Modified equilibrium constants

The equilibrium state is generated by minimizing the Gibbs free energy of the system at a given temperature and pressure. In [57], the method is described as the modified equilibrium constant approach. The reaction products are obtained from a data base that contains information on the enthalpy of formation, the heat capacity, the specific enthalpy, the specific entropy, and the specific volume of substances. The desired gaseous equation of state can be chosen. The conditions of the decomposition reaction are chosen by defining the value of a pair of variables (e.g., p and T, V and T). The requirements for input are ... 

The resultant (modified) equilibrium constant is called the acidity constant of phenol, and has the new symbol Ka, which has a value is 10-10 for phenol. Ka is also called the acid constant, the acid dissociation constant or just the dissociation constant. The value of Ka for phenol is clearly tiny, and quantifies just how small the extent is to which it dissociates to form a solvated proton. 

By applying the general cyclic process of Fig. 9.1 to this process, we obtain the modified equilibrium constant... 

Two logical questions to ask at this point are how one predicts in which direction an acid-base reaction lies and to what extent the reaction goes to completion. The common physical chemical measurement that contains this information is known as the pK,. The pK, is the negative logarithm of the modified equilibrium constant. K . for an acid-base reaction written so that water is the base or proton acceptor. It can be derived as follows ... 

Equation 2-7 is more commonly called the Henderson-Ha.sselbalch equation and is the basis for most calculations involving weak acids and bases. It is used to calculate the pH of solutions of weak acids, weak ba.ses. and buffers consisting of weak acids and their conjugate bases or weak bases and their conjugate acids. Because the pK is a modified equilibrium constant, it corrects for the fact that weak acids do itnl completely react with water. 

Originally, a modified equilibrium constant, the pKt, was derived following the same steps that produced Equation 2-2. It is now more commun to express the basicity of a chemical in terms of the pKa. using the relationship in Equation 2-12. 

Calculating the hydronium ion concentration in weak acid solutions isn t as straightforward as it is in strong solutions, because not all of the weak acid that dissolves initially has ionized. In order to calculate the hydronium ion concentration, you must use the equilibrium constant expression for the weak acid. Chapter 8 covers the K, expression that represents the equilibrium system. For weak acid solutions, you use a modified equilibrium constant expression called the K, — the acid ionization constant Take a look at the generalized ionization of some weak acid HA ... 

Like a weak acid, a weak base is only partially ionized. There s a modified equilibrium constant expression for weak bases — the K. You use it exactly the same way you use the K (see Weak acids for the details) except you solve for the [Off]. 

This reaction is an equilibrium reaction. A modified equilibrium constant, called the (which stands for water dissociation constant) is associated with this reaction. The K , has a value of 1.0 x lO and has the following form ... 

A comparison of equations 6.1 and 6.12 shows that a modified equilibrium constant can be written for the formation of the activated complex, i.e.,... 

This equilibrium constant is dimensionless, as are all equilibrium constants expressed in terms of activities, because the equilibrium constant is a ratio of products of activities to various powers (+ and —), and activities are all dimensionless. That was one of the reasons for defining the activity, to make all equilibrium constants dimensionless. However, the numerical value of the activity depends on the choice of standard states. If we change standard states we will change the computed numerical value of the equilibrium constant, but in a way that will not change the computed concentrations at equilibrium (if we pay careful attention to standard states ). As we will see later, we often use modified equilibrium constants that have built-in dimensions however, the basic equilibrium constant, which is defined by Eqs. 12.14and 12.15,is always dimensionless. 

Soave, G. (1972), Equilibrium constants from a modified Redlich-Kwong equation of state . Chem. Eng. Sci., Vol. 27, p. 1197. 

For biochemical reactions in which hydrogen ions (H ) are consumed or produced, the usual definition of the standard state is awkward. Standard state for the ion is 1 M, which corresponds to pH 0. At this pH, nearly all enzymes would be denatured, and biological reactions could not occur. It makes more sense to use free energies and equilibrium constants determined at pH 7. Biochemists have thus adopted a modified standard state, designated with prime ( ) symbols, as in AG°, AH°, and so on. For values determined... 

The overall direction of the reaction will be determined by the relative concentrations of ATP, ADP, Cr, and CrP and the equilibrium constant for the reaction. The enzyme can be considered to have two sites for substrate (or product) binding an adenine nucleotide site, where ATP or ADP binds, and a creatine site, where Cr or CrP is bound. In such a mechanism, ATP and ADP compete for binding at their unique site, while Cr and CrP compete at the specific Cr-, CrP-binding site. Note that no modified enzyme form (E ), such as an E-PO4 intermediate, appears here. The reaction is characterized by rapid and reversible binary ES complex formation, followed by addition of the remaining substrate, and the rate-determining reaction taking place within the ternary complex. 

