First-order absorption models solution

Even though the absorption rate constant (kf) defines the rate of absorption, its accurate determination is largely dependent on the adequacy of the plasma concentration-time data associated with the absorption phase of the drug. When a drug is administered orally, as a conventional (immediate-release) dosage form, or injected intramuscularly as an aqueous parenteral solution, the absorption and disposition kinetics can often be analysed in terms of a one-compartment pharmacokinetic model with apparent first-order absorption. The plasma concentration-time curve is described by the equation... 

A separate mass balance equation is written in the form of Section 10.6.2 for each compartment in the model. Thus a total of n mass balance equations must be written and solved for an n compartment model. The details of these equations and their solution are not provided in this chapter. However, it will be noted that absorption, distribution, and elimination rates are written in the same form as in the previous one- and two-compartment models. The absorption rate for instantaneous, zero-order, or first-order absorption is identical to the previous forms for one- and two-com-partment models. Distribution and elimination rates are written as first-order linear rate equations using micro rate constants. So the distribution rate from compartment 1 to compartment n is given by kj Aj, the distribution rate from compartment n back to compartment 1 equals k i A , and the elimination rate from any compartment is written k o A schematic diagram for the generalized n compartment model is illustrated in Figure 10.90. 

Emmert and Pigford (E2) have studied the reaction between carbon dioxide and aqueous solutions of monoethanolamine (MEA) and report that the reaction rate constant is 5400 liter/mole sec at 25°C. If it is assumed that MEA is present in excess, the reaction may be treated as pseudo first-order. This pseudo first-order reaction has been recently used by Johnson et al. (J4) to study the rate of absorption from single carbon dioxide bubbles under forced convection conditions, and the results were compared with their theoretical model. 

Most published data deals with model solutions to assess the major factors influencing betalain stability, among which pH and temperature are most frequently addressed. Until recently, total color loss was assessed by spectrophotometric monitoring of the decline at the wavelength of maximum absorption. To predict color fading over time, kinetic data were derived therefrom, most often obeying first-order decay principles. 

Suppose pure CO, (A) at 1 bar is absorbed into an aqueous solution of NaOH (B) at 20 C. Based on the data given below and the two-film model, how should the rate of absorption be characterized (instantaneous, fast pseudo-first-order, fast second-order), if cB = (a) 0.1 and (b) 6 mol L 1 ... 

Let us start with an example the Matlab function Data AB. m models the absorption spectra of a reacting solution as a function of time. They are stored as rows of the matrix Y. The reaction is a simple first order reaction A - B as introduced in Chapter 3.4.2, Rate Laws with Explicit Solutions. 

The enhancement factors are either obtained by fitting experimental results or are derived theoretically on the grounds of simplified model assumptions. They depend on reaction character (reversible or irreversible) and order, as well as on the assumptions of the particular mass transfer model chosen [19, 26]. For very simple cases, analytical solutions are obtained, for example, for a reaction of the first or pseudo-first order or for an instantaneous reaction of the first and second order. Frequently, the enhancement factors are expressed via Hatta-numbers [26, 28]. They can be used in combination with the HTU/NTU-method or with a more advanced mass transfer description method. However, it is generally not possible to derive the enhancement factors properly from binary experiments, and a theoretical description of reversible, parallel or consecutive reactions is based on rough simplifications. Thus, for many reactive absorption processes, this approach appears questionable. 

The two bubble class model is applied here to the absorption of CO2 in NaOH, which conforms to a fast pseudo-first order reaction under certain operating conditions (15). In the data reported by Schumpe et al. ( 7 ), COo was absorbed during cocurrent flow in NaOH solution in a 0.102 m diameter bubble column. The gas phase consisted of approximately 10 vol % of CO2 in N2. The gas velocities ranged from 0.025 to 0.15 m/s. Since the churn turbulent regime prevailed for gas velocities greater than approximately 0.07 m/s, only the data in the range 0.07 m/s to 0.15 m/s were considered. 

Thus the oxidation of aqueous Na2S03 solutions with C0SO4 as a catalyst proves to be a convenient model reaction for determining interfacial area in gas-liquid reactors. The kinetics of the reaction is not simple many variables influence the reaction rate but, provided the range of cobalt and sulfite concentrations, pH values, and temperatures previously indicated is satisfied, the reaction is zero-order in sulfite, first-order in cobalt, and second-order in O2. The specific rate of absorption is... 

Conceptual models of percutaneous absorption which are rigidly adherent to general solutions of Pick s equation are not always applicable to in vivo conditions, primarily because such models may not always be physiologically relevant. Linear kinetic models describing percutaneous absorption in terms of mathematical compartments that have approximate physical or anatomical correlates have been proposed. In these models, the various relevant events, including cutaneous metabolism, considered to be important in the overall process of skin absorption are characterized by first-order rate constants. The rate constants associated with diffusional events in the skin are assumed to be proportional to mass transfer parameters. Constants associated with the systemic distribution and elimination processes are estimated from pharmacokinetic parameters derived from plasma concentration-time profiles obtained following intravenous administration of the penetrant. 

The effect of humic materials on the photolytic micellar system was evaluated in DR s photodegradation. DR solubilized within Tween 80 micellar solution with or without humic materials was determined. In order to calculate the quantum yield, the molar absorptivity of DR was determined by spectrophotometry. The determination of the quantum yield and reaction rates was examined through a pseudo first-order decay rate expression. Quenching and catalytic effects resulting from the humic substances were examined through Stem-Volmer analysis. A reaction mechanism of photolytic decay of DR solubilized within surfactant micelles in the presence of various amount of humic materials was proposed for this purpose. The effect of high and low concentration of humic materials has been accounted for by a designed model. 

Case 3 Reaction Occurring Within fluid Film. As a third example consider die silantion when species A disappears by homogeneous reaction in Ihe fluid film. Such a model has been used to predict die effect of chemical reaction on gas absorption rales (Chapter 6) or on canier-faciliimed transport in membranes (Chapter 19). For simplicity, assume die reection to be first order and irreversible and the solution to be dilute so that bulk flow transport is negligible and Ihe total molar concentration constant. The standy-state belsuce for A is obtained from simplification of Eq, (2.3-14) ... 

We present an analysis for absorption and reaction of a pulse of reactant gas moving along a column containing a stationary liquid phase. For first order homogeneous reaction in the liquid film, measured moments of the effluent curve can be used to evaluate rate constants for gas-liquid reactions. This model has been applied to experimental data obtained for the absorption and reaction of carbon dioxide in aqueous Na GO - NaHCO, solutions. ... 

A graphical representation of these equations is given in Fig. 6.4-14. van Kievelen and Hoftijzer originally developed their correlation only for irreversible second-order reactions (first-order in each reactant) and for equal difiusivities of the two reactants. Danckwetts pointed out that the results also are applicable to the case where is not equal to Dg. Decoursey developed an approximate solution for absorption with irreversible second-order reaction based on the Danckwerts surface-renewal model. The resulting expression, which is somewhat easier to use than the van Krevelen-HofUJzer approach, is... 

On the basis of the analytical solutions, i.e. eq. (12) incorporating the appropriate expressions for Q, the influence of gas phase mixing on reactor performance can be studied. For a first order reaction it can be shown that the ratio of absorption rates and conversions, respectively, of the mixed (m) and unmixed (p) model is given by... 

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