Macroscopic constants

In Eq. (6) Ecav represents the energy necessary to create a cavity in the solvent continuum. Eel and Eydw depict the electrostatic and van-der-Waals interactions between solute and the solvent after the solute is brought into the cavity, respectively. The van-der-Waals interactions divide themselves into dispersion and repulsion interactions (Ed sp, Erep). Specific interactions between solute and solvent such as H-bridges and association can only be considered by additional assumptions because the solvent is characterized as a structureless and polarizable medium by macroscopic constants such as dielectric constant, surface tension and volume extension coefficient. The use of macroscopic physical constants in microscopic processes in progress is an approximation. Additional approximations are inherent to the continuum models since the choice of shape and size of the cavity is arbitrary. Entropic effects are considered neither in the continuum models nor in the supermolecule approximation. Despite these numerous approximations, continuum models were developed which produce suitabel estimations of solvation energies and effects (see Refs. 10-30 in 68)). 

We considered micro-pA), values in Section 3.6. A parallel concept applies to partition coefficients (of multiprotic molecules) namely, if an ionizable substance of a particular stoichiometric composition can exist in different structural forms, then it is possible for each form to have a different micro-log P [224,243,273,275], When logP is determined by the potentiometric method (below), the constant determined is the macro-log P. Other log/1 methods may also determine only the macroscopic constant. 

The constant a is the molecular polarizability, and the macroscopic constant X(1) is known as the linear susceptibility. The molecular polarizability is related... 

COOH group of the neutral acid. The situation is complicated, however, by the coexistence of the zwitter-ion with the neutral acid, so that the observed macroscopic constants K and K2 relate to equilibria as follows ... 

Thus, k is the intrinsic binding constant for the site a, while kab is the intrinsic binding constant for the pair of sites a and b. When the sites are identical, then one should take care to distinguish between the thermodynamic (sometimes referred to as macroscopic) constant Ky, which is the binding constant to the rsr site, and the intrinsic (sometimes referred to as microscopic) binding constant Ky/2, which refers to a specific single site. In the case of two different sites, the thermodynamic first constant, as measured from the total BI, is simply k + k. ... 

Observable constants, usually thermodynamic dissociation constants, for the association of a particular ligand with two or more sites on a larger molecular entity (e.g., a macromolecule). Macroscopic constants are composites of microscopic (i.e., intrinsic) constants. 

MACROMOLECULAR SEQUENCE ANALYSIS (Computer Methods) MACROSCOPIC CONSTANTS MICROSCOPIC CONSTANTS ZWITTERION... 

It connects the macroscopic constant D with the microscopic jumps of the particle. 

The dissociation constants fC — Kn for a multi-protic acid HrA are defined as stepwise or macroscopic constants (also called molecular constants). For some compounds, e.g. alanine, the pfCa values are far apart (pfC and pK2 are 2.4 and 9.8, respectively). The macroscopic constants can be assigned specifically, fCj to the carboxyl group and K2 to the protonated amino group. At the isoelectric pH of 6.1 the alanine exists almost entirely as the dipolar ion. However, for compounds in which the macroscopic pfCa values are closer together, they cannot be assigned to specific groups. We will consider some specific examples in the next section. 

In recent years, there has been an increased need for a proper treatment of the effects of a surrounding medium on the electronic properties of a probe atomic, molecular or biomolecular system. Great progress has been obtained with continuum models [1-5], wherein the solvent is described in some average way and represented by its macroscopic constants. Although successful in some cases, the statistical nature of the liquid environment is not considered in such models. 

Oxygen-binding curves can be analyzed directly to yield fonr macroscopic binding constants K, K2, M3, and K4. Usnally, the macroscopic constants are defined as prodnct constants, i.e., products of the stepwise macroscopic constant (where i... 

The constant Mi j.i+i is composed of microscopic constants, as each O2 binding step is composed of multiple microscopic reactions, which is illustrated by the reaction arrows in Fig. 1. Thus, 4 ways exist to bind the first O2, 12 ways to bind the second O2, 12 ways to bind the third O2, and 4 ways to bind the fourth O2. Each microscopic constant is designated by the notation ij of the species formed in the binding process (Fig. 1, Table 1). For each binding step i = 1,2,3, and 4, the macroscopic constant Mi j.i4i represents the average of the microstate constants A ij (i+i)j, with accompanying statistical factors that account for the different isomeric forms of the microstate tetramers, as shown in Table 1. 

Figure 2 The dependence of O2 binding on Hb concentration. Binding curves are shown (solid black lines) for Hb concentrations of 0.005, 0.04, 0.10, 0.27,1.0, 5.4, and 38 pM (from left to right). Theoretical binding curves (broken red lines) are shown for a pure tetramer solution and a pure dimer solution. The macroscopic, thermodynamic linkage scheme relates the dimer tetramer assembly constants to the O2 binding constants for free dimer and assembled tetramer. The brackets around figurines indicate that the O2 ligand may be bound at any one of the available deoxy hemesites. Thus, the macroscopic constants are average values for multiple microscopic processes.

