Quantitative relations

Chemists learn to use the thermodynamic probability almost instinctively in a qualitative manner it is quantitatively related to entropy through an equation due to Boltzmann ... 

SH Bryant, CE Lawrence. The frequency of lon-pair substructures m proteins is quantitatively related to electrostatic potential A statistical model for nonbonded interactions. Proteins 9 108-119, 1991. 

The bulk properties of mixed solvents, especially of binary solvent mixtures of water and organic solvents, are often needed. Many dielectric constant measurements have been made on such binary mixtures. The surface tension of aqueous binary mixtures can be quantitatively related to composition. ... 

The relative electron-donating ability of rings can be estimated by the change in the infrared stretching frequencies of attached amino, carbonyl, cyano, or methoxy groups, but the quantitative relation to chemical reactivity does not appear to be defined at... 

The first generalization is illustrated by the behavior of the 2- and 4-vs. the 3-derivatives of pyridine, the second by the reactivity of 4- vs. 2-substituted pyridines, the third by the relation of 4- vs. 2-derivatives of pyrimidine, and the fourth by the appreciable reactivity of 3-substituted pyridines or 5-substituted pyrimidines compared to that of their benzene analogs. Various combinations of azine-nitrogens in other poly-azines supply further examples. Theoretical aspects of (1), (2) and (3) are discussed in Section II, B, 2. The effect involved in (4) is believed to be more the result of the inductive stabilization of an adjacent negative chaise in the transition state (cf. 251) than of the electron deficiency created in the ground state (cf. 252). The quantitative relation between inductive stabihzation and resonance stabilization is not precisely defined by available data. However, a... 

Mengen-bestimmung, /. quantitative determination. -lehre, /. Math.) theory of sets, mengenm sig, a. quantitative. Mengen-messer, m. volume meter, volumeter, -regelung, /. quantity control, -untersu-chtmg, /. quantitative examination, -ver-h tnis, n. quantitative relation, proportion, composition. 

An overview of the superplastic behavior of aluminum alloys to demonstrate the grain-size effect is depicted in Fig. 1, in which the quantitative relation between the logarithm of the optimum strain rate for superplastic flow and the grain size (plotted as the logarithm of reciprocal grain size) is clearly shown [4]. The slope of the curve in Fig. 1 is noted to be about 3. 

To obtain a quantitative relation between cell voltage and concentration, it is convenient to start with the general expression for the free energy change discussed in Chapter 17 ... 

Rate of aging process It has been known for a long time as an empirical fact that many reactions approximately double or treble their rates with a 10°C rise in temperature. A more quantitative relation is given by the classical Arrhenius modified equation ... 

The quantitative relations have been tested by Braun, by Stackelberg (1896), and by Cohen, Inouye, and Euwen (1911). 

In 163—167 we have deduced some properties of systems of two components in two phases ( binary systems V) directly from the fundamental principles, and in 169—173 we have obtained quantitative relations in certain special cases. Here we shall j obtain some general equations relating to such systems with the i help of the thermodynamic potential (cf. 155)., ... 

This generalisation is due to Gibbs (1874), who at the same time showed how quantitative relations could be found. 

Cases in which a cell cools or warms itself in action had been investigated by Braun (1878—1888), and the quantitative relation was verified in a number of cases by Jahn (1886), who measured the latent heats by placing the cell in an ice calorimeter. 

For the region near the attachment point, Mullis found a strong effect of axial position on flux, but no satisfactory general correlation for this effect. In addition, he found no quantitative relation for the heat-transfer characteristics of jets directed toward the propellant surface. Under most conditions studied by Mullis, the radiation contribution is approximately 10% of the convective flux. The effects of solid-particle impingement were not investigated. 

Because the mechanisms governing mass transfer are similar to those involved in both heat transfer by conduction and convection and in momentum transfer (fluid flow), quantitative relations exist between the three processes, and these are discussed in Chapter 12. There is generally more published information available on heat transfer than on mass transfer, and these relationships often therefore provide a useful means of estimating mass transfer coefficients. 

There are strict limitations to the application of the analogy between momentum transfer on the one hand, and heat and mass transfer on the other. Firstly, it must be borne in mind that momentum is a vector quantity, whereas heat and mass are scalar quantities. Secondly, the quantitative relations apply only to that part of the momentum transfer which arises from skin friction. If form drag is increased there is little corresponding increase in the rates at which heat transfer and mass transfer will take place. 

