Modeling equilibrium constant

Criscenti and Sverjensky (1999, 2002) continued to build the internally consistent set of triple layer model equilibrium constants developed by Sverjensky and Saliai (1996) and Sahai and Sverjensky (1997a,b) by reexamining sets of adsorption edge and isotherm data for divalent metal cation adsorption onto oxide surfaces. In contrast to previous investigations, they found tliat the adsorption of transition and heavy metals on solids such as goethite, y-ALOs, corundum, and anatase, which have dielectric constants between 10 and 22, was best described by surface complexes of the metal with the electrolyte anion. Metal (M +j adsorption from NaNOs solutions is described by... 

Figure B2.5.7 shows the absorption traces of the methyl radical absorption as a fiinction of tune. At the time resolution considered, the appearance of CFt is practically instantaneous. Subsequently, CFl disappears by recombination (equation B2.5.28). At temperatures below 1500 K, the equilibrium concentration of CFt is negligible compared witli (left-hand trace) the recombination is complete. At temperatures above 1500 K (right-hand trace) the equilibrium concentration of CFt is appreciable, and thus the teclmique allows the detennination of botli the equilibrium constant and the recombination rate [54, M]. This experiment resolved a famous controversy on the temperature dependence of the recombination rate of methyl radicals. Wliile standard RRKM theories [, ] predicted an increase of the high-pressure recombination rate coefficient /r (7) by a factor of 10-30 between 300 K and 1400 K, the statistical-adiabatic-chaunel model predicts a...

Suppose now that the sites are not independent, but that addition of a second (and subsequent) ligand next to a previously bound one (characterized by an equilibrium constant K ) is easier than the addition of the first ligand. In the case of a linear receptor B, the problem is fonnally equivalent to the one-dimensional Ising model of ferromagnetism, and neglecting end effects, one has [M] ... 

Conformational free energy simulations are being widely used in modeling of complex molecular systems [1]. Recent examples of applications include study of torsions in n-butane [2] and peptide sidechains [3, 4], as well as aggregation of methane [5] and a helix bundle protein in water [6]. Calculating free energy differences between molecular states is valuable because they are observable thermodynamic quantities, related to equilibrium constants and... 

Decades of work have led to a profusion of LEERs for a variety of reactions, for both equilibrium constants and reaction rates. LEERs were also established for other observations such as spectral data. Furthermore, various different scales of substituent constants have been proposed to model these different chemical systems. Attempts were then made to come up with a few fundamental substituent constants, such as those for the inductive, resonance, steric, or field effects. These fundamental constants have then to be combined linearly to different extents to model the various real-world systems. However, for each chemical system investigated, it had to be established which effects are operative and with which weighting factors the frmdamental constants would have to be combined. Much of this work has been summarized in two books and has also been outlined in a more recent review [9-11]. 

The best-known equation of the type mentioned is, of course, Hammett s equation. It correlates, with considerable precision, rate and equilibrium constants for a large number of reactions occurring in the side chains of m- and p-substituted aromatic compounds, but fails badly for electrophilic substitution into the aromatic ring (except at wi-positions) and for certain reactions in side chains in which there is considerable mesomeric interaction between the side chain and the ring during the course of reaction. This failure arises because Hammett s original model reaction (the ionization of substituted benzoic acids) does not take account of the direct resonance interactions between a substituent and the site of reaction. This sort of interaction in the electrophilic substitutions of anisole is depicted in the following resonance structures, which show the transition state to be stabilized by direct resonance with the substituent ... 

The text listed below provides more details on how the potentiometric titration data may be used to calculate equilibrium constants. This text provides a number of examples and includes a discussion of several computer programs that have been developed to model equilibrium reactions. 

Although this experiment is written as a dry-lab, it can be adapted to the laboratory. Details are given for the determination of the equilibrium constant for the binding of the Lewis base 1-methylimidazole to the Lewis acid cobalt(II)4-trifluoromethyl-o-phenylene-4,6-methoxysalicylideniminate in toluene. The equilibrium constant is found by a linear regression analysis of the absorbance data to a theoretical equilibrium model. 

