Linear distribution

Liquid—Hquid equiHbria having more than three components caimot as a rule be represented on a two-dimensional diagram. Such systems are important in fractional extraction, for example, operations in which two consolute components C and D are separated by means of two solvents A and B. For the special case where A and B are immiscible, the linear distribution law can be appHed to components C and D independendy ... 

Whereas the linear distribution law can be appHed to the undissociated monomer, the interfacial distribution of total benzoic acid, as determined by analysis, is nonlinear. 

The solute concentrations very close to the interface, and are assumed to be in equiUbrium, in the absence of any slow interfacial reaction. According to the linear distribution law, Cg. = thus from equation 14 the mass-transfer flux can be expressed in terms of an overall... 

For many modeling purposes, Nhas been assumed to be 1 (42), resulting in a simplified equation, S = C, where is the linear distribution coefficient. This assumption usually works for hydrophobic polycycHc aromatic compounds sorbed on sediments, if the equdibrium solution concentration is <10 M (43). For many pesticides, the error introduced by the assumption of linearity depends on the deviation from linearity. 

First-order sehemes use a uniform distribution aeross an element and seeond-order sehemes use a linear distribution aeross the element as shown in Fig. 9.16. Higher-order adveetion sehemes use more eomplex distributions aeross an element [29]. The distributions aeross the donor eell must be eon-strained to prevent numerieal oseillations. As an illustration, for seeond-order van Leer sehemes, the slope is limited using (9.15) and Fig. 9.17. The slope... 

Fig. 1.3. (a) Linear distribution of masses, (b) surface distribution of masses. 

The first synthesized pentafluorophenylgold(I) complex was [Au(C6F5)PPh3[ obtained by the reaction of [AuClPPhs] and CgFsMgBr [30-32]. This complex was at the same time the first characterized [33] by X-ray studies showing the common nearly linear distribution of the phosphorous and the ipso carbon atom (178(1)°). Nowadays, there are many methods of obtaining this complex, in many cases as a by-product of some reaction and, in other cases, its formation is the driving force of the reaction. We will see examples of both in this chapter. 

The simplest type of isotherm is the linear-distribution coefficient, A d.12 It is also called the partition coefficient, 7. 58 The equation for calculating adsorption at different concentrations is... 

Major problems are associated with using the linear distribution coefficient for describing adsorption-desorption reactions in groundwater systems. Some of these problems include the... 

Retardation-factor models, which incorporate a simple retardation factor derived from a linear- or linearized-distribution coefficient... 

The problems with linear-distribution coefficients apply equally to any retardation factor derived from them. Field measurements can be made but are expensive to obtain and highly site specific. Nevertheless, retardation factors provide some insight into organic chemical transport. 

The introduction of alkyl phosphine complexes of cobalt carbonyl as hydroformylation catalysts was reported to have a significant effect on product composition (50, 51). Slaugh and Mullineaux (52) reported that hexanol with a 91% linear distribution was formed by the hydroformylation of 1-pentene at 150°C, 500 psi, H2/CO 2.0, catalyst [Co2(CO)8 + 2(n-C4H9)3P]. Under the same conditions except at a temperature of 190°C, the n-hexanol was 84% of the hexyl alcohol produced. 

The resistance of the electrolyte causes a potential drop that is linearly distributed in the electrolyte inside the pores and thus does not have an effect on the current distribution on the pore bottom. However, the potential drop in the electrolyte has an important effect in maintaining the flat growth front of the PS layer as will be discussed later. 

In papers , unsteady-state regime arising upon propagation of the stationary fundamental mode from linear to nonlinear section of a single-mode step-index waveguide was studied via numerical modeling. It was shown that the stationary solution to the paraxial nonlinear wave equation (2.9) at some distance from the end of a nonlinear waveguide has the form of a transversely stable distribution ( nonlinear mode ) dependent on the field intensity, with a width smaller than that of the initial linear distribution. 

The type of the distribution of T-values is a much discussed topic. Experience shows that, in a mono-exponential case, the values should spread over an interval of more than 3 x Ti but not much over 4 x Ti, and a linear distribution appears to be slightly better than a logarithmic one. This is probably due to the fact that in a three-parameter exponential fit, the points with large x-values play as crucial a role in determining the relaxation rate as the slope at small x-values, and one needs both to determine R. On the other hand, it is evident that in multi-exponential cases, logarithmic distribution is often better suited for the task, especially when the relaxation rates of different sample components differ by an order of magnitude or more. 

