In diluted systems

The tme driving force for any diffusive transport process is the gradient of chemical potential rather than the gradient of concentration. This distinction is not important in dilute systems where thermodynamically ideal behavior is approached. However, it becomes important at higher concentration levels and in micropore and surface diffusion. To a first approximation the expression for the diffusive flux may be written... 

Tbe mass-transfer coefficients k c and /cf by definition are equal to tbe ratios of tbe molal mass flux Na to tbe concentration driving forces p — Pi) and (Ci — c) respectively. An alternative expression for tbe rate of transfer in dilute systems is given by... 

In dilute systems the logarithmic-mean insoluble-gas and nonvolatile-hquid concentrations approach unity, and Eq. (5-261) reduces to the dilute-system formula. For equimolar counter diffusion (e.g., binary distillation), these log-mean factors should be omitted. See Eq. (5-189). 

An interesting problem arises when we consider solutions or colloidal sols where the diffusing component is much larger in size than the solute molecules. In dilute systems Equation (1.14) would give an adequate value of the Peclet number but not so when the system becomes concentrated, i.e. the system itself becomes a condensed phase. The interactions between the diffusing component slow the motion and, as we shall see in detail in Chapter 3, increase the viscosity. The appropriate dimensionless group should use the system viscosity and not that of the medium and now becomes... 

More recent work with cosolvency in dilute systems seems to indicate that the magnitude of the solubility enhancement is linear up to some 10-20% cosolvent fraction [55,172,184,250-262]. At very low concentrations of cosolvent, the assumption of non-interaction between the cosolvent and water cannot hold. In dilute solutions the individual cosolvent molecules will be fully hydrated and, as a result, will disrupt the water network structure. If the total volume disrupted is regarded as the extended hydration shell, and if Sc is the average solubility within this shell, then the overall solubility Sm in the water-cosolvent mixture will be approximated by... 

The rate of transfer for a homogeneous system of donors and acceptors has been shown to be linear with acceptor concentration in dilute systems.(43,44) This can be understood simply by presuming that the donor has a sphere of influence, the radius of which is equal to the Forster range R0.If an acceptor molecule lies inside this sphere, the excitation is transferred otherwise the donor deexcites by fluorescence. The probability that an acceptor will lie within the sphere of influence of an excited donor is directly proportional to the acceptor concentration, and so the transfer is linear with acceptor concentration in dilute systems. 

S. Matsumoto and M. Kohda The Viscosity of W/OAV Emulsions An Attempt to Estimate the Water Permeation Coefficient of the Oil Layer from the Viscosity Changes in Diluted Systems on Aging under Osmotic Pressure Gradients. J. Colloid Interface Sci. 73,13 (1980). 

Buswell JA. 1975. Metabolism of phenol and cresol by Bacillus stearothermophilus. J Bacteriol 124 1077-1083.Buzzell JC Jr., Young RHF, Ryckman DW. 1968. Behavior of organic chemicalsin the aquatic environment. Part II. -- Behavior in dilute systems. St.Louis, MO Environmental Sanitary Engineering Laboratories, WashingtonUniversity, 81. 

The bubble column and spray tower depend on nozzles to disperse the drop or bubble phase and thus provide the high area and small particle size necessary for a high rate. Drop and bubble coalescence are therefore problems except in dilute systems because coalescence reduces the surface area. An option is to use an impeller, which continuously redisperses the drop or bubble phase. For gases this is called a sparger reactor, which might look as shown in Figure 12-16. 

Bi , BiIV, Biv. All of these oxidation states are well known to chemists, with the exception of Biiv. We know that s1 cations such as BiIV are unstable when concentrated. Thus, we do not normally find discrete compounds where bismuth is entirely in the tetravalent state. However, we do find BiIV in dilute systems, both insulating and metallic systems (3). 

All these equations reduce to their dilute-system equivalents as the inert concentrations approach unity in terms of mole-fractions of inert concentrations in the fluids. In dilute systems, the logarithmic-mean insoluble-gas and non-volatile-liquid concentrations approach unity. 

It has been noted that the regular solution expression, Equation 2.41, holds only with nonpolar systems (Claramonte et al., 1993), or in dilute systems where the solute-solvent and solvent-solvent... 

