Solutions multicomponent

A variant of the zero average contrast method has been applied on a solution of a symmetric diblock copolymer of dPS and hPS in benzene [331]. The dynamic scattering of multicomponent solutions in the framework of the RPA approximation [324] yields the sum of two decay modes, which are represented by exponentials valid in the short time limit. For a symmetric diblock the results for the observable scattering intensity yields conditions for the cancellation of either of these modes. In particular the zero average contrast condition, i.e. a solvent scattering length density that equals the average of both 

N is the total number of monomers, (p the polymer volume fraction and Pi and Pi/2 the form factors of the total copolymer and of the single blocks respectively. 12=Vd=Vh is the excluded volume interaction parameter which relates to the second virial coefficient A2=vN/ 2Mc). 

Under the simplifying assumption that hydrodynamic interactions may be neglected, the only new parameter that controls the dynamics is a monomeric friction coefficient (Rouse model). Then the prediction for the rate Pj is given by  

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The use of Henry s constant for a standard-state fugacity means that the standard-state fugacity for a noncondensable component depends not only on the temperature but also on the nature of the solvent. It is this feature of the unsymmetric convention which is its greatest disadvantage. As a result of this disadvantage special care must be exercised in the use of the unsymmetric convention for multicomponent solutions, as discussed in Chapter 4. 

Most tanks store Hquid rather than gases or soHds. Characteristics and properties such as corrosiveness, internal pressures of multicomponent solutions, tendency to scale or sublime, and formation of deposits and sludges are vital for the tank designer and the operator of the tank and are discussed herein. Excluded from the discussion are the unique properties and hazards of aerosols (qv), unstable Hquids, and emulsions (qv). A good source of information for Hquid properties for a wide range of compounds is available (2). 

Thus for a binary solution, the partial properties are given directly as functions of composition for given T and P. For multicomponent solutions such calcufations are complex, and direc t use of Eq. (4-47) is appropriate. 

The unsymmetric convention of normalization is readily applicable to multicomponent solutions, but care must be taken to specify exactly the conditions that give y - 1. Whereas Eq. (35) is immediately applicable to solutions containing any number of components, Eq. (36) is not complete for a solution containing components in addition to 1 and 2. For a solute 2 dissolved in a mixture of solvents 1 and 3, the normalization conditions are completely specified if we write, for a fixed ratio xl/(x1 + x3),... 

In their correlation, Chao and Seader use the original Redlich-Kwong equation of state for vapor-phase fugacities. For the liquid phase, they use the symmetric convention of normalization for y and partial molar volumes which are independent of composition, depending only on temperature. For the variation of y with temperature and composition, Chao and Seader use the equation of Scatchard and Hildebrand for a multicomponent solution ... 

Navier-Stokes equations, 318, 386-387 Nitrocellulose, 31 Nitroglycerine, 31-32 Normalization binary solutions, 156-157 multicomponent solutions, 157-158 Nusselt number, 118... 

A similar expression can be derived for the polymer activity (11). The formalism can also be extended to multicomponent solutions (11). 

Unfortunately, relatively little work has been done on the solution thermodynamics of concentrated polymer solutions with "gathering". The definitive work on the subject Is the article of Yamamoto and White (17). The corresponding-states theory of Flory (11) does not account for gathering. We therefore restrict our consideration here to multicomponent solutions where the solvents and polymer are nonpolar. For such solutions, gathering Is unlikely to occur. 

Binary electrolyte solutions contain just one solute in addition to the solvent (i.e., two independent components in all). Multicomponent solutions contain several original solutes and the corresponding number of ions. Sometimes in multicomponent solutions the behavior of just one of the components is of interest in this case the term base electrolyte is used for the set of remaining solution components. Often, a base electrolyte is acmaUy added to the solutions to raise their conductivity. 

When both solutions are binary and identical in nature and differ only by their concentration and the component E of the held strength is given by Eq. (4.18), the diffusion potential 9 can be expressed by Eq. (4.19). An equation of this type was derived by Walther Nemst in 1888. Like other equations resting on Eick s law (4.1), this equation, is approximate and becomes less exact with increasing concentration. For the more general case of multicomponent solutions, the Henderson equation (1907),... 

Almost all methods of chemical analysis require a series of calibration standards containing different amounts of the analyte in order to convert instrument readings of, for example, optical density or emission intensity into absolute concentrations. These can be as simple as a series of solutions containing a single element at different concentrations, but, more usually, will be a set of multicomponent solutions or solids containing the elements to be measured at known concentrations. It is important to appreciate that the term standard is used for a number of materials fulfilling very different purposes, as explained below. 

A common failing of earlier work on multicomponent solutions was to consider the 3R,x to be a constant for each salt. As Bromley (5) pointed out, the equations developed from the... 

Thermodynamic Analysis of Multicomponent Solutions Edward F. Casassa and Henryk Eisenbero... 

Clegg, S.E. and Brimblecombe, P. Solubility of ammonia in pure aqueous and multicomponent solutions, / Phys. Chem., 93(20) 7237-7238, 1989. 

Yeh and Keeler 244) extended the method of laser-scattering spectroscopy to probe systems undergoing rapid chemical reactions. They observed the spectral line broadening in light from a singlemode He-Ne laser scattered from multicomponent solutions, as a function of time. The experiment employed a pressure-scanned Fabry-Perrot interferometer and photon counting techniques. 

