Quantitative formulation

A quantitative formulation of Hooke s law is facilitated by considering the rectangular sample shown in Fig. 3.1a. If a force F is applied to the face of area A, the original length of the block Lq will be increased by AL. Now consider the following variations ... 

The quantitative formulation of chemical exchange involves modification of the Bloch equations making use of Eq. (4-67). We will merely develop a qualitative view of the result." We adopt a coordinate system that is rotating about the applied field Hq in the same direction as the precessing magnetization vector. Let and Vb be the Larmor precessional frequencies of the nucleus in sites A and B. Eor simplicity we set ta = tb- As the frequency Vq of the rotating frame of reference we choose the average of Va and Vb, thus. 

According to this very simple derivation and result, the position of the transition state along the reaction coordinate is determined solely by AG° (a thermodynamic quantity) and AG (a kinetic quantity). Of course, the potential energy profile of Fig. 5-15, upon which Eq. (5-60) is based, is very unrealistic, but, quite remarkably, it is found that the precise nature of the profile is not important to the result provided certain criteria are met, and Miller " obtained Eq. (5-60) using an arc length minimization criterion. Murdoch has analyzed Eq. (5-60) in detail. Equation (5-60) can be considered a quantitative formulation of the Hammond postulate. The transition state in Fig. 5-9 was located with the aid of Eq. (5-60). 

If two neutral reactant molecules yield a polar product, then presumably the transition state will be intermediate in polarity, and we anticipate an increase in rate as the solvent polarity is increased. A quantitative formulation of this case is based on Kirkwood s expression for the free energy of transfer of a dipole of... 

A more quantitative formulation of the varying resonance effects in electrophilic nuclear substitution reactions has been suggested by Tsuno, who has proposed to use Eq. (2), where Aa+ is a resonance exaltation term, and r is a susceptibility constant. 

Identification of such universal relations between activation energies and heats of adsorption for particular classes of reaction can be seen as a more precise and more quantitative formulation of Sabatier s Principle. It is promising tool in the search for new materials on the basis of optimized interaction strength between relevant intermediates and the surface. 

Full development of electrochemical techniques in mass-transfer measurement had to await a quantitative formulation of the role played by convective diffusion in electrode processes, which was successfully provided in... 

The quantitative formulation of the electron-transfer paradigm in Scheme 1 as the free-energy relationship for electron transfer (FERET)22 in equation... 

Equations (3.75a)-(3.75j) constitute a more quantitative formulation of the relationship between hybrid p character and substituent electronegativity, generalizing Bent s rule. The accuracy of these approximations is generally of the order of a few percent, sufficient to determine hybrid angles within 1-2° as illustrated in the following examples. 

When Plateau s rule is applied to solids, it may be said that a crack advances whenever the corresponding decrease d (SE)/dx in the strain energy is at least as great as the work necessary to extend the material (in front of the crack) to its maximum strain, that is, to rupture. Two semi-quantitative formulations of this idea are possible. 

We can recognize four main periods in the history of the study of aqueous solutions. Each period starts with one or more basic discoveries or advances in theoretical understanding. The first period, from about 1800 to 1890, was triggered by the discovery of the electrolysis of water followed by the investigation of other electrolysis reactions and electrochemical cells. Developments during this period are associated with names such as Davy, Faraday, Gay-Lussac, Hittorf, Ostwald, and Kohlrausch. The distinction between electrolytes and nonelectrolytes was made, the laws of electrolysis were quantitatively formulated, the electrical conductivity of electrolyte solutions was studied, and the concept of independent ions in solutions was proposed. 

P. George and J. Griffith, in Enzymes, P. D. Boyer, H. Lardy and K. Myrback, eds., Vol. 1, p. 347, Academic Press, New York (1959). The first quantitative formulation of vibrational activation for redox reactions from first-layer ligands. 

This method has been developed for the research on the stereochemical factors influencing the sence of smell3 5,7) (i.e., the quantitative formulation for Ruzicka s stereochemical theory 6) of olfaction). The comparison of the molecular shape is made by means of molecular model silhouettes within a procedure which consists of ... 

A more quantitative measure of stability, known as the stability ratio, can be obtained by setting up and solving the equation for diffusive collisions between the particles. Quantitative formulations of stability, known as the Smoluchowski and Fuchs theories of colloid stability, are the centerpieces of classical colloid science. These and related issues are covered in Section 13.4. 

Changes in the qualitative or quantitative formulation, including inactive ingredients, as provided in the approved application, are considered major changes... 

The quantitative formulation of the site sorption mode, on the other hand, has the virtue of simplicity, but is undoubtedly rather highly idealised. Ideally, Eq. (9) refers to a collection of distinct, permanent and independent sites each accomodating one penetrant molecule Sq measures the concentration of these sites in the membrane and K2 their affinity for the penetrant assuming them to be isoenergetic 35,36). On this basis, the temperature dependence of K2 should yield a constant enthalpy for this sorption mode AH2 35,36). Consistency with the physical picture presented above requires moreover that AH2 be more exothermic than AH, U). 