In concentrated systems obtained in a thin uniform shape, the simplest way to record X-ray absorption data is the transmission mode in which the incident and transmitted photons are directly measured by means of ionisation chambers. However, in dilute systems or for surface characterisations, data are usually recorded using secondary effects resulting from the creation of the core hole during the absorption process and from its subsequent relaxation by radiative or non-radiative decays. These processes are the X-ray fluorescence emission and the total electron yield (TEY) emission, respectively. In these detection modes, the linear absorption coefficient is proportional to the ratio of the fluorescence or TEY intensity to... 

These two conditions (Eqs. (4.97) and (4.98)) are usually sufficient for assuming the medium as quiescent in dilute systems in which both cua.s and cda,oo are small. However, in nondilute or concentrated systems the mass transfer process can give rise to a convection normal to the surface, which is known as the Stefan flow [Taylor and Krishna, 1993]. Consider a chemical species A which is transferred from the solid surface to the bulk with a mass concentration cua.oo- When the surface concentration coa,s is high, and the carrier gas B does not penetrate the surface, then there must be a diffusion-induced Stefan convective outflux, which counterbalances the Fickian influx of species B. In such situations the additional condition for neglecting convection in mass transport systems is [Rosner, 1986]... 

The presence of the fourth-rank tensor in (7.127) and its absence in (7.11) suggests that the stress optical rule should not apply for dilute solutions of rigid rods. Unfortunately, because of the difficulty of acquiring truly rigid rods, and the problems of making measurements of stress in dilute systems, there are no data available on dilute rigid rod solutions where the stress optical rule can be investigated on this class of polymer liquids. 

Al/W/O ratio of 4/1/1. The monomer/catalyst ratio necessarily increased with initial monomer concentration. Figure 1 indicates that large amounts of the extractable fraction are formed in polymerizations carried out in dilute systems. 

In dilute systems consisting of a number of different species, the Gibbs free energy is computed from the sum of independent contributions from Ns different species. 

Particle interaction. Scattering theory does not take account of physical interactions between particles. In concentrated systems such as pastes or sediments, the particles are in contact with each other and different theories have to be used to describe their ultrasonic behaviour [64,65]. The ultrasonic properties can be affected, even in dilute systems, if flocculation occurs. The dependence of the ultrasonic properties of a material on the degree of particle interaction allows ultrasonics to be used to investigate this phenomenon. 

Studies like these may be considered as bridges between electrokinetics in dilute systems and in plugs (sec. 4.7). ... 

Adsorption and diffusion of alkanes in zeolites and in well-structured porous materials like MCM-41 materials are studied widely [1,5,8,9,11], The reported difusivities however differ sometimes by orders of magnitude. These differences are sometimes attributed to the use of microscopic techniques in stead of macroscopic techniques [12]. We, however, think that a major part of the found differences must be imputed to the use of a carrier gas. Adsorption is often studied in diluted systems with methods as ZLC [3], gaschromatography [4], inverse gas chromatography [10], gravimetry [12], > ile others are not using carriers gasses at all. 

Diluents and plasticizers in polymeric systems increase the steady-state compliance and decrease the zero shear rate viscosity. These two combined opposing effects give rise to a diminution in the value of Hence the critical value of the shear rate in dilute systems is shifted to higher values as the dilution increases (see Fig. 13.28). 

The ability to predict drop size is critical to determining both the interfadal area for mass transfer and the state of dispersion of the system. In dilute systems and in moderately concentrated systems where coalescence can be neglected, the following equation describes the maximum equilibrium (i.e., after a long time) drop diameter of an inviscid or low viscosity dispersed phase ... 

If the medium becomes more dilute, the rate of Amadori compoimd formation is in general drastically reduced. However, this does not apply to the cysteine reaction. As a consequence, cysteine becomes clearly the most reactive amino acid in dilute neutral media (Table II). A possible explanation for the relatively high reactivity of cysteine in dilute aqueous systems is the high concentration of the cysteine-sugar condensation products thanks to the assistance by the thiol group. Therefore, in dilute systems the concentration of the Schiff s base of cysteine 6 will be much higher than the concentrations of the corresponding Schiff s bases of the other amino acids. 

In Fig. 10 we have given some excitation spectra of the VO luminescence. In concentrated systems they peak in the region between 28000 and 32000 cm , followed by a second band around 35000 cm L In diluted systems the latter band dominates and there is only a shoulder at lower energy. 

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