Concentration dependences

When diffusivity is concentration dependent, namely D = D(ni), the field (3.6) is modified to read 

Several methods for the solution of (5.12) were presented in the past (Crank 1980) and explored for specific assumed forms of D(m) - mostly linear and exponential. In a more recent work (Lee and Peppas 1993), simulations for the latter case were evaluated numerically and exhibited in detail, together with the associated residual stresses. The characteristic features of concentration dependence were listed by Rogers (1965). Among those it is important to note that M(t) vs. t depends on the specimen s thickness and that desorption varies significantly from absorption. While the foregoing features may serve to identify the presence of concentration dependence diffusivity, the actual task of expressing that dependence is formidable indeed. 

In general the concentration dependence of viscosity of dispersed spheres may be described with a form of equation given by Eilers [19]. 

Many equations similar to Eilers equation have been proposed for example, Maron s [21] equation is, 

In applying these equations other important considerations, namely the non-Newtonian nature of the flow also needs to be considered. A similar form of equation may be nsed to represent the modulus of the dispersed systems. However, for the modulus both strain-and strain-rate dependence must be considered. 

Montes, J. L. White and N. Tlakayrma., Journal of Non-Newtonian Fluid Mechanics, 1988,28, 183. 

Standard Test Method for Carbon Black-n-Dibutyl Phthalate Absorption Number. 

The potential generated by a cell is dependent on the concentration of the components present. The relationship is given by the equation  

These expressions are identical to those for the equilibrium constant given in Section 8.1.2, except that they apply to any concentrations or partial pressures, not just those when the system is at equilibrium. Pure liquids or solids, or water in solutions, appearing in the reaction equation do not appear in the equations for Q. (The quantity of importance is the activity rather than the concentration. Activity and concentration are equal in dilute solutions see Section S3.2). 

It is convenient to separate the term relating to the equilibrium constant and write the equation for the cell voltage as 

When all species are in the standard state, Q=, In (2 = 0, and the cell voltage is E°, hence  

Determination of the dependence of the fractional activity on the concentration of the monomer forming the uni- or supramolecular active structure is required to obtain the fc Cso (Section 11.2.2). The shape of the Cm profile contains further information on the thermodynamics and, in some cases, the stoichiometry of the self-assembly of supramolecular ion channels and pores (Fig. 11.7) [21]. 

Class I charmels and pores follow the Hill Eq. (11.2) applied to self-assembly (Fig. 11.7a) 

The Hill coefficient n obtained from the curve fit of the Cm profile of Class I channels and pores (Fig. 11.7a) corresponds to the number of monomers in the active supramolecule (if self-assembly indeed occurs from an excess monomer in solution. With self-assembly from excess dimer, the number of monomers per active supramolecule is 2n, and so on). The compatibility with the Hill equation further demonstrates the presence of excess monomer besides a small population of active supramolecule. The presence of excess monomer, in turn, reveals that the self-assembly of the channel or pore is an endergonic process. Structural studies of unstable n 1 supramolecules at concentrations near the EC o by conventional methods are therefore meaningless. For example, NMR or IR measurements will report on the inactive monomers, whereas the unstable active structure of Class I channels and pores is invisible (see Section 11.4 for methods to selectively detect and study minority populations of active supramolecules). In BLMs, the thermodynamic instability of Class I channels and pores is expressed in low open probabilities Po (Fig. 11.4). The n 1 of Class I channels and pores is unrelated to the kinetic stability expressed in short lifetimes for labile Class lA and long lifetimes for inert Class IB supramolecules. 

Tire self-assembly of Class II channels and pores is exergonic. Their Cm profile is, therefore, incompatible with the Hill equation, revealing either n = 1 (Fig. 11.7b) 

It is important to note that Hill analysis of supramolecular synthetic ion channels and pores is not fully developed and deserves appropriate caution. The usefulness of chemical or thermal denaturation [12-14, 24] to temporarily transform exergonic Class II into endergonic Class I supramolecules for clear-cut demonstration of their true Hill coefficient as well as their supramolecular nature, for example, remains to be explored. 

Alloush et al. (1982) represented, with an uncertainty of 0.8%, the concentration dependence of aqueous LiBr solutions by an equation of the form 

Chiquillo (1967) proposed the following equation for the concentration dependence of the thermal conductivity of aqueous solutions 

In this equation, A, were adjustable parameters tabulated for several electrolyte solutions. A very similar relation was also proposed by Losenicky (1969) and Zaytsev and Aseyev (1992). 

