Curve complexes

Figure 5.8 Schematic representation of the dissolution-dissociation-recrystallisation process of a cyclodextrin complex with a poorly soluble guest. The complex rapidly dissolves, and a metastable oversaturated solution is obtained. The anomalously high level of dissolved guest drops bock but remains higher than the level that can be obtained with noncomplexed drug. Solid curve = complexed drug broken curve = noncomplexed drug.

Fig. 11. Calculated holdback factor, normalized to Gd=1.0, for (a) lanthanide (solid curves) and Y (dashed curve) complexes with hiba at pH 4.5, and (b) lanthanide (solid curves) and Y (dashed curve) complexes with...

Figure 6. Thermal denaturation of complexes of chicken ovoinhibitor (01) with both o-chymo-trypsin (XT) and subtilisin BPN (Su). XT-OI-Su (top curve), complex formed by direct mixing of components (mole ratio XT 0I Su = 1.1 1.0 1.4). Lower solid curve shows displacement of XT from a 3 1 mixture with 01 by Su at 1 mol/mol 01. Dashed lines at the bottom show positions of peaks for free XT and the 1 1 complexes (XT-OI, Su-OI). 

Figure 1 Measured (symbols) and fitted (curves) complex viscosity of PC1030 and PC3820 at various temperatures.

To extend the applicability of the characterization factor to the complex mixtures of hydrocarbons found in petroleum fractions, it was necessary to introduce the concept of a mean average boiling point temperature to a petroleum cut. This is calculated from the distillation curves, either ASTM or TBP. The volume average boiling point (VABP) is derived from the cut point temperatures for 10, 20, 50, 80 or 90% for the sample in question. In the above formula, VABP replaces the boiling point for the pure component. 

The generalized use of computers makes seemingly complex calculations quite easy to perform however, curves and tables are still invaluable when one needs to obtain approximate values or to take into account the sensitivity of a property to operating conditions or to a mixture s characteristics. 

Figure A3.7.6. Photoelectron spectrum of. Here the F is complexed to para-R - Solid curve experimental results. Dashed curve simulated spectrum from scattering calculation on ab initio surface.

Figure B3.4.12. A schematic ID vibrational pre-dissociation potential curve (wide flill line) with a superimposed plot of the two bound fimctions and the resonance fimction. Note that the resonance wavefiinction is associated with a complex wavevector and is slowly increasing at very large values of R. In practice this increase is avoided by iismg absorbing potentials, complex scaling, or stabilization.

Fig. 2. Time-evolution of the methyl/ethyl C-C distances for both the zirconocene and the corresponding titanocene catalyst. The two curves starting at around 3.2 A represent the distance between the methyl carbon atom and the nearest-by ethylene carbon atom in the zirconocene-ethylene and the titanocene-ethylene complex, respectively. The two curves starting at around 1.35 A reflect the ethylene internal C-C bond lengths in the two complexes.

Fig. 3. Time evolution of the distance between the Zr atom and each of the three hydrogen atoms belonging to the methyl group (the original methyl group bonded to the Zr) in the zirconocene-ethylene complex. The time-evolution of one of the hydrogen atoms depicted by the dotted curve shows the development of an a-agostic interaction. Later on in the simulation (after about 450 fs) one of the other protons (broken curve) takes over the agostic interaction (which is then a 7-agostic interaction).

The curious shape of the computed parts of the curves suggests that there is, in each case, a discontinuity of slope. However, examination of the results shows that there is, in fact, a switch in the dominant eigenvalue as the Lewis number changes. Above a certain value of the Lewis number this is real and moves to the right as decreases. But eventually it crosses with a pair of complex eigenvalues moving to the left and these, which become the dominant eigenvalues for smaller values of, cause... 

