Mathematical Standpoint

The mechanism of the buffer effect can be studied from two points of view a chemical standpoint and a mathematical standpoint. We will examine both viewpoints here. 

In the limits that prevail when setting up Henderson-Hasselbach s relation (see above), it has been found that after the addition of C mol/L of a strong acid, the concentrations of the base A and of the acid HA (in the case of a buffer of the kind HA/A ) have become 

These results are issued from a comparison of Eqs. (6.1) and (6.3). C mol/L of base A have disappeared and C mol/L of acid HA have appeared, whereas C mol/L of strong acid have also disappeared. The result is that the complete reaction 

The C moles (per solution liter) ofhydroxonium ions added have been completely transformed into C moles of the weak acid HA. Briefly, due to the buffer effect, the addition of C mol/L of the strongest acid that can exist in water (H3O+) has been commuted to the addition of C mol/L of a weak acid that is very poorly dissociated. This is the chemical mechanism of the buffer effect. 

The weak pH change also originates in the mathematical properties of the function pH/C. The corresponding curve clearly shows the following property (see Fig. 6.2) between the asymptotes its slope is weak. This is not the case close to the asymptotes C = Cha = Ca. However, this last point is not of great significance since we must not calculate pH from Henderson-Hasselbach s equation, as it is not legitimate near the asymptotes. 

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From the mathematical standpoint, this model can be formulated as follows. Neglecting the intrinsic conductance, the condition of electrical neutrality of the semiconductor may be... 

In order to understand the nature and mechanisms of foam flow in the reservoir, some investigators have examined the generation of foam in glass bead packs (12). Porous micromodels have also been used to represent actual porous rock in which the flow behavior of bubble-films or lamellae have been observed (13,14). Furthermore, since foaming agents often exhibit pseudo-plastic behavior in a flow situation, the flow of non-Newtonian fluid in porous media has been examined from a mathematical standpoint. However, representation of such flow in mathematical models has been reported to be still inadequate (15). Theoretical approaches, with the goal of computing the mobility of foam in a porous medium modelled by a bead or sand pack, have been attempted as well (16,17). 

From a mathematical standpoint the various second-order reversible reactions are quite similar, so we will consider only the most general case—a mixed second-order reaction in which the initial system contains both reactant and product species. 

Most standard chemical engineering tests on kinetics [see those of Car-berry (50), Smith (57), Froment and Bischoff (19), and Hill (52)], omitting such considerations, proceed directly to comprehensive treatment of the subject of parameter estimation in heterogeneous catalysis in terms of rate equations based on LHHW models for simple overall reactions, as discussed earlier. The data used consist of overall reaction velocities obtained under varying conditions of temperature, pressure, and concentrations of reacting species. There seems to be no presentation of a systematic method for initial consideration of the possible mechanisms to be modeled. Details of the methodology for discrimination and parameter estimation among models chosen have been discussed by Bart (55) from a mathematical standpoint. 

Finally a few words would be perhaps not superfluous about the relation of our results to the quantum mechanics of wave fields. It might appear that all calculations based on p.m. in the latter theory are not justifiable from the mathematical standpoint. But it must be remarked that even the operators themselves used in this theory are quite singular and at present do not accept rigorous mathematical taeatment. In such a state of affairs it would be premature to discuss the validity of p,m. in particular. 

The area of reactor design has been widely studied, and there are many excellent textbooks that cover this subject. Most of the emphasis in these books is on steady-state operation. Dynamics are also considered, but mostly from the mathematical standpoint (openloop instability, multiple steady states, and bifurcation analysis). The subject of developing effective stable closedloop control systems for chemical reactors is treated only very lightly in these textbooks. The important practical issues involved in providing reactor control systems that achieve safe, economic, and consistent operation of these complex units are seldom understood by both students and practicing chemical engineers. 

Although column separation constraints usually consist of primary and derived variable specifications as discussed in Sections 4.2.1 and 4.2.2, in general, specifications could be any function of these variables. One could define the g functions of Equation 4.3 as, for instance, sums, differences, or ratios of component rates, recoveries, and so on. The only restrictions on the specification functions, at least from the mathematical standpoint, are that they be independent and feasible. 

The reciprocal lattice, which from a mathematical standpoint is equivalent to the dual space of the direct space, is very often used in X-ray diffraction, since it makes it possible to associate each family of planes with its normal. 

