Equilibrium constants calculated from

Equilibrium constants calculated from the composition of saturated solutions are dependent on the accuracy of the thermodynamic model for the aqueous solution. The thermodynamics of single salt solutions of KC1 or KBr are very well known and have been modeled using the virial approach of Pitzer (13-15). The thermodynamics of aqueous mixtures of KC1 and KBr have also been well studied (16-17) and may be reliably modeled using the Pitzer equations. The Pitzer equations used here to calculate the solid phase equilibrium constants from the compositions of saturated aqueous solutions are given elsewhere (13-15, 18, 19). The Pitzer model parameters applicable to KCl-KBr-l O solutions are summarized in Table II. 

It is easily shown that equation 2.65 is a fair approximation only if the ideal solution model is valid and A, B, and C are in very low concentrations. By accepting unity activity coefficients for all the species, equation 2.63 leads one to the equilibrium constant calculated from the molalities (Km) ... 

Free energy changes and equilibrium constants calculated from the enthalpy and entropy values estimated by the group-contribution method generally are reliable only to the order of magnitude. For example, Andersen et al. [1] have found that their estimated enthalpies and entropies usually differ from experimental values [7]... 

Figure 15.2 Graph of log K, the thermodynamic equilibrium constant calculated from molecular data (solid line), against 1 /T. Also shown is a prediction of log K obtained by Nernst (dashed-dotted line).

Reaction (15.1) is known as the Haber reaction in recognition of the major role of Fritz Haberf in characterizing this process early in the twentieth century. At that time neither the molecular data nor the mathematical relationships were available for calculating the equilibrium condition, so that Haber had to rely upon experimental measurement. He determined the equilibrium concentration of NH3 in the (N2 + 3H2) mixture8 as a function of temperature. His measurements, graphed as mole percent NH3, were made at a total pressure of 1 atm (1.01 bar), and are also shown in Figure 15.3.1 The agreement with the prediction from the thermodynamic equilibrium constant calculated from the molecular parameters (solid line) is excellent. 

In molecular reaction schemes, only stable molecular reactants and products appear short-lived intermediates, such as free radicals, are not mentioned. Nearly all the reactions written are considered as pseudo-elementary processes, so that the reaction orders are equal to the mol-ecularities. For some special reactions (such as cocking) first order or an arbitrary order is assumed. Pseudo-rate coefficients are written in Arrhenius form. A systematic use of equilibrium constants, calculated from thermochemical data, is made for relating the rate coefficients of direct and reverse reactions. Generally, the net rate of the reversible reaction... 

The agreement between theory and experiment holds good for other gas reactions, and is not merely accidental. Another example is the dissociation of carbon dioxide according to the equation 2002 = 200 + 02- In the following table iCexp. is the degree of dissociation (per cent.) of CO2 at a total pressure of 1 atmosphere,t and log / cxp. is the equilibrium constant calculated from iCexp. The heat of reaction at the ordinary temperature (300° abs.) is 13600 cal., and Ncp = 2x8 — 2x7-7= — 5. The constant J" is calculated from the value of ajexp. at 1443°, viz. a = 2 5 X10 2. Thus... 

The conversion of NO(g) to N20(g) plus NOiig) is spontaneous under standard conditions. The forward reaction under these conditions is scarcely observed because its rate is so slow. Nonetheless, its equilibrium constant can be calculated Such calculations often have enormous impact in evaluating proposed solutions to practical problems. For example, the calculation shows that this reaction could be used to reduce the amount of NO in cooled exhaust gases from automobiles. The fundamental reaction tendency is there, but successful application requires finding a route to increasing the reaction rate at standard conditions. Had the equilibrium constant calculated from thermodynamics been small, this proposed application would be doomed at the outset and investment in it would not be justified. 

