Dilution approximation

Two hundred grams of eleaned and dried crab shells (Note 1) ground to a fine powder is placed in a 2-1. beaker, and an excess of dilute (approximately 6 N) commercial hydrochloric acid is added slowly to the powdered material until no further action is evident. Much frothing occurs during the addition of the acid, and care must be exercised to avoid loss of material due to foaming over the sides of the beaker. After the reaction has subsided, the reaction mixture is allowed to stand from 4 to 6 hours to ensure complete removal of calcium carbonate. The residue is then filtered, washed with water until neutral to litmus, and dried in an oven at 50-60°. The weight of dried chitin is usually about 70 g., but with some lots of crab shells it may be as low as 40 g. 

Fig. 51 Phase diagram for PS-PI diblock copolymer (Mn = 33 kg/mol, 31vol% PS) as function of temperature, T, and polymer volume fraction, cp, for solutions in dioctyl ph-thalate (DOP), di-n-butyl phthalate (DBP), diethyl phthalate (DEP) and M-tetradecane (C14). ( ) ODT (o) OOT ( ) dilute solution critical micelle temperature, cmt. Subscript 1 identifies phase as normal (PS chains reside in minor domains) subscript 2 indicates inverted phases (PS chains located in major domains). Phase boundaries are drawn as guide to eye, except for DOP in which OOT and ODT phase boundaries (solid lines) show previously determined scaling of PS-PI interaction parameter (xodt

Fig. 7. The effective free-energy potentials for retraction of the free end of arms in a mon-odisperse star polymer melt. The upper curve assumes no constraint-release, the lower two curves take the dynamic dilution approximation with the assumptions (Ball-...

This chapter is concerned with these phases, where a substantial amount of the experimental work has been on poly(oxyethylene)-containing block copolymers in aqueous solution. From another viewpoint, the phase behaviour in concentrated block copolymer solutions has been interpreted using the dilution approximation, which considers concentrated solution phases to be simply uniformly swollen melt phases. Work on styrenic block copolymers in concentrated solution has been interpreted in this framework. There is as yet no unifying theory that treats ordered micellar phases and diluted melt phases coherently. 

Fig. 4.43 Volume fraction at the ODT versus %N for a range of PEP-PEE diblocks in squalene (Lodge et al. 1995). The best fit to the data (solid line) gives a slope of -0.81, compared to -1 for the dilution approximation and -0.626 observed for PS-PI diblocks in toluene (Fig. 4.42).

For cases in which the dilute approximation cannot be used, alternative expressions to equation IV.6 have been investigated for combustion systems. Such alternative expressions may also have computational advantages for CVD systems (86, 210, 211). The solution of equations IV.5 and IV.6, along with the following constraint... 

The increase of the solvent concentration in SB41 films on raising the partial pressure of chloroform vapor, and the related loss of long-range order, can be explained in terms of the so-called dilution approximation for the bulk block copolymer phases [167, 168], The above results clearly demonstrate the high sensitivity of the polymer-polymer interactions towards solvent content. Therefore, the microphase-separated structures in swollen block copolymer films can be used as a qualitative measure of the degree of swelling of the films [49, 166],... 

The applicable concentration range of the ideally dilute approximation depends on the size of the solute molecules, because, obviously, large polymer molecules will interact at much lower concentrations than will smaller species. For ionic solutes, the range of applicable concentrations is so small that it is practically useless, and even in this range, allowance must be made for the... 

A few values of Kf and Kb are given in Table 2. For a macromolecular solution, the ideally dilute approximation holds only up to such low molality that freezing-point depression and boiling-point elevation are useless for determining... 

Pure benzene freezes at 5.50°C and has a density of 0.876 g/mL. A solution of 1.7 g of nitrobenzene in 250 mL benzene freezes at 5.18°C. What is the molality-based freezing-point depression constant of benzene and at what temperature does a solution containing 3.2 g of bromobenzene in 250 mL of benzene freeze (You may make the ideally dilute approximation for both these solutions.)... 

For concentrations sufficiently high that the infinite dilution approximation breaks down, Eq. 2.17 must be modified by the incorporation of an activity coefficient y, which compensates for additions to the chemical potential from such nonidealities as solute-solute interactions. We have... 

Rate of Plasmin Hydrolysis. It may be concluded from the above electrophoretic and chromatographic data that partial methylation of a protein does not interfere with the site specificity of plasmin hydrolysis or result in the production of artifactual hydrolytic products. It remained to show what effect methylation had on the rate of hydrolysis by plasmin. M-/3-C was diluted approximately 300-fold with unlabeled protein and subjected to plasmin hydrolysis. The reaction mixture was sampled at various intervals up to 70 min, by which time most of the /3-casein was transformed (see Figure 7, Slots 1-5). The extent of transfer of radioactivity to the reaction products was examined to determine... 

Once thawed, stock virus is diluted approximately 1/100 with sterile saline and this dilution is kept on ice. 

