Fractional extent and rate

Take F as the component (element of stmcture) which leaves the solid it can be either a main constituent in the case of the variation of stoichiometry or a foreign element in the case of the variation of composition of a solid solution. The reaction can be written as  

If the gas G does not exist, G is either the product of a decomposition of F (in the cases of complex elements of stractures) or the vapor of F. The reaction 

Take o as the initial F concentration in the sohd and a as its average concentration (in the btrlk) at arty time t. As the voltrme of the solid remains constant, the fractional extent will be defined by  

If F is a main component of the solid and if the initial state is nearly the stoichiometric composition, thenoo= 1. 

At the end of the experiment the fractional extent does not reach 1 because F concentration is given by the equilibriitm condition of the preceding reaction and we 

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We saw that we can define as many fractional extents and rates as there are components in the total reactioa... 

And thus the fractional extent and rate take the forms ... 

Drugs may be formulated as their salt forms (i.e., hydrochloride salt for base, sodium salts for acids) that dissociate in the body, or they may be formulated as the free acid or base. The fraction of the drug absorbed can be difficult to predict, as it is influenced by many factors. The extent and rate of absorption are partly determined by the physicochemical properties of the drug. Favorable absorption is related to lipid solubility, nonpolarity, and small molecular size. Reduced absorption is often observed for highly polar, non-lipid-soluble, and large-molecular size drugs. 

If the liquid medium is chemically inert but is a solvent for the polymers, its presence can reduce markedly the polymer viscosity. Low molecular weight fractions in the polymer can cause a related effect, acting as a plasticizer [34]. Consequently, chain mobility is increased and the actual mechanical forces applied to molecules are reduced [35,36]. Thus, the nature, amount, and distribution of solvent in the polymer can influence the extent and rate of reaction (Fig. 3.34). 

There is no experimental method that makes it possible to directly measure the extent, the fractional extent, and the speed or the rate of a system while it evolves. It... 

We have already seen the direct measurement of the composition of a solid phase by X-ray diffraction. This measiuement can be used to quantify the disappearance or the appearance of sohd phases, that is, the fractional extent and the rates relative to A or B (see sections 11.2.4 and 11.2.5). 

Important remark.- In all the cases (see tables of Appendix 3) of this kind of relationship between the rate and the fractional extent, we note that the space function depends only on the fractional extent and thus does not depend on the reactivity of growth, whatever is the past of the sample, and the space function is completely determined by the fractional extent. The past of the sample has no influence on speed. Consequently, the variations of the rate with an intensive variable will give (with a coefficient which will be the space function) those of the rate of growth on the condition of studying these variations with constant space function, that is, with constant fractional extent. 

Remark - The space function is completely determined by the fractional extent, and thus, at a given time, the rate is a function only of the physicochemical state (pressures, temperature) and of the fractional extent at this time. 

We are able to calculate at each fractional extent the rate, adopting the good value of (f>, on the condition of knowing the variations of with time and the variations of a with time (these last data ate the direct result of the experiment). 

As an example, we will consider again the case of temperature break (Figure 10.12) and we note now that for a given fractional extent, the rate depends on the former values of the reactivity of growth and the frequency of nucleation, that is, depends on the past of the sample. 

In the cases of two-process models, it is initially advisable to determine the relationship between real time t and dimensionless time 9. For this, we determine the correspondence between t and the fractional extent on the ejqierimental eurve, on the one hand, and the correspondence between the fractional extent and dimensionless time given by the model, on the other hand. We also plot dimensionless time versus real time (for the various fractional extents). In accordance with relation [10.2], we must obtain a line whose slope allows us to calculate the reactivity of growth (/o = 1 for diffusion as the rate determining step and xb for interface reaction as the rate determining step) as follows ... 

Thns, the fractional extent and the rate are, respectively, proportional to the mass change and its derivative with respect to time. 

Remark.- It is important to note that e varies with the thermodynamic condihons of pressures and temperature it means that the smdy of the influence of these variables on the reaction rate cannot be made by a simple comparison of the curves of mass change but it is imperahve to remrn in each case to the fractional extent and the rate. 

Establish the expressions of the fractional extent and the rate according to time (from the starting time to) the molar volume of the solid A, the grain sizes, and the leachvity of growth. 

As we are in pseudo-steady state modes and that extent is enough to define the rate, this means that we can do with a one-process model of instantaneous nucleation and slow growth or slow nucleation and instantaneous growth. The observation of the metal-sulfide interface after partial sufturization shows that we probably have an instantaneous nucleation and slow growth. The increase in mass is thus proportional to the fractional extent and the experiment gives the rate of growth, which is separable, therefore,... 

This appendix gives the functions used for the calculation of the fractional extent and the dimensionless reactance as functions of dimensionless time, in the case of two-process models with isotropic growth with inward development and for internal reaction as rate-determining step for various grain shapes. 

These tables are drawn up for two shapes of powder samples that have an internal interface reaction as a rate-determining step and inward development. They give, for various values of the model parameter, the remarkable properties of the kinetic curves fractional extents and dimensionless times at the points of iirflection (if it exists) and the dimensionless time at the fractional extent 0.5 in two-process models with isotropic growth. 

The experiment aims to derive a number of kinetic properties, such as the extent, fractional extent and speed (speed, specific speed or rate), which are not directly measurable. The results are grouped into two families ... 

Having identified the reaction mechanism and stoichiometric coefficients, the experimental values (measured rate, measured fractional extent) can be used to determine the kinetic values, including the fractional extent and the rate (or reactivity). To do this, it is crucial for a reaction to undergo the pseudo-steady state test. Indeed, it is only when this test has been satisfied that the reaction will be defined by the reaction rate only, whatever the component or measurement analyzed. If this test is not satisfied, we will have to remember that each measurement is defined by its own fractional extent or a combination of several fractional extents. In the search for a kinetic mode, several methods are used. These will now be reviewed. 

The rate (mass/time) and fractional extent of drug absorption (mass/dose = fa) from the intestinal lumen in vivo at any time t is shown schematically in Eq. (2) [4] ... 

Too often results are compromised by a poor experimental set-up of the studies and nontransparent data. Even essential information such as the relevant physicochemical characteristics of the drug in relation to the chosen aerosol system or the fraction that is deposited in the alveoli is often not provided. This makes it impossible to evaluate the impact of such studies. As a result, it is unclear until now to what extent and at what rate macromolecular drugs (> 20 kDa) can be absorbed by the lung. Moreover, the routes by which macromolecules pass through the different pulmonary membranes, especially the alveolar membrane, are unknown. Appropriate experiments and models that provide adequate answers to these questions are required in the coming years. 

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