The equilibrium constants calculated from rate measurements are in agreement with those obtained directly from equilibrium measurements [218]. For the decomposition of phosgene, a similar rate law is given with the signs reversed [387]. The full mechanism is essentially similar to that developed for the photochemical reaction (apart from initiation), and is discussed in more detail in Section 5.1.1. 

The spectroscopic (UV, IR, NMR) studies of 2-aminopyridine, 2-(phenylami-no)pyridine and 5-nitro-2-(phenylamino)pyridine indicated the presence of only amino tautomer in a solution. The tautomeric equilibrium constants, calculated from the acidity of the fixed derivatives, were found to be 2 x 105 for 2-(phenylamino)pyridine and 8 x 103 for 5-nitro-2-(phenylamino)pyridine, indicating the shift of the equilibrium toward the imino tautomer on introduction of a nitro group (76BCJ2770, 80BCJ717). The molar fractions of 2-aminopyridine and 4-aminopyridine in acetone were estimated as 92+6% and 94+6%, respectively, by 14N NMR spectroscopy... 

Similar PR experiments performed on P. denitrificans COX (50) have also revealed a rapid CuA-heme-a electron equilibration step with an observed rate constant of 30,430 2,300 s and with an equilibrium constant of 2.0 0.1 at 25°C, pH 7.0 (cf. Table VII) corresponding to a difference in reduction potentials between the heme-a [Fe /Fe )] and Cua [Cu VCu ] couples of + 18 + 1 mV. For Cua in this enzyme, a midpoint potential of 213 mV versus SHE was found under the aforementioned conditions (157), while the potential of heme-a was reported to be 428 mV versus SHE (158). The observed equilibrium constant thus disagrees considerably with an equilibrium constant calculated from these potential differences (K = 4300). This is not surprising. 

Reaction (2) is basically the production of hydrogen from carbon gasification followed by Reaction (1). The water-gas shift reaction may provide additional hydrogen. At 427°C, the equilibrium constant calculated from thermodynamic information in the JANAF tables (JL6) is K2 = 2.6 x 10 2 atm, where K2 is the equilibrium constant for Reaction (2). Assuming unit solid activities and typical retort gas compositions, we find that... 

Figures 3 and 4 are the predicted profiles of vapor and liquid composition along the column with 43 ml of catalyst and a reflux flow rate of 22 g/tnin. It is important to note that both the liquid and vapor concentration profiles for acetone in the column are relatively high and hence it is favorable for the formation of DAA. The equilibrium constants calculated from the equilibrium conversion data [9,10] are given in Figure 5, which indicates that at 54 °C, the Ac conversion at equilibrium conversion is only 4.3 wt %. In order to carry out the aldol condensation of acetone in the CD column, the temperature at the reaction zone of the CD column will be near the boiling point of Ac in order to maintain liquid vapor equilibrium. Our CD experimental results show that a maximum concentration of 55 wt% of DAA concentration was obtained which clearly exceeds the equilibrium conversion. The aldol condensation of Ac to produce DAA is an excellent example to demonstrate that in situ separation in a CD column results in an increased yield for equilibrium limited reactions.

The numerical values for the standard state heat of reaction A nW°(7 ) and the equilibrium constant calculated from these equations are plotted in Fig. 1. 

Dunken and Fritzsche (6) have summarized all the equilibrium constant calculations from infrared data used by earlier workers. They have shown that within the limits of accuracy in measurement, agreement can be reached between the results of two methods of calculation which contradict each other in their assumptions. By making infrared measurements at different temperatures, they also show that these chance agreements can come about at one temperature and not at another. Dunken and Fritzsche consider that simplified treatments of the type enumerated by Lussan are only approximations to the truth, and one should always employ a general model of association in which all the associated species (up to a certain maximum size) are present. The authors discuss the evidence for the cyclic dimer form but do not refer specifically to it in their calculations nor do they suggest that any of the higher multimers are cyclic. 

Dimensionless equilibrium constant calculated from adsorption isotherms Equilibrium based on the pressure, kmol m Pa" ... 

