Isochore curve with

Figure 25. Upper panel dielectric relaxation times for PCB62 experimental data for 0.1 MPa other isobars and isochoric curve at V 0.6131 ml/g were calculated. Dotted line indicates the average of logits) = —5.9 for the different curves. Lower panel Stickel function, with low- and high-Tlinear fits, done over the range —5.42 < log10(r[i]) < 2.169 and —8.87< log10(r[i]) < -5.98, respectively. The vertical dotted lines in both panels represent the dynamic crossover.

Using eq. (3) the temperatures are determined that will yield a value of the quantity TV at the higher P equal to values for the atmospherie pressure data. The resulting curves are shown in Figure 2, along with the corresponding derivative plots. From the intersection of the linear segments, we obtain the Ta for each pressure and hence ta(P). The latter, listed in Table 1, is invariant to pressure. We also include in Fig. 2 an isochoric curve at an arbitrary V. Likewise, the transition to Arrhenius behavior under constant volume is associated with the same value of xa- A similar analysis was applied to... 

Figure 1. Coexistence curve with isochores and the isochoric slope, j/0. At points A and B the curvature of the isochores is zero as well as along the locus indicated by the dotted line (the locus of the isochoric heat...

With rcgaid to these other coefficients, an interpolation method is used, which we will describe by calculating the coefficient (dF dP)j. Close to point M (Figure 2.1b) we see a first couple, an isobar at P and an isochoric curve at Vi, which intersect at the temperature of point M. As close as possible, on the other side of point M, we see a second couple, an isobaric curve at P2 and an isochoric curve at V2, which also intersect at the temperature of point M and we write ... 

Two isotherms, isochores, adiabatics, or generally any two thermal lines of the same kind, never cut each other in a surface in space representing the states of a fluid with respect to the three variables of the characteristic equation taken as co-ordinates, for a point of intersection would imply that two identical states had some property in a different degree (e.g., two different pressures, or temperatures). Two such curves may, however,... 

Figure 5.17 The solubility s of a partially soluble salt is related to the equilibrium constant (partition) and obeys the van t Hoff isochore, so a plot of In s (as y ) against 1 IT (as x ) should be linear, with a slope of AF lution -t- R . Note how the temperature is expressed in kelvin a graph drawn with temperatures expressed in Celsius would have produced a curved plot. The label KIT on the x-axis comes from l/T -t- 1/K...

FIGURE 3.20 Successive cooling curves for hydrate formation with successive runs listed as Sj < S2 < S3. Gas and liquid water were isochorically cooled into the metastable region until hydrates formed in the portion of the curve labeled Sj. The container was then heated and hydrates dissociated along the vapor-liquid water-hydrate (V-Lyy-H) line until point H was reached, where dissociation of the last hydrate crystal was visually observed. (Reproduced from Schroeter, J.R, Kobayashi, R., Hildebrand, M.A., Ind. Eng. Chem. Fundam. 22, 361 (1983). With permission from the American Chemical Society.)... 

Figure 6. Evolution of isochors in the P - 7 phase diagram for the core softened potential with third critical point in metastable region. Cl - gas + liquid, C2 - LDL + LIDL, and C3 - HDL + VHDL critical points. Red lines (online) are coexistence curves. Blue curves (online) are isochors. Critical point location na = 0.0064, Xa = 0.1189, ya =0.0998 nc2 = 0.1423, Xc2 = 0.3856, yc2 = 0.33 Ties = 0.07487, xcs = 0.2398, yes = 0.6856. Model parameter set a = 6.962, bh =2.094, Ur/Ua=3, b,=7.0686.

Another alternative, particularly for high-pressure measurements, is the isochoric method. A cell of known volume is filled with the fluid, and then the pressure is measured as the temperature of the cell is changed. This provides data along curves of approximately constant density, known as isochors. Corrections are made for the expansion of the vessel due to temperature and pressure. This method is most useful for supercritical fluids and other situations where the fluid is fairly compressible. Uncertainties with this method can be on the order of 0.1%, but are often higher at elevated temperatures and pressures. 

Figure 26. Characterization of the inherent structures for the model calamitic system GB(3,5,2, 1) ( = 256). (a) Parallel radial distribution function g (/ ) for the inherent structures at all temperatures considered along the isochor at density p = 0.32. Note that the curves for the highest five temperatures are nearly superposed on each other. For others, the amplitude of the peaks gradually increases as the temperature drops, (b) Evolution of the 6-fold bond orientational order parameter 4>6 for the inherent stmctures with temperature at three densities. (Reproduced from Ref. 144.)...

An isochoric equation of state, applicable to pure components, is proposed based upon values of pressure and temperature taken at the vapor-liquid coexistence curve. Its validity, especially in the critical region, depends upon correlation of the two leading terms the isochoric slope and the isochoric curvature. The proposed equation of state utilizes power law behavior for the difference between vapor and liquid isochoric slopes issuing from the same point on the coexistence cruve, and rectilinear behavior for the mean values. The curvature is a skewed sinusoidal curve as a function of density which approaches zero at zero density and twice the critical density and becomes zero slightly below the critical density. Values of properties for ethylene and water calculated from this equation of state compare favorably with data. 

In 1976, Hall and Eubank (12,13) published two papers which have direct bearing upon the present equation of state. In the first paper, they noted the rectilinear behavior for the mean of the vapor and liquid isochoric slopes issuing from the same point on the vapor pressure curve near the critical point and the power law behavior for the difference in these slopes. The second paper presented an empirical description of the critical region which generally agreed with the scaling model but differed in one significant way—the curvature of the vapor pressure curve. 

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 ... 

Fig. 4.12. The isochoric thermal pressure cx>efiicient, yy, of fluid mercury versus density along the liquid-vapor coexistence curve (open circles) derived from isochores of Fig. 4.10. Solid lines represent predictions of the soft-sphere equation of state, Eq. (4.11) with = 12, appropriate for rare gases, and n = 7. Points (x) denote predictions of Eq. (4.11) for n = 15.5.

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. ... 

As can be expected, g(7 uz) is rather complex function of both variables, T and/w. InFig. 2.26 are plotted changes of isochoric heat capacity witli volume, g(T=const m.) for water and citric acid solutions. As can be observed, for temperatures lower than about 30 °C, g T m) curves have curvature concave downward (g"(7 = const m) < 0) with the maximum (at about 2.0 mol kg at 15 °C) which is shifted to higher concentrations when temperature increases. After this, the cnrvature changes gradually to concave upward (g"(T= const m)>0). After about 4.0 mol kg , the change in temperature has an inverse effect on the isochoric heat capacities of citric acid solutions. 

This is the leading term in a power series expansion of Sv in St. Notice that the coexistence curve is not symmetric with respect to reflection across the critical isochore—except very close to the critical point (cf. below). 

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