Calculation of the Equilibrium Constant

Upon the thermodynamic condition of the equilibrium for a reaction, the well-known relationship, -RTlnK = AX V yW° is derivable. Here, K is the 

By the application of the equal values of the chemical potentials for the reactants and products in the equilibrium state, Eqs. 6.2-6.4 are derivable and hold for both the gas-phase and the solution. Subscripts f and b stand for the forward and backward reactions, respectively. 

In the gas phase, the activity a, =fjp , where is the fugacity of the component at T (not necessarily T = 298 K) and / = 1 bar, the standard state pressure. This is a dimensionless term. Similarly, if a,- is set to for a real solution, where is the molar fraction and is the related dimensionless activity coefficient for the ith component, all a/s and thus K remain dimensionless quantities. 

As proven in basic physical chemistry handbooks, an important feature of any y,- is that y, 1 with increasing dilution. For tautomeric equilibria, K = Y- x-gly x dilute solution, where the values of the activity 

Although the experimental determination of K is possible, see Chapters 2 and 5 in [1], the theoretical calculations raise important questions. If Eq. 6.4 is applied for a real solution, the standard chemical potential is hypothetical, and does not correspond to any real physical state. Thus, if one wants to calculate K theoretically as exp(-( g - fi )/RT), no direct modeling of the individual values is possible. When a continuum solvent approach is applied, the customary way to calculate K is setting the in-solution determined AGt t equal to —RT In K. This assumption triggers a very important question. Through the calculations of the individual Gs, it is assumed that the solute-solute interactions, including the interactions of the tautomeric forms should not be taken into consideration. Thus, for these very dilute solution models, only one tautomer is considered as a solute. Then the derived for the solutes in the modeled dilute solutions is 

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It is of interest to consider the calculation of the equilibrium constant of the general redox reaction, viz. ... 

First we will review investigations on IR, UV, and NMR spectra because they were used as analytical tools for calculation of the equilibrium constant K. 

If we include the presence of the enzyme (E) in the calculation of the equilibrium constant for a reaction,... 

By methods analogous to those used for the tetrahedral intermediates related to carboxylic acid derivatives, Guthrie proceeded from the heat of formation of pentaeth-oxyphosphorane to free energies of the P(OEt) (OH)5 species. °° This allowed the calculation of the equilibrium constants for addition of water or hydroxide to simple alkyl esters of phosphoric acid see Table 1.7. 

The substitution of Eqs. (16-18) into Eq. (15) yields a simple equation that readily permits the calculation of the equilibrium constant, A", ... 

Thermochemical properties of gas-phase, surface, and bulk species are assumed to be available. This information is used in the calculation of the equilibrium constant, Eq. 11.110, and thus the reverse rate constant, Eq. 11.108. There is not a great deal of thermochemical data for species on surfaces, but techniques are becoming available for their calculation (e.g., Pederson et al. [310]). If surface reactions are specified to be irreversible, or if Arrhenius coefficients for the reverse rate constant are explicitly supplied, then the thermochemical data are not actually used. 

The stability conditions displayed in Figures 2 and 3 suggest that the processes discussed above are thermodynamically possible. To permit more exact calculations of the equilibrium constants for the processes, the reactions can be rewritten as follows ... 

The concentrated solution viscosity measurement yields the weight-average degree of association of active chain ends rather than the more conventional number-average (mole fraction) value. However, the calculation of the equilibrium constant for association, K, can be accomplished if Mw and the heterogeneity index of the polymer sample are known. The latter parameter can be determined via postpolymerization characterization. 

The calculation of the equilibrium constant is as follows. Equation (a) can be written as ... 

To apply equation (3) for calculation of the equilibrium constant K waves Ia and ic must both be limited by diffusion. To prove this the current is measured under conditions when it is 15% or less of the total limiting current and its dependence on the mercury pressure is followed. A diffusion current must, under these conditions, show a linear dependence on the square root of the height of the mercury column. Whenever possible, polarographic dissociation curves should be compared with data on dissociation obtained by other methods, e.g. potentiometry, N.M.R. or spectrophotometry. In the latter case it is important to show that the species responsible for a given polarographic wave is identical with that responsible for the observed absorption peak. 

For less reactive ketones where an excess of mercaptan is needed to shift equilibrium (9 b) sufficiently to the right, the measurement of the anodic mercaptan wave is not advantageous, because its height changes only slightly. In the calculation of the equilibrium constant K, the dissociation of the mercaptan according to equilibrium (9 a) must be taken into consideration. 

This calculation using equcalcc involves the problem that Af G° H2 O) is used in the calculation of the equilibrium constant K, but the expression for the equilibrium constant does not involve the concentration of H2 O. Thus in effect oxygen atoms are not conserved, because in dilute aqueous solutions they are drawn for th essentially infinite reservoir of the solvent. Therefore, the further transformed Gibbs energy G has to be used. The conservation matrix with C and P as components and GlcP2-, Glc, and HP042"as species is given by... 

