Coexistence isochores

Finally, we consider the isothennal compressibility = hi V/dp)y = d hi p/5p) j, along tlie coexistence curve. A consideration of Figure A2.5.6 shows that the compressibility is finite and positive at every point in the one-phase region except at tlie critical point. Differentiation of equation (A2.5.2) yields the compressibility along the critical isochore ... 

Equation (11.163) shows how the isochoric heat capacity of a heterogeneous two-phase system can be evaluated from known isobaric properties (CP, aP) of the individual phases and the direction y(T of the coexistence coordinate cr. 

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... 

If, on the other hand, the substrates are sufficiently attractive, one notices from the plots in Fig. 4.7 that F (T, pb) may either vary continuously or discontinuously depending on whether the (bulk) isochoric path is super-or subcritical, respectively, with regard to the critical point of the confined fluid. Hence, discontinuities in the plots in Fig. 4.7 indicate capillary condensation (evaporation) in the model pore prior to condensation in the bulk, which would, of course, occur at bulk gas-liquid coexistence, i.e., at T - Tzb) /Tzb = 0. 

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.

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. 

Here 2 is the exponent for the heat capacity measured along the critical isochore (i.e. in the two-phase region) below the critical temperature, while is the exponent for the isothermal compressibility measured in the one-phase region at the edge of the coexistence curve. These inequalities say nothing about the exponents a and y in the one-phase region above the critical temperature. 

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. 

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...

Figure 3. Mean reduced isochoric slopes at the coexistence curve for ethylene as a function of the reduced...

Figure 5. Curvature of the isochores at the coexistence curve vs. density. Points A and B correspond to...

Figure 6. The reduced curvatures of isochores at the coexistence curve of ethylene vs. the molar density. CP indicates the critical point while the dotted curves show the trend of >a as predicted by the revised scaling...

R = value divided by its critical value cr = value at the coexistence curve p = value along an isochore... 

An isochoric equation has been developed for computing thermodynamic functions of pure fluids. It has its origin on a given liquid-vapor coexistence boundary, and it is structured to be consistent with the known behavior of specific heats, especially about the critical point. The number of adjustable, least-squares coefficients has been minimized to avoid irregularities in the calculated P(p,T) surface by using selected, temperature-dependent functions which are qualitatively consistent with isochores and specific heats over the entire surface. Several nonlinear parameters appear in these functions. Approximately fourteen additional constants appear in auxiliary equations, namely the vapor-pressure and orthobaric-densities equations, which provide the boundary for the P(p,T) equation-of-state surface. 

Iteration for Coexisting Densities. Orthobaric densities near the critical point generally cannot be obtained accurately from isochoric PpT data by extrapolation to the vapor-pressure curve because the isochore curvatures become extremely large near the critical point. The present, nonanalytic equation of state, however, can be used to estimate these densities by a simple, iterative procedure. Assume that nonlinear parameters in the equation of state have been estimated in preliminary work. For data along a given experimental isochore (density), it is necessary merely to find the coexistence temperature, Ta(p), by trial (iteration) for a best, least-squares fit of these data. 

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