Equilibrium conditions are determined by the chemical reactions that occur in a system. Consequently, it is necessary to analyze the chemistry of the system before doing any calculations. After the chemistry is known, a mathematical solution to the problem can be developed. We can modify the seven-step approach to problem solving so that it applies specifically to equilibrium problems, proceeding from the chemistry to the equilibrium constant expression to the mathematical solution. 

In contrast to the reactions of the cycloamyloses with esters of carboxylic acids and organophosphorus compounds, the rate of an organic reaction may, in some cases, be modified simply by inclusion of the reactant within the cycloamylose cavity. Noncovalent catalysis may be attributed to either (1) a microsolvent effect derived from the relatively apolar properties of the microscopic cycloamylose cavity or (2) a conformational effect derived from the geometrical requirements of the inclusion process. Kinetically, noncovalent catalysis may be characterized in the same way as covalent catalysis that is, /c2 once again represents the rate of all productive processes that occur within the inclusion complex, and Kd represents the equilibrium constant for dissociation of the complex. 

To obtain the attachment reaction efficiency in the quasi-free state, we denote the specific rates of attachment and detachment in the quasi-free state by kf and kf respectively and modify the scavenging equation (10.10a) by adding a term kfn on the right-hand side, where is the existence probability of the electron in the attached state. From the stationary solution, one gets kf/kf = (kfk ikfkf), or in terms of equilibrium constants, K(qf) = Kr.Kr, where k, and k2 are the rates of overall attachment and detachment reactions, respectively. Furthermore, if one considers the attachment reaction as a scavenging process, then one gets (see Eq. 10.11) = k f fe/(ktf + kft) = fe,f/(l + Ku) and consequently k2 = kfKJ(l + KJ. 

Poisoning is caused by chemisorption of compounds in the process stream these compounds block or modify active sites on the catalyst. The poison may cause changes in the surface morphology of the catalyst, either by surface reconstruction or surface relaxation, or may modify the bond between the metal catalyst and the support. The toxicity of a poison (P) depends upon the enthalpy of adsorption for the poison, and the free energy for the adsorption process, which controls the equilibrium constant for chemisorption of the poison (KP). The fraction of sites blocked by a reversibly adsorbed poison (0P) can be calculated using a Langmuir isotherm (equation 8.4-23a) ... 

Equation 10.2-9 is known as the Michaelis-Menten equation. It represents the kinetics of many simple enzyme-catalyzed reactions which involve a single substrate (or if other substrates are in large excess). The interpretation of Km as an equilibrium constant is not universally valid, since the assumption that step (1) is a fast equilibrium process often does not hold. An extension of the treatment involving a modified interpretation of Km is given next. 

It is the reaction in the box that is of analytical importance. The position of equilibrium is conveniently expressed by a modified or conditional equilibrium constant K UL in which allowance has been made for the competing side reactions. The fraction of M"+ which has not reacted with L and remains present as... 

The dependences of pH and C-potential on the adsorbed amount of M(H20)2+ at the total metal ion concentrations of 3 x10-3 mol dm-3 are shown in Figures 7 and 8, respectively. The amount adsorbed for each M2+ increases with the pH, and the inflection points are shifted toward the lower pH region in the order of Co2+, Zn2+, Pb2+, Cu2+, which corresponds to the order of the hydrolysis constant of metal ions. To explain the M2+-adsorption/desorption, Hachiya et al. (16,17) modified the treatment of the computer simulation developed by Davis et al. (4). In this model, M2+ binds coordina-tively to amphoteric surface hydroxyl groups. The equilibrium constants are expressed as... 

At very low ligand concentrations, Ka2bi (pco)2/Q. 1 becomes the predominant expression in the inhibition term and Eq. 6 can eventually be simplified to (kg modified constant including the equilibrium constants)... 

Exhaustive catalytic hydrogenation of triptycene affords an equilibrium mixture of perhydrotriptycene isomers. As expected, Boyd s force field (37) calculations, with a modified torsional constant, reproduced the observed composition fairly well (Table 6). All important conformations were taken into account for each isomer. The most stable conformations agree with the results of the X-ray analysis (131) and have the characteristic that the cyclohexane rings are invariably either boat or deformed chair. The most stable conformation of all is 20 (ttt). The predominant conformation of ccc, in which all cyclohexane rings are boat, has an enthalpy only 2.56 kcal/mol above that of 20. The difference is virtually all due to angle and torsional terms. 

A drawback of Gran plots is the fact that all deviations from the theoretical slope value cause an error and that side reactions are not considered. The method was modified by Ingman and Still [63], who considered side reactions to a certain degree, but the equilibrium constants and the concentrations of the components involved must be known. The Gran method is, however, advantageous for determinations in the vicinity of the determination limit The extrapolation of the linear dependence yields the sum + c, where c, is the residual concentration of the test component produced by impurities, dissolution of the ISE membrane, etc. 

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