Dilution curves, theoretical

Stream-discharge quality trends are usually displayed in log-log plots (cf. Gang and Langmuir 1974 Lerman 1979 Levinson 1980 Hem 1985). Such plots are often linear or nearly so. Figure 8.1 6(b) shows dissolved silica versus discharge for a river in British Columbia (Kleiber and Erle-bach 1977). Theoretical dilution curves based on Eq. (8.33) have been added to the figure. The difference between a dilution curve and data regression line is a measure of the amount of additional dissolved silica contributed to the stream by increased flow. 

Stream composition is a function of stream discharge. What is a theoretical dilution curve How is stream chemistry related to discharge ... 

Figure 8.3b shows that phase separation in polymer mixtures results in two solution phases which are both dilute with respect to solute. Even the relatively more concentrated phase is only 10-20% by volume in polymer, while the more dilute phase is nearly pure solvent. The important thing to remember from both the theoretical and experimental curves of Fig. 8.3 is that both of the phases which separate contain some polymer. If it is the polymer-rich or precipitated phase that is subjected to further work-up, the method is called fractional precipitation. If the polymer-poor phase is the focus of attention, the method... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

The indicators numbered 1 and 2 at the bottom of Table 39 both have vacant proton levels low enough for use in dilute solution the circles in Fig. 67 give the experimental results obtained in aqueous solutions of HC1. In each case the slope of the line does not differ from the theoretical slope of (218) by as much as 5 per cent. Reading off the constant vertical distance between the two curves (the length of the vertical arrow in Fig. 67), we find... 

In Sections 10.11-10.16 it is shown how the change in pH during acid-base titrations may be calculated, and how the titration curves thus obtained can be used (a) to ascertain the most suitable indicator to be used in a given titration, and (b) to determine the titration error. Similar procedures may be carried out for oxidation-reduction titrations. Consider first a simple case which involves only change in ionic charge, and is theoretically independent of the hydrogen-ion concentration. A suitable example, for purposes of illustration, is the titration of 100 mL of 0.1M iron(II) with 0.1M cerium(IV) in the presence of dilute sulphuric acid ... 

In the case of dynamic mechanical relaxation the Zimm model leads to a specific frequency ( ) dependence of the storage [G ( )] and loss [G"(cd)] part of the intrinsic shear modulus [G ( )] [1]. The smallest relaxation rate l/xz [see Eq. (80)], which determines the position of the log G (oi) and log G"(o>) curves on the logarithmic -scale relates to 2Z(Q), if R3/xz is compared with Q(Q)/Q3. The experimental results from dilute PDMS and PS solutions under -conditions [113,114] fit perfectly to the theoretically predicted line shape of the components of the modulus. In addition l/xz is in complete agreement with the theoretical prediction based on the pre-averaged Oseen tensor. 

Fig. 14. Data (points) for G (co) and G (co) for a range of compositions of a blend of two polyisoprene stars of molecular weights 28 and 144 kg mol The fractions of the bigger star are in order 0.0,0.2,0.5,0.8 and 1.0. Curves are theoretical predictions of the tube model with co-operative constraint release treated by dynamic dilution [56]. The choice of 2.0 rather than 7/3 for the dilution exponent p is compensated for by taking M = 5500 kg mol" ...

Fig. 5. Kinetics of brain microtubule depolymerization following rapid dilution. (A) Time course of the disassembly reaction with experimental data represented by the data points and the theoretical progress curve indicated by the solid line. (The inset to A shows that the process can be fitted to a simple decaying exponential for part of the depolymerization reaction.) (B) Microtubule length distribution for the sample subjected to rapid dilution in A. (Reproduced from Karr et al. (1980)./. Biol. Chm. 255, 8560-8566.)...

A theoretical treatment of combination titrations with an ideal indicator electrode was given by Meites et al. [89-91 ]. They have shown that the dilution effect causes a deviation of the titration curve inflection point from the equivalence point. However, this deviation is small compared with the error... 

When the second derivative of (5.32) is calculated and set equal to zero, the inflection point of the titration curve is obtained [23, 24, 133, 134). It has been found that the theoretical titration error generally increases with decreasing sample concentration, with increasing value of the solubility product or of the dissociation constant, with increasing value of the dilution factor and with increasing concentration of the interferents. Larger errors are obtained with unsymmetrical titration reactions. The overall error is a combination of these factors the greatest effect is exerted by the sample concentration, a smaller one by the equilibrium constant and the interferents, and the smallest by dilution. To obtain errors below 1%, it must approximately hold that eg, > 10 2 i,K< 10 , < 10 to 10" and r < 0.3. 

Since the decay follows an exponential function, the similarity between the simulated decay curve slopes and the theoretical, infinite dilution ideal is even more apparent when the plots are compared in log space, as shown in Fig. 3.14B. 

Figure 3.14D shows the degree of correlation for the rate of decay of the protein-ligand complex in a modeled ALIS quench experiment and the theoretical decay curve expected from infinite dilution. The modeled decay curve is shown for ks-off = 0.01 s and theoretical curves are shown for dissociation rates +10% of this value. The results indicate that the measured dissociation rate is well within 10% of the actual value, a very good approximation of the actual dissociation rate given the simplicity of this experimental method. 

Figure 2. Progress curve for dilution-induced microtubule depolymerization. Inset Polymer length distribution prior to dilution-induced disassembly. The data points are experimentally determined, and the solid line is based on the theoretical treatmenF. ...

Fig. 3. Monitoring of the variant formation in a continuous culture of Bacillus stearothermo-philus PV72 (as in Fig. 2) by SDS-PAGE. The organisms differed in the S- (surface) layer proteins. The wild type formed an S-layer protein of 130 kDa apparent molecular mass, the variant S-layer protein appears at 97 kDa molecular mass. The wild-type S-layer protein was quantified relative to the band of the altered protein by densitometry. Numbers on the curve represent samples harvested at distinct points of time during variant formation. The decrease of the wild-type S-layer protein followed the theoretical washout curve in a stirred tank reactor at the set dilution rate of 0.3 h 1 (Reprinted from J. Biotechnol. 54, K. C. Schuster et al, p. 20,1997, with permission from Elsevier Science)...

Here we discuss a simple theoretical molecular model of triflic acid dissociation in dilute aqueous solution along the gas-liquid saturation curve to elevated... 

The -maxima and minima on viscosity-composition curves are reminiscent of those on vapour pressure-composition curves of binary, mixtures. 5 The vapour pressures and viscosities are equal at some temperatures, say T and To, and T and To respectively. Then To/T—To7T =C(T —T), where C is a constant. A plot of TojT—To IT against T—T gives a straight line in many cases, both for vapour pressure and viscosity in other cases, the vapour pressure shows a minimum and the viscosity a maximum, and the vapour pressure a maximum and the viscosity a minimum. Prasad, 6 from the relation with vapour pressure deduced the equation rj =rjjrio= +ac, where c=conc. of non-electrolyte. The theoretical value of a is 0 00652 the observed values were glucose 0 44, fructose 0 44, sucrose 0 78, independent of temperature. According to Errera, the curves depend on the electric dipolarity of the liquids if both are nonpolar, the curve is concave to the composition axis whilst if both are polar, it is convex. Wolkowa found that the viscosity of a solution is approximately proportional to its heat of dilution. There seems to be no relation between the viscosity and surface tension of a mixture of acetic acid and water (cf. salt solutions, 13.VIII E). Mixtures of isomorphous substances obey an approximately linear relation. 

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