Equilibrium constants intrinsic

In pure and stoichiometric compounds, intrinsic defects are formed for energetic reasons. Intrinsic ionic conduction, or creation of thermal vacancies by Frenkel, ie, vacancy plus interstitial lattice defects, or by Schottky, cation and anion vacancies, mechanisms can be expressed in terms of an equilibrium constant and, therefore, as a free energy for the formation of defects, If the ion is to jump into a normally occupied lattice site, a term for... 

The acid-base equilibrium constant for the Me residue can be determined by spectroscopic pH titration. An example for the titration is shown in Fig. 2. The electrostatic potential

difference between the apparent pK on the charged surface (pKobs) and that on an intrinsic neutral surface (pK1) by... 

A wide range of nitroxidcs and derived alkoxyamincs has now been explored for application in NMP. Experimental work and theoretical studies have been carried out to establish structure-property correlations and provide further understanding of the kinetics and mechanism. Important parameters are the value of the activation-deactivation equilibrium constant K and the values of kaa and (Scheme 9.17), the combination disproportionation ratio for the reaction of the nilroxide with Ihe propagating radical (Section 9.3.6.3) and the intrinsic stability of the nitroxide and the alkoxyamine under the polymerization conditions (Section 9.3.6.4). The values of K, k3Cl and ktieact are influenced by several factors.11-1 "7-"9 ... 

As the pre-equilibria in Schemes 5-10 and 5-11 are not identical and their equilibrium constants are therefore likely to be different from one another, the rate constants k - and are not intrinsic rate constants of the corresponding slow dissociation steps, but are dependent in addition on the constants of these pre-equilibria. 

In addition to the influence of the complexation equilibrium constant K, the observed reaction rate of arenediazonium salts in the presence of guest complexing reagents is influenced by the intrinsic reaction rate of the complexed arenediazonium ion. This system of reactions can be rationalized as in Scheme 11-1. Here we are specifically interested in the numerical value of the intrinsic rate constant k3 of the complexed diazonium ion relative to the rate constant k2 of the free diazonium ion. 

CeHot. From thermodsmamic measurements, it was found that the intrinsic defects were Anti-Frenkel in nature, i.e.- (H + Vh). An equilibrium constant was calculated as ... 

We have given defect-equations for edl nine types of defects, and the Equilibrium Constants thereby associated. However, calculation of these equilibria would require values in terms of energy at each site, values which are difficult to determine. A better method is to convert these EC equations to those involving numbers of each Qrpe of intrinsic defect, as a ratio to an intrinsic cation or einion. This would allow us to calculate the actual number of intrinsic defects present in the crystal, at a specified temperature. 

Given ksh = 3 x 10-3 for CaQ2, calculate the number of intrinsic defects present in this crystal. If CaCl2 is face-centered cubic, use the same equilibrium constant to calculate the intrinsic Frenkel, Anti-Frenkel and Interstial defects expected in this crystal. 

The overall dissociation constant obtained comprises the intrinsic dissociation constants for the ligands and the equilibrium constant for the conformational transition [68],... 

The material balance was calculated for EtPy, ethyl lactates (EtLa) and CD by solving the set of differential equation derived form the reaction scheme Adam s method was used for the solution of the set of differential equations. The rate constants for the hydrogenation reactions are of pseudo first order. Their value depends on the intrinsic rate constant of the catalytic reaction, the hydrogen pressure, and the adsorption equilibrium constants of all components involved in the hydrogenation. It was assumed that the hydrogen pressure is constant during... 

Both these methods require equilibrium constants for the microscopic rate determining step, and a detailed mechanism for the reaction. The approaches can be illustrated by base and acid-catalyzed carbonyl hydration. For the base-catalyzed process, the most general mechanism is written as general base catalysis by hydroxide in the case of a relatively unreactive carbonyl compound, the proton transfer is probably complete at the transition state so that the reaction is in effect a simple addition of hydroxide. By MMT this is treated as a two-dimensional reaction proton transfer and C-0 bond formation, and requires two intrinsic barriers, for proton transfer and for C-0 bond formation. By NBT this is a three-dimensional reaction proton transfer, C-0 bond formation, and geometry change at carbon, and all three are taken as having no barrier. 

By convention, [HA(s)] = [B(s)] = 1. Eqs. (6.1) represent the precipitation equilibria of the uncharged species, and are characterized by the intrinsic solubility equilibrium constant, Sq. The zero subscript denotes the zero charge of the precipitating species. In a saturated solution, the effective (total) solubility S, at a particular pH is defined as the sum of the concentrations of all the compound species dissolved in the aqueous solution ... 

We have already seen (p. 56) that the Ka, and hence pKa, value for an acid is not an intrinsic attribute of the species itself, because it varies from one solvent to another the value depending on the overall system of which the acid is a constituent. Values are normally quoted for aqueous solution, unless otherwise specified, because most data are available for that solvent. Most values are also quoted as at 25°, again because most data were obtained at this temperature. A constant temperature has to be specified as Ka, an equilibrium constant, varies with temperature. We have been concerned above with the relative... 

