Concentration-time diagram

In this section the properties of linear systems relevant to kinetic analysis are discussed. Using this approach information can be extracted from the characteristics of diverse graphical representations. By looking for extrema and points of inflection in the concentration-time diagrams one can draw conclusions with respect to the mechanism - a prerequisite to determine the number of exponential functions of the overall rate law. 

If all eigenvalues are different, the following equations are valid in the concentration-time diagrams for extrema... 

If at least one of the s parameters differs from zero, neither extrema nor points of inflection are found in concentration-time diagrams. For example, assuming the values p, and p,2 are different from zero, eqs. (2.64) and (2.65) can be solved with respect to t. As soon as the concentrations either reach a minimum or a maximum (index m) one obtains... 

In consequence, one finds, for any component Ai for which exactly two of the parameters differ from zero as the distance between the extrema and the point of inflection within a concentration-time diagram in general. 

Further information can be obtained, if instead of concentration-time diagrams, other types of graphical representations are used ... 

In many cases both these representations characterise a system in a better way than concentration-time diagrams. A special advantage of these diagrams is their limited length in their axes. For example, it represents the total advancement of the reaction in contrast to the time axis of concentrationtime diagrams which in principle extends to infinity. 

These points of inflection ate typical for the K-diagram. They are not found in concentration-time diagrams. 

The related p-values can be determined by use of Table 2.6. The concentration-time diagrams are given in Fig. 2.10. They have been calculated with the program presented in Section 2.2.3. 

Fig. 5.72. Concentration-time diagram for the photoreaction of the Z-isomcr in the polymer...

In the same manner that the target constants and Vkt are calculable from just one concentration-time diagram by differentiating the values obtained as outlined above, it is also feasible to utilize the integrated Michaelis-Menten equation with the time-dependent values directly. Eqn (4.8) and its detailed integration from eqn (4.1) can be found in the literature... 

Figure 5.48. Multisubstrate kinetics for a two-substrate reaction (a) Concentration/ time diagram for strictly sequential substrate utilization (diauxic growth), (b) Partly overlapping and partly sequential substrate use. (c) Simultaneous substrate utilization. Sum-type kinetics are often applied in (b) and (c).

Figure 5.81. Schematic representation of a bioprocess in a multicomponent foodprocessing system in a normalized concentration/time diagram. The kinetic rate constant, k, of reactions 1-7 is (A. Moser, et al. 1980b) ...

Figs 5.4-34 to 5.4-37 show results of the measurements and calculations. In Figs 5.4-34 and 5.4-35 the results of temperature and heat flow measurements are shown. Isothermal operation was quite easy to reach due to the relatively low heat of reaction and the high value of the product of the heat-transfer coefficient and the heat-exchange surface area Art/ in relation to the volume of the reaction mixture. Peaks in the heat flow-versus-time diagram correspond to the times at which isothermal operation at the next temperature level started. After each peaks the heat flow decreased because of the decrease in the concentrations of the reactants. 

However, by examining the adsorption behavior of polypeptides and proteins with comparable porous and nonporous particles in finite baths, packed columns and expanded or fluidized beds, an iterative simulation approach based on the heuristic principles described earlier and along the lines of the flow diagram shown in Fig. 32 can be developed, leading ultimately to the implementation of useful scale-up criteria. Along the way, computer simulations, generated from the analysis of the concentration-time... 

Here, rheology is used to characterize the gel state, whose stability, as measured thermodynamically or kinetically, can be described by temperature-concentration phase diagrams or simply time. The structural features of gelator aggregates at nanoscopic scales are described via data from the complementary techniques of electron microscopy and scattering techniques. Finally, the optical properties, including absorption and luminescence, are detailed. 

Figure 6-6 Temperature versus Concentration State Diagram Illustrating the Sol-Gel Boundao for BSA in which Threshold Times for Gelation are shown circles, 100 s A triangles up, 1,000 s triangles down, 10,000 s. Horizontal broken line is boiling point of solvent. Curves shown are based on Equation 6.26 (Tobitani and Ross-Murphy, 1997).

