Modeling titration curves

Figure A-19 Modelled titration curves against experimental titration data (curve (e) in Fig. 2 of [69FAU/DER]) for the system 20 mL KHCO3/KOH 1 M + 10 mL KOH 1 M + 2 mL ZrOCb IM (diluted with pure water to 100 mL), assuming formation of different Zr-carhonate complexes. The assumed formation constants (logn, ) are given in parentheses in the legend.

In summary, our re-evaluation of [80MAL/CHU] leads us to the following conclusions (i) it is not possible to fit the titrations with a single Zr-carbonate species mixtures of Zr-carbonate complexes are required (ii) Zr(C03)5 can be excluded as significant species in most of the titrated solutions (iii) some, but not all of the modelled titration curves are consistent with the formation of Zr(C03)4. ... 

Modelled titration curves against experimental titration data (curve (e) in Fig. 2 of [69FAU/DER]) for the system 20 mL KHCO3/KOH 1 M + 10... 

Fig. 16-4 pH sensitivity to SO4- and NH4. Model calculations of expected pH of cloud water or rainwater for cloud liquid water content of 0.5 g/m. 100 pptv SO2, 330 ppmv CO2, and NO3. The abscissa shows the assumed input of aerosol sulfate in fig/m and the ordinate shows the calculated equilibrium pH. Each line corresponds to the indicated amoimt of total NH3 + NH4 in imits of fig/m of cloudy air. Solid lines are at 278 K, dashed ones are at 298 K. The familiar shape of titration curves is evident, with a steep drop in pH as the anion concentration increases due to increased input of H2SO4. (From Charlson, R. J., C. H. Twohy and P. K. Quinn, Physical Influences of Altitude on the Chemical Properties of Clouds and of Water Deposited from the Atmosphere." NATO Advanced Research Workshop Acid Deposition Processes at High Elevation Sites, Sept. 1986. Edinburgh, Scotland.)... 

Tanford, C, Theory of Protein Titration Curves. 11. Calculations for Simple Models at Low Ionic Strength, Journal of the American Chemical Society 79, 5340, 1957. 

In the practice of potentiometric titration there are two aspects to be dealt with first the shape of the titration curve, i.e., its qualitative aspect, and second the titration end-point, i.e., its quantitative aspect. In relation to these aspects, an answer should also be given to the questions of analogy and/or mutual differences between the potentiometric curves of the acid-base, precipitation, complex-formation and redox reactions during titration. Excellent guidance is given by the Nernst equation, while the acid-base titration may serve as a basic model. Further, for convenience we start from the following fairly approximate assumptions (1) as titrations usually take place in dilute (0.1 M) solutions we use ion concentrations in the Nernst equation, etc., instead of ion activities and (2) during titration the volume of the reaction solution is considered to remain constant. 

Figure 10-3. Theoretical titration curves for the model compounds of Asp and His obtained from REX-CPHMD simulations [41]. Solid curves are the obtained by fitted the computed deprotonated fraction to the generalized Henderson-Hasselbach equation. The dashed lines indicate the computed pKa values...

Figure 3.16 Two series of STM images of 37nm x27nm continuously taken at RT under a nominal CO pressure of 1 x 10 8Torr for clean (a-d) and C-containing (e-g) Ag(l 1 0) (2 x l)-0 surfaces (/, = 0.2 nA, V tip=1.4V). Schematic models ofthe regions are also shown for (a-d). (h) Titration curves obtained for both clean (red solid circles) and C-containing...

First of all, the mesomerism of HBI is rendered complex by the presence of several protonable groups actually, HBI might exist, depending on pH, under cationic, neutral, zwitterionic, anionic, and possibly enolic forms (Fig. 3a). The experimental p/sTa s of model analogs of HBI in aqueous solutions have been studied. Titration curves follow two macroscopic transitions at pH 1.8 and pH 8.2, each corresponding to a single proton release [69]. Comparison of theoretical... 

An assumed 1 1 stoichiometry for a complex can be confirmed or invalidated by the fit of the titration curves described above for this case. If the fit is not satisfactory, a model of formation of two successive complexes can be tried. 

Whatever the aim of a particular titration, the computation of the position of a chemical equilibrium for a set of initial conditions (e.g. total concentrations) and equilibrium constants, is the crucial part. The complexity ranges from simple 1 1 interactions to the analysis of solution equilibria between several components (usually Lewis acids and bases) to form any number of species (complexes). A titration is nothing but a preparation of a series of solutions with different total concentrations. This chapter covers all the requirements for the modelling of titrations of any complexity. Model-based analysis of titration curves is discussed in the next chapter. The equilibrium computations introduced here are the innermost functions required by the fitting algorithms. 

PbOj anode, 40 155-156 oxygen evolution, 40 109-110 PCE, catalytic synthesis of, l,l,l-trifluoro-2,2-dischloroethane, 39 341-343 7t complex multicenter processes of norboma-diene, 18 373-395 PdfllO), CO oxidation, 37 262-266 CO titration curves, 37 264—266 kinetic model, 37 266 kinetic oscillations, 37 262-263 subsurface oxygen phase, 37 264—265 work function and reaction rate, 37 263-264 Pd (CO) formation, 39 155 PdjCrjCp fCOljPMe, 38 350-351 (J-PdH phase, Pd transformation, 37 79-80 P-dimensional subspace, 32 280-281 Pdf 111) mica film, epitaxially oriented, 37 55-56... 

