Quantitative. . . Relationship

To understand how to derive a quantitative relationship between properly and structure... 

Two approaches to quantify/fQ, i.e., to establish a quantitative relationship between the structural features of a compoimd and its properties, are described in this section quantitative structure-property relationships (QSPR) and linear free energy relationships (LFER) cf. Section 3.4.2.2). The LFER approach is important for historical reasons because it contributed the first attempt to predict the property of a compound from an analysis of its structure. LFERs can be established only for congeneric series of compounds, i.e., sets of compounds that share the same skeleton and only have variations in the substituents attached to this skeleton. As examples of a QSPR approach, currently available methods for the prediction of the octanol/water partition coefficient, log P, and of aqueous solubility, log S, of organic compoimds are described in Section 10.1.4 and Section 10.15, respectively. 

A balanced chemical reaction indicates the quantitative relationships between the moles of reactants and products. These stoichiometric relationships provide the basis for many analytical calculations. Consider, for example, the problem of determining the amount of oxalic acid, H2C2O4, in rhubarb. One method for this analysis uses the following reaction in which we oxidize oxalic acid to CO2. 

Quantitative Calculations In acid-base titrimetry the quantitative relationship between the analyte and the titrant is determined by the stoichiometry of the relevant reactions. As outlined in Section 2C, stoichiometric calculations may be simplified by focusing on appropriate conservation principles. In an acid-base reaction the number of protons transferred between the acid and base is conserved thus... 

To develop a more quantitative relationship between particle size and T j, suppose we consider the melting behavior of the cylindrical crystal sketched in Fig. 4.4. Of particular interest in this model is the role played by surface effects. The illustration is used to define a model and should not be taken too literally, especially with respect to the following points ... 

To develop this model into a quantitative relationship between T j, and the thickness of the crystal, we begin by realizing that for the transition crystal liquid, AG is the sum of two contributions. One of these is AG , which applies to the case of a crystal of infinite (superscript °o) size the other AG arises specifically from surface (superscript s) effects which reflect the finite size of the crystal ... 

What we seek next is a quantitative relationship between the extent of the polymerization reaction, the composition of the monomer mixture, and the point of gelation. We shall base our discussion on the system described by reaction (5.U) other cases are derived by similar methods. To further specify the system we assume that A groups limit the reaction and that B groups are present in excess. Two parameters are necessary to characterize the reaction mixture ... 

Melting and recrystallization behavior of virgin PTEE has been studied by dsc (64). A quantitative relationship was found between and the heat of crystallization (A/T) in the molecular weight range of 5.2 x 10 to 4.5 X 1 0, where is heat of crystallization in J/g, which is independent of cooling... 

The quantitative relationship between the degree of adsorption at a solution iaterface (7), G—L or L—L, and the lowering of the free-surface energy can be deduced by usiag an approximate form of the Gibbs adsorption isotherm (eq. 9), which is appHcable to dilute biaary solutions where the activity coefficient is unity and the radius of curvature of the surface is not too great ... 

Dose—response evaluation is used in describing the quantitative relationship between the amount of exposure to a substance and the extent of toxic injury or disease. Data may be derived from animal studies or from studies in exposed human populations. Dose—response toxicity relationship for a substance varies under different exposure conditions. The risk of a substance can not be ascertained with any degree of confidence unless... 

Quantitative Relationship of Conductivity and Antistatic Action. Assuming that an antistatic finish forms a continuous layer, the conductance it contributes to the fiber is proportional to the volume or weight and specific conductance of the finish. As long as the assumption of continuity is fulfilled it does not matter whether the finish surrounds fine or coarse fibers. Assuming a cylindrical filament of length 1 cm and radius r, denoting the thickness of the finish layer as Ar and the specific conductance of the finish k, the conductance R of the finish layer is given by the equation (84) ... 

Mote quantitative relationships of the CSD obtained from batch operations can be developed through formulation of a population balance. Using a population density defined in terms of the total crystallizer volume rather than on a specific basis (n = nU), the general population balance given by equation 42 can be modified in recognition of there being no feed or product streams ... 

Many different approaches to QSAR have been developed since Hansch s seminal work. These include both two-dimensional (2D) and 3D QSAR methods. The major differences among these methods can be analyzed from two viewpoints (1) the strucmral parameters that are used to characterize molecular identities and (2) the mathematical procedure that is employed to obtain the quantitative relationship between a biological activity and the structural parameters. 

