The location of phase boundaries

Thermodynamics provides us with a way of predicting the location of the phase boundaries and relating their location and shape to the thermodynamic properties of the system. For instance, the shape of the vapor pressme cmve (the liquid-vapor boundary) is related to the enthalpy of vaporization of the hquid. 

S The cooling curve for the B-E section of the horizontal line in Fig. 3.5. The halt at D corresponds to the pause in cooling while the hquid freezes and releases its enthalpy of transition. The halt lets us locate Tf even if the transition carmot be observed visually. 

For the liquid-vapor boundary (the vapor pressure curve), both the enthalpy and volume of vaporization are invariably positive, so the vapor pressure invariably increases with temperature (dp is positive if dT is positive). However, we have to be cautious because although the enthalpy of vaporization is not very sensitive to temperature, the volume of vaporization depends strongly on the temperature (through the effect of temperature on the volume of a gas). If we suppose that the vapor behaves as a perfect gas, then we show in the following Justification that the relation between a change in temperature and a change in vapor pressure is given by the Qausius-Clapeyron equation  

9 At equilibrium, two phases have the same molar Gibbs energy. When the temperature is changed by dT, for the two phases to remain in equilibrium, the pressiue must be changed by dp so that the Gibbs energies of the two phases remain equal. 

The liquid-vapor boundary of the phase diagram for water has a positive slope, and we shall see in Section 3.4(d) that the slope of the liquid-vapor phase boundary is much less steep than the slope of the ice-water phase boundary. 

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Sample phase diagrams collected using the two gradient methods are presented in Fig. 9 and 11. It is important to realize that in both cases only the location of phase boundaries are shown, while in fact, phase identification and structural characterization is available continuously along each isotherm and isopleth in the respective plots. 

Although any of the designs mentioned above will provide the location of phase boundaries (versus temperature and pressure), it is also important to know the compositions of the two phases in equilibrium. Note that while tie lines (lines connecting phases in equilibrium on T-x or p-x diagrams) are horizontal for simple binary mixtures, this is not true for phase separation in multicomponent systems (most notably polymer-fluid systems where the polymer sample contains chains of various lengths). Consequently, ports which allow withdrawal of samples following phase separation and equilibration are an important feature of view cells. Such ports also allow for the measurement of partition coefficients of solutes between, for example, aqueous and CO2 phases. 

The study of simple liquids can be said to be the beginning of Molecular Dynamics and Monte Carlo in the 1950s and 60s. Although the scope of molecular simulation, as a field or discipline, has widened dramatically since then, there is nevertheless a continual interest in simple liquids. In fact, this is partly due to the fact that the so-called simple liquids are far from simple One of the motivations for the continual interest in the simple liquids is that, because of the basic nature of the interparticle interactions, an improved understanding of these systems should lead to better theoretical models, which can be extended to more complex molecular liquids. Also, the rapid growth of interest in colloids and polymers (so-called complex liquids) in recent years has provided new areas where the theories of simple liquids can be applied, especially those associated with local structure and thermodynamics. In the latter case, phase equilibria and the location of phase boundaries feature prominently. In this section, some of the recent advances in our understanding of simple liquids are covered. 

Differential thermal analysis (DTA) is a commonly used method to determine the location of phase boundaries. Figure 8.3 shows a hypothetical binary phase diagram and the DTA curves that would be expected for several compositions within the system. The endothermic melting and solidi-solid2 first-order transformations during heating are clearly evident. These peaks are due to the temporary... 

One important point is that the phase boundary which separates the dilute phase from the dense phase is somewhat arbitrary to define and is located with difiiculty. Instead of defining the phase boundary, it might be more reasonable to place a transition zone between the above two phases. In what follows, however, the concept of phase boundary has been... 

Criticism of the phase diagrams The phase diagrams appear consistent with expectations from die theory of mixtures, and reversibility experiments show that their boundaries are equilibrium boundaries. Still, the locations of these boundaries may be inaccurate for 2 reasons. First, the two-phases (sol and floe) may not separate properly and one of them may take an excessive amount of one component. We have checked this by making sols located at the boundary through direct mixing, and we have found that the location of the "sol" boundary is accurate. Second, the volume mesured for the floes may not reflect their concentrations. Indeed, the floes are turbid, which indicates that they must be heterogeneous, i.e. made of lumps and voids. 

From Table 9.1, it is seen that at a reduction of the interdiffusion coefficient in the ternary y phase, the growth rate constant of the binary phase decreases, the concentration in the P phase at the fi-y interphase boundary approaches the boundary concentration the concentration in the y phase at the same interphase boundary becomes more and more different from Cgg(OO), and the coefficient S, which characterizes the diffusivity of the phase, decreases. Thus, as the interdiffusion coefficient in the ternary y phase diminishes (which is achieved due to introduced Pt or other additions), the possibility of growth suppression of the binary intermediate p phase becomes evident. At that, the magnitudes of the boundary concentrations at the fi-y interphase boundary correspond to the conode, which shifts along the phase diagram toward eg" , and the respective value determines the concentration interval of the phase, which allows one to find that the effective diffusivity of the phase is less than the maximal one D, by 1/5. The concentration distributions and locations of phase boundaries yr and y obtained from the calculations for the model Si-Ni-Pt system are given in Figure 9.7. 

