Concentration simplifying approximations

We shall now consider a simplifying approximation for the system A 2P. The reaction proceeds at a rate expressed in terms of S by Eq. (3-33). If the shift resulting from the concentration jump is small, the term AK l82 is negligible in comparison to (1 + 4 l [P](,)S. In that case, the solution is... 

Equations (8.1)-(8.13) can be solved to provide transient- or steady-state profiles of O2 and CH4 concentration, reaction rates and surface fluxes for any combination of the controlling variables 9q,0], v,k, a, Vm,Vq and Vr. Where, as is usual, one or more of the controlling variables may be further simplified, approximated or neglected, process-based simulation of CH4 emission becomes possible using a relatively limited set of input data. 

The original equation for salt effect in vapor-liquid equilibrium, proposed by Furter (7) and employed subsequently by Johnson and Furter (8), described the effect of salt concentration on equilibrium vapor composition under the condition of a fixed ratio of the two volatile components in the liquid phase. The equation, derived from the difference in effects of the salt on the chemical potentials of the two volatile components, with simplifying approximations reduces to the form... 

A kinetic scheme is most easily worked out for the pure polymerization.106 It is useful first to make certain simplifying approximations and definitions. We replace Scheme 6 by Scheme 8, where In is an initiator producing radicals R , M is the monomer, Mn is the growing polymer chain, and M —Mm is combination product and Mn( H) are disproportionation products. If all radicals R produced by the initiator were available to start chains, we could write, from the first two reactions, kinetic Equation 9.40 for rate of change of R-concentration. Because of cage recombination, only some fraction f of the... 

Note that our simplifying approximation, 0.10 — x 0.10, is valid because x is only 7.0 X 10-6 and the initial [HCN] is 0.10. It s important to check the validity of the simplifying approximation in every problem because x is not always negligible compared to the initial concentration of the acid. Worked Example 15.10 illustrates such a case. Step 7. The concentrations of the species involved in the principal reaction are the "big" concentrations. The species involved in the subsidiary reaction(s) are present in smaller concentrations that can be calculated from equilibrium equations for the subsidiary reaction(s) and the big concentrations already determined. In... 

The Bodenstein approximation recognises that, after a short initial period in the reaction, the rate of destruction of a low concentration intermediate approximates its rate of formation, with the approximation improving as the maximum concentration of intermediate decreases (see Chapters 3 and 4). Equating rates of formation and destruction of a non-accumulating intermediate allows its concentration to be written in terms of concentrations of observable species and rate constants for the elementary steps involved in its production and destruction. This simplifies the kinetic expressions for mechanisms involving them, and Scheme 9.3 shows the situation for sequential first-order reactions. The set of differential equations... 

FIGURE 5-8 Blood concentrations of rapidly cleared chemical to which there is frequent and nearly uniform exposure. Highlighted line ( — ) is mean blood concentration. Under these exposure conditions, biomarker concentration will be within a factor of 2 of mean after first several hours. Simplifying assumption of pseudosteady state (mean concentration is approximated by concentration found at any sampling time) may suffice for estimating exposure dose from blood concentration under these circumstances. 

The assumption of equilibrium sorption has been supported by the long anticipated residence times in an installed barrier (e.g., days for a barrier thickness of 1-2 m), and by batch kinetic data reported by Cantrell (1996) that indicate near-equilibrium is achieved on the order of one day. Similar assumptions have been applied to the analysis of GAC barriers (e.g., Schad and Gratwohl, 1998). Cantrell (1996) also observed linear isotherms for Sr concentrations below approximately 0.1 mg/L, although this result should be viewed as particular to the specific experimental conditions. Although these results lend support to the simplified modeling approach, more data are clearly needed to better evaluate the key assumption of linear equilibrium sorption. 

A plastic bar of infinite length (e.g. 1 m or longer) is uniformly colored with an initial color concentration of c0 (Fig. 7-9) except for a thin layer in the middle with thickness d = 2 1 (approximately 1 cm). As a simplified approximation it is assumed that the concentration of the color at the location x = 1 remains constant at cq. The boundary conditions are expressed mathematically as ... 

The original equation for the evaporation of a droplet as a function of time was first derived by James Maxwell in 1877. Although his derivation contains a number of simplifying approximations, Maxwell s equation gives reasonable results for fairly large droplets of pure substances. For this equation, it is assumed that the vapor pressure at the droplet temperature is equal to the partial pressure at the surface of the drop, that is, psurfoce = p,(Teurface) = ps. In terms of concentrations of molecules, this means that the vapor concentration at the surface just equals the concentration of saturated vapor, the saturation determined at the droplet temperature. This assumption is valid when the droplet size is not too small compared to the mean free path of the vapor molecules. 

We formulate the problem in general terms first, and then consider several simplifying approximations in the subsequent subsections. Hence we begin by assuming that the surfactant may be soluble in both of the bulk fluids, that it adsorbs to and from the bulk fluids following Langmuir kinetics, and that the interfacial tension is related to the interface concentration of surfactant by means of the Langmuir limit of (2-155),... 