The partition data for the dilute solution of the extractants TBP, TBPO, TOPO, TEHPO, TPPO and some betadiketones were reported39-44) and some of these data are given in Table 1. The distribution of TPrP between water and several diluents was studied45) and the data were used for calculation of the equilibrium constant for the reaction represented by... 

Thus, in vacuum, heating of Ca(I03)2 would produce Cal2. In air, the first reaction is not displaced appreciably to the right below 510°C.22 For this reason the calculation of the equilibrium constant for reaction (a) is somewhat misleading since it suggests appreciable decomposition of Ca(I03)2... 

It should be kept in mind that the calculation of the equilibrium constant is subject to high sensitivity to small errors in thermochemical data due to its exponential dependency to the standard Gibbs free energy variation, as expressed by the relation ... 

Calculation of the Equilibrium Constant from AG ° Calculate the equilibrium constant K eq for each of the following reactions at pH 7.0 and 25 °C, using the AG ° values in Table 13-4. 

Table 19 Example for the calculation of the equilibrium constant of a reaction using the equilibrium constants of partial reactions...

When the values of the concentrations of the species that appear in the equilibrium expression are known, the calculation of the equilibrium constant is an easy task. Likewise, when the equilibrium constant is known, the equilibrium concentrations can often be obtained. See Examples 2.1-2.4. 

Their calculation of the equilibrium constant was based on spectroscopic measurements of the Br2 concentration. They were able to observe a time variation of the Br2 concentration after mixing but report no rates. In ether and formaldehyde solutions equilibrium was achieved so rapidly that the variation with time of [Br2] could not be detected. 

As long as we are ignorant of the values of the entropy constants S, the calculation of the equilibrium constant K from thermal data can only be carried out by actually determining K at a chosen temperature T, and then calculating the integration constant J from equation (9). When this has been done, however, we are in a position to predict the position of the chemical equilibrium, and hence the reactivity of the gases for all other temperatures. 

The calculation of the equilibrium constant by equation (9) is usually a lengthy proceeding. For this purpose it is necessary to know the true specific heats, or the variation of the specific heats with the temperature. For the dissociation of water vapour, Nernst calculates from the mean specific heats of HgO, Hg, and Og an equation which, after the necessary transformation 2 X10 "... 

Calculation of the equilibrium constant for the proton transfer reaction is easy. Proton transfers go toward the formation of the weaker base. Let s use the reaction of hydroxide with hydrochloric acid as an example. The reaction goes from the stronger base, hydroxide, p/fabH 15 7, to the weaker base, chloride, pifabH 7. 

Tables of the standard Gibbs free energy of formation allow the calculation of the equilibrium constant at standard conditions (i.e., T = 25°C and p = Ibar). However, in most situations, we require the equilibrium constant at other conditions. In the next section, we discuss how to estimate the equilibrium constant at other temperatures.

This conformational freeze is exceptional. The only other example among the tetraacetylated pentopyranoses is the -D-lyxo derivative. The calculation of the equilibrium constants from average spectra requires certain extrapolations. We cannot observe any regular effect from the nature and the polarity of the solvent. Let us now examine different types of derivatives. 

After removal of Se(0) the solution was treated with hypochlorous acid and liberated iodine titrated with hydrazine. This was intended to yield total iodine plus iodide in the equilibrium mixture. This analytical step is not understood. It may be mentioned, however, that this determination was not used in the calculation of the equilibrium constant or elsewhere in the paper. 

Dilute solutions, 0.001 to 0.05 M, of hydrochloric acid were added to aliquots of a solution of KSeCN in flasks. The reaction flasks were purged with nitrogen, capped and left in a thermostated bath for one day to reach equilibrium. The precipitate of red selenium was weighed, which together with a pH determination yields sufficient information for the calculation of the equilibrium constant. 

The gas phase reaction 2SeOBt2(g) Se02(g) + SeBr2(g) +Br2(g) was measured in the temperature range 570 to 770 K. No primary data are presented and it is not understood why an iterative process was necessary for the calculation of the equilibrium constant. The data were evaluated by the third law to yield Af//° (SeOBr2, g. 

Although as already pointed out m several instances the theorem directly applies only to solid or liquid systems, it will be shown that it can also be extended to the calculation of the equilibrium constant K in a gaseous system, provided we know beforehand the heat of the reaction at a single temperature, the molecular heats of the gaseous substances at a few temperatures, and the integration constants of the integrated form of the Clapeyron vapour pressure expression, which has been discussed in an earlier part of this book Suppose the reaction under discussion is the general one—... 

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