There are some limitations to the use of DSC in phase studies in addition of its inability to identify phases [5], They are that (1) there are difficulties in locating very steep phase boundaries in heterogeneous systems from DSC data alone because heat capacity depends strongly on the slope of phase boundaries and so the heat capacity jumps are small [10] and (2) the determination of phase boundaries in systems with slow nucleation rate, interfacial transport problems, or inherently slow phase changes may not be possible [11], This is why it is hard to discover a liquid miscibility gap by DSC measurements. 

Unlike Sn-Pb joints, which have a dual phase structure and block the path of corrosion due to the existence of phase boundaries, the SAC305 joint is basically pure Sn with coarse islands of A n and CueSns intermetallic precipitate (Fig. 5). A corrosion crack can propagate and lead to additional corrosion along the way, without interruption from the Sn phase structure. Although both materials show strong resistance to corrosion, the localized nature of the corroded area at critical locations causes significant degradation in Sn-Ag-Cu solder joints[40]. 

Fig. 5. Metastable Fe—Ni—Cr "temary"-pliase diagram where C content is 0.1 wt % and for alloys cooled rapidly from 1000°C showing the locations of austenitic, duplex, ferritic, and martensitic stainless steels with respect to the metastable-phase boundaries. For carbon contents higher than 0.1 wt %, martensite lines occur at lower ahoy contents (43). A is duplex stainless steel, eg. Type 329, 327 B, ferritic stainless steels, eg. Type 446 C, 5 ferrite + martensite D, martensitic stainless steels, eg. Type 410 E, ferrite + martensite F, ferrite + pearlite G, high nickel ahoys, eg, ahoy 800 H,...

FIG. 16-9 General scheme of adsorbent particles in a packed bed showing the locations of mass transfer and dispersive mechanisms. Numerals correspond to mimhered paragraphs in the text 1, pore diffusion 2, solid diffusion 3, reaction kinetics at phase boundary 4, external mass transfer 5, fluid mixing. 

Fig. 7.77 Thermodynamic stability diagram for the Fe-Ni-Cr system at 1 143 K, assuming metal activities to be unity.-, phase boundaries involving Fe —phase boundaries involving Ni ----, phase boundaries involving Cr. The location of environments 1, 2, 3, and 4 are...

By covalently attaching reactive groups to a polyelectrolyte main chain the uncertainty as to the location of the associated reactive groups can be eliminated. The location at which the reactive groups experience the macromolecular environment critically controls the reaction rate. If a reactive group is covalently bonded to a macromolecular surface, its reactivity would be markedly influenced by interfacial effects at the boundary between the polymer skeleton and the water phase. Those effects may vary with such factors as local electrostatic potential, local polarity, local hydrophobicity, and local viscosity. The values of these local parameters should be different from those in the bulk phase. 

A triple point is a point where three phase boundaries meet on a phase diagram. For water, the triple point for the solid, liquid, and vapor phases lies at 4.6 Torr and 0.01°C (see Fig. 8.6). At this triple point, all three phases (ice, liquid, and vapor) coexist in mutual dynamic equilibrium solid is in equilibrium with liquid, liquid with vapor, and vapor with solid. The location of a triple point of a substance is a fixed property of that substance and cannot be changed by changing the conditions. The triple point of water is used to define the size of the kelvin by definition, there are exactly 273.16 kelvins between absolute zero and the triple point of water. Because the normal freezing point of water is found to lie 0.01 K below the triple point, 0°C corresponds to 273.15 K. 

SOLUTION Although the phase diagram in Fig. 8.6 is not to scale, we can find the approximate locations of the points. Point A is at 5 Torr and 70°C so it lies in the vapor region. Increasing the pressure takes the vapor to the liquid-vapor phase boundary, at which point liquid begins to form. At this pressure, liquid and vapor are in equilibrium and the pressure remains constant until all the vapor has condensed. The pressure is increased further to 800 Torr, which takes it to point B, in the liquid region. 

The location of the phase boundary as a function of time is therefore given... 

Often the experimentalist will have an approximate idea of the location of a phase boundary from data taken at other temperatures or from measurements on closely related compounds. In such cases, a reasonably accurate guess about the location of the phase boundary may allow one to choose a starting composition that crosses the boundary with only a single titration. This can be considerably more efficient than starting with a neat compound and performing titrations over successive compositional ranges until the phase boundary is found. 

The determination of the character and location of phase transitions has been an active area of research from the early days of computer simulation, all the way back to the 1953 Metropolis et al. [59] MC paper. Within a two-phase coexistence region, small systems simulated under periodic boundary conditions show regions of apparent thermodynamic instability [60] simulations in the presence of an explicit interface eliminate this at some cost in system size and equilibration time. The determination of precise coexistence boundaries was usually done indirectly, through the... 

Liquid-liquid multiphasic catalysis with the catalyst present in the ionic liquid phase relies on the transfer of organic substrates into the ionic liquid or reactions must occur at the phase boundary. One important parameter for the development of kinetic models (which are crucial for up-scaling and proper economic evaluation) is the location of the reaction. Does the reaction take place in the bulk of the liquid, in the diffusion layer or immediately at the surface of the ionic liquid droplets ... 

Fig. 2 Location of reaction within the bulk of the catalyst phase or at the L/L-phase boundary...

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