Discontinuous isotherms

Well-Mixed (Discontinuous) Isothermal Batch Reactor... 

Figure 4. Quantization and probabilities of compositions, scalar case. In a) the composition profile c(x) (solid solution composition as a function of space at a given time) is represented. There is a sharp fi ont (corresponding compositions have zero probability) and a continuous evolution, wherein the spatial spreading of a specific composition 2, lying between compositions I and 3 is represented its probability p is proportional to f (c), i.e. the difference of the neighbouring velocities. In b) and c) the application of this rule is given for a continuous isotherm the envelope f between the extreme points is shown. The probability distribution is given in c). In d) and e) the same method is applied for a discontinuous isotherm (isotherm is given in d) and probability distribution in e)) Guy, 1993, wifii permission from Eur. J. Mineral.

Temperature profiles in continuous isothermal exposure testing (a), discontinuous isothermal exposure testing (b) and thermal cycling oxidation testing (c). 

Fig. 2.35 Compact calorimeter and reflux condenser for discontinuous, isothermal reaction...

Fig. 17 shows the adsorption isotherms of all (undimerized and dimerized) particles. Except for a very fast increase of adsorption connected with filling of the first adlayer, the adsorption isotherm for the system A3 is quite smooth. The step at p/k T 0.28 corresponds to building up of the multilayer structure. The most significant change in the shape of the adsorption isotherm for the system 10, in comparison with the system A3, is the presence of a jump discontinuity at p/k T = 0.0099. Inspection of the density profiles attributes this jump to the prewetting transition in the... 

James Thomson (1871) suggested that these prolongations of the 0i isotherm of a liquid and vapour might ultimately bend round and join, so that the discontinuous flat part of the isotherm is replaced by a continuous curve (Fig. 84). aft corresponds to... 

The predominant gaseous products of the decomposition [1108] of copper maleate at 443—613 K and copper fumarate at 443—653 K were C02 and ethylene. The very rapid temperature rise resulting from laser heating [1108] is thought to result in simultaneous decarboxylation to form acetylene via the intermediate —CH=CH—. Preliminary isothermal measurements [487] for both these solid reactants (and including also copper malonate) found the occurrence of an initial acceleratory process, ascribed to a nucleation and growth reaction. Thereafter, there was a discontinuous diminution in rate (a 0.4), ascribed to the deposition of carbon at the active surfaces of growing copper nuclei. Bassi and Kalsi [1282] report that the isothermal decomposition of copper(II) adipate at 483—503 K obeyed the Prout—Tompkins equation [eqn. (9)] with E = 191 kJ mole-1. Studies of the isothermal decompositions of the copper(II) salts of benzoic, salicylic and malonic acids are also cited in this article. 

It has to be noticed that no isothermal volume contraction on cooling or volume expansion on heating is associated with Tg, contrary to crystallisation or melting. The latter is a true first-order transition exhibiting a discontinuity in... 

The film balance may be regarded as a two-dimensional piston, and the most commonly studied property is the surface pressure (n) versus area (A) isotherm. The analogy to a PV isotherm is so appropriate that in the gaseous monolayer regime the two-dimensional analogue of the ideal gas law pertains 114 = nRT. It is therefore reasonable to relate discontinuities in n/A isotherms as the monolayer film is compressed in two dimensions to... 

When a reversible transition from one monolayer phase to another can be observed in the 11/A isotherm (usually evidenced by a sharp discontinuity or plateau in the phase diagram), a two-dimensional version of the Gibbs phase rule (Gibbs, 1948) may be applied. The transition pressure for a phase change in one or both of the film components can be monitored as a function of film composition, with an ideally miscible system following the relation (12). A completely immiscible system will not follow this ideal law, but will... 

Nontrace isothermal systems give the adsorption effect (i.e., significant change in fluid velocity because of loss or gain of solute). Criteria for the existence of simple waves, contact discontinuities, and shocks are changed somewhat [Peterson and Helfferich, J. Phys. Chem., 69,1283 (1965) LeVan et al., AIChE J. 34, 996 (1988) Frey, AIChE L, 38,1649(1992)]. 

Pressure-induced amorphization of solids has received considerable attention recently in physical and material sciences, although the first reports of the phenomenon appeared in 1963 in the geophysical literature (actually amorphization on reducing the pressure [18]). During isothermal or near isothermal compression, some solids, instead of undergoing an equilibrium transition to a more stable high-pressure polymorph, become amorphous. This is known as pressure-induced amorphization. In some systems the transition is sharp and mimics a first-order phase transition, and a discontinuous drop in the volume of the substance is observed. Occasionally it is strictly not an amorphous phase that is formed, but rather a highly disordered denser nano-crystalline solid. Here we are concerned with the situation where a true amorphous solid is formed. 

Different models determine A in different ways. Nation exhibits a water-uptake isotherm as shown in Figure 7. The dashed line in the figure shows the effects of Schroeder s paradox, where there is a discontinuous jump in the value of A. Furthermore, the transport properties have different values and functional forms at that point. Most models used correlate A with the water-vapor activity, since it is an easily calculated quantity. An exception to this is the model of Siegel et al., ° which assumes a simple mass-transfer relationship. There are also models that model the isotherm either by Flory—Huggins theory" or equilibrium between water and hydrated protons in the membrane and water vapor... 

Figure 6. Set ofjump discontinuities (kinks and shocks) compatible with the isothermal viscosity-capillarity model W=2.5.

Equation (92) gives the negative of the slope of AGM with respect to x at x = and constant T. The equation is valid for all degrees of dissociation and shows that the slope at x = j depends on fiA, / 13, and/ 14 — 034 and is zero if these are all zero. Approximate treatments for the case of complete association (z = 1) have sometimes been given that incorrectly yield a V shape to the AGM isotherms and hence a discontinuity in jil — fi3 at x =... 

Figure 2.9 Phase diagram for C02, showing solid-gas (S + G, sublimation ), solid-liquid (S + L, fusion ), and liquid-gas (L + G, vaporization ) coexistence lines as PT boundaries of stable solid, liquid, or gaseous phases. The triple point (triangle), critical point (x), and selected 280K isotherm of Fig. 2.8 (circle) are marked for identification. Note that the fusion curve tilts slightly forward (with slope 75 atm K-1) and that the sublimation and vaporization curves meet with slightly discontinuous slopes (angle < 180°) at the triple point. The dotted and dashed half-circle shows two possible paths between a liquid (cross-hair square) and a gas (cross-hair circle) state, one discontinuous (dashed) crossing the coexistence line, the other continuous (dotted) encircling the critical point (see text).

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