Coefficients critical

The ratio 6K/pK = k18 called the critical coefficient of a substance (Guye). 

The critical coefficients derived above correspond to the case when the metal ion m is the minority ion, and the cation + and anion - of the supporting electrolyte form the majority ions. The critical potential coefficients can be also obtained when the cation + of the supporting electrolyte is taken as the minority ion and the metal ion m and the anion - form the majority ions. Such equations are easily obtained if the subscripts m and + are converted to + and m, respectively, in Eqs. (50) and (51) for constant Qm and M+-,... 

Critical distances [32] of ion stabilization in organic solids for the rectangular, Rrcr, and the Coulomb, Rccr, forms of the barrier for electron tunneling, and critical coefficients, -D, of ion diffusion as functions of the ionization energy, 7d, of the donor, and the static dielectric permeability of a solid,... 

A similar procedure as indicated above shows that any functions A t) and B(t) are allowed, provided A(l) = 3, Zf(l) = 1/3. So also the equation of Berthelot, with A f) = 3/f, B f) = 1/3 is included immediately into the general consideration. Various gas equations in reduced form are shown in Table 4.3. The field is still open to research. A reduced equation of state based on the critical coefficient and still other approaches have been derived [6]. 

Substance Density (p/kg.m" ) Refractive Critical Coefficient Abbe Refrac- Specific Molar... 

Heating and skin supersolidity create gradients of thermal diffusion with a critical coefficient ratio of as/ B > p /ps — 4/3 in the liquid for heat conduction. Convection alone raises only the skin temperature. 

Fig. 6.2 Stability diagrams in terms of critical coefficient h and inlet velocity Fin p = 5 bar (squares Tin = 700 K, triangles Tin = 700 K without gas-phase chemistry, circles Tin = 600 K, crosses Tin = 600 K without gas-phase chemistry), p = 1 bar (filled diamonds Tin = 700 K). The stable regimes for the 700 K cases are shown by the shaded areas...

When the pressure is low and mixture conditions are far from critical, activity coefficients are essentially independent of pressure. For such conditions it is common practice to set P = P in Equations (18) and (19). Coupled with the assumption that v = v, substitution gives the familiar equation... 

To use Equation (10b), we require virial coefficients which depend on temperature. As discussed in Appendix A, these coefficients are calculated using the correlation of Hayden and O Connell (1975). The required input parameters are, for each component critical temperature T, critical pressure P, ... 

Figure 1 shows second virial coefficients for four pure fluids as a function of temperature. Second virial coefficients for typical fluids are negative and increasingly so as the temperature falls only at the Boyle point, when the temperature is about 2.5 times the critical, does the second virial coefficient become positive. At a given temperature below the Boyle point, the magnitude of the second virial coefficient increases with... 

Equations (2) and (3) are physically meaningful only in the temperature range bounded by the triple-point temperature and the critical temperature. Nevertheless, it is often useful to extrapolate these equations either to lower or, more often, to higher temperatures. In this monograph we have extrapolated the function F [Equation (3)] to a reduced temperature of nearly 2. We do not recommend further extrapolation. For highly supercritical components it is better to use the unsymmetric normalization for activity coefficients as indicated in Chapter 2 and as discussed further in a later section of this chapter. 

In some cases, the temperature of the system may be larger than the critical temperature of one (or more) of the components, i.e., system temperature T may exceed T. . In that event, component i is a supercritical component, one that cannot exist as a pure liquid at temperature T. For this component, it is still possible to use symmetric normalization of the activity coefficient (y - 1 as x - 1) provided that some method of extrapolation is used to evaluate the standard-state fugacity which, in this case, is the fugacity of pure liquid i at system temperature T. For highly supercritical components (T Tj,.), such extrapolation is extremely arbitrary as a result, we have no assurance that when experimental data are reduced, the activity coefficient tends to obey the necessary boundary condition 1... 

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. 

The coefficient can also be estimated starting with the critical constants by the following formula ... 