In thermodynamic calculation are usually used constants expressed in molality. Nevertheless, the concentration equilibrium constant calculated from molarities always may be converted into constant by molality and... 

The evaluation and checking of the objectivity of the results are to a certain extent facilitated by the fact that whereas the molar absorptivity values are dependent on the wavelength, the equilibrium constants naturally are not. Hence, comparison of the equilibrium constants calculated from the concentration dependence of the absorbances measured at the different wavelengths is of assistance in the calculation of the correct constants and correct molar absorptivities. When the real equilibrium data are known, the distribution curves describing the system can be calculated and from these one can read off directly how the concentrations of the various species in solution vary with the change of the analytical concentrations (cf., Figs 8.1 and 8.2). 

The equilibrium constant calculated from the AH values for the equilibrium phenol — 2,4-cyclohexa-dienone was 2 x 10 , i.e., pAr = 12.7, in agreement with other experimental data. The calculation was for the gas phase however, the equilibrium constant for the tautomerization agreed with experimental results for aqueous solution. This example confirms... 

Table I shows the effects on the equilibrium constant calculated from equations 1 and 2. It should be noted that, not only does die "equilibrium constant" not remain constant as the concentrations of both the environment substance and the racemic complex increase (while keeping their ratio constant), but this "constant" also increases when the ratio of the concentrations of the environment substance to the complex increases. This implies that it may be possible for more than one molecule of the environment substance to undergo hydrogen-bonding simultaneously to one molecule of the complex, which is a matter that is currently undergoing careful scrutiny in this laboratory.

The equilibrium constant calculated from these solubility data agrees reasonably well with that found by Roller and Ervin (23) in a study of the association of silicate ions in the CaO-SiO -HjO system. These authors found at 30 C a value of 1.5 x 10. Thus it is clear that the solubility of silica increases at high pH because of the formation of silicate ion in addition to SKOH) in solution. 

X-xan = 0-substituted xanthate) in benzene at 25°C. This reaction represents the addition of a bidentate ligand to give an octahedral complex. The latter presumably has the cis configuration so a substitution process is involved in the reaction. Values of kf and K are dependent upon the nature of the substituent X, as are the equilibrium constants (calculated from kfikr as they are too large to be measured directly with accuracy). For the reaction in which X = cyclohexyl the activation parameters are A/// = 5.4 0.6 kcal mol = 24 1 kcal mol-, Sf = -21 4 cal K mol and A5f = 26 3 cal K" moF. These values reflect the nature of the processes involved and are similar to values found for analogous reactions of Ni(II) substrates when steric factors were absent. The substituent effects are discussed in terms of Taft s a parameters. Reactions... 

A novel methodology based on SEC-ESI-MS was developed in the stndy reported by Schmidt et al. [62] for the qnalitative and quantitative analysis of arsenic interactions with peptides and proteins. While the suitability of the new method has to be tested for the investigation of other protein-ligand bindings, the approach was successfully used by the authors to study reactions of phenylarsine oxide with cysteine residues of biomolecules of different mass (300-14,000g mol ) and compared equilibrium constants calculated from SEC/ESl-MS with ESI-MS-binding experiments without SEC coupling. 

It should be noted that (i) equation 13 corresponds to the Bjerrum equation 4, (ii) from equation 12, K is independent of solvent (if the value of a is independent of solvent). Some of the implications of these equations have been recently discussed by Hemmes [44]. Table 4 shows values of , and the corresponding rate and equilibrium constants, calculated from equations 10,11,12,13 for a = 0.5 nm and Da = Da- = I>BH+ = Db = 2 X 10" cm s . 

The equilibrium constant calculated from thermochemical polynomials [Eq. (6)] is Kp. Misidentifying with Kp can cause confusion (Golden, 1971) since = Kp only when the number of moles of products is equal to the number of moles of reactants i.e., when An = 0. When this is not the case... 

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