The extent to which the effect of changing substituents on the values of ks and kp is the result of a change in the thermodynamic driving force for the reaction (AG°), a change in the relative intrinsic activation barriers A for ks and kp, or whether changes in both of these quantities contribute to the overall substituent effect. This requires at least a crude Marcus analysis of the substituent effect on the rate and equilibrium constants for the nucleophile addition and proton transfer reactions (equation 2).71-72... 

Table 2 Rate constants, equilibrium constants, and estimated Marcus intrinsic barriers for the formation and reaction of ring-substituted l-phenylethyl carbocations X-[6+] (Scheme 8)°... 

A value of kjkp = 17 000 has been determined for partitioning of the acetophenone oxocarbenium ion [12+] in water.15,16 It is not possible to estimate an equilibrium constant for the addition of water to [12+], because of the instability of the hemiketal product of this reaction. However, kinetic and thermodynamic parameters have been determined for the reaction of [12+] with methanol to form protonated acetophenone dimethyl ketal [12]-OMeH+ and for loss of a proton to form a-methoxystyrene [13] in water (Scheme 10).15,16 Substitution of these rate and equilibrium constants into equation (3) gives values of AMeoH = 6.5 kcal mol-1 and Ap = 13.8 kcal mol-1 for the intrinsic... 

The building blocks of supramolecular systems are held together by intramolecular interactions and these systems are reversible. This intrinsic property is not only a consequence of the more labile interactions within supermolecules, compared to covalent bonds in molecules, but reversibility is essential for the function expressed by supramolecular systems. Kinetics can never be inferred from thermodynamic studies. For example, the knowledge of a host-guest equilibrium constant does not... 

The compound will be stoichiometric, with an exact composition of MX10ooo when the number of metal vacancies is equal to the number of nonmetal vacancies. At the same time, the number of electrons and holes will be equal. In an inorganic compound, which is an insulator or poor semiconductor with a fairly large band-gap, the number of point defects is greater than the number of intrinsic electrons or holes. To illustrate the procedure, suppose that the values for the equilibrium constants describing Schottky disorder, Ks, and intrinsic electron and hole numbers, Kc, are... 

Therefore further progress in this area depends on the measurement of equilibrium constants. At this stage I simply cannot say how much of the difference of two powers of 10 between the k+Bpl of the alkenes and the styrenes is to be attributed to an intrinsic difference in reactivity and how much to the existence of the P+ G complexes. The negative temperature coefficient of the rate constant for a-methyl styrene found by Chawla Huang (1975) is a strong indication in favour of my view that the propagation is not a simple bimolecular reaction. 

We return to the complex formation equilibria described in Chapter 2 (Eqs. 2.1 -2.10). The equilibrium constants as given in these equations are essentially intrinsic constants valid for a (hypothetically) uncharged surface. In many cases we can use these constants as apparent constants (in a similar way as non-activity corrected constants are being used) to illustrate some of the principal features of the interdependent variables that affect adsorption. Although it is impossible to separate the chemical and electrical contribution to the total energy of interaction with a surface without making non-thermodynamic assumptions, it is useful to operationally break down the interaction energy into a chemical and a Coulombic part ... 

The intrinsic equilibrium constants are postulated to be independent of the composition of the solid phase They remain conditional in the sense of the constant ionic medium reference state if interacting ions such as H+ are expressed as concentrations. 

Using the TLM, the intrinsic equilibrium constant is defined using the activity coefficients, as described previously, i.e.,... 

To calculate the concentration terms in Equations 22 and 23, the intrinsic equilibrium constants Rint and K nt, 3, and the equilibrium concentrations of surface species have to be obtained. These values were determined by solving a set of simultaneous equations, i.e., Equations 4-14, 20 and 21. In a series of calculations, the values, K nton = 80, Nation = 50, and C2 = 20 yF/cm2, which have been reported by Davis et al. (5), were used. 

Here r and r refer to the predominant chloroquine, K E is the product of the intrinsic equilibrium constant for ethidium binding and an experimental efficiency factor,(53) Cbp is the total concentration of base pairs, and... 

Temperature reduction generally provides a severalfold enhancement of nonequivalence magnitude (15,16,19). Cocaine (3) at 20°C shows methylphenylcarbinol-induced nonequivalence in Hj and Hj, and in the A-methyl and 0-methyl resonances of 0.14, 0.03, 0.01, and 0.05 ppm, respectively (15). On lowering the temperature to -40°C, these differences increase to 0.47, 0.06, 0.12, and 0.17 ppm. Only the nonequivalence for Hs changes in sense (zero nonequivalence is observed for H5 at 0°C). Although the increase in nonequivalence magnitude with a reduction of temperature can be attributed in some cases to an increase in the equilibrium constants for CSA-solute association, such enhancement is observed even when the CSA is present in such excess as to cause essentially complete solvation of the enantiomeric solutes (doubtless 3 is such an example). Here, temperature reduction must also increase the intrinsic spectral differ-... 

This is the BI expressed in terms of the measurable equilibrium constants K. Since we require that the BI as represented by the thermodynamic constants be the same as that represented by the intrinsic constant, we can make the identifications ... 

Here kgr is the surface reaction rate constant, Rr is the adsorption equilibrium constant for product R, Pr is the partial pressure of R and Kp is the reaction equilibrium constant. At low loading the reaction rate simply becomes proportional to the product of the intrinsic rate constant and the Henry coefficient. 

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