Barrier plastics will be used as examples in this paper. Barrier and selectivity are two sides of the same coin. Both properties are determined by the same types of transport phenomena. [3] Similar techniques can, therefore, also be applied to study separation membranes. The same approximations are valid if the mixture flowing through the membrane is sufficiently dilute that (i) it does not significantly affect the structure and properties of the membrane, and (ii) the components of the mixture can be treated as independent penetrants. The same general approaches can also be applied to concentrated mixtures, but only provided that certain simplifying approximations are not made. 

If the Peclet number is small, Pe l, two simplifying approximations are possible (Dukhin, 1965, 1981). As it will be shown in the following, if Pe l the surface concentrations differ negligibly from their equilibrium values,... 

One simplifying approximation can be made for all non-zero concentrations of NaOH, the pH should be basic and we can omit from the charge balance (19.21). The above 7 equations can then be reduced to 1 non-linear equation in 1 unknown, which can be solved by a numerical technique such as Newton-Raphson iteration. Suitable numerical equation solvers are now available as software for personal computers. The range of solutions to these equations for different NaOH concentrations and temperatures is illustrated in Figure 19.1. At very high NaOH concentrations we would also have to consider the doubly deprotonated species H2Si04. ... 

We next make the simplifying approximation that x, the amount of acid that dissociates, is small compared with the initial concentration of acid, 0.20 — x — 0.20. Thus,... 

Figure 8(a) shows absorption spectra and refractive index spectra for P(MMA-co-GMA-PNCA) before and after the photochromic reaction. The dye concentration is approximately 2.4 mol/l assuming that the density of this polymer is 1.0 g/cm. The actual concentration could be higher. Refractive indices were measured at wavelengths where they showed normal dispersion. Therefore, we simplified eq. (3) to the form which does not depend on (oi as shown below. 

We use the hydrolysis of A into P and Q as an illustration. Examples are the hydrolysis of benzylpenicillin (pen G) or the enantioselective hydrolysis of L-acetyl amino acids in a DL-mixture, which yields an enantiomerically pure L-amino acid as well as the unhydrolysed D-acetyl amino acid. In concentrated solutions these hydrolysis reactions are incomplete due to the reaction equilibrium. It is evident that for an accurate analysis of weak electrolyte systems, the association-dissociation reactions and the related phase behaviour of the reacting species must be accounted for precisely in the model [42,43]. We have simplified this example to neutral species A, P and Q. The distribution coefficients are Kq = 0.5 and Kp = K = 2. The equilibrium constant for the reaction K =XpXQ/Xj = 0.01, where X is a measure for concentration (mass or mole fractions) compatible with the partition coefficients. The mole fraction of A in the feed (z ) was 0.1, which corresponds to a very high aqueous feed concentration of approximately 5 M. We have simulated the hydrolysis conversion in the fractionating reactor with 50-100 equilibrium stages. A further increase in the number of stages did not improve the conversion or selectivity to a significant extent. Depending on the initial estimate, the calculation requires typically less than five iterations. 

Our initial concentration was given only to two significant figures, so our simplifying approximation is reasonably valid. 

X 10 is much smaller than 5% of 0.20, the initial HCN concentration. Our simplifying approximation is therefore appropriate. We now calculate the pH of the solutkm ... 

Once you know the value of for an acid HA, you can calculate the equilibrium concentrations of species HA, A , and H30 for solutions of different molarities. The general method for doing this was discussed in Chapter 15. Here we illustrate the use of a simplifying approximation that can often be used for weak acids. In the next example, we will look at an amplification of the first question posed in the chapter opening. 

This quadratic equation can be solved without undue labor, but a simplifying approximation (Appendix 1.16) is often applicable. The acid is weak, meaning that only a smaii fraction of it ionizes. We thus expect thaty, representing the number of moles of acid that has ionized per liter, will be much less than the initial concentration of acid. If this approximation is valid, 0.20 —y can be replaced by 0.20 without serious error, and Equation (1) becomes... 

While we have demonstrated how quantities of interest, such as permeability, porosity, hydrocarbon viscosity, and pore pressure, can be uniquely obtained, at least from invasion depth data satisfying our equations for piston-like fluid displacement, the actual problem is far from solved even for the simple fluid dynamics model considered here. For one, the tacit assumption that invasion depths can be accurately inferred from resistivity readings is not entirely correct invasion radii are presently extrapolated from resistivity charts that usually assume concentric layered resistivities, which are at best simplified approximations. And second, since tool response and data interpretation introduce additional uncertainties, not to mention unknown three-dimensional geological effects in the wellbore, time lapse analysis is likely to remain an iterative, subjective, and qualitative process in the near future. With these disclaimers said and done, we now demonstrate via numerical examples how formation parameters might be determined from front radii in actual field runs. 

With the four equilibrium consfant expressions, a material balance equation and a charge balance equation, we have six equations involving six unknowns. In principle, this system of equations can be solved to find the six unknown concentrations, either by making appropriate simplifying approximations or by computerized calculation. 

The full concentration equation for the contaminant may be simplified in the same manner as the Navier-Stokes equations to derive a boundary-layer approximation for the concentration, namely,... 

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

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