Pseudosteady-state approximation

Equation (6.89) is identical with Equation (6.72) derived under the pseudosteady-state approximation. The error caused by Higuchi s equation increases with decreasing values of A/Cs. 

Since the molar concentration of Li in the electrolyte is no necessarily small, neglecting the second term on the right side requires that we invoke the pseudosteady-state approximation for the dLjdt term. Since electrons and Li are consumed at the same rate, Eq. (27), combining Eqs. (28) and (29) yields... 

When the chemical reaction at the interface presents a negligible resistance to the progress of reaction compared with diffusion through the ash layer, the overall rate is controlled by the latter. In order to obtain the rate of diffusion (hence the overall rate), we make the pseudosteady state approximation [7,8] as far as diffusion is concerned the reaction interface may be considered stationary at any time due to the high density of a solid compared to that of a gas.t... 

The pseudosteady state approximation is appropriate for describing the concentration of the gaseous reactants within the pellet [40, 41]. 

The energy balance in the pellet, assuming the pseudosteady state approximation for heat conduction, is given as... 

From an assumed mechanism, after having verified the adequacy of the pseudosteady state approximation, we will seek the kinetic laws applicable to these pseudosteady state modes. The results can then be compared with those from the experiment. 

This type of pseudosteady-state approximation is common and underlies most mass transfer coefficients discussed later in this book. 

During each run, the products are analyzed to determine "pseudosteady state conditions" and the temperature is varied to obtain an approximate measure of activity changes. These measurements are made during a span of several hours. 

Consequently, while I jump into continuous reactors in Chapter 3, I have tried to cover essentially aU of conventional chemical kinetics in this book. I have tried to include aU the kinetics material in any of the chemical kinetics texts designed for undergraduates, but these are placed within and at the end of chapters throughout the book. The descriptions of reactions and kinetics in Chapter 2 do not assume any previous exposure to chemical kinetics. The simplification of complex reactions (pseudosteady-state and equilibrium step approximations) are covered in Chapter 4, as are theories of unimolecular and bimolecular reactions. I mention the need for statistical mechanics and quantum mechanics in interpreting reaction rates but do not go into state-to-state dynamics of reactions. The kinetics with catalysts (Chapter 7), solids (Chapter 9), combustion (Chapter 10), polymerization (Chapter 11), and reactions between phases (Chapter 12) are all given sufficient treatment that their rate expressions can be justified and used in the appropriate reactor mass balances. 

Several useful approximate analytical solutions to Eq. (4.8) were developed. A well-known example is Higuchi s equation, based on a pseudosteady-state approach7 ... 

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. 

In summary, conversion of biomonitoring data to exposure dose requires knowledge of chemical elimination rate and Vd and requires that conditions be approximately pseudosteady state. It may be useful to estimate dose from body burden however, this cannot be used to interpret an individual s biomonitoring result, because the elimination rate and Vd would not be known. Reasonable bounds on elimination rate and Vd could be used to calculate an upper end of daily dose that is still compatible with the biomonitoring results (for example, when both Vd and elimination rate are high). 

The Higuchi model is an approximate solution in that it assumes a pseudosteady state , in which the concentration profile from the dispersed drug front to the outer surface is linear. Paul and McSpadden [24] have shown that the correct expression can be written as ... 

Three methods which do not require solution of the nonlinear partial differential equation are presented for estimating extractor performance. The choice of method depends on the value of the dimensionless outlet solute concentration, oj. If (oj + 1)/oJ is close to 1, the reaction is effectively irreversible and the pseudosteady-state solution of the advancing front model satisfactorily predicts performance after normalization to include solute solubility in the globule. If (oj + 1)/oJ is not close to 1, the advancing front results will still apply, provided that the amount of solute extracted by reaction is small and membrane solubility controls. When oj is small enough so that (oj + 1) is close to 1, then the reversible reaction model can be reduced to a linear equation with an analytical solution. Otherwise, for oj values when neither (oj + 1) nor (oj + 1)/o is nearly 1, a reasonable first approximation is made by adjusting the actual concentration of internal reagent to an effective concentration which equals the amount consumed to reach equilibrium. 

To switch from the solutions obtained in the pure cases to the general pseudosteady state solution, it is enough to apply the law of slownesses since all the multiplying coefficients are equal (section 7.7). For that we can make the assumption that the energy of activation of the elementary step of expulsion of a building unit does not depend on the size of the clusters. Again, we conceive well that this approximation is justified only for rather large clusters, in which the chemical bond of the heart is identical to that of the infinite crystal, and thus if the... 

In deriving (25), we also set the transient concentration term on the left in (10) to zero, thus considering only steady-state (or pseudosteady-state) processes. One may still apply the approximation of the thin boundary layer, as stated by (25), to transient problems, allowing the concentration within the thin boundary layer to vary with time. Detailed discussion of this class of problems is, however, outside the scope of this review and can be found in publications focusing on transients (e.g.. Refs. [9-12]). 

Lastly, we can consider systems under steady-state (or pseudosteady-state) mass transport control with no flow (stagnant). Here, within the region of varying concentrations, where the approximation is applied (usually the boundary l er), (44) simplifies to... 

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