At low temperatures, using the original function/(T ) could lead to greater error. In Tables 4.11 and 4.12, the results obtained by the Soave method are compared with fitted curves published by the DIPPR for hexane and hexadecane. Note that the differences are less than 5% between the normal boiling point and the critical point but that they are greater at low temperature. The original form of the Soave equation should be used with caution when the vapor pressure of the components is less than 0.1 bar. In these conditions, it leads to underestimating the values for equilibrium coefficients for these components. 

Based o the test data, the parameter a6 is correlating with the residual resistance (table 1). It is discovered that the less resistible samples have much higher value of a6. On the base of collected data it is possible to identify the critical value of the accumulation coefficient (which is a defective sign of the material (if aG> AiScR-the sample is defected if aG< a6cr - the sample is without defects). 

This definition is in terms of a pool of liquid of depth h, where z is distance normal to the surface and ti and k are the liquid viscosity and thermal diffusivity, respectively [58]. (Thermal diffusivity is defined as the coefficient of thermal conductivity divided by density and by heat capacity per unit mass.) The critical Ma value for a system to show Marangoni instability is around 50-100. 

The SPC/E model approximates many-body effects m liquid water and corresponds to a molecular dipole moment of 2.35 Debye (D) compared to the actual dipole moment of 1.85 D for an isolated water molecule. The model reproduces the diflfiision coefficient and themiodynamics properties at ambient temperatures to within a few per cent, and the critical parameters (see below) are predicted to within 15%. The same model potential has been extended to include the interactions between ions and water by fitting the parameters to the hydration energies of small ion-water clusters. The parameters for the ion-water and water-water interactions in the SPC/E model are given in table A2.3.2. 

Iiifomiation about the behaviour of the 3D Ising ferromagnet near the critical point was first obtained from high- and low-temperatnre expansions. The expansion parameter in the high-temperatnre series is tanli K, and the corresponding parameter in the low-temperatnre expansion is exp(-2A ). A 2D square lattice is self-dual in the sense that the bisectors of the line joining the lattice points also fomi a square lattice and the coefficients of the two expansions, for the 2D square lattice system, are identical to within a factor of two. The singularity occurs when... 

Figure A2.5.17. The coefficient Aias a fimction of temperature T. The line IRT (shown as dashed line) defines the critical point and separates the two-phase region from the one-phase region, (a) A constant K as assumed in the simplest example (b) a slowly decreasing K, found frequently in experimental systems, and (c) a sharply curved K T) that produces two critical-solution temperatures with a two-phase region in between.

Povodyrev et aJ [30] have applied crossover theory to the Flory equation ( section A2.5.4.1) for polymer solutions for various values of N, the number of monomer units in the polymer chain, obtaining the coexistence curve and values of the coefficient p jj-from the slope of that curve. Figure A2.5.27 shows their comparison between classical and crossover values of p j-j for A = 1, which is of course just the simple mixture. As seen in this figure, the crossover to classical behaviour is not complete until far below the critical temperature. 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

Collisional ionization can play an important role in plasmas, flames and atmospheric and interstellar physics and chemistry. Models of these phenomena depend critically on the accurate detennination of absolute cross sections and rate coefficients. The rate coefficient is the quantity closest to what an experiment actually measures and can be regarded as the cross section averaged over the collision velocity distribution. 

Furtliennore, since tlie bifurcation must occur from a stable homogeneous steady state we must have D ID < 1 i.e. tlie diffusion coefficient of tlie inhibitor is greater tlian tliat of tlie activator. The critical diffusion ratio at tlie bifurcation is... 

G is a multiplier which is zero at locations where slip condition does not apply and is a sufficiently large number at the nodes where slip may occur. It is important to note that, when the shear stress at a wall exceeds the threshold of slip and the fluid slides over the solid surface, this may reduce the shearing to below the critical value resulting in a renewed stick. Therefore imposition of wall slip introduces a form of non-linearity into the flow model which should be handled via an iterative loop. The slip coefficient (i.e. /I in the Navier s slip condition given as Equation (3.59) is defined as... 

Our interest from the outset has been in the possibility of crosslinking which accompanies inclusion of multifunctional monomers in a polymerizing system. Note that this does not occur when the groups enclosed in boxes in Table 5.6 react however, any reaction beyond this for the terminal A groups will result in a cascade of branches being formed. Therefore a critical (subscript c) value for the branching coefficient occurs at... 

---