The most interesting example of a quantum mechanical object is the photon itself. By using the relativistic and quantum mechanical definition of the photon energy, we can obtain a quantitative formulation of the concepts just described. The relativistic form of the total energy of a particle with rest mass m and momentum p is ... 

Thermodynamics and Equilibrium Second We present chemical equilibrium from the viewpoint of thermodynamics. We believe that the quantitative formulation of equilibrium should rest on an understanding of free energy and entropy. To this end, we introduce the laws of thermodynamics before equilibrium, and we formulate equilibrium concepts in terms of standard free energies. This approach allows us to present a unified treatment of a wide range of chemical processes. 

Quantitative Formulations. Computer simulations (213) have been used to put the Gurney-Mott mechanism on a more quantitative basis. Hamilton s recent formulation (214) uses "a more analytical approach. ., that gives a maximum insight into the concepts involved." The method is based on the principle that when there is a branch in a sequence of events allowing two or more possible pathways, a particular event j will be selected with probability p given by... 

Formulation and components. The specific quantitative formulation and components should be listed, along with identification or company code numbers. The amounts per can, per batch, and percentages should be listed here. Additional formulation information also may be enumerated including the following ... 

The second classification concerns the adiabaticity of the e.t. reaction. Conceptually, the reaction is adiabatic if the probability of reaction for each passage through the intersection region (point X of Fig. 2) between the potential surfaces of the reactants and products is close to unity [16]. If this probability is small, then the system remains on the initial state potential surface and the reaction is nonadiabatic . In the quantitative formulation of Jortner and Bixon [16], the pre-exponential factor in the rate constant equation is given as follows, for the case of an adiabatic e.t. ... 

The classical (or semiclassical) equation for the rate constant of e.t. in the Marcus-Hush theory is fundamentally an Arrhenius-Eyring transition state equation, which leads to two quite different temperature effects. The preexponential factor implies only the usual square-root dependence related to the activation entropy so that the major temperature effect resides in the exponential term. The quadratic relationship of the activation energy and the reaction free energy then leads to the prediction that the influence of the temperature on the rate constant should go through a minimum when AG is zero, and then should increase as AG° becomes either more negative, or more positive (Fig. 12). In a quantitative formulation, the derivative dk/dT is expected to follow a bell-shaped function [83]. 

Equation 15.1 offers a quantitative formulation of this statement. This equation makes a statement about the stabilization AEJS of the transition state of a reaction between substrate I (reacting at its reactive center 1, where its frontier orbitals have the coefficients Cj HOMOl and C i LUMOj) and substrate II (reacting at its reactive center 1 as well, where its frontier orbitals have the coefficients C] H0M0[l and Cj LUMOn) owing to the two frontier orbital interactions ... 

The calculations in this case are clearly analogous to those required to prove the Bernoulli theorem. In order to show the first part of the statement, all we have to do is to determine the maximum of Eq. (36), i.e., the minimum of Eq. (43), given the auxiliary condition of Eq. (45). Boltzmann makes use of the second half of the statement in all those cases when he calls the Maxwell velocity distribution overwhelmingly the most probable one." A more quantitative formulation and derivation of this part of the statement is sketched by Jeans in [2, 22-26] and in Dynamical Theory, 44-46 and 56. 

It is widely accepted that localized adsorption is a result of large variation in potential energy from point to point on the adsorbent surface, and that the lateral interaction in such systems must be affected to some extent by the existence of these variations (3). This paper consists in a more quantitative formulation of these concepts than has been available previously. Furthermore, it is shown that, for adsorption on crystallographically perfect surfaces, a considerable amount of information about the nature of the potential field at the surface of the solid can be obtained from a proper treatment of the experimental data. This approach to the adsorption problem can also be extended to give the formal equations applicable to adsorption on a heterogeneous surface however, it seems unlikely that much practical use could be made of these equations because of the very large number of unknown parameters which appear. 

It is well known from structural and kinetic studies that enzymes have well-defined binding sites for their substrates (3), sometimes form covalent intermediates, and generally involve acidic, basic and nucleophilic groups. Many of the concepts in catalysis are based on transition state (TS) theory. The first quantitative formulation of that theory was extensively used in the work of H. Eyring (4, 5 ). Noteworthy contributions to the basic theory were made by others (see (6) for review). As an elementary introduction, we will apply the fundamental assumptions of the TS theory in simple enzyme catalysis as follows. 

Quantitative formulation of the spur model was given by Tao [2l]. It is based on the following assumptions ... 

These two requirements provide upper and lower limits on the solubility of the oxides. They are quantitatively formulated by using the thermodynamic properties of the oxides in the phosphate solution, which is discussed in Section 5.5. 

---