The goal of a kinetic study is to establish the quantitative relationship between the concentration of reactants and catalysts and the rate of the reaction. Typically, such a study involves rate measurements at enough different concentrations of each reactant so that the kinetic order with respect to each reactant can be assessed. A complete investigation allows the reaction to be described by a rate law, which is an algebraic expression containing one or more rate constants as well as the concentrations of all reactants that are involved in the rate-determining step and steps prior to the rate-determining step. Each concentration has an exponent, which is the order of the reaction with respect to that component. The overall kinetic order of the reaction is the sum of all the exponents in the... 

Dispersion equations, typically the van Deemter equation (2), have been often applied to the TLC plate. Qualitatively, this use of dispersion equations derived for GC and LC can be useful, but any quantitative relationship between such equations and the actual thin layer plate are likely to be fraught with en or. In general, there will be the three similar dispersion terms representing the main sources of spot dispersion, namely, multipath dispersion, longitudinal diffusion and dispersion due to resistance to mass transfer between the two phases. 

Assuming the work of adhesion to be measurable, one must next ask if it can be related to practical adhesion. If so, it may be a useful predictor of adhesion. The prospect at first looks bleak. The perfect disjoining of phases contemplated by Eq. 1 almost never occurs, and it takes no account of the existence of an interphase , as discussed earlier. Nonetheless, modeling the complex real interphase as a true mathematical interface has led to quantitative relationships between mechanical quantities and the work of adhesion. For example, Cox [22] suggested a linear relationship between Wa and the interfacial shear strength, r, in a fiber-matrix composite as follows ... 

Dose-Response Cune A graphical representation of the quantitative relationship between the administered, applied, or internal dose of a chemical or agent, and a specific biological response to that chemical or agent. 

Of all the heteroaromatic compounds that have been examined qualitatively and quantitatively for covalent hydration, the pteridines constitute the largest series. Most of the quantitative relationships which were used in earlier discussions were first derived for the hydroxypteridines. Also most of the known examples of hydration in anions were found in this series. 

Composition of the liquid environment The ionic composition, arising from dissolved salts and gases, has a considerable influence on the performance of inhibitors. In near-neutral aqueous systems the presence of certain ions tends to oppose the action of inhibitors. Chlorides and sulphates are the most common examples of these aggressive ions, but other ions, e.g. halides, sulphides, nitrates, etc. exert similar effects. The concentration of inhibitor required for protection will depend on the concentrations of these aggressive ions. Laboratory tests " have given some quantitative relationships... 

Although there is no simple quantitative relationship between the stability of a carbocation intermediate and the rate of its formation, there is an intuitive relationship. It s generally true when comparing two similar reactions that the more stable intermediate forms faster than the less stable one. The situation is shown graphically in Figure 6.13, where the reaction energy profile in part (a) represents the typical situation rather than the profile in part (b). That is, the curves for two similar reactions don t cross one another. 

One of the most important parameters that defines the structure and stability of inorganic crystals is their stoichiometry - the quantitative relationship between the anions and the cations [134]. Oxygen and fluorine ions, O2 and F, have very similar ionic radii of 1.36 and 1.33 A, respectively. The steric similarity enables isomorphic substitution of oxygen and fluorine ions in the anionic sub-lattice as well as the combination of complex fluoride, oxyfluoride and some oxide compounds in the same system. On the other hand, tantalum or niobium, which are the central atoms in the fluoride and oxyfluoride complexes, have identical ionic radii equal to 0.66 A. Several other cations of transition metals are also sterically similar or even identical to tantalum and niobium, which allows for certain isomorphic substitutions in the cation sublattice. 

Chemistry is a quantitative science. This means that a chemist wishes to know more than the qualitative fact that a reaction occurs. He must answer questions beginning How much. . . The quantities may be expressed in grams, volumes, concentrations, percentage composition, or a host of other practical units. Ultimately, however, the understanding of chemistry requires that amounts be related quantitatively to balanced chemical reactions. The study of the quantitative relationships implied by a chemical reaction is called stoichiometry. 

We have all of this familiar experience to build upon, but it is all qualitative. We need a quantitative relationship. How much energy is carried by light The answer is simple in form, but not in concept. Light, too, comes in packages. Each package, called a photon, contains an amount of energy determined by the frequency. This statement is contained in the famous equation... 

A. Precipitation of sulphides. In order to understand fully the separations dependent upon the sulphide ion, we shall consider first the quantitative relationships involved in a saturated solution of hydrogen sulphide. The following equilibria are present ... 

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