Together with pyridones, the tautomerism of pyrazolones has been studied most intensely and serves as a model for other work on tautomerism (76AHC(Sl)l). 1-Substituted pyrazolin-5-ones (78) can exist in three tautomeric forms, classically known as CH (78a), (DH (78b) and NH (78c). In the vapour phase the CH tautomer predominates and in the solid state there is a strongly H-bonded mixture of OH and HN tautomers (Section 4.04.1.3.1). However, most studies of the tautomerism of pyrazolones correspond to the determination of equilibrium constants in solution (see Figure 20). 

Figure 2-1. Chemical equilibrium constants as a function of temperature. (Source M. Modell and R. C. Reid, Thermodynamics and its Applications, Prentice-Hall, Inc., Englewood Cliffs, NJ.)...

It is reasonable to expeet that models in ehemistry should be capable of giving thermodynamic quantities to chemical accuracy. In this text, the phrase thermodynamic quantities means enthalpy changes A//, internal energy changes AU, heat capacities C, and so on, for gas-phase reactions. Where necessary, the gases are assumed ideal. The calculation of equilibrium constants and transport properties is also of great interest, but I don t have the space to deal with them in this text. Also, the term chemical accuracy means that we should be able to calculate the usual thermodynamic quantities to the same accuracy that an experimentalist would measure them ( 10kJmol ). 

The critical hydrogen content for the ductility loss increased with increasing hydrogen solubility in the alloy. The fracture surfaces were not characteristic of those found under conditions of SCC. In terms of hydrogen and deuterium solubility in a similar series of bcc alloys, the equilibrium constants were determined at infinite dilution as a function of temperature The free energy function was expressed in terms of the bound-proton model. 

Two-state model, a model of proteins that coexists in two states controlled by an equilibrium constant. Molecules with selective affinity for one of the states will produce a bias in that state upon binding to the system. Two-state theory was conceived to describe the function of ion channels but also has relevance to receptors (see Chapter 3.7). 

Therefore, we must find a new equilibrium constant, K , that Is analogous to Ks. We have considered (21) several methods of evaluating the model parameters (the equation that gives Kjj, as a function of temperature) from the available data. 

Figure 6.18 shows how the model predicts the four main types of r vs O global behaviour (electrophobic, electrophilic, volcano, inverted volcano) for fixed XD and IA, Pd and pA, by just varying the adsorption equilibrium constants kD and kA. Note that in Figure 6.18 and till the end of this chapter we omit the units of Pd and pA (e.g. kPa) and kD,kA (e.g. kPa 1), unless we refer to experimental data. This is because one is free to use any consistent set of units, since only the dimensionless products kApA and kDpD enter the calculations. 

What Do We Need to Know Already Much of this chapter stands alone, but it would be helpful to review the kinetic model of gases (Section 4.10) and equilibrium constants (Section 9.2). 

The composition of sodium polysulfide solutions saturated with sulfur of zero oxidation number (S°) has also been studied at 25 and 80 °C (solutions in contact with elemental sulfur) [76]. In this case the ratio 8° 8 per polysulfide ion increases with increasing alkahnity. The maximum average number of sulfur atoms per polysulfide molecule was obtained as 5.4 at 25 °C and 6.0 at 80 °C and pH values of >12. Equilibrium constants for reactions as in Eqs. (26) and (27) have been derived assuming various models with differing numbers of polysulfide ions present. 

Hamilton [13] assumed the presence of all ions with n ranging from 1 to 8 in aqueous polysulfide solutions which is by far the most acceptable model but since there is insufficient experimental data available this model cannot be worked out quantitatively without additional assumptions. The general idea is that those species are most abundant which are close to the average composition of the particular solution, e.g., 84 and 85 for a solution of composition Na284.5, and that the larger and smaller ions are symmetrically less abundant. Equilibrium constants for the various reactions... 

