Isochoric line

Figure 1. Pure water phase diagram in (P,T) coordinates calculatedfrom the IAPWS-95 equation of state, extrapolated at negative liquid pressures in the superheat domain. The outer lines starting from the critical point are the thermodynamic limits of metastability (spinodal). The dotted line is one of the proposed kinetic metastability limii (see text). Three isochoric lines (950, 900 and 850 kg m ) are also calculated by extrapolation of the IAPWS-95 equation.

Fig. 2.16. Temperature dependences of the dielectric relaxation times for PVAc at atmospheric pressure ( ) and at a constant volume equal to 0.847 mlg (A), 0.849 ml ( ), and 0.852 ml g (V). The slopes at the intersection of the iso-baric and isochoric lines yield values for the respective activation energies at constant pressure and constant volume a = 238 and 448kJmol (r = 2.5 s) and = 166 and 293 kJ mol (r = 0.003 s). The ratio of the isochoric and isobaric activation energies is a measure of the relative contribution of thermal energy and volume that is, this ratio would be unity if the molecular motion were thermally activated, and zero if it were strictly dominated by density. For PVAc, the ratio is 0.6, indicating that both contributions are significant. From Roland and Casalini by permission [132].

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,... 

Equation (2.18) is another example of a line integral, demonstrating that 6q is not an exact differential. To calculate q, one must know the heat capacity as a function of temperature. If one graphs C against T as shown in Figure 2.8, the area under the curve is q. The dependence of C upon T is determined by the path followed. The calculation of q thus requires that we specify the path. Heat is often calculated for an isobaric or an isochoric process in which the heat capacity is represented as Cp or Cy, respectively. If molar quantities are involved, the heat capacities are C/)m or CY.m. Isobaric heat capacities are more... 

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.)... 

Of course, the vapour pressure is very temperature dependent, and reaches P° = 101.325 kPa at the normal boiling point, Tb. The isochoric thermal pressure coefficient, dp/dT)v = otp/KT, can be obtained from the two quantities on the right hand side listed in Table 3.1. Except at T it does not equal the coefficient along the saturation line, (dp/dT)a, which is the normal vapour pressure curve. The latter temperature dependence is often described by means of the Antoine equation ... 

The results of Knoblauch, Linde, and Klebe showed that the isochores ( =const.) of steam are practically straight lines and can be represented by ... 

An important consideration in the existence of a spinodal is the prescribed experimental conditions. In a monodisperse melt, liquid liquid coexistence can only occur along a line in the pressure-temperature/>—T plane. Hence, liquid liquid phase separation under isobaric conditions can only be transient, before the entire phase reverts to the dense liquid. On the other hand, an isochoric quench would be expected to yield true spinodal-like behaviour. The true system is probably something between the two extremes, with volume leaving the system on some timescale. Based on estimates of thermal diffusivity in melts, the time to shrink is of order 10 s (based on a 1 m sample thickness). If... 

Figure 3. Constant-volume heat capacity, Cy, for the CHD fluid on the critical isochore as obtained from NVT MC simulations fluctuation formula (6) (points) from a [5,5] Pade approximant (9) fitted to the energy (solid line). The kinetic contribution is not included.

Figure 5. Constant-volume heat capacity, Cy, of the AHS fluid with a = 6 as obtained from NVT (filled circles and solid line) and pVT (open circles and dashed line) MC simulations of systems with L = 10a along the bulk critical isochore. The kinetic contribution (equal to SNIcb/Z) is included.

Figure 24. Upper panel Dielectric relaxation time for BMMPC experimental data for 0.1 MPa, other isobars (200 and 600 MPa), and the isochore at V = 0.9032 ml/g were calculated. Dotted line indicates the average of logjo(is) = —6.1 for the different curves. Lower panel Shekel function, with low- and high-71 linear fits, done over the range —4.68 < log10(x[i]) < 3.85 and —8.55 < log10(x[j]) < —6.4, respectively. Vertical dotted lines indicate the dynamic crossover.

A key result of the sorption experiments conducted 1 Thommes and Findenegg concerns the pore condensation line T p (pb) > T b (Pb) at which pore condensation occurs along a subcritical isochoric path Pb/Pch < 1 in the bulk (/ b and peb arc the density of tliis isochore and the bulk critical density, respectively). Experimentally, Txp (pb) is directly inferred from the temperature dependence of F (T), which changes discontinuously at n, (Pb) (see Ref. 31 for detaiLs). The pore condensation line ends at the pore critical temperature Tep (rigorously defined only in the ideal single slit-pore case) [31]. Because of confinement Tep is shifted to lower values with decreasing pore size. If, on the other hand, the pore becomes large, Tep — (the bulk... 

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.

Figure. 6 presents water isotherms. From experimental data follows the smoothness of extension of isotherms, isochores, isobars from the stable into the metastable region and the absence of singularities, at least for the first two derivatives of the thermodynamic potential, on the phase-equilibrium line. As distinct from isotherms and isobars, which are essentially nonlinear, isochores are close to straight lines in the metastable region up to the critical point. 

Isochores with v < v, are above the Lh-G two-phase coexistence line isochores with v > are below the line. The isochore with v = v, coincides with the Lh-G coexistence line over the entire range from the triple point t to the critical point c. Beyond the critical point it is a continuous extension of the coexistence line. The isochores are remarkable by their near linearity. The curvature, generally small, is the largest at states close to the two-phase line. 

Figure 16. Orientational dynamics of the ellipsoids of revolution in the 50 50 binary mixture (N = 256) at several temperatures along the isochor at density p = 0.8. (a) Time evolution of the single-particle second-rank orientational time correlation function in a log-log plot. Temperature decreases from left to right (4.997 >T> 0.498). (b) Time dependence of the OKE signal in a log-log plot at temperatures T = 0.574,0.550,0.529, and 0.498. The continuous lines are linear fits, showing the power law decay. Temperature decreases from top to bottom across the linear regime. (Reproduced from Ref. 126.)...

Figure 20. (a) Orientational correlation time t in the logarithmic scale as function of the inverse of the scaled temperature, with the scaling being done by the isotropic to nematic transition temperature with Ti-N. For the insets, the horizontal and the vertical axis labels read the same as that of the main frame and are thus omitted for clarity. Along each isochor, the solid line is the Arrhenius fit to the subset of the high-temperature data and the dotted line corresponds to the fit to the data near the isotropic-nematic phase boundary with the VFT form, (b) Fragility index m as a function of density for different aspect ratios of model calamitic systems. The systems considered are GB(3, 5, 2, 1), GB(3.4, 5, 2, 1), and GB(3.8, 5, 2, 1). In each case, N = 500. (Reproduced from Ref. 136.)... 

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