Let us assume that titration curves for the feed streams are known. These can be the typical sharp curves for strong acids or the gradual curves for weak acids, with or without buffering. The dynamie model keeps traek of the amount of each stream that is in the tank at any point in time. Let C be the concentration of the nth stream in the tank, F be the flow rate of that stream into the tank, and Fou, be the total flow rate of material leaving the tank. 

In addition to the two examples above, I have developed TKISolver models for the ideal gas, for two-component mixture concentrations, for acid base chemistry (including the generation of titration curves), for transition metal complex equilibria, for genei gaseous and solution equilibria, and for linear regression (121. 

The pH dependence of could be due to changes in A-B loop disorder rates, perhaps the chemical exchange phenomenon observed for NPl-ImH (Section ll,E,2,b), or to changes in ligand bond strength. The change in lies in the off-rates (Tables I-Ill) consistent with the loop disorder model. Plots of vs pH display an excellent fit with the equation for a titration curve (Fig. 21), indicating that the transition... 

Scheme 3.4-1. Simulated titration curves for the catalytic model system described above. The change in the steady-state concentrations following the ligand association process is schematically depicted (the species present at relatively high concentration is underlined).

An apparently straightforward method for the determination of number and strength of acid sites consists of the determination of the amount of base required to poison catalytic activity for a model reaction. By means of plots of activity versus amount of added base, the number of acid sites is obtained from the threshold amount of base required to remove catalytic activity acid strength is gauged from the slope of the titration curve. This method can therefore be called a catalytic titration. 

Figure 3. ITC-measurement of the complexation of (-)camphor by a-cyclodextrin in water. Left Primary heat pulse trace (CFB = cell feedback current) the saw-tooth shape arises from changing the aliquots of titrand solution. Right The time integral of the heat pulses furnishes the titration curve. The solid line represents the best fit to a 2 1 host-guest sequential binding model.

Figure 21.5 Potential errors introduced by the assumption that Ar2+ is constant in the derivation of the linkage Eq. (21.30). The dependence of Ar2+ on [Mg2 1J was modeled by a polynomial fit to the solid gray data points in Fig. 21.4D. The polynomial was used in the integration of Eq. (21.30) to give expressions for In Kobs and 0 with [Mg2+]-dependent AT2 (in contrast to Eqs. (21.33) and (21.34), which assume AT2+ is constant). 9 is plotted for the calculated titration curve when the midpoint of the titration, [Mg2+]0, is 10 jiM (circles), 30 uA / (squares), or 100 li.M (diamonds). AT2+ (as used to calculate the displayed curves) at the titration midpoints (9 = 0.5) is 1.45, 2.30, and 2.73, respectively. The simulated data points have been fit to either a modified version of Eq. (21.34) that assumes the y-intercept of the curve has the value 6 = 0, 9 = 90 + (1 - 0o)([Mg2+]/[Mg2+]o)"/[l + ([Mg2+]/[Mg2+j0)"], or to an equation that allows a nonzero y-intercept, Eq. (21.35). The residuals of the fits are shown in the lower three panels closed symbols correspond to Eq. (21.34) and open symbols to Eq. (21.35). For the curve with a midpoint of [Mg2+]0 = 100 fiM, Eq. (21.35) could not be fit to the data because the value of Co became vanishingly small. The values of A IT. at the titration curve midpoints obtained from the modified Eq. (21.34) are 1.99, 2.39, and 2.73, in order ofincreasing [Mg2+]0. AT2+ obtained by fitting ofEq. (21.35) and application of Eq. (21.36) are 1.43 and 2.29 (10 and 30 fiM transition midpoints, respectively).

FIGURE 2.3 Potentiometric titration curve of copper-montmorillonite in 0.1 mol dm-3 NaC104 solution, m = 50 mg, V = 20 cm3 (upper left). Vs are the experimental points, line is the plotted curve by the surface complexation model. The concentration of surface sites—lower left interlayer cations upper right silanol sites lower right aluminol sites (Nagy and Konya 2004). 

Madrid, L. and Diaz-Barrientos, E., Description of Titration curves of mixed materials with variable and permanent surface charge by a mathematical model. 1. Theory. 2. Application to mixtures of lepidocrocite and montmorillonite, J. Soil Sci., 39, 215, 1988. 

The reaction site at oxide electrodes, the oxide-electrolyte interface, differs from the metal-electrolyte interface in several respects and its structure and properties are of the utmost importance for the understanding of reaction kinetics at oxide electrodes. Most of the information available on the properties of the interfacial region in oxides comes from colloid chemistry, i.e. electrokinetic, or zeta, potentials and surface titration curves. Several models developed by Lyklema, Berube and De Bruyn, and Levine and Smith to explain these experimental results have been reviewed elsewhere [7-9],... 

Table V shows that the vast majority of the titratable groups of the smaller protein molecules have pK nt values which are quite close to the values predicted from the pK s of model compounds. This feature of protein titration curves has been well known for a long time, and is accepted as normal. It is however really an astonishing result, for it implies that most of the titratable groups of the smaller protein molecules are in as intimate contact with the solvent as similar groups on smaller molecules, and that they are able to accept or release hydrogen ions in this location without requiring any modification of the protein conformation in the vicinity of the titratable group. Since most of the proteins examined have been globular proteins, tightly folded so as to exclude solvent from most of the interior portions, the titratable groups must be nearly always at the surface.

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