Figure 3. Arylation of n-butyl acrylate with 4-bromoanisole concentration vs. time diagram at r= 140 °C, Pd(OAc)2/4 Co(Pd) = 2 mol % (taken from Ref. [50, 51]).

Even in an ideal coltunn, the reorganization of the distribution of the component concentrations between the injection and the formation of the isotachic train requires a certain time, i.e., it cannot be achieved in less than a minimum migration distance. This distance can be derived from the distance-time diagram. 

Distance Time diagram In displacement chromatography, diagram indicating the trajectory of the concentration shocks and the regions where diffuse boimd-aries appear. 

We have carried out similar computations covering the entire duration of three similar experiments that were carried out by Duncan and Toor (1962). The results of these calculations are shown in the triangular diagram, Figure 5.5, along with the data of Duncan (1960). We see that for all three experiments theoretical profiles are in good agreement with the data. This experiment (and others like it) provides support for the theoretical considerations of earlier chapters and the successful prediction of the concentration time history in the two bulb diffusion cell is a valuable test of the linearized theory of multicomponent diffusion. ... 

Figure 6.1 shows the concentration time history in the diffusion cell for the experiment of Duncan and Toor that was described in detail in Example 5.4.1. The mole fraction of hydrogen predicted by the effective diffusivity model compares well with the experimental data of Duncan (1960). However, the effective diffusivity model suggests that the mole fraction of nitrogen should remain almost constant at approximately 0.5. This is in stark contrast to the experimental data (Fig. 6.1). The results obtained with the effective diffusivity method for nitrogen are completely different from those obtained with the linearized theory. Additional comparisons between the data of Duncan and Toor and the predictions of both the linearized equations and the effective diffusivity models are shown in the triangular diagram in Figure 6.2.

Figure 3. Concentrations vs time diagram of the MNP hydrogenation, on Pd/C 5% (Johnson Matthey 487),...

The conversion vs. time diagram (Figure 1) illustrates the dependence of the reaction rate on the chain length of the alkene. The reaction proceeds according to first-order kinetics, i.e., the consumption rate of the substrate alkene is proportional to the concentration of the substrate. 

In Figures 10.1(d) and (h), we show the distribution of solutes along the whole column (cells 1 10) at a specific time, here after 40 shifts. These diagrams are called concentration profile diagrams. 

The computer is programmed according to the timing diagram in Fig. 2.12b. Once a calibration graph has been run for each analyte, the determination of the respective concentrations requires programming ... 

For differential methods, moderate mathematical efforts are required. However, the accuracy in calculations of the rate is low since numerical or graphical differentiation is used to determine the rate values from concentration vs time diagrams (Figure 10.10). 

Fig. S.17. Absorbance-time diagram of the photoreaction of 2-chloro-anthraquinone in neutral methanolic solution, irradiated at 313 nm with /q= 1.26 x KT Einstein cm s", initial product concentration oq = 8.24 x 10 mol T. ...

Figure 4, Ce(IV) concentration vs. time diagrams in a BZ system calculated with the MBM mechanism. A) Unperturbed system. Initial concentrations [MA] = 0.1 moldm [KBrOs] = 0.03 moldm, [Ce(lV)J = 410 moldm [H2SO4] = / mohdm (See Fig. lA for comparison). B) The same system as in A) but perturbed with 0.015 mol dm polymer (regarding backbone reactions only). kp2= 400 dm mot s It is assumed that the rate determining step of process (P3)+(P4) leading to HOBr is a diffusion controlled radical-radical recombination reaction (kp3=l(f dm mol s ). C) The same system as in A) but perturbed with the alcoholic endgroup reaction exclusively. Bromous acid inflow = 6 10 dm mor s due to (P5) was regarded to be independent of time. D) Combination of the two perturbations shown in B) and C).

Detailed investigation of observation and measurement methods [23, 78-80] has shown that the standard approach to Tj = Tj (P, T) measurements using pressure-time, concentration-time, impulse-time / = I(t) diagrams might result in potential errors. 

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