Concentration linear independent

Due to the linear Independence of occupation indices Tji the concentration-dependent parameters can be determined from the concentration-independent ones. In the special case of interactions limited to the fourth order and nearest neighbors one finds for fcc-based alloys... 

If AW AW the process of finding a linear-mixture basis can be tedious. Fortunately, however, in practical applications Nm is usually not greater than 2 or 3, and thus it is rarely necessary to search for more than one or two combinations of linearly independent columns for each reference vector. In the rare cases where A m > 3, the linear mixtures are often easy to identify. For example, in a tubular reactor with multiple side-injection streams, the side streams might all have the same inlet concentrations so that c(2) = = c(iVin). The stationary flow calculation would then require only AW = 1 mixture-fraction components to describe mixing between inlet 1 and the Nm — I side streams. In summary, as illustrated in Fig. 5.7, a turbulent reacting flow for which a linear-mixture basis exists can be completely described in terms of a transformed composition vector ipm( defined by... 

Choosing the concentration of ADP as the dependent variable, the network is described by a set of four linearly independent differential equations. The link matrix L is... 

Fixed-size-window-EFA plots reveal the number of different species that coexist in the particular window. More precisely, it is the number of species with linearly independent concentration profiles. 

In this model, the rate of migration of each solute along with the mobile phase through the column is obtained on the assumptions of instantaneous equilibrium of solute distribution between the mobile and the stationary phases, with no axial mixing. Ihe distribution coefficient K is assumed to be independent of the concentration (linear isotherm), and is given by the following equation ... 

In addition, if each calibration sample contains only one analyte, then R contains already the spectra of the pure components and C K x K) is a diagonal matrix (i.e. non-zero values only in the main diagonal). Hence eqn (3.20) simply calculates S by dividing the spectrum of each pure calibration sample by its analyte concentration. To obtain a better estimate of S, the number of calibration samples is usually larger than the number of components, so eqn (3.19) is used. We still have two further requirements in order to use the previous equations. First, the relative amounts of constituents in at least K calibration samples must change from one sample to another. This means that, unlike the common practice in laboratories when the calibration standards are prepared, the dilutions of one concentrated calibration sample carmot be used alone. Second, in order to obtain the necessary number of linearly independent equations, the number of wavelengths must be equal to or larger than the number of constituents in the mixtures (7>A). Usually the entire spectrum is used. 

As can be observed from these curves, the rate of variation of linear and real diffusion layer thickness with time increases with k°, being maximum for A° > 0.1 cm s 1. which corresponds to the reversible case. From Fig. 3.1a, it can be seen that for reversible processes the surface concentration is independent of time in agreement with Eq. (2.20) (see also Fig. 2.1 in Sect. 2.2.1). However, for non-reversible processes (Fig. 3.1b and c), the time has an important effect on the surface concentration, such that csQ decreases as I increases, with this behavior being more marked for intermediate k° values (quasi-reversible processes). So, for k° = 10 3 cm s 1. the surface concentration decreases by 19 % from t = 0.1 to 0.4 s, whereas for k° = 10 4 cm s 1 it only varies 7 %. It is also worth noting that for the reversible case (Fig. 3.1a), the diffusion control (cf, > 0) has practically been reached at the selected potential. 

It is easy to demonstrate that in the reactive system with an arbitrary set of monomolecular (or reduced to monomolecular) reactions, the station ary state with respect to the intermediate concentration corresponds to the minimum in the value of functional (3.6) even under conditions that are far from equilibrium of the system. In other words, the functional 0( Ao( ) is, by definition, the Lyapunov function of this system. In fact, for a system that consists of monomolecular reactions in its stationary state, in respect to linearly independent (i.e., not related via mass balance with other intermediates) intermediate A , the following expression is valid ... 

Here the rate is linear in developer concentration but independent of developer potential, halide ion concentration and rate of scavenging of oxidized developer. Such cases have not been seen experimentally. 

The rate of drug release from microspheres dictates their therapeutic action. Release is governed by the molecular structure of the drug and polymer, the resistance of the polymer to degradation, and the surface area and porosity of the microspheres. " " Reservoir delivery systems extend the residence time of drug within the systemic circulation and were originally focused on zero-order dissolution kinetics. This mathematical expression describes a linear relationship between rate of appearance in plasma and time. Ideally, the plasma drug concentration is independent of time for most of the dissolution period and is optimally maintained in the therapeutic window. 

