Solution invariant

Industrial solutions invariably contain dissolved impurities that can increase or decrease the solubility of the prime solute considerably, and it is important that the solubility data used to design crystallisation processes relate to the actual system used. Impurities can also have profound effects on other characteristics, such as nucleation and growth. 

The y values of aqueous solutions generally increase with different electrolyte concentrations. The magnitude of y of aqueous solutions containing organic solutes invariably decreases. As mentioned earlier, the surface of a liquid is where the density of the liquid changes to that of a gas by a factor 1000 (Chapter 1, Figure 1.1). 

Next, for the solution invariant under the subalgebra Mb the orthogonality condition leads to the following set of algebraic equations to be satisfied by the parameters Ci,..., C(, ... 

A band of solute invariably spreads as it travels through a chromatography column (Figure 23-11) and emerges at the detector with a standard deviation o. Each individual mechanism... 

Nonpolar solute in a nonpolar solvent. In this case, only dispersion forces contribute to the solvation of the solute. Dispersion forces, operative in any solution, invariably cause a small bathochromic shift, the magnitude of which is a function of the solvent refractive index n, the transition intensity, and the size of the solute molecule. The function (n — l)/(2n - -1) has been proposed to account for this general red shift [69, 70]. Corresponding linear correlations between this function of n and Av have been observed for aromatic compounds e.g. benzene [22], phenanthrene [71]), polyenes e.g. lycopene [23], y9-carotene [464]), and symmetrical polymethine dyes e.g. cyanines [26, 27, 292, 293]). 

Using Flory-Huggins theory it is possible to account for the equilibrium thermodynamic properties of polymer solutions, particularly the fact that polymer solutions show major deviations from ideal solution behavior, as for example, the vapor pressure of solvent above a polymer solution invariably is very much lower than predicted from Raoult s law. The theory also accounts for the phase separation and fractionation behavior of polymer solutions, melting point depressions in crystalline polymers, and swelling of polymer networks. However, the theory is only able to predict general trends and fails to achieve precise agreement with experimental data. 

S-PS, isolated by freeze drying from dilute solution, invariably is more soluble under such conditions than S-PS isolated via freeze drying at 4% concentration. This behavior is summarized in Table III below. 

Polymer Solutions. All film-forming liquids are polymers or polymer solutions. The viscosity of these systems is constant only at extremely high dilution where the measurements leading to knowledge of molecular dimensions are made. At practical concentrations polymer solutions invariably become shear sensitive. 

A procedure of this kind is practically obligatory for the preparation of UF3 because the uranium(III) oxidation state is only barely stable in aqueous solution attempts to precipitate solid compounds from a U8+ solution invariably yield compounds of uranium (IV). For the transuranium elements, however, either aqueous or high-temperature dry-way procedures... 

In flow injection analysis (FIA), this solution invariably detracts from throughput while in SIA, the loss can be minimized by stopping the flow at the reaction coil, located behind the switching valve, and starting processing of the next sample. When the second sample reaches the reaction coil, it will push the first, which by then will have reacted to an adequate extent, to the detector. 

It should be noted that the transformation matrix becomes unbounded for / 7t/2. This is the reason for taking other parameterizations of the rotation matrix if /3 tends towards tt/2. Such a reparameterization introduces discontinuities which can be avoided when using a redundant set of rotation coordinates. One typically uses quaternions often also called Euler parameters. These are four coordinates instead of the three angles and one additional normalizing equation, see Ex. 5.1.10. This normalizing equation describes a property of the motion, a so-called solution invariant. Differential equations with invariants will be discussed in Sec. 5.3. 

Gear86] Gear C. W. (1986) Maintaining solution invariants in the numerical solution of ODEs. SIAM J. Sci. Stat. Comput. 7(3) 734-743. 

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