The location and momentum of a particle are complementary that is, both the location and the momentum cannot be known simultaneously with arbitrary precision. The quantitative relation between the precision of each measurement is described by the Heisenberg uncertainty principle. 

In Section 4.4, we used a molecular model of a gas to explain qualitatively why the pressure of a gas rises as the temperature is increased as a gas is heated, its molecules move faster and strike the walls of their container more often. The kinetic model of a gas allows us to derive the quantitative relation between pressure and the speeds of the molecules. 

The constant-volume and constant-pressure heat capacities of a solid substance are similar the same is true of a liquid but not of a gas. We can use the definition of enthalpy and the ideal gas law to find a simple quantitative relation between CP and Cv for an ideal gas. 

The challenge that we now face is to justify these remarks and to derive a quantitative relation between the Gibbs free energy and the maximum nonexpansion work that a system can do. 

To find the quantitative relation between the equilibrium constants for the same reaction at two temperatures T, and T2, we note that Eq. 8 allows us to write the relation... 

In order to obtain information as to the extent to which the bond strength S of an orbital can be considered a qualitative measure of its bondforming power and also as to the quantitative relation between the energy of a bond and the strengths of the bond orbitals involved in its formation, we have now carried out the thorough discussion of various sp and spd one-electron bonds between identical atoms. 

Jenny (1941) attempted to quantitatively relate the factors of soil formation to soil properties such as N, C, or clay content, depth of leaching... 

The number of degrees of freedom is represented by/. These are chosen from the list of all quantitatively related aspects of a system that can change. This includes T, P, and the concentrations of c components in each phase, c is the minimum number of components necessary to reproduce the system (ingredients), and p is the number of phases present at equilibrium. A phase is a domain with uniform composition and properties. Examples are a gas, a liquid solution, a solid solution, and solid phases. 

In order to derive a quantitative relation between emission Intensity as measured by EMI and actual metal content, cell samples were subjected to graphite furnace atomic absorption (GFAA) analysis (14). Atomic absorption experiments were performed both on cells which had been stained with a fluorescent reagent and on cells not exposed to a lumlnophore. After EMI analysis, 50 fiL of cell suspension were withdrawn from the 0.30 mL of sample used for EMI and were digested In 150 iiL of concentrated HNO3 for 90 minutes at 85° . These solutions were then diluted to 1/10 of their concentration with deionized water, and the 150 liL of these diluted... 

Qualitative and quantitative relations between enthalpy and entropy were observed several times in the 1920 s, and their importance was rightly recognized by some authors. However, some ideas from this early work seem to have been overlooked later, perhaps because they were connected with obsolete theories or because they were developed independently in the fields of organic chemistry, catalysis, and pure physical chemistry. For this reason, a brief historical survey seems appropriate. 

According to modem views, the basic points of the theory of electrolytic dissociation are correct and were of exceptional importance for the development of solution theory. However, there are a number of defects. The quantitative relations of the theory are applicable only to dilute solutions of weak electrolytes (up to 10 to 10 M). Deviations are observed at higher concentrations the values of a calculated with Eqs. (7.5) and (7.6) do not coincide the dissociation constant calculated with Eq. (7.9) varies with concentration and so on. For strong electrolytes the quantitative relations of the theory are altogether inapplicable, even in extremely dilute solutions. 

Today, electrochemistry is a rigorous science concerned with the quantitative relations among the chemical, surface, and electrical properties of systems. Electrochemistry has strong finks to many other fields of science. Electrochemical concepts proved particularly fruitful for studying and interpreting a number of very important biological processes. 

According to the mode of parameter correlation, Ej(30) was introduced in the group of parameters that describe the acidity of solvent and partially its polarity, and n in the group of parameters that present the dielectric properties of solvents. Quantitative relations between different parameters of polarity, such as correlation between the n scale or and the dielectric constant and refractive index of... 

An important aspect of all methods to be discussed concerns the choice of the model complexity, i.e., choosing the right number of factors. This is especially relevant if the relations are developed for predictive purposes. Building validated predictive models for quantitative relations based on multiple predictors is known as multivariate calibration. The latter subject is of such importance in chemo-metrics that it will be treated separately in the next chapter (Chapter 36). The techniques considered in this chapter comprise Procrustes analysis (Section 35.2), canonical correlation analysis (Section 35.3), multivariate linear regression... 

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