For detailed understanding of many chemical reactions, more knowledge about ionization constants is needed than has been considered so far. The macroscopic constants K yKi,--. of polybasic acids can have several components arising from the detailed ionization processes. It has been shown that, if a dibasic add has two equivalent and independent ionization processes, the minimum value of KJK2 is 4. 

The vector r describes the overall changes of the chromophore s Cartesian coordinates upon interaction with light. The macroscopic constant x is the linear susceptibility, related to the dielectric constant e according to Eq. (8) ... 

Also the question How polar are ionic liquids has been addressed by many methods that previously have been used to characterize the polarity of common molecular solvents. The macroscopic constant, generally used by the chemists to evaluate the solvent power of a molecular liquid, the dielectric constant, has been evaluated in the case of ILs initially using indirect metliods. and more recently by microwave dielectric spectroscopy. Generally, the values found for the investigated ILs are moderate and, at least those obtained by microwave dielectric spectroscopy, insignificantly affected by the IL structure. 

Under specific restriction and for given values of macroscopic constants KI and K2 the microscopic constant and henceforth all can be extracted from . The restriction is the indicator spin to be monitored is sensitive to one type of protonation process only, long-range protonation effects have to be zero or negligible. For example, the chemical shift of Cl in phenylephrine is identical in the pairs of microscopic dissociation species (AA and AB) and in (BA and BB) respectively. Analogous reasoning has to hold for other indicator spins. 

In a zwitterion, each ionizing group has two constants, one for the fraction that has the other group ionized and one for the remaining fraction. These are called microscopic constants, and the macroscopic constants (as determined by spectrometry or potentiometric titration) are gross composites of these. Historically, tyrosine was one of the first zwitterions to be examined for microconstants (Edsall, Martin and Hollingworth, 1958) and the same methods are still used. See Section 12.4 for the pursuit of microconstants in the catecholamine series. 

Here are the results for norepinephrine. The macroscopic constants were 8.63 (phenolic) and 9.73 (basic). With the help of some ultraviolet spectrometry, the following microscopic constants were calculated pA iz 8.78, pi 2Z 9.58, p- iN 9.16, and 9.20, where Z and N refer to the zwitterion and the neutral species respectively. From these four constants, the top line of Table 12.1 was... 

In addition, the analysis of the association constants derived from the fitting suggests a positive cooperative effect on the binding of the second molecule of cytochrome C3 molecule, upon binding of the firt one (i.e. the macroscopic constants for the formation of the binary complex, Kl, is always smaller the K2, the macroscopic contant for the formation of the ternary complex (TABLE... 

When condition (dai/dt)react (dai/dt)rei is not satisfied, the perturbation of equilibrium distribution is substantial. However, in this case realization of the quasi-steady-state condition is possible, provided the overall rate dai/dt is low compared to partial rates (daj/dt)rei and (dai/dt)i.eact Then, the microscopic kinetic equation can be solved by the quasi-steady-state approximation. The approximation implies that the non-equilibrium distribution functions depend on time implicitly via the total concentration of reactants rather than explicitly. This also means that the macroscopic reaction rates are low compared to microscopic reaction and relaxation rates. Since the distribution functions in this approximation depend on the total concentration only, the reaction rates, according to Eq. (8.50) also depend on the total concentration. Hence, we come to macroscopic kinetic equations that involve only the total concentration of reactants and certain combinations of microscopic rate constants that have the meaning of macroscopic constants. Note that these macroscopic equations need not be consistent with the macroscopic kinetic law as, besides elementary reactive processes, they involve unreactive processes. 

FRIES - In your treatment of the bimolecular reactions, you have used a macroscopic constant, and isotropic diffusion tensor in your Smoluchowski equation. Don t you believe that for those separations of the reactants which are smaller than one solvent diameter, the diffusion tensor becomes space dependent, especially for charged species which are coupled by very strong electrostatic forces In this case, would it be possible to employ space dependent diffusion tensors obtained by molecular dynamics studies ... 

The macroscopic constants are the ones most frequently mentioned in the literature. 

Even in this particular case, the macroscopic constants are not equal to the microscopic constants. We can notice, however, that in this case the microscopic constants are accessible. This is an exception with regard to the previous assertion. 

The study of microscopic constants permits us to demonstrate that the second macroscopic constant, K 2, is necessarily lower than the first one in the case of a diacid. Let s consider, for example, the case of a symmetric diacid HAH in which A is a polymethylenic chain, (CH2) , in which n tends toward infinity. We can admit that the microscopic constants and k2i (and evidently ku = k22) are equal. The electrical field due to the electrical negative charge of the intermediate species AH (identical to HA ) tends to be null due to the distance between the groups A and HA. In these conditions, the triple equality... 

The constants Kai and Ka2 are called macroscopic constants (see Chap. 4). The knowledge of their values and that of their analytical concentration is sufficient to calculate the solution s pH value. However, it is interesting to make a further study of the ionization constants of species A° and A + taken separately. By definition, we can write... 

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