Thermodynamic equilibrium curve

In general, the selectivity of pervaporation is high, as demonstrated in Figure 6.41 for the system "benzene-cyclohexane, polyethylene membranes". The separation characteristic is shifted to much more favorable figures compared to the thermodynamic equilibrium curve the azeotropic point can be suppressed.28-30 Parameter in the diagram is the ratio of total pressure p at the permeate side to saturation pressure Pl = TjXjP + TjXjpj. 

Figure 6.41 shows that the selectivity decreases with increasing pressure at the permeate side. For a ratio of p/p = 1 finally, the separation characteristic is identical to the thermodynamic equilibrium curve (for = constant). Pervapo-ration will always be a process which is relatively expensive compared to other membrane processes for two reasons ... 

Constitutive relation An equation that relates the initial state to the final state of a material undergoing shock compression. This equation is a property of the material and distinguishes one material from another. In general it can be rate-dependent. It is combined with the jump conditions to yield the Hugoniot curve which is also material-dependent. The equation of state of a material is a constitutive equation for which the initial and final states are in thermodynamic equilibrium, and there are no rate-dependent variables. 

In the region of pure CH4, the equilibrium is governed by Equation 4. For this reaction, the equilibrium constant increases with temperature so that at high enough temperatures there will be appreciable dissociation of CH4 to H2 and graphite. In the lower temperature range considered here, the thermodynamic equilibrium indicates only a very small amount of dissociation so the intersection of the graphite deposition curve with the H2-CH4 line occurs at almost pure CH4. As the temperature increases, the point of intersection will move toward pure H2 on the H2-CH4 line. 

Monolayers of distearoylphosphatidylcholine spread on the water-1,2-dichloro-ethane interface were studied by Grandell et al. [52] in a novel type of Langmuir trough [53]. Isotherms of the lipid were measured at controlled potential difference across the interface. Electrocapillary curves derived from the isotherms agreed with those measured under the true thermodynamic equilibrium. Weak adsorption or a stable monolayer was found to be formed, when the potential of the aqueous phase was positive or negative respectively, with respect to the potential of the 1,2-dichloroethane phase [52]. This result... 

The states produced by compression of monolayers are not necessarily at thermodynamic equilibrium, but may be metastable. In some cases, this is manifested clearly by different force-area curves being produced at different rates of compression. Slow reorganization of monolayer molecules is also apparent as hysteresis when films are repeatedly compressed and expanded. In chiral monolayers, the rates of molecular reorganization may be stereospecific as well as the thermodynamic behavior. 

In their subsequent works, the authors treated directly the nonlinear equations of evolution (e.g., the equations of chemical kinetics). Even though these equations cannot be solved explicitly, some powerful mathematical methods can be used to determine the nature of their solutions (rather than their analytical form). In these equations, one can generally identify a certain parameter k, which measures the strength of the external constraints that prevent the system from reaching thermodynamic equilibrium. The system then tends to a nonequilibrium stationary state. Near equilibrium, the latter state is unique and close to the former its characteristics, plotted against k, lie on a continuous curve (the thermodynamic branch). It may happen, however, that on increasing k, one reaches a critical bifurcation value k, beyond which the appearance of the... 

Graphically, the conditions for thermodynamic equilibrium are equal to two points which have a common tangent. These points give the composition of a polymer-rich phase (I) and a solvent-rich phase (II) that can coexist in thermodynamic equilibrium. The summation of such points is also called the coexistence curve or binodal line. 

For continuous systems, molar flow rates Q can be used instead of n. The thermodynamic activity (ax) can be calculated according to Equation 2, but requires knowledge of the saturation pressure of the pure compound (Ppsatx). This data can be obtained from the saturation curves (vapor-liquid equilibrium curves) and is taken at the working temperature of the gas stream. The thermodynamic activity is then calculated using the following equation ... 

Note, however, that EM is not determined by the thermodynamic equilibrium potentials Ef and E2 but by the kinetics of the respective reactions, i.e. by the respective anodic and cathodic component curves in Fig. 13(a) with the condition indicated by eqn. (190). These curves may be altered by mass transport conditions, surface area and, specific properties and consequently the mixed potential EM may be susceptible to those kinetic factors, unlike the equilibrium potential of each partial electrode reaction which is fixed by thermodynamics and the activities in the bulk solution. 