Equality between the 1, 2 wave function and the modulus of the 2, 1 wave function, v /(j2, i), shows that they have the same curve shape in space after exchange as they did before, which is necessary if their probable locations are to be the same. The phase factor orients one wave function relative to the other in the complex plane, but Eq. (9-17) is simplified by one more condition that is always true for particle exchange. When exchange is canied out twice on the same particle pair, the operation must produce the original configuration of particles... 

Understanding how the force field was originally parameterized will aid in knowing how to create new parameters consistent with that force field. The original parameterization of a force field is, in essence, a massive curve fit of many parameters from different compounds in order to obtain the lowest standard deviation between computed and experimental results for the entire set of molecules. In some simple cases, this is done by using the average of the values from the experimental results. More often, this is a very complex iterative process. 

Equation 8.7 explains the solubility curve for AgCl shown in Figure 8.1. As Ch is added to a solution of Ag+, the solubility of AgCl initially decreases because of reaction 8.1. Note that under these conditions, the final three terms in equation 8.7 are small, and that equation 8.1 is sufficient to describe the solubility of AgCl. Increasing the concentration of chloride, however, leads to an increase in the solubility of AgCl due to the soluble chloro-complexes formed in reactions 8.3-8.5. ... 

Examples of titration curves for (a) a complexation titration, (b) a redox titration, and (c) a precipitation titration. 

The approach that we have worked out for the titration of a monoprotic weak acid with a strong base can be extended to reactions involving multiprotic acids or bases and mixtures of acids or bases. As the complexity of the titration increases, however, the necessary calculations become more time-consuming. Not surprisingly, a variety of algebraic and computer spreadsheet approaches have been described to aid in constructing titration curves. 

The equivalence point of a complexation titration occurs when stoichiometri-cally equivalent amounts of analyte and titrant have reacted. For titrations involving metal ions and EDTA, the equivalence point occurs when Cm and Cedxa are equal and may be located visually by looking for the titration curve s inflection point. 

Spectrophotometric titration curve for the complexation titration of a mixture. 

To evaluate a redox titration we must know the shape of its titration curve. In an acid-base titration or a complexation titration, a titration curve shows the change in concentration of H3O+ (as pH) or M"+ (as pM) as a function of the volume of titrant. For a redox titration, it is convenient to monitor electrochemical potential. 

Sketch the spectrophotometric titration curve for the titration of a mixture of 5.00 X 10 M Bi + and 5.00 X 10 M Cu + with 0.0100 M EDTA. Assume that only the Cu +-EDTA complex absorbs at the selected wavelength. 

Both the method of continuous variations and the mole-ratio method rely on an extrapolation of absorbance data collected under conditions in which a linear relationship exists between absorbance and the relative amounts of metal and ligand. When a metal-ligand complex is very weak, a plot of absorbance versus Ay or n-J m may be curved, making it impossible to determine the stoichiometry by extrapolation. In this case the slope ratio may be used. 

Values of sx are a complex function of transmittance when indeterminate errors are dominated by the noise associated with photon transducers. Curve B in Figure 10.35 shows that the relative uncertainty in concentration is very large for low absorbances, but is less affected by higher absorbances. Although the relative uncertainty reaches a minimum when the absorbance is 0.96, there is little change in the relative uncertainty for absorbances between 0.5 and 2. This source of inde-... 

Sensitivity Sensitivity in flame atomic emission is strongly influenced by the temperature of the excitation source and the composition of the sample matrix. Normally, sensitivity is optimized by aspirating a standard solution and adjusting the flame s composition and the height from which emission is monitored until the emission intensity is maximized. Chemical interferences, when present, decrease the sensitivity of the analysis. With plasma emission, sensitivity is less influenced by the sample matrix. In some cases, for example, a plasma calibration curve prepared using standards in a matrix of distilled water can be used for samples with more complex matrices. 

In the absence of Fe +, the membrane is colorless, but when immersed in a solution of Fe + and C, the membrane develops a red color as a result of the formation of a Fe +-bathophenanthroline complex. A calibration curve determined using a set of external standards with known molar concentrations of Fe + gave a standardization relationship of... 

---