Note that from the mathematical standpoint the condition (6.178) implies that Vrp is parallel to the tangent to the surface S. Instead of examining the expression occurring in (6.178), the density Laplacian can be analysed, since substituting into the volume integral in (6.176) for i Vp, the condition (6.178) may be replaced by the requirement... 

From a mathematical standpoint, McLean notes that there is little danger of actually running into your parallel-world doppelganger, as the number of sub-Earths where you are not bom so vasdy outweighs the ones where you are bom, (Still, an infinite number of your equivalents exist on an Infinity World.) And, although the infinity effect means that some extremely bizarre sub-Earths will exist, some version of the central limit theorem (the statistical notion that repeated combinations of random observations tend to cluster around a single mean) will cause Earths to cluster around some average behavior. 

This is much easier to deal with from a mathematical standpoint The influence of reactants and products depends once again on the nature of the rate hmiting process. To be convinced of such a thing, we only have to compare the expressions of step 1 and step n in the hypothesis that these are limiting processes ... 

To reach a simple expression of Vm using /i and i2, it is necessary, from a mathematical standpoint, to suppose that the forward global reaction is distinctly advanced. We will therefore only use the first term of I2 and ii while expressing this condition ... 

From the mathematical standpoint, the main problem may well lie in defining the significance and dependability of the various expansion and function-fitting procedures which have been outlined in 4-6. 

From a mathematical standpoint, the types of problems we usually deal with in process design are LP, MILP, and most typically NLP and MINLP. The main characteristics of the different types of problems and solvers that can be used are briefly described in Chapter 10. 

From a mathematical standpoint the present proposition amounts to the demonstration that the thermodynamic temperature T is an integrating denominator for dg in a reversible change, i.e. dSs(dg/T) is an exact differential. The possibility of defining T depended, in its turn, on the use of Statement A, concerning the... 

From equation [32] it is clear that x = -a is not a collocation point and an inconsistency with Reynolds equation is thus avoided. This additional constraint may be interpreted as requiring that pressure gradients in the neighbourhood of the inlet point x = -a are small and it should be noted that the use of a Chebyshev series expansion necessitates that pressure gradient at an end point is either zero or infinite. The constraint is therefore appropriate from both physical and mathematical standpoints. 

For the three solutions presented in this chapter, can we assert that these are the right solutions From a mathematical standpoint, it could be argued that these are exact solutions to Navier-Stokes equations, since they verify them and also verify the boundary conditions. Nonetheless, we carmot assert that these are necessarily the ones the experimenter will observe, even when achieving the experiment with the highest care. 

Many authors, among whom the authors of the following chapters of this volume, have addressed the relevance of various aspects of topology in chemistry (see for example Ayers et al. [25]). From the abstract mathematical standpoint, however, a topology is defined within the framework of set theory given a set X, a topology T on X is a family of parts of X, called m open sets, i.e. a subset of P X) = 2, such that ... 

From a strict mathematical standpoint, Ca and Xa are not in a linear relationship since E and En Mi do change with xa. However, the linear relationship appears when the solution is sufficiently diluted. In this case, indeed,... 

These relationships are interesting and must be carefully considered. Clearly, it appears that the standard state is the one in which the solute is at the concentration 1 mol/L (or 1 mol/kg) and in which it is endowed with an ideal behavior. In this case, indeed, YcX Ym,i are equal to unity. It is rare that at I mol/L (or 1 mol/kg), a solution exhibits an ideal character. As a result, solute standard states are very often virtual ones. The preceding four relationships also show that the solute activity coefficient alone measures the chemical— potential change arising from interactions among the solute particles. Finally, they show that the logarithm arguments are dimensionless. This is necessary from a mathematical standpoint. 

It is difficult to study the titrations of mixtures easily any further because, from a mathematical standpoint, a sequential titration depends on several parameters. For a binary mixture, for example, the titration curve depends on the four parameters Ci, Kai, C2, and a2-... 

The five simultaneous Eqs. (16.2)-(16.6) constitute a system in five unknowns. It can be solved from a mathematical standpoint. When activities are assumed to be equal to the corresponding concentrations, we find Eq. (16.7) after reducing the system by substitution and elimination ... 

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