Johnson, K. S. Pytkowicz, R. M. Ion association and activity coefficients in multicomponent solutions. In Activity Coefficients in Electrolyte Solutions, Pytkowicz, R. M., Ed., Vol. II, CRC Press, Boca Raton, Florida, 1979 1-62. Millero, F. J. Effects of pressure and temperature on activity coefficients. In Activity Coefficients in Electrolyte Solutions, P5dkowicz, R. M., Ed., Vol. II, CRC Press, Boca Raton, Florida, 1979, pp. 63-151. 

Currie K. L. and Curtis L. W. (1976). An application of multicomponent solution theory to jadeitic pyroxenes. J. Geol, 84 179-194. 

Graves J. (1977). Chemical mixing in multicomponent solutions. In Thermodynamics in Geology, D. E. Eraser, ed. Reidel, Dordrecht-Holland. 

Technically, one includes solvent as one of the components when expressing mole fraction in chemical thermodynamics, but in describing dilute biological solutions, solvent is oftcn Omitted. With multicomponent solutions, one may chose to analyze the fractional composition of any two (or more) substances while not including others held constant in the experiment (e.g., concentrations of buffer components, proton, supporting electrolyte(s), enzyme, eta). For two components,... 

Kulapina, E. G. and Mikhaleva, N. M. (2005). The analysis of multicomponent solutions containing homologous ionic surfactant with sensor arrays. Sens. Actuators B 106(1), 271-277. 

Table 3. Definitions of the parameters appearing in Eqs. (26)—(28) for multicomponent solutions...

Recently Sato et al. [144,145] have extended the viscosity equation, Eq. (74), to multicomponent solution containing stiff-chain polymer species with different lengths. They showed a favorable comparison of the extended theory with the viscosity data for the quasi-ternary xanthan solutions presented in Fig. 21. 

It is not difficult to generalize the above formulation of the effective diffusion coefficient to the case in which there appear r different kinds of barriers or perturbation elements in the diffusion space. The result can be used to formulate the effective diffusion coefficient in a multicomponent solution. See [144] for a detailed explanation. 

According to their analysis, if c is zero (practically much lower than 1), then the fluid-film diffusion controls the process rate, while if ( is infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the mechanical parameter represents the ratio of the diffusion resistances (solid and fluid-film). This equation can be used irrespective of the constant pattern assumption and only if safe data exist for the solid diffusion and the fluid mass transfer coefficients. In multicomponent solutions, the use of models is extremely difficult as numerous data are required, one of them being the equilibrium isotherms, which is a time-consuming experimental work. The mathematical complexity and/or the need to know multiparameters from separate experiments in all the diffusion models makes them rather inconvenient for practical use (Juang et al, 2003). 

The chemical and petrochemical industries have utilized distillation, freezing, ion exchange, electrodialysis, selective membrane, and hydrate processes for a number of years to separate certain species or components from a multicomponent solution in their refining operations. Recent emphasis has been placed on developing and modifying these basic processes to obtain fresh water from brackish and sea water supplies. 

The last two equations offer an important means in calculating the activity coefficient of a dilute solute in multicomponent solutions. 

In more complicated cases, different combinations of phase transitions of stratification and ordering are possible. For example, a system may pass over to the disordered and ordered phases or to two different ordered phases. The type of the phase transitions and the regions of their realization are determined by the concentrations of the components, the temperature, and the potentials of particle interaction. Similar transitions also occur in multicomponent solutions. An increase in the number of components increases the number of different combinations of the phase transitions [29]. 

The theoretical tools for the interpretation of solution experiments are the thermodynamics of multicomponent solutions (Casassa and Eisenberg, 1964 Eisenberg, 1976, 1990) and the theory of small-... 

The convective diffusion equation is analogous to equations commonly used in dealing with heat and mass transfer. Similarly, if migration can be neglected in a multicomponent solution, then the convective diffusion equation can be applied to each species,... 

The method of Blanc [16] permits calculation of the gas-phase effective multicomponent diffusion coefficients based on binary diffusion coefficients. A conversion of binary diffusivities into effective diffusion coefficients can be also performed with the equation of Wilke [54]. The latter equation is frequently used in spite of the fact that it has been deduced only for the special case of an inert component. Furthermore, it is possible to estimate the effective diffusion coefficient of a multicomponent solution using a method of Burghardt and Krupiczka [55]. The Vignes approach [56] can be used in order to recalculate the binary diffusion coefficients at infinite dilution into the Maxwell-Stefan diffusion coefficients. An alternative method is suggested by Koijman and Taylor [57]. 

J. Grover, Chemical mixing in multicomponent solutions. An introduction to the use of Margules and other thermodynamics excess functions to represent nonideal behavior, pp. 67-97 in Thermodynamics in Geology, ed. by D. G. Fraser, D. Reidel, Dordrecht, The Netherlands, 1977. It follows from Eqs. 5.17 and 5.19 that, in general, In fA = Xjat and In fBA = Xjbj. Equation 5.20a is a special case of this relation for a third-order Margules expansion. 

Equation 5.46 was analyzed incorrectly by K. L. Currie and L. W. Curtis, An application of multicomponent solution theory to geodetic pyroxenes, J. Geol. 84 179 (1976) and by J.-J. Gruffat and J.-L. Bouchardon, Coefficients d activite d une solution subreguliere ddduits d un modele d interaction par triplets, Compt. Rend. Acad. Sci. Paris, 300 259 (1985). The error lay in assuming q, = 0 (see Eq. 5.49). Otherwise, the calculations in these two papers are correct. 

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