Butler equation

Consider a system with more than one equivalent exchangeable site, each of which contributes a factor of this form in the numerator for such sites on the transition state and in the denominator for those on the reactant. That is, for n such sites the factor is (l - d + xni)"/(1 - d + d< r) Then, if there are other sites inequivalent to these, corresponding factors for them also appear. The final expression, referred to as the Gross-Butler equation,23 is... 

Gross-Butler equation is that the reactant is in isotopic equilibrium with the solvent. Given that the process under consideration occurs on an exceptionally short time scale, the assumption is not necessarily valid. A very thorough analysis of the isotopic possibilities was used to show that the interpretation presented here is nonetheless correct.25... 

Experimental studies of electrode kinetics resulted in the formnlation of the basic empirical relationship, the Volmer-Butler equation, (6.10) or (6.13), describing the dependence of the electric current on the electrode potential. This eqnation involves the potential E, the rate constants, and the concentrations. 

Important parameters involved in the Volmer-Butler equation are the transfer coefficients a and (1. They are closely related to the Bronsted relation [Eq. (14.5)] and can be rationalized in terms of the slopes of the potential energy surfaces [Eq. (14.9)]. Due to the latter, the transfer coefficients a and P are also called symmetry factors since they are related to the symmetry of the transitional configuration with respect to the initial and final configurations. 

Figure 3a is an illustration of the effect of surface overpotential on the limiting-current plateau, in the case of copper deposition from an acidified solution at a rotating-disk electrode. The solid curves are calculated limiting currents for various values of the exchange current density, expressed as ratios to the limiting-current density. Here the surface overpotential is related to the current density by the Erdey Gruz-Volmer-Butler equation (V4) ... 

We commence with the adsorption of nonionic surfactants, which does not require the consideration of the effect of the electrical double layer on adsorption. The equilibrium distribution of the surfactant molecules and the solvent between the bulk solution (b) and at the surface (s) is determined by the respective chemical potentials. The chemical potential /zf of each component i in the surface layer can be expressed in terms of partial molar fraction, xf, partial molar area a>i, and surface tension y by the Butler equation as [14]... 

Combining the obtained expressions for and pij, we can deduce the respective form of the Butler equation see Equation 5.16 below. 

We can check that Equation 5.13 is equivalent to the Frumkin s surface tension isotherm in Table 5.2 for a nonionic surfactant. Furthermore, eliminating ln(l - 0) between Equations 5.13 and 5.14, we obtain the Butler equation in the form ... 

By substituting pi from the Butler Equation 5.16 into Equation 5.26 and integrating, we can derive the sought-for expression ... 

Measurements in mixtures of H2O and D2O give information about the number of protons ehanging their fraetionation faetors between ground state and transition state. The analysis is eondueted in terms of the Gross-Butler equation, whose modern form is given as equation 1.17, where kn is the rate eonstant at atom fraetion deuterium n, that in pure H2O, and and are the fractionation factors of the fth exchangeable proton in the transition state and the ground state, respectively. 

For more detailed discussion of the Gross-Butler equation see refs. 7, 18 and 19. 

For the description of mixed monolayers, the choice of the dividing surface proposed by Lucassen-Reynders (see Eqs. 2.18, 2.19) is superior [58, 59]. The results obtained using the Butler equation (2.7) and Lucassen-Reynders dividing surface model for the description of mixed monolayers of non-ionic or ionic surfactants, and proteins assuming reorientation or aggregation of adsorbed molecules were presented and discussed in overviews [58, 59]. In this chapter, these concepts are discussed and further developed. 

More rigorous thermodynamic relations valid for adsorption layers which undergo a phase transition could be derived based on the requirement that the chemical potentials in either phase should be equal to each other. The phases are represented by the surfactant bulk solution, the non-condensed (surface solution) and the condensed part of the surface layer. The dependence of p- on the composition of a surface layer is given by the Butler equation (2.7). The chemical potential of the i component in the condensed phase comprised of the given component only (f x = 1) can be derived from Eq. (2.7) as... 

The main feature of the theoretical model given by Eqs. (2.124)-(2.128) is the self-regulation of both the state of the adsorbed molecules and the adsorption layer thickness via the surface pressure. The theory is based on the concept first formulated by Joos [19, 21]. The mechanism of self-regulation is inherent in the Butler equation (2.7), from which all main equations are derived. Of course, surface pressure cannot be regarded as the only self-regulating parameter, but for the solution/fluid interface this is possibly the main factor. From Eq. (2.128) one can calculate the portion of adsorbed molecules which exist in the state coj. The dependence of the... 

The main guidelines for the application of the Butler equation (2.7) and numerous examples for mixtures of soluble surfactants were presented above in Sections 2.4-2.8. It should be noted that, as the solubility or insolubility of the i component does not affect Eq. (2.7), it can be used for the analysis of penetration processes in a way quite similar to how it was employed for mixed soluble components only, however, the expression for the chemical potentials of the components in the bulk solution (2.8) is applicable to soluble components only. Therefore, adsorption isotherms can be derived only for the soluble components of the monolayer. 

Kreevoy studied in depth the deoxymercuration of CH3CH(OR)CH2HgI with perchloric and acetic acids in methanol, and found that the process was pseudo-first order in mercurial, with ka. [HA]. Specific hydronium ion catalysis was involved, and solvent HOH/DOD isotope effects were those predicted by the Butler equations for a pre-rate-determining proton transfer. Further studies on what had been termed - and /9-2-methoxy-cyclohexylmercuric iodides under similar conditions, led to similar findings concerning the solvent isotope effects, and correlation of log k with — Hq. This suggested that the transition state differs from substrate only by a proton. 

The kinetics of the simplest case, the liquid-liquid interface are well known and follow the Volmer-Butler equation which can be written, with emphasis on the... 

In the special case of Langmuir isotherm we have p = 0, and then = 1.) The Butler equation is used by many authors [12,22-24] as a starting point for the development of thermodynamic adsorption models. It should be kept in mind that the specific form of the expressions for and Yu, which are to be substituted in Equation 4.16, is not arbitrary, but must correspond to the same thermodynamic model (to the same expression for F —in our case Equation 4.11). At last, substituting Equation 4.16 into Equation 4.9, we derive the Frumkin adsorption isotherm in Table 4.2, where K is defined by Equation 4.3. 

Hu Y-F, Lee H (2004) Prediction of the surface tension of mixed electrolyte solutions based on the equation of Patwardhan and Kumar and the fundamental Butler equations. J Colloid Interface Sci 269 442 48... 

Cold wash, distributed over the wax cake by drip pipes or sprays, will displace the cake liquids, reducing the Oil in Wax and increasing the yield. This occurs in two steps. The first step is a piston displacement where the wash liquid pushes out the cake liquids. In the second step, oil from within the wax crystal diffuses into the low oil concentration wash liquid. The theoretical reduction in oil content may be predicted by the Butler equation . 

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