignored, equation (A2.5.4) is the Taw of the rectilinear diameter as evidenced by the straight line that extends to the critical point in figure A2.5.10 this prediction is in good qualitative agreement with most experiments. However, equation (A2.5.5). which predicts a parabolic shape for the top of the coexistence curve, is unsatisfactory as we shall see in subsequent sections. 

Jungst S, Knuth B and Hensel F 1985 Observation of singular diameters in the coexistence curves of metals Phys. Rev. Lett. 55 2160-3... 

Pestak M W, Goldstein R E, Chan M H W, de Bruyn J R, Balzarini D A and Ashcroft N W 1987 Three-body interactions, scaling variables, and singular diameters in the coexistence curves of fluids Phys. Rev. B36 599-614... 

Now we discuss the diameter (Mi + Mf) /2 of the coexistence curve. For a long time, the diameter anomaly in nonionic systems was a matter of controversy, because the deviations from rectilinear behavior are small, and there is an additional spurious 2(5 contribution, when an improper order parameter is chosen in data evaluation [104], The investigations of the picrate systems [72] and of the IL solutions [103] both yielded a substantial anomaly, consistent with an (1 — a) anomaly. Large diameter anomalies are expected, when the intermolecular interactions depend on the density [84], In the systems considered here, the dilute phase is essentially composed of ion pairs, while the concentrated phase is an ionic melt, which may explain the rather pronounced deviation from the rectangular diameter in the ionic systems. 

Fig. 40. Schematic description of unstable thermodynamic fluctuations in the two-phase regime of a binary mixture AB at a concentration cb (a) in the unstable regime inside the two branches tp of the spinodal curve and (b) in the metastable regime between the spinodal curve tp and the coexistence curve The local concentration c(r) at a point r = (x. y, z.) in space is schematically plotted against the spatial coordinate x at some time after the quench. In case (a), the concentration variation at three distinct times t, ti, u is indicated. In case (b) a critical droplet is indicated, of diameter 2R , the width of the interfacial regions being the correlation length Note that the concentration profile of the droplet reaches the other branch ini, of the coexistence curve in the droplet center only for weak supersaturations of the mixture, where cb -

It is customary to cast equations like the above in reduced form hence the variable is not (T — T ), but rather (T — T )jT°, which we shall later symbolize by t.] The exponent j8 is approximately 1/3 and almost certainly lies between 0.31 and 0.36, well below the classical (i.e. analytic) 1/2. The other property of the coexistence curve, the midpoint of the two-phase region or diameter (x + x")l2 describes a very nearly straight line, but recent theoretical studies suggest a small curvature close to T as described by the equation ... 

Up to now, much less effort has focussed on the question of the rectilinear diameter in mixtures than in gas-liquid one-component systems. However, Stein and Allen have examined this as part of their analysis of coexistence data on nine systems and have been unable to draw meaningful conclusions about the possible singularity. For example, they find that analysis of the coexistence curve results of Wims, McIntyre, and Hynne on 3-methylpentane 4- nitroethane yields (1 — a) = 0.62 0.21 if volume fractions are used, 0.67 0.20 if... 

N. W. Ashcroft, Three-body Interactions, Scaling Variables, and Singular Diameters in the Coexistence Curves of Fluids. Phys. Rev. B, 36, 599-614 (1987). 

Fisher and Wortis have shown that Tohnan s length is zero for symmetric fluid coexistence and non-zero for asymmetric fluid coexistence. " Symmetric fluids are represented by the lattice-gas (Ising) model in which the shape of the coexistence curve is perfectly symmetric with respect to the critical isochore. Real fluids always possess some degree of asymmetryAsymmetry in the vapour-liquid coexistence in helium, especially in He, is very small, but not zero. In the mean-field approximation, the asymmetry in the vapour-liquid coexistence is represented by the rectilinear diameter ... 

Guided by some theoretical models that predicted a singular behaviour of the coexistence-curve diameter pd as a function of temperature, a revised-scaling approximation has been proposed in which the scaling fields are defined by ... 

To estimate the critical density, an eqnation for the diameters, (pi + pf/2, of the coexistence curve is used (Sengers and Levelt-Sengers 1978) ... 

The diameter p of the coexistence curve is the average value of the densities of the coexisting liquid and vapor phases. It is equal to the critical density at T = Tc and changes mainly regulary with r. In the close vicinity of the critical point, diameter of fluids shows a critical anomaly, which may behave as [14] or [15], or as superposition of... 

The shape of the coexistence curve is determined by the temperature dependences of the order parameter Ap and of the diameter pd. 

Figure A2.5.10. Phase diagram for the van der Waals fluid, shown as reduced temperature versus reduced density p. . The region under the smooth coexistence curve is a two-phase liquid-gas region as indicated by the horizontal tie-lines. The critical point at the top of the curve has the coordinates (1,1). The dashed line is the diameter, and the dotted curve is the spinodal curve.

Figure 3 Diagram illustrating the various directions of approach to the critical point and the exponents appropriate to the various quantities. Negative exponents (e.g. —a) indicate the quantity becomes infinite at the critical point positive exponents (e.g. jS) indicate the quantity vanishes at the critical point. The coexistence curve and the diameter correspond to = 0.35 1 — a = 0.90 note that the curvature of the diameter is barely noticeable. is the long-range correlation length it and its exponent v are discussed in Section 6)...

A real fluid, of course, has no such natural and exact symmetry. Consequently, in analysing real data to extract critical exponents, it is essential to select variables in such a way that at least an approximate symmetry appears over as wide a range as possible only then can the asymptotic behaviour be examined. It is well known that the coexistence curve is much more nearly symmetric in a T, p representation than in a r,Im representation only the former yields, at least approximately, the rectilinear diameter. 

FIG. 10 The coexistence curve for a two-dimensional Lennard-Jones fluid, showing the estimated critical point as the filled circle ( ). The points are simulation results, and the solid lines are fits to the 2D scaling law and the rectilinear diameter law the dashed line is from an earlier equation of state. (From Ref, 75a.)... 

Figure 10.2 Diameter of the vapour-liquid coexistence curve for nitrogen (a) and...

These effective diameters are simply the values of cr needed to account for the isochore slope at each point along the coexistence curve. 

It is difficult to determine directly the liquid-vapor coexistence curves of metals by measurement of the coexisting densities of the two phases. Rather, the curves have been established indirectly from the intercepts of measured isochores (Figs. 3.17 and 4.10) with the vapor pressure curve Psat versus T. The coexistence curves determined in this way for cesium and rubidium are presented in Fig. 6.1. This figure shows a plot of the reduced densities of the coexisting liquid, p lPc, and vapor, PvIPc as a function of the reduced temperature TjT. The plot also shows the mean densities, = (l/2)(pi, + py), known as the diameters. ... 

The coexistence curves of cesium, rubidium, and mercury violate a century-old empirical rule known as the Law of Rectilinear Diameters (Cailletet and Mathias, 1886). According to this rule, a plot of the diameter, versus T, should be linear right up to the critical point. In contrast, renormalization group theory predicts that the temperature derivative of the diameter, dp /dT, should diverge at least as fast as the constant-volume specific heat c . Specifically, as the reduced temperature T = (T — T)/Tc goes to zero, the diameter varies as... 

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