Figure 33-10 Dose-response curves. L/ne A illustrates the linear relationship between serum drug concentration and total daily dose of a drug that displays first-order kinetics typical of most drugs. Line B illustrates the dose-response relationship for a drug that displays capacity-limited kinetics because of a saturable enzyme or transport mechanism in this situation, serum concentration becomes independent of total daily dose, and the relationship of drug concentration to dose becomes nonlinear. (Modified from Pippenger CE. Practical pharmacokinetic appiications. Syvo Monitor, Son Jose Syva Co, January, i 979 1-4.)...

Under linear conditions, the velocity associated with a concentration is independent of the concentration, so shocks are neither stable nor unstable. In SMB chromatography, however, the cycle-average velocity of a given concentration on a diffuse profile depends on the concentration or, more exactly, on the position of this concentration at the beginning of a cyde. 

For a given reaction nemork with nt linearly independent reactions, any steady state that is achievable by any reactor-separator design with total reactor volume V is achievable by a design with not more than nt + I CSTRs, also of total, reactor volume V. Moreover the concentrations, temperatures and pressures in the CSTRs are arbitrarily close to those occurring in the reactors of the original design. 

Figure2 Ohnishi et al. (1985) and Chijimatsu et al. (2000)) and reactive-mass transport model (inside the box named Chemical in Figure 2). This is a system of governing equations composed of Equations (l)-(9), which couple heat flow, fluid flow, deformation, mass transport and geochemical reaction in terms of following primary variables temperature T, pressure head y/, displacement u total dissolved concentration of the n master species C< > and total dissolved and precipitated concentration of the n" master species T,. Here we set master species as the linear independent basis for geochemical reactions, and speciation in solution and dissolution/precipitation of minerals are calculated by a series of governing equations for geochemical reaction. Now we adopt equilibrium model for geochemical reaction (Parkhurst et al. (1980)), mainly because of reliability and abundance of thermodynamic data for geochemical reaction.

As in thermal kinetics, the changes in concentration are usually monitored in photokinetics by spectroscopic measurements. Under these conditions, a distinction between different mechanisms is not possible for many reactions. Just the number of spectroscopically linear independent steps of reaction can be determined (see Chapter 4). 

Assuming that the concentrations of the intermediate and of the final products equal zero at the beginning of the reaction (bo = Cq = do = eg = 0), the following three linear independent equations with mass-balance result ... 

There exist some other possibilities to reduce the number of linear independent reactions, degrees of advancement, or concentrations at conditions specifically obtained in kinetics. Further information is given in Section 2.3. 

This differential equation represents a set of linear independent concentrations. The matrix Kc becomes a regular sxs matrix. 

If either the Bodenstein hypothesis can be applied to a reaction system or one of the steps is an isomeric equilibrium, the number of linear independent steps of reaction is reduced. The concentration of an unstable intermediate is according to the Bodenstein hypothesis with z = 0 ... 

Based on the fundamental considerations in Sections 1.3 and 1.4, the principle of quantum yield was introduced in Section 2.1.2 to allow a treatment of photochemical reactions in a way comparable to thermal reactions. The difference from thermal reactions has been demonstrated by taking account of the photophysical steps in Sections 2.1.3.3 and 2.1.4.3. These have to be considered in detail to find out whether the partial photochemical quantum yield depends on the intensity of the irradiation source or even on concentrations. Furthermore the definitions derived in Chapter 2.1 and summarised in Table 2.2 are used. In particular the definitions for the degrees of advancement for partial steps in general x, for photophysical steps in special x-, and for linearly independent steps x of the reaction procedure have to be remembered. 

The amount of light absorbed is substituted according to eq. (1.36) and the local average according to eq. (3.32) is formed. The result is an explicit expression for the change of the degree of advancement x of the Arth partial reaction step with time. This relationship can be used to set up the differential equations for the concentrations a, of the reactants A, according to eq. (2.5). This procedure is discussed in Section 2.2.1 and its subsections. The results for photoreactions are compared to the thermal Jacobi matrices in Section 2.2.1.4. There the results for the two linear independent steps of a consecutive photoreaction... 

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