Polymers in solution or as melts exhibit a shear rate dependent viscosity above a critical shear rate, ycrit. The region in which the viscosity is a decreasing function of shear rate is called the non-Newtonian or power-law region. As the concentration increases, for constant molar mass, the value of ycrit is shifted to lower shear rates. Below ycrit the solution viscosity is independent of shear rate and is called the zero-shear viscosity, r 0. Flow curves (plots of log r vs. log y) for a very high molar mass polystyrene in toluene at various concentrations are presented in Fig. 9. The transition from the shear-rate independent to the shear-rate dependent viscosity occurs over a relatively small region due to the narrow molar mass distribution of the PS sample. 

Qualitatively, the same behaviour is observed for the flow curves at a fixed polymer concentration with various molar masses (Fig. 10). The shear rate depend- 

Viscosity h vs. shear rate y for polystyrene (MW—23.60106) in toluene for various concentrations at 25 °C 

Viscosity vs. shear rate for polystyrene samples of various molar masses in toluene at 25 °C 

The fact that viscosity is independent of molar mass at high shear rates is of fundamental importance, since it follows that it is impossible to distinguish between different samples if the viscosity is measured in the power-law region. 

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It must be kept in mind that both pictures are modelistic and invoke extrather-modynamic concepts. Except mathematically, there is no such thing as a two-dimensional gas, and the solution whose osmotic pressure is calculated is not uniform in composition, and its average concentration depends on the depth assumed for the surface layer. 

Figure Bl.20.11. Force curves of DMPC/DPPE (dimyristoyl phosphatidylcholine and dipalmitoyl phosphatidylethanolainine) bilayers across a solution of PEG at different concentrations. Clearly visible is a concentration-dependent depletion attraction, with pennission from [17],...

In equation (C2.6.14) it can be seen that tire required salt concentration depends strongly on tire valency of tire ions... 

Samples can be concentrated beyond tire glass transition. If tliis is done quickly enough to prevent crystallization, tliis ultimately leads to a random close-packed stmcture, witli a volume fraction (j) 0.64. Close-packed stmctures, such as fee, have a maximum packing density of (]) p = 0.74. The crystallization kinetics are strongly concentration dependent. The nucleation rate is fastest near tire melting concentration. On increasing concentration, tire nucleation process is arrested. This has been found to occur at tire glass transition [82]. 

The chemical shifts of O—H and N—H protons are temperature and concentration dependent... 

Although the terms solute and solution are often associated with liquid samples, they can be extended to gas-phase and solid-phase samples as well. The actual units for reporting concentration depend on how the amounts of solute and solution are measured. Table 2.4 lists the most common units of concentration. 

Source Adapted from Baker, D. R. Capillary Electrophoresis. Wiley-Interscience New York, 1995. "Concentration depends on the volume of sample injected. 

As noted above, all of the partial molar quantities are concentration dependent. It is convenient to define a thermodynamic concentration called the activity aj in terms of which the chemical potential is correctly given by the relationship... 

The change in interaction energy per 1,2 pair is thus h. Aw. Next we must consider how this scales up for a large array of molecules, and particularly how to describe the concentration dependence of the result. 

At equilibrium, these concentration and pressure effects must be equal and opposite for Eq. (8.75) to apply. Equation (8.13) describes the concentration dependence of jui, and Eq. (8.12) describes the pressure effect. Assembling these results, we write... 

By describing the concentration dependence of an observable property as a power series, Eq. (9.9) plays a comparable role for viscosity as Eq. (8.83) does for osmotic pressure. 

As with the diffusion coefficient, sedimentation coefficients are frequently corrected for concentration dependence and reduced to standard conditions ... 

The concentration dependence of s is eliminated by making measurements at several different concentrations and then extrapolating to zero concentration. The limiting value is given by the symbol s°. This is the sedimentation analog of D°. 

Thus we have finally established how light scattering can be used to measure the molecular weight of a solute. The concentration dependence of r enters Eq. (10.54) through an expression for osmotic pressure, and this surprising connection deserves some additional comments ... 

A rapid increase in diffusivity in the saturation region is therefore to be expected, as illustrated in Figure 7 (17). Although the corrected diffusivity (Dq) is, in principle, concentration dependent, the concentration dependence of this quantity is generally much weaker than that of the thermodynamic correction factor d ap d a q). The assumption of a constant corrected diffusivity is therefore an acceptable approximation for many systems. More detailed analysis shows that the corrected diffusivity is closely related to the self-diffusivity or tracer diffusivity, and at low sorbate concentrations these quantities become identical. 