In most liquid- and solid-phase systems, the dilute approximation is typically invalid, and, as a result, many body effects play a significant role. Many body effects are manifested through the effect of solvent or catalyst on reactivity and through concentration-dependent reaction rate parameters. Under these conditions, the one-way coupling is inadequate, and fully coupled models across scales are needed, i.e., two-way information traffic exists. This type of modeling is the most common in chemical sciences and will be of primary interest hereafter. In recent papers the terms multiscale integration hybrid, parallel, dynamic,... 

Two samples are generally prepared, diluted approximately 35 to 40 times by weight in flux powder. Use of two different dilution factors allows the effectiveness of the attenuation... 

The stock solution of peroxide was freshly prepared for each experiment. Peroxide (30%) was diluted approximately 1 to 1000 in 0.05 M phosphate buffer, pH 7.4, to produce a 1 x 10 M solution. The concentration was established using a millimolar extinction coefficient of 72 at 230 nm. 

Solubility of a Solid. For the solubilities of poorly soluble crystalline nonelectrolytes in a multicomponent mixed solvent, one can use the infinite-dilution approximation and consider that the activity coefficient of a solute in a mixed solvent is equal to the activity coefficient at infinite dilution. Therefore, one can write the following relations for the solubility of a poorly soluble crystalline nonelectrolyte in a ternary mixed solvent and in two of its binaries i2,i3... 

Eq. (18) allows one to calculate the protein solubility in a wide range of cosolvent concentrations if information regarding (i) the composition dependence of the preferential binding parameter and (ii) the properties of the protein-free mixed solvent such as the molar volume and the activity coefficients of the components are available. In addition, one should mention that Eq. (18) was obtained for ternary mixtures (water (l)-protein (2)-cosolvent (3)). However, those mixtures contain also a buffer, the effect of which is taken into account only indirectly through the preferential binding parameter. Another limitation of Eq. (18) is the infinite dilution approximation, which means that the protein solubility is supposed to be small enough to satisfy the infinite dilution approximation (y2 = where is the activity coefficient of a protein at infinite dilution). 

Solubility of drugs in aqueous solutions. Part 4 Drug solubility by the dilute approximation. 

The limitations of the proposed method are directly related to the simplifications made. The two most important ones are (1) the ideality of the mixed solvents and (2) the infinite dilution approximation. Our next papers will be focused on nonideal mixed solvents and on the effect of the finite concentration of a solute. 

Eq. (6) is a rigorous equation for the solubility of poorly soluble solids in a mixed solvent. The only approximation involved is that the solubilities of the solid in either of the pure solvents and in the mixed solvent are very small (infinite dilution approximation). It is not applicable when at least one of these solubilities has an appreciable value. Indeed (see Table 1), when the solubility of a solute in a nonaqueous solvent exceeds about 5 mol%, such as the solubilities of drugs in propylene glycol (Rubino and Obeng, 1992), the deviation from the experimental data is about 11.5% for the Wilson equation, whereas the average deviation for all 32 mixtures of Table 1 is only 7.7%. 

The results obtained previously by Ruckenstein and Shulgin [Int. J. Pharm. 258 (2003a) 193 Int. J. Pharm. 260 (2003b) 283] via the fluctuation theory of solutions regarding the solubility of drugs in binary aqueous mixed solvents were extended in the present paper to multicomponent aqueous solvents. The multicomponent mixed solvent was considered to behave as an ideal solution and the solubility of the drug was assumed small enough to satisfy the infinite dilution approximation. 

In the above papers, the solubility of drugs in mixed solvents was assumed to be low enough for the infinite dilution approximation to be applicable. Let us examine this approximation in more detail. The solubility of solid substances in pure and mixed solvents... 

The infinite dilution approximation implies that the activity coefficients in Eqs. (l)-(3) can be replaced by their values at infinite dilution of the solute y2 °°, and However, the solubilities of drugs in aqueous mixed solvents are not always very low. While the solubilities of various drugs in water (only poorly soluble drugs are considered in the present paper) do not exceed l-2mol%, the solubilities of the same drugs in the popular cosolvents ethanol and 1,4-dioxane can reach 5-20 mol%, and the solubilities in the water/l,4-dioxane and water/ethanol mixtures are often appreciable and can reach 8-30 mol%. Therefore, the effect of the infinite dilution approximation on the accuracy of the predictions of the solubilities of poorly soluble drugs deserves to be examined. 

The paper is organized as follows first, an equation for the activity coefficient of a low concentration solute in individual and binary solvents will be written. This equation will be combined with the flucmation theory of solutions and with Eqs. (l)-(3) to derive an expression for the drug solubility. Further, the expression obtained will be compared with experimental data and with the infinite dilution approximation (Ruckenstein and Shulgin, 2003a,b). 

The parameters L13 and L31 were also determined from the experimental solubility data. Therefore, Eq. (28) can be considered as a five parameters equation. The results of the calculations as well as a comparison with those obtained under the infinite dilution approximation are listed in Table 1. 

Table 1 shows that Eq. (28) provides slightly better results that the correlation based on the infinite dilution approximation. However, it is not clear whether this improvement was caused by the use of the more realistic dilute approximation, or of a larger number of adjustable parameters (five in the present case instead of four in the equation based on the infinite dilution approximation). 

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