Rough equilibria

The boundary system Xq itself may have a finite number of rough equilibria and periodic orbits, or infinitely many of them. For the latter case an example is given by a period-doubling cascade. 

The parameter space here is partitioned into three regions Z o, D2 and i 4, joined at the origin. When /i G >4, the Eq, (11.2.27) has four rough equilibrium states, two of which are stable and two are unstable for /x >2, the equation has two rough equilibria, one stable and the other unstable for Do, there is no equilibrium states at all. 

The balance between evaporation, precipitation, glaciers, and oceans, known as the hydrological cycle, is usually considered to be in rough equilibrium over the Earth, although there is evidence that the... 

Rough equilibrium vapor pressures for H20 above a variety of common desiccants are given in Table 3.1 along with comments on other properties such as H20 capacity. Additional factors that are particularly important in the drying of... 

It follows from the above theorem that a rough system on the plane may possess only rough equilibrium states (nodes, foci and saddles) and rough limit cycles. As for separatrices of saddles, they either tend asymptotically to a node, a focus, or a limit cycle in forward or backward time, or leave the region G after a finite interval of time. 

The phase portraits for systems of dimension two and higher are illustrated in Figs. 11.2.4-11.2.7, respectively. Here, when l2 < 0, there are two rough equilibrium states a stable node and a saddle that approach each other as i2M... 

Consider a one-parameter family of (r > 2) smooth dynamical systems in (n > 1). Suppose that when the parameter vanishes the system possesses a non-rough equilibrium state at the origin with one characteristic exponent equal to zero and the other n exponents lying to the left of the imaginary axis. We suppose also that the equilibrium state is a simple saddle-node, namely the first Lyapimov value I2 is not zero (see Sec. 11.2). Without loss of generality we assume /2 > 0. 

Summarizing, we can classify the rough equilibrium states in as follows ... 

Shilnikov, L. P. [1970] A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type, Math. USSR Sbornik 10, 91-102. 

We have repeatedly observed that the slowly converging variables in liquid-liquid calculations following the isothermal flash procedure are the mole fractions of the two solvent components in the conjugate liquid phases. In addition, we have found that the mole fractions of these components, as well as those of the other components, follow roughly linear relationships with certain measures of deviation from equilibrium, such as the differences in component activities (or fugacities) in the extract and the raffinate. 

We may now understand the nature of the change which occurs when an anhydrous salt, say copper sulphate, is shaken with a wet organic solvent, such as benzene, at about 25°. The water will first combine to form the monohydrate in accordance with equation (i), and, provided suflScient anhydrous copper sulphate is employed, the effective concentration of water in the solvent is reduced to a value equivalent to about 1 mm. of ordinary water vapour. The complete removal of water is impossible indeed, the equilibrium vapour pressures of the least hydrated tem may be taken as a rough measure of the relative efficiencies of such drying agents. If the water present is more than sufficient to convert the anhydrous copper sulphate into the monohydrate, then reaction (i) will be followed by reaction (ii), i.e., the trihydrate will be formed the water vapour then remaining will be equivalent to about 6 mm. of ordinary water vapour. Thus the monohydrate is far less effective than the anhydrous compound for the removal of water. 

Enthalpy of Vaporization (or Sublimation) When the pressure of the vapor in equilibrium with a liquid reaches 1 atm, the liquid boils and is completely converted to vapor on absorption of the enthalpy of vaporization ISHv at the normal boiling point T. A rough empirical relationship between the normal boiling point and the enthalpy of vaporization (Trouton s rule) is ... 

An important characteristic of pervaporation that distinguishes it from distillation is that it is a rate process, not an equilibrium process. The more permeable component may be the less-volatile component. Perv oration has its greatest iitihty in the resolution of azeotropes, as an acqiinct to distillation. Selecting a membrane permeable to the minor corTiponent is important, since the membrane area required is roughly proportional to the mass of permeate. Thus pervaporation devices for the purification of the ethanol-water azeotrope (95 percent ethanol) are always based on a hydrophihc membrane. 

With a reactive solvent, the mass-transfer coefficient may be enhanced by a factor E so that, for instance. Kg is replaced by EKg. Like specific rates of ordinary chemical reactions, such enhancements must be found experimentally. There are no generalized correlations. Some calculations have been made for idealized situations, such as complete reaction in the liquid film. Tables 23-6 and 23-7 show a few spot data. On that basis, a tower for absorption of SO9 with NaOH is smaller than that with pure water by a factor of roughly 0.317/7.0 = 0.045. Table 23-8 lists the main factors that are needed for mathematical representation of KgO in a typical case of the absorption of CO9 by aqueous mouethauolamiue. Figure 23-27 shows some of the complex behaviors of equilibria and mass-transfer coefficients for the absorption of CO9 in solutions of potassium carbonate. Other than Henry s law, p = HC, which holds for some fairly dilute solutions, there is no general form of equilibrium relation. A typically complex equation is that for CO9 in contact with sodium carbonate solutions (Harte, Baker, and Purcell, Ind. Eng. Chem., 25, 528 [1933]), which is... 

Owing to the stability of the uranyl carbonate complex, uranium is universally present in seawater at an average concentration of ca. 3.2/rgL with a daughter/parent activity ratio U) of 1.14. " In particulate matter and bottom sediments that are roughly 1 x 10 " years old, the ratio should approach unity (secular equilibrium). The principal source of dissolved uranium to the ocean is from physicochemical weathering on the continents and subsequent transport by rivers. Potentially significant oceanic U sinks include anoxic basins, organic rich sediments, phosphorites and oceanic basalts, metalliferous sediments, carbonate sediments, and saltwater marshes. " ... 

Ox and Red are general symbols for oxidation and reduction media respectively, and n and (n-z) indicate their numerical charge (see Section 2.2.2). Where there is no electrochemical redox reaction [Eq. (2-9)], the corrosion rate according to Eq. (2-4) is zero because of Eq. (2-8). This is roughly the case with passive metals whose surface films are electrical insulators (e.g., A1 and Ti). Equation (2-8) does not take into account the possibility of electrons being diverted through a conductor. In this case the equilibrium... 

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