P-V-T-x curves

The type of correlation between experimentally measured variable parameters in the system is changing as a result of phase transformations within that system. The indirect methods are used to identify the changes in those correlations. For example, the p-V-T-x curves (pressure as a function of temperature (p-T curves) pressure or temperature as a function of volume (p-V or T-V curves) pressure drop during salt dissolution as a function of solution composition (p-x curves)) show a break at the location of the phase transformation. Not only pressure or temperature or composition could be considered as variables. The properties of the solution or the whole system itself as well as heating or cooling period in thermal analysis can be used as variables to find the phase transitions. 

We shall consider in detail some versions of p-V-T-x curve methods using schemes of typical isothermal and isobaric sections (Figure 1.10) of phase diagrams for binary systems. 

Figure 1.10 Behavior of p-V-T-x curves at phase transformations during experiments. FI - fluid phase S - solid phase (The Experimental Determination of Solubilities, Vol 6, Hefter Tomkins).

Sampl Sampl Therm, anal. Sampl Sampl p-x p-V T-V curves Vis.obs. Vap.pr. 

Fig. 98. Fragments of X-ray diffraction patterns of an U2CO3 and NbC>2F mixture performed in situ during synthesis of Li4NbOfF at several temperatures. Curve numbers correspond to 580 (1) 730 (2) 830 (3) 900 (4) and 1150X1 (5). Reproduced from [419], S. Y. Stefanovich, B. A. Strukov, A. P. Leonov, A. I. Agulyansky, V. T. Kalinnikov, Jap. J. Appl. Phys., 24 (1985) 630, Copyright 1985, with permission of Institute of Pure and Applied Physics, Tokyo, Japan.

Fig. 8. Recording of t—E curves during the course of a reaction Reaction of S-ben-zyl-isothiuronium chloride with hydroxide and carbonate ions in carbonate buffer pH 9.7. Initial concentration of S-benzylisothiuronium chloride 4 x 10 4M. Anodic waves of benzylmercaptan formed were recorded after intervals given in the polarogram. Curves starting at — 1.0 V towards more positive potentials, S.C. E., 200 mV/absc., h = 70 cm, full scale sensitivity 2.2 p.A...

FIG. 2-7 Enthalpy-concentration diagram for aqueous ammonia. From Thermodynamic and Physical Properties NH3-H20, Int Inst. Refrigeration, Paris, France, 1994 (88 pp.). Reproduced by permission. In order to determine equilibrium compositions, draw a vertical from any liquid composition on any boiling line (the lowest plots) to intersect the appropriate auxiliary curve (the intermediate curves). A horizontal then drawn from this point to the appropriate dew line (the upper curves) will establish the vapor composition. The Int. Inst. Refrigeration publication also gives extensive P-v-xtah es from —50 to 316°C. Other sources include Park, Y. M. and Sonntag, R. E., ASHRAE Trans., 96,1 (1990) 150-159 x, h, s, tables, 360 to 640 K) Ibrahim, O. M. and S. A. Klein, ASH E Trans., 99, 1 (1993) 1495-1502 (Eqs., 0.2 to 110 bar, 293 to 413 K) Smolen, T. M., D. B. Manley, et al.,/. Chem. Eng. Data, 36 (1991) 202-208 p-x correlation, 0.9 to 450 psia, 293-413 K)i Ruiter, J. P, 7nf. J. R rig., 13 (1990) 223-236 gives ten subroutines for computer calculations. 

In P-T projections, the composition axis is collapsed into the pressure-temperature plane. The vapor pressure curve for component A is labeled LV(A) and that for component B is labeled LV(B). These curves terminate at the component critical points (L = V) designated as hollow circles. In Fig. 2, dew pressure and bubble pressure curves for an intermediate composition x intersect at a point on the (L = V) critical locus where the liquid and vapor phases become critically identical. Normally, dew and bubble pressure curves are not shown in projections. They are shown here so that the construction of the related P-x at fixed T, and T-x at fixed P, phase diagrams is clearly illustrated. Each critical point on the critical locus corresponds to a fixed composition. Points close to the critical point of component A are critical points for mixtures with high concentrations of A, whereas points closer to the critical point of... 

The idea that a and p are functions of time is based on the fact that the surface potential value, and therefore the curve V = f(x,t), changes during the shift towards equihbrium. Therefore, the coefficients a and p which describe this curve, may from now on be associated with such an evolution type. 

Figure 3 MFLG calculated p (T) and T (x) projections of the critical curve(s) (L = V) for the system ethylene/naphtalene, using the critical determinant (x = mole % Ciq s)-0 Data by Van Welie and Diepen.

Nonvariant points of pure components can be omitted on the T-X projection because they do not take part in a formation of ternary monovariant curves. Various points on the ordinates of T-X diagram show temperature of all nonvariant equilibria (stable and metastable) in binary subsystems A-B and A-C, and are the starting points for ternary monovariant curves, corresponding to the equihbria spreading from the binary subsystems into three-component region of composition. The type of binary subsystems is uniquely determined by a combination of these binary non-variant points on the ordinate of T-X diagram. For instance, the set of nonvariant points N L M N" corresponds to type lb, V A - type lb. A L M - type lb". A A A i -type Ic, A Z, M A A - type Ic, A ZJ - type Id, A L M R- type Id, A L p R - type Id", A p R" Q - type 2d, A L M p R" Q - type 2d (where A and are the metastable nonvariant points A and R, respectively) etc. 

The plots in Figure 7-41c and d arc conventional spectra of two monradiant power P v) is plotted with respect to the frequency in hertz. The symbol in parentheses is added to emphasize the frequency dependence of the power time-domain power is indicated by P(t). 

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