Figure 1. Thermodynamic equilibrium in atmospheres of varying elemental proportions. The ternary diagram shows all compositions of systems containing carbon, hydrogen, and oxygen (each point represents 100% of the three components). Lower curves indicate the potential formation of solid carbon if equilibrium could be attained. Dashed curve holds at 500°K., the continuous one at 700°K. The upper lines indicate the asphalt threshold, the dashed one at 500° K., and the continuous one at 700° K. Above this threshold, thermodynamic equilibrium favors the formation of large proportions of polycyclic aromatic compounds ( asphalt ) ana a lesser increase of most of the other families of compounds. The dots through points A to C indicate the points used in the computations for Figure 2 (6).

As there exists a phase equilibrium both phases must have reached in the internal thermodynamic equilibrium with respect to the arrangement and distribution of the molecules the measuring time. Therefore, no time effects or path dependencies of the thermodynamic properties in the liquid crystalline phase should be expected. To check this point for the l.c. polymer, a cut through the measured V(P) curves at 2000 bar has been made (Fig. 6) and the volume values are inserted at different temperatures in Fig. 7, which represents the measured isobaric volume-temperature curve at 2000 bar 38). It can be seen from Fig. 7 that all specific volumes obtained by the cut through the isotherms in Fig. 6 he on the directly measured isobar. No path dependence can be detected in the l.c. phase. From these observations we can conclude that the volume as well as other properties of the polymers depend only on temperature and pressure. The liquid crystalline phase of the polymer is a homogeneous phase, which is in its internal thermodynamic equilibrium within the normal measuring time. 

Fig. 5. Several metals have been shown to be volatilized by carbon monoxide. A map of the safe (no metal loss) operating conditions ( ) and unsafe ( ) operating conditions for Ni/Al2Oj catalysts. The equilibrium curves for various equilibrium partial pressures of Ni(CO)4 were calculated by using thermodynamic data from the literature (72).

This behavior can be understood by the assumption that two different types of ion pairs exist in a thermodynamic equilibrium and add the monomer with different rate constants. The less reactive contact ion pair (tight-ion pair) and the more reactive solvent separated ion pair (loose-ion pair), which is more stable at lower temperatures (31). The curved lines show the transition from one ion pair species to the other. Thus, the polymerization mechanism can be described by this scheme ... 

Fig. 7. PDC-calibration curves for polystyrene in cyclohexane measured 3> at eight temperatures, as indiciated, and an overall rate of the column liquid of 15 cm3/h (ordinate is normalized as indicated). The 15 °C-calibration curve dyn is measured, whereas the dashed curve 15 °C therm is extrapolated from the measured part of the dyn curve (cf. Fig. 8), and corresponds to reversible-thermodynamic equilibrium of the PDC-column. The difference between both curves shows a pronounced PDC-effect at 15 °C for P = 1082. Elution volume V = Ve and zero volume V0 = are expressed in counts (1 count = 0.51423 cm3). For the definition of r0 see Eq. (5 b)...

The pronounced discrepancy between the measured dynamic 15 °C-elution curve and its extrapolated reversible-thermodynamic part, shown in Fig. 7, represents a direct proof of the inadequacy of the reversible Eq. (3) in the dynamic region of the column (PDC-effect). Moreover, the experiment shows immediately that the polymer of the mobile phase has to dissolve in the gel layer within the transport zone to a considerably higher extent than is allowed by the partition function (4) in a reversible-thermodynamic equilibrium between the gel and the sol at the same column temperature. As a consequence, a steady state, i.e. a flow-equilibrium, must be assumed in the system sol/gel within the considered transport zone, governing the polymer trans-... 

The calculation of the phenomenological function a(P T) of the flow-equilibrium from the measured calibration curves, shown in Figs. 7 and 8, is based on a nonlinear fit of Eq. (19) to these curves. It proceeds by the same method 4) as applied to the calculation of the reversible-thermodynamic data from Table 2 in Section 3.1 the phenomenological function a(P T), obtained in this way, is shown in Table 3 b. With this, the relative perturbation, 8Q/K, of the thermodynamic equilibrium by the transport can be calculated according to Eq. (20). 

In this chapter we get to know the second essential equation of surface science — the Kelvin5 equation. Like the Young-Laplace equation it is based on thermodynamic principles and does not refer to a special material or special conditions. The subject of the Kelvin equation is the vapor pressure of a liquid. Tables of vapor pressures for various liquids and different temperatures can be found in common textbooks or handbooks of physical chemistry. These vapor pressures are reported for vapors which are in thermodynamic equilibrium with liquids having planar surfaces. When the liquid surface is curved, the vapor pressure changes. The vapor pressure of a drop is higher than that of a flat, planar surface. In a bubble the vapor pressure is reduced. The Kelvin equation tells us how the vapor pressure depends on the curvature of the liquid. 

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