Prom the diffusion equation, it can be seen that if d Cldx = 0, then dCldt = 0, meaning that the concentration at the position would not vary with time. Hence, if the initial concentration is uniform, the concentration would not change with time. If the initial concentration profile is linear and the concentrations at the two ends are not changed from linear distribution, because... 

If 3-D distribution can be obtained accurately, uncertainty may still arise because some of the six assumptions are not necessarily satisfied. Sometimes the CSD does not follow a log-linear relation. Then the above theory cannot be applied. Even if the CSD follows a log-linear distribution, whether the crystal growth rate can be reliably inferred has not been verified experimentally. Higgins (2002) and Pan (2002) pointed out other possible uncertainties in applying the CSD theory. 

Fig. 9.14. Double piezoelectric response of a tube scanner with asymmetric connections. (A) The yi quadrant is connected to an ac voltage source. The current from other quadrants to the ground is measured. (B) Stress in the cross section has a linear distribution with respect to y. (Reproduced from Chen, 1992a, with permission.)...

One usefiil way of describing the ability of apatites to take up and immobilize contaminants is the linear distribution coefficient, Ka, the mass of contaminant sorbed to the mass of apatite (Af,-(ads>) relative to the mass of contaminant in... 

Both a simplified continuous and discrete model, describing the behaviour of single component mass transport in chromatographic columns with non-linear distribution isotherm, were developed and simulated by Smit et al. Studies of more complex but still relatively simple (multicomponent) transport models have been published (see e.g. ... 

Suppose we want to know the mass per unit area mg, momentum g(x) of the detonation products expanding in the direction of the metal plate in a 1-D system. The Gurney assumption of linear distribution of gas velocities enables us to compute all these quantities. The derivations of the necessary equations are given in Appendix G. Here we present only the final results, namely ... 

Molecular distribution in polycondensation involving unlike reactants. II. Linear distributions. J. Polymer Sci. 29, 455 (1958). 

This equation describes the linear distribution of data points on a plot of 8D versus 8lsO that is commonly referred to as the global meteoric-water line (GMWL). The zero intercept for this line, defined as the deuterium excess... 

This points up the desirability of using the dimensionless index variable x and of having the liberty of choosing h x) in Eq. 12. If h x) = x then k, the first-order constant, is being linearly distributed. If it is desirable to distribution kn linearly then h(x) = x[/(x)] 1 1. 

Problems in chemical reactor analysis with stochastic features Control of linearized distributed systems on discrete and corrupted observations (with T.M. Pell, Jr.). Ind. Eng. Chem. Fund. 9,15-20 (1970). 

Tropomyosin is a two-stranded, o-helical coiled-coil molecule that aggregates head-to-tail with others to form long filamentous ropes. These lie in each of the two long period grooves of the actin microfilaments where, in vertebrate skeletal muscle, they play an important part in the Ca2+-mediated regulation of actin via troponin (a tropomyosin-associated protein). An important feature of tropomyosin is its 39.2-residue period— that is also quasi-halved (19.6 residues)—in the linear distribution of the acidic residues and, to a lesser extent, the apolar residues (McLachlan and Stewart, 1976 Parry, 1975). The number of residues in tropomyosin (284 residues), and the head-to-tail overlap (nine residues) that allows axial... 

The slope of this line is the distribution coefficient (Kd), which is the ratio of the arsenic concentration on the adsorbent (Cads) to the concentration of the associated remaining arsenic in the aqueous solution (Csdn). With each linear adsorption isotherm, Kd has only one value. That is, a linear distribution indicates that the partitioning of arsenic between the adsorbent and the solution is constant over the given range of arsenic concentrations (Eby, 2004), 221. If both concentrations (Cads and Csoin) are in the same units (such as molal), Kd is unitless. However, if the adsorbed concentration is given in molal and the dissolved concentration is molar, then Kd has the units of liter/kilogram. 

The Freundlich equation is similar to a linear equation, expect for the presence of the exponent n. For linear distributions, n = 1. With Freundlich isotherms usually have n < 1, which causes the adsorption isotherm to curve downward at higher concentrations as the readily available adsorption sites are filled and lower proportions of the arsenic from the aqueous solutions are adsorbed (Figure 2.8). The distribution coefficient for a Freundlich isotherm is often written as Kt to stress that the isotherm is not linear (Drever, 1997), 89. 

A theoretical study of ring closure as applied to the cyclic linear distribution in polydimethylsiloxanes assuming a three-rotational-state model resulted in good correlations between experimental and theoretical equilibrium constants (54). 

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