Figure 11.3 is a flow model representing in extremely simple form the main relevant features of nitrogen metabolism. It is not difficult to propose a sufficient explanation why Bprot is isotopically heavier than the diet. We might expect that the net effect of transamination and deamination of amino acids is to remove isotopically lighter N (Macko et al. 1987). That is to say, we may expect that the equilibrium constant for the reaction ... 

This redefinition establishes the effective activation/deactivation equilibrium constant K=k k 2 = k lk2. (Note that in the cellular automata models, the rate constants k are expressed as transition probabilities per iteration Pi) Using the above redefinition, the mechanism of Eq. (9.1) becomes a set of first-order reactions... 

The experimental results imply that the main reaction (eq. 1) is an equilibrium reaction and first order in nitrogen monoxide and iron chelate. The equilibrium constants at various temperatures were determined by modeling the experimental NO absorption profile using the penetration theory for mass transfer. Parameter estimation using well established numerical methods (Newton-Raphson) allowed detrxmination of the equilibrium constant (Fig. 1) as well as the ratio of the diffusion coefficients of Fe"(EDTA) andNO[3]. 

Here we illustrate how to use kinetic data to establish a power rate law, and how to derive rate constants, equilibrium constants of adsorption and even heats of adsorption when a kinetic model is available. We use the catalytic hydrodesulfurization of thiophene over a sulfidic nickel-promoted M0S2 catalyst as an example ... 

Hinshelwood model, and the corresponding values of the equilibrium constants. [From A. Borgna and J.W. Niemantsverdriet, to be published (2003).]... 

The fits indicate that the Langmuir-Hinshelwood model describes the measurements very well. The equilibrium constants point to a relatively strong adsorption of thiophene and, in particular, H2S, while adsorption of hydrogen is weak. Hence the term K may safely be ignored in Eq. (32). The order in H2 is 0.93, i.e. close to one, which is another indication that hydrogen adsorbs only weakly. 

Table 10.4 lists the rate parameters for the elementary steps of the CO + NO reaction in the limit of zero coverage. Parameters such as those listed in Tab. 10.4 form the highly desirable input for modeling overall reaction mechanisms. In addition, elementary rate parameters can be compared to calculations on the basis of the theories outlined in Chapters 3 and 6. In this way the kinetic parameters of elementary reaction steps provide, through spectroscopy and computational chemistry, a link between the intramolecular properties of adsorbed reactants and their reactivity Statistical thermodynamics furnishes the theoretical framework to describe how equilibrium constants and reaction rate constants depend on the partition functions of vibration and rotation. Thus, spectroscopy studies of adsorbed reactants and intermediates provide the input for computing equilibrium constants, while calculations on the transition states of reaction pathways, starting from structurally, electronically and vibrationally well-characterized ground states, enable the prediction of kinetic parameters. 

Another problem which can appear in the search for the minimum is intercorrelation of some model parameters. For example, such a correlation usually exists between the frequency factor (pre-exponential factor) and the activation energy (argument in the exponent) in the Arrhenius equation or between rate constant (appears in the numerator) and adsorption equilibrium constants (appear in the denominator) in Langmuir-Hinshelwood kinetic expressions. 

Assuming that the equilibria between the adjacent layers are depicted by the same equilibrium constant and that K is the equilibrinm constant between the first layer and the active sites, the two-layer isotherm model can be expressed as ... 

Consequences of the Snyder and Soczewinski model are manifold, and their praetieal importance is very signifieant. The most speetaeular conclusions of this model are (1) a possibility to quantify adsorbents ehromatographic activity and (2) a possibility to dehne and quantify chromatographic polarity of solvents (known as the solvents elution strength). These two conclusions could only be drawn on the assumption as to the displacement mechanism of solute retention. An obvious necessity was to quantify the effect of displacement, which resulted in the following relationship for the thermodynamic equilibrium constant of adsorption, K,, in the case of an active chromatographic adsorbent and of the monocomponent eluent ... 

H. Gampp, M. Maeder, C.J. Meyer and A.D. Zuberbuhler, Calculation of equilibrium constants from multiwavelength spectroscopic data. Ill Model-free analysis of spectrophotometric and ESR titrations. Talanta, 32 (1985) 1133-1139. 

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