The Du Pont HaskeU Laboratory for Toxicology and Industrial Medicine has conducted a study to determine the acute inhalation toxicity of fumes evolved from Tefzel fluoropolymers when heated at elevated temperatures. Rats were exposed to decomposition products of Tefzel for 4 h at various temperatures. The approximate lethal temperature (ALT) for Tefzel resins was deterrnined to be 335—350°C. AH rats survived exposure to pyrolysis products from Tefzel heated to 300°C for this time period. At the ALT level, death was from pulmonary edema carbon monoxide poisoning was probably a contributing factor. Hydrolyzable fluoride was present in the pyrolysis products, with concentration dependent on temperature. 

Concentrations depend on severity of pyrolysis. At a high severity (- 2000° C) acetylene/ethylene ratio is 1, but at lower severity acetylene concentration is reduced and ethylene is increased. 

Manganate(VI) formed in the initial oxidation process must first be dissolved in a dilute solution of potassium hydroxide. The concentrations depend on the type of electrolytic cell employed. For example, the continuous Cams cell uses 120 150 g/L KOH and 50 60 g/L K MnO the batch-operated Bitterfeld cell starts out with KOH concentrations of 150 160 g/L KOH and 200 220 g/L K MnO. These concentration parameters minimize the disproportionation of the K MnO and control the solubiUty of the KMnO formed in the course of electrolysis. 

The volume of extracellular fluid is direcdy related to the Na" concentration which is closely controlled by the kidneys. Homeostatic control of Na" concentration depends on the hormone aldosterone. The kidney secretes a proteolytic enzyme, rennin, which is essential in the first of a series of reactions leading to aldosterone. In response to a decrease in plasma volume and Na" concentration, the secretion of rennin stimulates the production of aldosterone resulting in increased sodium retention and increased volume of extracellular fluid (51,55). 

When sulfuric acid is present in the mixed acids, the following ionisation reactions occur. These ionic reactions are rapid, and equiHbrium concentrations of NO2 are likely to be present at all times in the acid phase. NO2 concentrations depend mainly on the composition of the mixed acids but decrease to some extent as the temperature increases (3). 

R is hydrogen, alkenyl, or alkyne. In remote tropospheric air where NO concentrations ate sometimes quite low, HO2 radicals can react with ozone (HO2 + O3 — HO + 2 O2) and result in net ozone destmction rather than formation. The ambient ozone concentration depends on cloud cover, time of day and year, and geographical location. 

The role of specific interactions in the plasticization of PVC has been proposed from work on specific interactions of esters in solvents (eg, hydrogenated chlorocarbons) (13), work on blends of polyesters with PVC (14—19), and work on plasticized PVC itself (20—23). Modes of iateraction between the carbonyl functionaHty of the plasticizer ester or polyester were proposed, mostly on the basis of results from Fourier transform infrared spectroscopy (ftir). Shifts in the absorption frequency of the carbonyl group of the plasticizer ester to lower wave number, indicative of a reduction in polarity (ie, some iateraction between this functionaHty and the polymer) have been reported (20—22). Work performed with dibutyl phthalate (22) suggests an optimum concentration at which such iateractions are maximized. Spectral shifts are in the range 3—8 cm . Similar shifts have also been reported in blends of PVC with polyesters (14—20), again showing a concentration dependence of the shift to lower wave number of the ester carbonyl absorption frequency. 

GVD Coatings. As in PVD, the stmcture of the deposited material depends on the temperature and supersaturation, roughly as pictured in Figure 8 (12). In the case of CVD, however, the effective supersaturation, ie, the local effective concentration in the gas phase of the materials to be deposited, relative to its equiUbrium concentration, depends not only on concentration, but on temperature. The reaction is thermally activated. Because the effective supersaturation for thermally activated reactions increases with temperature, the opposing tendencies can lead in some cases to a reversal of the sequence of crystalline forms Hsted in Figure 8, as temperature is increased (12). 

In the same class of polymers, an antistat can exhibit different degrees of effectiveness. As seen in Table 6 the performance of ethoxylated oleyl amine varies among polyolefins. The data for polypropylene (PP) also shows the concentration dependence of antistats. 

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