Liquid isochore

If density (p) is constant, the fluid is referred to as isochoric (i.e., a given mass occupies a constant volume), although the somewhat more restrictive term incompressible is commonly used for this property (liquids are normally considered to be incompressible or isochoric fluids). If gravity (g) is also constant, the only variables in Eq. (4-5) are pressure and elevation, which can then be integrated between any two points (1 and 2) in a given fluid to give... 

This says that the sum of the local pressure (P) and static head (pgz), which we call the potential (4>), is constant at all points within a given isochoric (incompressible) fluid. This is an important result for such fluids, and it can be applied directly to determine how the pressure varies with elevation in a static liquid, as illustrated by the following example. 

Several techniques are available for measuring values of interaction second virial coefficients. The primary methods are reduction of mixture virial coefficients determined from PpT data reduction of vapor-liquid equilibrium data the differential pressure technique of Knobler et al.(1959) the Bumett-isochoric method of Hall and Eubank (1973) and reduction of gas chromatography data as originally proposed by Desty et al.(1962). The latter procedure is by far the most rapid, although it is probably the least accurate. 

A first study refers to liquid water [77]. The signals AS q,x) and A5[r,r] were measured using time-resolved X-ray diffraction techniques with 100 ps resolution. Laser pulses at 266 and 400 nm were employed. Only short times x were considered, where thermal expansion was assumed to be negligible and the density p to be independent of x. To prove this assumption, the authors compared their values of AS q, x) to the values of AS q) obtained from isochoric (i.e., p = const) temperature differential data [78-80]. Their argument is based on the fact that liquid H2O shows a density maximum at 4 °C. Pairs of temperatures Ti, T2 thus exist for which the density p is the same constant density conditions can thus be created in this unusual way. The experiment confirmed the existence of the acoustic horizon (Fig. 8). 

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.)... 

The adiabatic compression of saturated vapours was considered by Bruhat, who also calculated the angle between the liquid and vapour phase isochores in the entropy-temperature diagram. Amagat investigated the discontinuity in specific heats where an isothermal cuts the saturation curve. Hausen, from a complicated formula for the specific heat of steam involving two Einstein terms ( 2.IX N), calculated the heat content and entropy of steam. Leduc found the value of n for dry steam in Rankine s equation for adiabatic... 

Fig 3 The isochoric tenuperatuie derivative to liquid water from Sufi et al [7] shovring systematic vaiiation over the density maximum. 

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 4 Cartoon for hidden liquid-liquid spinodal in a polymer melt, calculated for a Flory Rotational Isomeric State chain with a simple coupling between density and conformational order A is the trans-gauche energy gap and a dimensionless density. Shown is the path of an isochoric quence into the unstable regime. r, is the spinodal temperature, T the melting temperature, T the liquid-liquid critical point, and Tp the temperature at which the harrier between dense liquid and crystal is of order ksT ...

Cp = isobaric specific heat c = isochoric specific heat e = specific internal energy h = enthalpy k = thermal conductivity p = pressure s = specific entropy t = temperature T = absolute temperature u = specific internal energy 4 = viscosity V = specific volume / = subscript denoting saturated liquid g = subscript denoting saturated vapor... 

Typical uncertainties in density are 0.02% in the liquid phase, 0.05% in the vapor phase and at supercritical temperatures, and 0.1% in the critical region, except very near the critical point, where the uncertainty in pressure is 0.1%. For vapor pressures, the uncertainty is 0.02% above 180 K, 0.05% above 1 Pa (115 K), and dropping to 0.001 mPa at the triple point. The uncertainty in heat capacity (isobaric, isochoric, and saturated) is 0.5% at temperatures above 125 K and 2% at temperatures below 125 K for the liquid, and is 0.5% for all vapor states. The uncertainty in the liquid-phase speed of sound is 0.5%, and that for the vapor phase is 0.05%. The uncertainties are higher for all properties very near the critical point except pressure (saturated vapor/liquid and single pliase). The uncertainty in viscosity varies from 0.4% in the dilute gas between room temperature and 600 K, to about 2.5% from 100 to 475 K up to about 30 MPa, and to about 4% outside this range. Uncertainty in thermal conductivity is 3%, except in the critical region and dilute gas which have an uncertainty of 5%. 

Fig. 2.26 Extraction circuit A-G in the t,s-diagram of carbon dioxide CP critical point, V isochors with density indication, g gas, I liquid, f supercritical...

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.

Abstract A synthetic pure water fluid inclusion showing a wide temperature range of metastability (Th - Tn 50°C temperature of homogenization Th = 144°C and nucleation temperature of Tn = 89°C) was selected to make a kinetic study of the lifetime of an isolated microvolume of superheated water. The occluded liquid was placed in the metastable field by isochoric cooling and the duration of the metastable state was measured repetitively for 7 fixed temperatures above Tn. Statistically, metastability lifetimes for the 7 data sets follow the exponential reliability distribution, i.e., the probability of non nucleation within time t equals. This enabled us to calculate the half-life periods of metastability r for each of the selected temperature, and then to predict i at any temperature T > Tn for the considered inclusion, according to the equation i(s) = 22.1x j Hence we conclude that... 

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