Equilibrium constants, corrections for

Fig. 30. Equilibrium constant in the hydrolytic polymerization of caprolactam. Apparent equilibrium constants (220—265 C) and equilibrium constant corrected for the water activity coefficient (270°C). Data from refs. [2], [14] and [233].

Starting with the equilibrium constant expressions for reactions 8.1, and 8.3-8.5, verify that equation 8.7 is correct. 

Which of these is the correct way to express the equilibrium constant expression for the reaction above ... 

The capacity factor takes into account the fact that the observed retention will be determined by the equilibrium distribution constant corrected for the relative volumes of the two phases. 

Calorimetric measurements on a reaction like the hydrolysis of ATP yields Ar // ° at the experimental T, pH, and ionic strength. The calorimetric heat of reaction Aj/Zc must be corrected for the heat effect of the hydrogen ions produced by the enzyme-catalyzed reaction on the acid dissociation of the buffer, as described in Chapter 15. If Zf is measured at several temperatures and the acid dissociation constants of all the reactants are known at these temperature, the equilibrium constant K for the reference reaction can be calculated at each temperature. Plotting InAT versus 1/Tyields, which is given by... 

For computational approaches, it is possible to use thermodynamic equilibrium constants in conjunction with activities. In order to do this and to relate the constants to concentrations, the values of single-ion activity coefficients must be known. Alternatively, apparent equilibrium constants valid for the medium of particular interest (or a closely similar one) or constants that have been corrected for the medium under consideration can be used in conjunction with concentrations. Nonthermodynamic assumptions are involved in either case. 

Fig. 4.10 Progress and titration curves for the titration with 0.1 M NaOH of 0.1 M orthophosphoric acid (top) and citric acid (bottom) respectively, calculated with (colored open circles) or without (black solid circles) taking into account activity corrections according to the Davies equation. Electrochemical pH detection is assumed (i.e., with a glass electrode), as well as the absence of other salts, so that the ionic strength varies during the experiment. Equilibrium constants used for orthophosphoric acid pATai = 2.15, pK 9 = 7.20, pK,J= 12.15 for citric acid pKal = 3.13, p Ka2 = 4.76, p Ka3 = 6.40.

Modeling programs deal exclusively with equilibrium constants defined for the aqueous solution, so the intrinsic or surface equilibrium constants. surface obtained from the literature (e.g., Dzombak and Morel, 1990) are corrected using Equations (7.17) or (7.18). 

Experimental determinations made in terms of concentrations give concentration quotients which are non-ideal constants. Corrections for non-ideality are made in terms of the calculated ionic strength and the various Debye-Huckel expressions. However, emf experiments, including pH measurements, can sometimes furnish equilibrium constants directly in terms of activities, and as such these will be ideal equilibrium constants. 

The numerator (see equations 12.4 and 12.5) consists of a term for the gas phase concentration of the precursor of a reacting species and an equilibrium constant, K, for the adsorption of the resultant surface species. The first is outside the realm of catalyst design, while the second can be expressed as a product of a pre-exponential and an exponential energy (more correctly, enthalpy) term. 

The value of the equilibrium constant K5 for R5 is in doubt. Values as high as 2.2 x 10 were measured by pulse radiolysis (11) and as low as 3.3 x 10 measured in flash studies ( ). The ratio of [Br2 ] to [Br] in seawater should nevertheless be in the range of 2.6 - 180 to 1. Thus, even if the smallest value of K5 is correct, only 251 of the oxidized Br would be present as the free atom (spectroscopically transparent at accessible wavelengths) while higher values would reduce the stoichiometric importance of this species to negligible proportions. However, the facile equilibrium R5 ensures that monitorina the ion-radical concentration also monitors the effect of reactions of the free atom and other species in fast equilibrium with it (see (9) for discussion). 

We have noted that the equilibrium constant Kj for reaction depends only on the system temperature T and the standard state. Often, we need to determine how the equilibrium constant changes with temperature. For example, during a reactor design we routinely want to know whether product yield can be improved by an increase or decrease in operating temperature. Furthermore, many tables (discussed at the end of 10.4.2) give values for equilibrium constants only at selected temperatures then we must correct those values to the temperature of our situation. 

Strictly speaking, a correction for hydration ought to be applied to many of the other rates and "equilibrium constants. However, for most of the compounds concerned the degree of hydration is unknown, and in those instances where it is known the correction is not large. 

Select the correctly written equilibrium constant expression for the following reaction, and explain your choice ... 

What is the correct expression for the equilibrium constant (K ) for the reaction between carbon and hydrogen gas to form methane shown here ... 

The equihbrium lies to the right, on the side of the weaker acid-base pair. Recall that > 1 and AG° < 0 for a reaction that is thermodynamically favorable as written from left to right use this information to be sure to get the magnitude of K and the sign of AG° correct. The equilibrium constant, K, for the process is the ratio of the values, (10 /10 ) = l(f (not 10 ). With reference to Table 2-1, a K value of 100 corresponds to a AG° of —2.73 kcal mol (not +2.73). If the reaction were written in the opposite direction, with the equihbrium lying to the left, the correct values would be those in parentheses. For more practice with acids and bases, see Problem 28. 

We have assessed the qualitative correctness of reaction (21.19). To calculate the quantitative extent of the dissolution of CaCOg(s) in rainwater is somewhat more difficult. The calculation centers on the combined equilibrium constant expression for reaction (21.19), X = 2.6 X 10 , and is affected by the partial pressure of atmospheric CO2 in equilibrium with rainwater. Typical data are given in Chapter 16, Practice Example A, page 782. 

Reactions at surface functional groups are typically modeled via surface com-plexation models (SCMs), which are simply equilibrium chemical models modified to correct for surface electrostatic effects. SCMs model acid-base reactions at surface functional groups via intrinsic equilibrium constants and ionic solutirai species concentrations corrected to account for the electric field around the interface. Thus, the effective equilibrium constants account for both chemical and electrostatic effects, and continuously change as surface charging progresses. 

VPLQFT is a computer program for correlating binary vapor-liquid equilibrium (VLE) data at low to moderate pressures. For such binary mixtures, the truncated virial equation of state is used to correct for vapor-phase nonidealities, except for mixtures containing organic acids where the "chemical" theory is used. The Hayden-0 Connell (1975) correlation gives either the second virial coefficients or the dimerization equilibrium constants, as required. 

The quasi-equilibrium assumption in the above canonical fonn of the transition state theory usually gives an upper bound to the real rate constant. This is sometimes corrected for by multiplying (A3.4.98) and (A3.4.99) with a transmission coefifiwient 0 < k < 1. 

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